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7,000 | 777 | Bayesian Modeling and Classification of
Neural Signals
Michael S. Lewicki
Computation and Neural Systems Program
California Institute of Technology 216-76
Pasadena, CA 91125
lewickiOcns.caltech.edu
Abstract
Signal processing and classification algorithms often have limited
applicability resulting from an inaccurate model of the signal's underlying structure. We present here an efficient, Bayesian algorithm for modeling a signal composed of the superposition of brief,
Poisson-distributed functions. This methodology is applied to the
specific problem of modeling and classifying extracellular neural
waveforms which are composed of a superposition of an unknown
number of action potentials CAPs). Previous approaches have had
limited success due largely to the problems of determining the spike
shapes, deciding how many are shapes distinct, and decomposing
overlapping APs. A Bayesian solution to each of these problems is
obtained by inferring a probabilistic model of the waveform. This
approach quantifies the uncertainty of the form and number of the
inferred AP shapes and is used to obtain an efficient method for
decomposing complex overlaps. This algorithm can extract many
times more information than previous methods and facilitates the
extracellular investigation of neuronal classes and of interactions
within neuronal circuits.
590
Bayesian Modeling and Classification of Neural Signals
1
INTRODUCTION
Extracellular electrodes typically record the activity of several neurons in the vicinity of the electrode tip (figure 1). Most electrophysiological data is collected by
isolating action potentials (APs) from a single neuron by using a level detector or
window discriminator. Methods for extracting APs from multiple neurons can, in
addition to the obvious advantage of providing more data, provide the means to
investigate local neuronal interactions and response properties of neuronal populations. Determining from the voltage waveform what cell fired when is a difficult,
ill-posed problem which is compounded by the fact that cells frequently fire simultaneously resulting in large variations in the observed shapes.
There are three major difficulties in identifying and classifying action potentials
(APs) in a neuron waveform. The first is determining the AP shapes, the second is
deciding the number of distinct shapes, and the third is decomposing overlapping
spikes into their component parts. In general, these problems cannot be solved
independently, since the solution of one will affect the solution of the others.
2:
rn_Cl.
Figure 1: Each neuron generates a stereotyped action potential (AP) which is observed
through the electrode as a voltage fluctuation. This shape is primarily a function of
the position of a neuron relative to the tip. The extracellular waveform shows several
different APs generated by an unknown number of neurons. Note the frequent presence of
overlapping APs which can completely obscure individual spikes.
The approach summarized here is to model the waveform directly to obtain a probabilistic description of each action potential and, in turn, of the whole waveform.
This method allows us to compute the class conditional probabilities of each AP.
In addition, it is possible to quantify the certainty of both the form and number of
spike shapes. Finally, we can use this description to decompose overlapping APs
efficiently and assign probabilities to alternative spike sequences.
2
MODELING SINGLE ACTION POTENTIALS
The data from the event observed (at time zero) is modeled as resulting from a
fixed underlying spike function, s(t), plus noise:
(1)
591
592
Lewicki
where v is the parameter vector that defines the spike function. The noise,
modeled as Gaussian with zero mean and standard deviation u1]'
1],
is
From the Bayesian perspective, the task is to infer the posterior distribution of the
spike function parameters (assuming, for the moment, that u1] and Uw are known):
P( ID
v
'O"1]'O"w,
M) - P(Dlv, 0"'1' M) P(vluw, M)
P(DIO"1],O"w,M)
.
(2)
The two terms specifying the posterior distribution of v are 1) the probability of
the data given the model:
(3)
and 2) the prior assumptions of the structure of s(t) which are assumed to be of
the form:
(4)
The superscript (m) denotes differentiation which for these demonstrations we assumed to be m = 1 corresponding to linear splines. The smoothness of s(t) is
controlled through Uw with small values of Uw penalizing large fluctuations.
The final step in determining the posterior distribution is to eliminate the dependence of P(vID, 0"1]' O"w, M) on 0"1] and O"w. Here, we use the approximation:
(5)
The most probable values of 0"1] and O"w were obtained using the methods of MacKay
(1992) in which reestimation formulas are obtained from a Gaussian approximation
of the posterior distribution for 0"1] and O"w, P(O"1] , O"wID, M). Correct inference of O"w
prevents the spike function from overfitting the data.
3
MODELING MULTIPLE ACTION POTENTIALS
When a waveform contains multiple types of APs, determining the component spike
shapes is more difficult because the classes are not known a priori. The uncertainty
of which class an event belongs to can be incorporated with a mixture distribution.
The probability of a particular event, D n , given all spike models, M 1 : K , is
K
P(Dnlvl:K' 1r, 0"1]' M 1 : K) =
L 1I"k P (D nlvk, 0"'1' Mk),
(6)
k=l
where 1I"k is the a priori probability that a spike will be an instance of M k , and
E 1I"k = l.
As before, the objective is to determine the posterior distribution for the parameters
defining a set of spike models, P(V 1 :K, 1rID 1 :N , 0"1]1 trw, M 1 : K) which is obtained again
using Bayes' rule.
Bayesian Modeling and Classification of Neural Signals
Finding the conditions satisfied at a posterior maximum leads to the equation:
(7)
where 'Tn is the inferred occurrence time (typically to sub-sample period accuracy) of
the event Dn. This equation is solved iteratively to obtain the most probable values
of V l : K ? Note that the error for each event, D n , is weighted by P(Mk IOn, Vk, 1r, 0''7)
which is the probability that the event is an instance of the kth spike model. This is
a soft clustering procedure, since the events are not explicitly assigned to particular
classes. Maximizing the posterior yields accurate estimates of the spike functions
even when the clusters are highly overlapping.
The techniques described in the previous section are used to determine the most
probable values for 0''7 and rTw and, in turn, the most probable values of V l : K and 1r.
4
DETERMINING THE NUMBER OF SPIKE SHAPES
Choosing a set of spike models that best fit the data, would result eventually in a
model for each event in the waveform. Heuristics might indicate whether two spike
models are identical or distinct, but ad hoc criteria are notoriously dependent on
particular circumstances, and it is difficult to state precisely what information the
rules take into account.
To determine the most probable number of spike models, we apply probability theory.
{MHJ} denote a set of spike models and H denote information known
Let Sj
a priori. The probability of Sj, conditioned only on H and the data, is obtained
using Bayes' rule:
=
(8)
The only data-dependent term is P(OI:NISj, H) which is the evidence for Sj
(MacKay, 1992). With the assumption that all hypotheses SI :3 are equally probable
a priori, P(D l : NISj, H) ranks alternative spike sets in terms of their probability.
The evidence term P(OI :N[Sj, H) is convenient because it is the normalizing constant for the posterior distribution of the parameters defining the spike set. Although calculation of P(O I :N ISj ,H) is analytically intractable, it is often wellapproximated with a Gaussian integral which was the approximation used for these
demonstrations.
A convenient way of collapsing the spike set is to compare spike models pairwise.
Two models in the spike set are selected along with a sampled set of events fit by
each model. We then evaluate P(DISl) and P(D[S2)' S1 is the hypothesis that
the data is modeled by a single spike shape, S2 says there are two spike shapes. If
P(D[S1) > P(D[S2), we replace both models in S2 by the one in S1. The procedure
terminates when no more pairs can be combined to increase the evidence.
593
594
Lewicki
5
DECOMPOSING OVERLAPPING SPIKES
Overlaps must be decomposed into their component spikes for accurate inference
of the spike functions and accurate classification of the events. Determining the
best-fitting decomposition is difficult becaus(~ of the enormous number of possible
spike sequences, not only all possible model combinations for each event but also
all possible event times.
A brute-force approach to this problem is to perform an exhaustive search of the
space of overlapping spike functions and event times to find the sequence with
maximum probability. This approach was used by Atiya (1992) in the case of two
overlapping spikes with the times optimized to one sample period. Unfortunately,
this is often computationally too demanding even for off-line analysis.
We make this search efficient utilizing dynamic programming and k-dimensional
trees (Friedman et al., 1977). Once the best-fitting decomposition can be obtained,
however, it may not be optimal, since adding more spike shapes can overfit the
data. This problem is minimized by evaluating the probability for alternative decompositions to determine the most probable spike sequence (figure 2) .
a
..,,'
. b'
c
Figure 2: Many spike function sequences can account for the same region of data. The
thick lines show the data, thin lines show individual spike functions. In this case, the bestfitting overlap solution is not the most probable: the sequence with 4 spike functions is
more than 8 time& more probable than the other solutions, even though these have smaller
mean squared error. Using the best-fitting overlap solution may increase the classification
error. Classification error is minimized by using t he overlap solution that is most probable.
6
PERFORMANCE
The algorithm was tested on 40 seconds of neurophysiological data. The task is
to determine the form and number of spike ~hapes in a waveform and to infer the
occurrence times of each spike shape. The output of the algorithm is shown in
figure 3. The uniformity of the residual error indicates that the six inferred spike
shapes account for the entire 40 seconds of data. The spike functions M2 and M3
appear similar by eye, but the probabilities calculated with the methods in section 4
indicate that the two functions are significantly different. When plotted against each
Bayesian Modeling and Classification of Neural Signals
~~-- --r----r_--_r----r_--~
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TIIT.'mI)
Figure 3: The solid lines are the inferred spike models. The data overlying each model
is a sample of at most 40 events with overlapping spikes subtracted out. The residual
errors are plotted below each model. This spike set was obtained after three iterations of
the algorithm, decomposing overlaps and determining the most probable number of spike
functions after each iteration. The whole inference procedure used 3 minutes of CPU
time on a Sparc IPX. Once the spike set is infe! red, classification of the same 40 second
waveform takes about 10 seconds.
595
596
Lewicki
other, the two populations of APs are distinctly separated in the region around the
peak with M3 being wider than M 2 ?
The accuracy of the algorithm was tested by generating an artificial data set composed of the six inferred shapes shown in figure 3. The event times were Poisson
distributed with frequency equal the inferred firing rate of the real data set. Gaussian noise was then added with standard deviation equal to 0"'1. The classification
results are summarized in the tables below.
Table 1: Results of the spike model inference algorithm on the synthesized data set.
I Model /I
I b.max/O"fJ II
1
0.44
I 2 I 3 I 4 I 5 I 6 II
I 0.36 I 1.07 I 0.78 I 0.84 I 0.40 II
The number of spike models was correctly determined by the algorithm with the six-model
spike set was preferred over the most probable five-model spike set byexp(34) : 1 and over
the most probable seven-model spike set by exp(19) : 1. The inferred shapes were accurate
to within a maximum error of 1.0717'1. The row elements show the maximum absolute
difference, normalized by 17'1' between each true spike function and the corresponding
inferred function.
Table 2: Classification results for the synthesized data set (non-overlapping events).
True
Models
1
2
3
4
5
6
1
17
0
0
0
0
0
Inferred Models
2
3
4
5
0
0
0
0
25
0
0
1
0 15
0
0
0
0 116
0
0 56
0
0
0
0
0
0
6
0
0
0
0
0
393
Missed
Events
0
0
0
1
17
254
Total
Events
17
26
15
117
73
647
Table 3: Classification results for the synthesized data set (overlapping events).
True
Models
1
2
3
4
5
6
1
22
0
0
0
0
0
Inferred Models
2
4
5
3
0
0
0
0
36
1
0
0
0
0
0 20
0
1
0 116
0
1 61
0
2
0
0
3
6
0
0
0
1
1
243
Missed
Events
0
0
0
3
19
160
Total
Events
22
37
20
121
82
408
Tables 2 and 3: Each matrix component indicates the number of times true model i was
classified as inferred model j. Events were missed if the true spikes were not detected
in an overlap sequence or if all sample values for the spike fell below the event detection
threshold (417'1). There was 1 false positive for Ms and 7 for M 6 ?
Bayesian Modeling and Classification of Neural Signals
7
DISCUSSION
Formulating the task as having to infer a probabilistic model made clear what was
necessary to obtain accurate spike models. The soft clustering procedure accurately
determines the spike shapes even when the true underlying shapes are similar. Unless the spike shapes are well-separated, commonly used hard clustering procedures
will lead to inaccurate estimates.
Probability theory also allowed for an objective means of determining the number of
spike models which is an essential reason for the success of this algorithm. With the
wrong number of spike models overlap decomposition becomes especially difficult .
The evidence has proved to be a sensitive indicator of when two classes are distinct .
Probability theory is also essential to accurate overlap decomposition. Simply fitting data with compositions of spike models leads to the same overfitting problem
encountered in determining the number of spike models and in determining the
spike shapes. Previous approaches have been able to handle only a limited class of
overlaps, mainly due to the difficultly in making the fit efficient. The algorithm used
here can fit an overlap sequence of virtually arbitrary complexity in milliseconds.
In practice, the algorithm extracts many times more information from a neural
waveform than previous methods. Moreover, this information is qualitatively different from a simple list of spike times. Having reliable estimates of the action
potential shapes makes it possible to study the properties of these classes, since
distinct neuronal types can have distinct neuronal spikes. Finally, accurate overlap decomposition makes it possible to investigate interactions among local neurons
which were previously very difficult to observe.
Acknowledgements
I thank David MacKay for helpful discussions and Jamie Mazer for many conversations and extensive help with the development of the software. This work was
supported by Caltech fellowships and an NIH Research Training Grant.
References
A.F. Atiya. (1992) Recognition of multiunit neural signals. IEEE Transactions on
Biomedical Engineering 39(7):723-729.
J .H. Friedman, J.L. Bently, and R.A. Finkel. (1977) An algorithm for finding best
matches in logarithmic expected time. ACM Trans. Math. Software 3(3):209-226.
D. J. C. MacKay. (1992) Bayesian interpolation. Neural Computation 4(3):415-445.
597
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7,001 | 778 | A Learning Analog Neural Network Chip
with Continuous-Time Recurrent
Dynamics
Gert Cauwenberghs*
California Institute of Technology
Department of Electrical Engineering
128-95 Caltech, Pasadena, CA 91125
E-mail: gertalcco. cal tech. edu
Abstract
We present experimental results on supervised learning of dynamical features in an analog VLSI neural network chip. The recurrent network, containing six continuous-time analog neurons and 42
free parameters (connection strengths and thresholds), is trained to
generate time-varying outputs approximating given periodic signals
presented to the network. The chip implements a stochastic perturbative algorithm, which observes the error gradient along random
directions in the parameter space for error-descent learning. In addition to the integrated learning functions and the generation of
pseudo-random perturbations, the chip provides for teacher forcing and long-term storage of the volatile parameters. The network
learns a 1 kHz circular trajectory in 100 sec. The chip occupies
2mm x 2mm in a 2JLm CMOS process, and dissipates 1.2 mW.
1
Introduction
Exact gradient-descent algorithms for supervised learning in dynamic recurrent networks [1-3] are fairly complex and do not provide for a scalable implementation in
a standard 2-D VLSI process. We have implemented a fairly simple and scalable
?Present address: Johns Hopkins University, ECE Dept., Baltimore MD 21218-2686.
858
A Learning Analog Neural Network Chip with Continuous-Time Recurrent Dynamics
learning architecture in an analog VLSI recurrent network, based on a stochastic
perturbative algorithm which avoids calculation of the gradient based on an explicit
model of the network, but instead probes the dependence of the network error on
the parameters directly [4]. As a demonstration of principle, we have trained a
small network, integrated with the learning circuitry on a CMOS chip, to generate outputs following a prescribed periodic trajectory. The chip can be extended,
with minor modifications to the internal structure of the cells, to accommodate
applications with larger size recurrent networks.
2
System Architecture
The network contains six fully interconnected recurrent neurons with continuoustime dynamics,
d
6
(1)
T dtXi = -Xi +
W ij U(Xj - (Jj) + Yi ,
L
j=l
with Xi(t) the neuron states representing the outputs of the network, Yi(t) the
external inputs to the network, and u(.) a sigmoidal activation function. The 36
connection strengths W ij and 6 thresholds (Jj constitute the free parameters to be
learned, and the time constant T is kept fixed and identical for all neurons. Below,
the parameters Wij and (Jj are denoted as components of a single vector p.
The network is trained with target output signals x[(t) and xf(t) for the first two
neuron outputs. Learning consists of minimizing the time-averaged error
?(p)
1
= lim 2T
T-+oo
jT L Ixf(t) - Xk(t)IVdt
2
,
(2)
-T k=l
using a distance metric with norm v. The learning algorithm [4] iteratively specifies
incremental updates in the parameter vector p as
p(k+l) = p(k) _ J1, t(k) 7r(k)
(3)
with the perturbed error
t(k)
=
~
(?(p(k)
+ 7r(k?) _
?(p(k) _ 7r(k?))
(4)
obtained from a two-sided parallel activation of fixed-amplitude random perturbations '1ri(k) onto the parameters p/k); '1ri(k) = ?u with equal probabilities for both
polarities. The algorithm basically performs random-direction descent of the error
as a multi-dimensional extension to the Kiefer-Wolfowitz stochastic approximation
method [5], and several related variants have recently been proposed for optimization [6,7] and hardware learning [8-10].
To facilitate learning, a teacher forcing signal is initially applied to the external
input y according to
(5)
Yi(t) = .x ,(xi(t) - Xi(t)) , i = 1,2
providing a feedback mechanism that forces the network outputs towards the targets [3]. A symmetrical and monotonically increasing "squashing" function for ,(.)
serves this purpose. The teacher forcing amplitude .x needs to be attenuated along
the learning process, as to suppress the bias in the network outputs at convergence
that might result from residual errors.
859
860
Cauwenberghs
3
Analog VLSI Implementation
The network and learning circuitry are implemented on a single analog CMOS chip,
which uses a transconductance current-mode approach for continuous-time operation. Through dedicated transconductance circuitry, a wide linear dynamic range
for the voltages is achieved at relatively low levels of power dissipation (experimentally 1.2 m W while either learning or refreshing). While most learning functions,
including generation of the pseudo-random perturbations, are integrated on-chip in
conjunction with the network, some global and higher-level learning functions of
low dimensionality, such as the evaluation of the error (2) and construction of the
perturbed error (4), are performed outside the chip for greater flexibility in tailoring
the learning process. The structure and functionality of the implemented circuitry
are illustrated in Figures 1 to 3, and a more detailed description follows below.
3.1
Network Circuitry
Figure 1 shows the schematics of the synapse and neuron circuitry. A synapse cell of
single polarity is shown in Figure 1 (a). A high output impedance triode multiplier,
using an adjustable regulated casco de [11], provides a constant current Iij linear in the
voltage Wij over a wide range. The synaptic current Iij feeds into a differential pair,
injecting a differential current hj a(xj - OJ) into the diode-connected Id:.t and I;;"t output
lines. The double-stack transistor configuration of the differential pair offers an expanded
linear sigmoid range. The summed output currents Itut and I;;;"t of a row of synapses are
collected in the output cell, Figure 1 (b), which also subtracts the reference currents I;"c
and I;;c obtained from a reference rOw of "dummy" synapses defining the "zero-point"
synaptic strength Wolf for bipolar operation. The thus established current corresponds to
the summed synaptic contributions in (1). Wherever appropriate (i = 1,2), a differential
transconductance element with inputs Xi and xT is added to supply an external input
current for forced teacher action in accordance with (5).
I~U~
~
1",,/
Xi
f-Vc
(a)
(b)
Figure 1 Schematics of synapse and neuron circuitry. (a) Synapse of single polarity.
(b) Output cell with current-to-voltage converter.
The output current is converted to the neuron output voltage Xi, through an active resistive
element using the same regulated high output impedance triode circuitry as used in the
synaptic current source. The feedback delay parameter T in (1) corresponds to the RC
A Learning Analog Neural Network Chip with Continuous-Time Recurrent Dynamics
product of the regulated triode active resistance value and the capacitance Gout. With
= 5 pF, the delay ranges between 20 and 200jLsec, adjustable by the control voltage
of the regulated cascode. Figure 2 shows the measured static characteristics of the synapse
and neuron functions for different values of Wij and ()j ( i = j = 1), obtained by disabling
the neuron feedback and driving the synapse inputs externally.
Gout
~
'-
~
0.0
~
.-
-0.2
~
CII
CII
....~
0
;>
....~
.&
<5
-0.4
O.OV
-0.6
- 0.8V
....~
0
;>
....
~
<5
-0.8
-1.0
-0.5
0.0
Input Voltage x j
0.5
-1.0
1.0
-0.5
0.0
Input Voltage x j
(V)
0.5
1.0
(V)
(b)
(a)
Figure 2 Measured static synapse and neuron characteristics, for various values of
(a) the connection strength Wij, and (b) the threshold ()j.
3.2
Learning Circuitry
Figure 3 (a) shows the simplified schematics of the learning and storage circuitry, replicated
locally for every parameter (connection strength or threshold) in the network. Most of
the variables relating to the operation of the cells are local, with exception of a few global
signals communicating to all cells. Global signals include the sign and the amplitude
of the perturbed error t and predefined control signals. The stored parameter and its
binary perturbation are strictly local to the cell, in that they do not need to communicate
explicitly to outside circuitry (except trivially through the neural network it drives), which
simplifies the structural organization and interconnection of the learning cells.
The parameter voltage Pi is stored on the capacitor Gstore, which furthermore couples
to capacitor G pert for activation of the perturbation. The perturbation bit 7ri selects
either of two complementary signals V+<T and V-<T with corresponding polarity. With
the specific shape of the waveforms V+<T and V-<T depicted in Figure 3 (b), the proper
sequence of perturbation activations is established for observation of the complementary
error terms in (4). The obtained global value for
is then used, in conjunction with
the local perturbation bit 7ri, to update the parameter value Pi according to (3). A fineresolution charge-pump, shown in the dashed-line inset of Figure 3 (a), is used for this
purpose. The charge pump dumps either of a positive or negative update current, of equal
amplitude, onto the storage capacitor whenever it is activated by means of an EN_UPD
high pulse, effecting either of a given increment or decrement on the parameter value Pi
respectively. The update currents are supplied by two complementary transistors, and are
switched by driving the source voltages of the transistors rather than their gate voltages
in order to avoid typical clock feed-through effects. The amplitude of the incremental
update, set proportionally to Itl, is controlled by the VUPD nand VUPD p gate voltage
levels, operated in the sub-threshold region. The polarity of the increment or decrement
action is determined by the control signal DECR/INCR, obtained from the polarities of
t
861
862
Cauwenberghs
the perturbed error t and the perturbation bit 11"; through an exclusive-or operation. The
learning cycle is completed by activating the update by a high pulse on EN_UPD. The
next learning cycle then starts with a new random bit value for the perturbation 11";.
I
1t;
-
I
ii
X
X
E>O
El'CUPD
v+o
v-(J
: :
I
I
II
I
II
I
I
I
-"1t
-2+
--1_
:: rL
I I
I I
TI I
III
T I
I I_
I;;/"
I
E(p) E(p + It) E(p - It)
(a)
(b)
Figure 3 Learning cell circuitry. (a) Simplified schematics.
(b) Waveform and timing diagram.
The random bit stream 1I";(k) is generated on-chip by means of a set of linear feedback
shift registers [12]. For optimal performance, the perturbations need to satisfy certain
statistical orthogonality conditions, and a rigorous but elaborate method to generate a
set of uncorrelated bit streams in VLSI has been derived [13]. To preserve the scalability
of the learning architecture and the local nature of the perturbations, we have chosen a
simplified scheme which does not affect the learning performance to first order, as verified
experimentally. The array of perturbation bits, configured in a two-dimensional arrangement as prompted by the location of the parameters in the network, is constructed by an
outer-product exclusive-or operation from two generating linear sets of uncorrelated row
and column bits on lines running horizontally and vertically across the network array.
In the present implementation the evaluation of the error functional (2) is performed
externally with discrete analog components, leaving some flexibility to experiment with
different formulations of error functionals that otherwise would have been hardwired. A
mean absolute difference (/I = 1) norm is used for the metric distance, and the timeaveraging of the error is achieved by a fourth-order Butterworth low-pass filter. The
cut-off frequency is tuned to accommodate an AC ripple smaller than 0.1 %, giving rise to
a filter settling time extending 20 periods of the training signal.
3.3
Long-Term Volatile Storage
After learning, it is desirable to retain ("freeze") the learned information, in principle
for an infinite period of time. The volatile storage of the parameter values on capacitors
undergoes a spontaneous decay due to junction leakage and other drift phenomena, and
needs to be refreshed periodically. For eight effective bits of resolution, a refresh rate of
10 Hz is sufficient. Incidentally, the charge pump used for the learning updates provides
for refresh of the parameter values as well. To that purpose, probing and multiplexing
circuitry (not shown) are added to the learning cell of Figure 3 (a) for sequential refresh.
In the experiment conducted here, the parameters are stored externally and refreshed
sequentially by activating the corresponding charge pump with a DECR/INCR bit defined
by the polarity of the observed deviation between internally probed and externally stored
A Learning Analog Neural Network Chip with Continuous-Time Recurrent Dynamics
values. The parameter refresh is performed in the background with a 100 msec cycle,
and does not interfere with the continuous-time network operation. A simple internal
analog storage method obliterating the need of external storage is described in [14], and
is supported by the chip architecture.
4
Learning Experiment
As a proof of principle, the network is trained with a circular target trajectory
defined by the quadrature-phase oscillator
xi (t)
{ xr(t)
A cos (27rft)
A sin (27rft)
(6)
with A = o.SV and f = 1kHz. In principle a recurrent network of two neurons
suffices to generate quadrature-phase oscillations, and the extra neurons in the
network serve to accommodate the particular amplitUde and frequency requirements
and assist in reducing the nonlinear harmonic distortion.
Clearly the initial conditions for the parameter values distinguish a trivial learning
problem from a hard one, and training an arbitrarily initialized network may lead
to unpredictable results of poor generality. Incidentally, we found that the majority
of randomly initialized learning sessions fail to generate oscillatory behavior at convergence, the network being trapped in a local minimum defined by a strong point
attractor. Even with strong teacher forcing these local minima persist. In contrast,
we obtained consistent and satisfactory results with the following initialization of
network parameters: strong positive diagonal connection strengths W ii = 1, zero
off-diagonal terms W ij = 0 ; i f. j and zero thresholds (}i = O. The positive diagonal connections Wii repel the neuron outputs from the point attractor at the
origin, counteracting the spontaneous decay term -Xi in (1). Applying non-zero
initial values for the cross connections Wij ; i f. j would introduce a bias in the
dynamics due to coupling between neurons. With zero initial cross coupling, and
under strong initial teacher forcing, fairly fast and robust learning is achieved.
Figure 4 shows recorded error sequences under training of the network with the target oscillator (6), for five different sessions of 1, 500 learning iterations each starting
from the above initial conditions. The learning iterations span 60 msec each, for a
total of 100 sec per session. The teacher forcing amplitude .A is set initially to 3 V,
and thereafter decays logarithmically over one order of magnitude towards the end
of the sessions. Fixed values of the learning rate and the perturbation amplitude
are used throughout the sessions, with J.L = 25.6 V-I and (J' = 12.5 mV. All five sessions show a rapid initial decrease in the error under stimulus of the strong teacher
forcing, and thereafter undergo a region of persistent flat error slowly tapering off
towards convergence as the teacher forcing is gradually released. Notice that this
flat region does not imply slow learning; instead the learning constantly removes
error as additional error is adiabatically injected by the relaxation of the teacher
forcing.
863
864
Cau wenberghs
3.0
25
~
...0
(J
= 12.5 mV
2.0
t:
~
::I
0...
-1
Jl =25.6 V
8
15
1.0
05
0.0
0
20
40
60
80
100
Time (sec)
Figure 4 Recorded evolution of the error during learning,
for five different sessions on the network.
Near convergence, the bias in the network error due to the residual teacher forcing
becomes negligible. Figure 5 shows the network outputs and target signals at convergence, with the learning halted and the parameter refresh activated, illustrating
the minor effect of the residual teacher forcing signal on the network dynamics.
The oscillogram of Figure 5 (a) is obtained under a weak teacher forcing signal,
and that of Figure 5 (b) is obtained with the same network parameters but with
the teacher forcing signal disabled. In both cases the oscilloscope is triggered on
the network output signals. Obviously, in absence of teacher forcing the network
does no longer run synchronously with the target signal. However, the discrepancy
in frequency, amplitude and shape between either of the free-running and forced
oscillatory output waveforms and the target signal waveforms is evidently small.
(a)
(b)
Figure 5 Oscillograms of the network outputs and target signals after learning,
(a) under weak teacher forcing, and (b) with teacher forcing disabled.
Top traces: Xl(t) and Xl T(t). Bottom traces: X2(t) and X2 T (t).
A Learning Analog Neural Network Chip with Continuous-Time Recurrent Dynamics
5
Conclusion
We implemented a small-size learning recurrent neural network in an analog VLSI
chip, and verified its learning performance in a continuous-time setting with a simple
dynamic test (learning of a quadrature-phase oscillator). By virtue of its scalable
architecture, with constant requirements on interconnectivity and limited global
communication, the network structure with embedded learning functions can be
freely expanded in a two-dimensional arrangement to accommodate applications of
recurrent dynamical networks requiring larger dimensionality. A present limitation
of the implemented learning model is the requirement of periodicity on the input
and target signals during the learning process, which is needed to allow a repetitive
and consistent evaluation of the network error for the parameter updates.
Acknowledgments
Fabrication of the CMOS chip was provided through the DARPA/NSF MOSIS service.
Financial support by the NIPS Foundation largely covered the expenses of attending the
conference.
References
[1] B.A. Pearlmutter, "Learning State Space Trajectories in Recurrent Neural Networks,"
Neural Computation, vol. 1 (2), pp 263-269, 1989.
[2] RJ. Williams and D. Zipser, "A Learning Algorithm for Continually Running Fully
Recurrent Neural Networks," Neural Computation, vol. 1 (2), pp 270-280, 1989.
[3] N .B. Toomarian, and J. Barhen, "Learning a Trajectory using Adjoint Functions and
Teacher Forcing," Neural Networks, vol. 5 (3), pp 473-484, 1992.
[4] G. Cauwenberghs, "A Fast Stochastic Error-Descent Algorithm for Supervised Learning
and Optimization," in Advances in Neural Information Processing Systems, San Mateo,
CA: Morgan Kaufman, vol. 5, pp 244-251, 1993.
[5] H.J. Kushner, and D.S. Clark, "Stochastic Approximation Methods for Constrained
and Unconstrained Systems," New York, NY: Springer-Verlag, 1978.
[6] M.A. Styblinski, and T.-S. Tang, "Experiments in Nonconvex Optimization: Stochastic
Approximation with Function Smoothing and Simulated Annealing," Neural Networks,
vol. 3 (4), pp 467-483, 1990.
[7] J.C. Spall, "Multivariate Stochastic Approximation Using a Simultaneous Perturbation
Gradient Approximation," IEEE Trans . Automatic Control, vol. 37 (3), pp 332-341, 1992.
[8] J. Alspector, R. Meir, B. Yuhas, and A. Jayakumar, "A Parallel Gradient Descent
Method for Learning in Analog VLSI Neural Networks," in Advances in Neural Information
Processing Systems, San Mateo, CA: Morgan Kaufman, vol. 5, pp 836-844, 1993.
[9] B. Flower and M. Jabri, "Summed Weight Neuron Perturbation: An O(n) Improvement over Weight Perturbation," in Advances in Neural Information Processing Systems,
San Mateo, CA: Morgan Kaufman, vol. 5, pp 212-219, 1993.
[10] D. Kirk, D. Kerns, K. Fleischer, and A. Barr, "Analog VLSI Implementation of
Gradient Descent," in Advances in Neural Information Processing Systems, San Mateo,
CA: Morgan Kaufman, vol. 5, pp 789-796, 1993.
[11] J.W. Fattaruso, S. Kiriaki, G. Warwar, and M. de Wit, "Self-Calibration Techniques
for a Second-Order Multibit Sigma-Delta Modulator," in ISSCC Technical Digest, IEEE
Press, vol. 36, pp 228-229, 1993.
[12] S.W. Golomb, "Shift Register Sequences," San Francisco, CA: Holden-Day, 1967.
[13] J. Alspector, J.W. Gannett, S. Haber, M.B. Parker, and R. Chu, "A VLSI-Efficient
Technique for Generating Multiple Uncorrelated Noise Sources and Its Application to
Stochastic Neural Networks," IEEE T. Circuits and Systems, 38 (1), pp 109-123, 1991.
[14] G. Cauwenberghs, and A. Yariv, "Method and Apparatus for Long-Term Multi-Valued
Storage in Dynamic Analog Memory," U.s. Patent pending, filed 1993.
865
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7,002 | 779 | Address Block Location with a Neural Net System
Eric Cosatto
Hans Peter Graf
AT&T Bell Laboratories
Crawfords Corner Road
Holmdel, NJ 07733, USA
Abstract
We developed a system for finding address blocks on mail pieces that can
process four images per second. Besides locating the address block, our
system also determines the writing style, handwritten or machine printed, and
moreover, it measures the skew angle of the text lines and cleans noisy
images. A layout analysis of all the elements present in the image is
performed in order to distinguish drawings and dirt from text and to separate
text of advertisement from that of the destination address.
A speed of more than four images per second is obtained on a modular
hardware platform, containing a board with two of the NET32K neural net
chips, a SPARC2 processor board, and a board with 2 digital signal
processors. The system has been tested with more than 100,000 images. Its
performance depends on the quality of the images, and lies between 85%
correct location in very noisy images to over 98% in cleaner images.
1
INTRODUCTION
The system described here has been integrated into an address reading machine developed for
the 'Remote Computer Reader' project of the United States Postal Service. While the actual
reading of the text is done by other modules, this system solves one of the major problems,
namely, finding reliably the location of the destination address. There are only a few constraints
on how and where an address has to be written, hence they may appear in a wide variety of
styles and layouts. Often an envelope contains advertising that includes images as well as text.
785
786
Graf and Cosatto
Sometimes. dirt covers part of the envelope image. including the destination address. Moreover.
the image captured by the camera is thresholded and the reader is given a binary image. This
binarization process introduces additional distortions; in particular. often the destination address
is surrounded by a heavy texture. The high complexity of the images and their poor quality make
it difficult to find the location of the destination address. requiring an analysis of all the elements
present in the image. Such an analysis is compute-intensive and in our system it turned out to
be the major bottleneck for a fast throughput. In fact. finding the address requires much more
computation than reading it. Special-purpose hardware in the form of the NET32K neural net
chips (Graf. Henderson. 90) is used to solve the address location problem.
Finding address blocks has been the focus of intensive research recently. as several companies
are developing address reading machines (United States Postal Service 92). The wide variety
of images that have to be handled has led other researchers to apply several different analysis
techniques to each image and then try to combine the results at the end. see e.g. (palumbo et a1.
92). In order to achieve the throughput required in an industrial application. special purpose
processors for finding connected components and/or for executing Hough transforms have been
applied.
In our system we use the NET32K processor to extract geometrical features from an image. The
high compute power of this chip allows the extraction of a large number of features
simultaneously. From this feature representation. an interpretation of the image's content can
then be achieved with a standard processor. Compared to an analysis of the original image. the
analysis of the feature maps requires several orders of magnitude less computation. Moreover.
the feature representation introduces a high level of robustness against noise. This paper gives
a brief overview of the hardware platfOlm in section 2 and then describes the algorithms to find
the address blocks in section 3.
2 THE HARDWARE
The NET32K system has been designed to serve as a high-speed image processing platform.
where neural nets as well as conventional algorithms can be executed. Three boards form the
whole system. Two NET32K neural net chips are integrated with a sequencer and data
formatting circuits on one board. The second board contains two digital signal processors
(DSPs). together with 6 Mbytes of memOly. Control of the whole system is provided by a board
containing a SPARC2 processor plus 64 Mbytes of memory. A schematic of this system is
shown in Figure 1.
Image buffering and communication with other modules in the address reader are handled by
the board with the SPARC2 processor. When an image is received. it is sent to the DSP board
and from there over to the NET32K processor. The feature maps produced by the NET32K
processor are stored on the DSP board. while the SPARC2 starts with the analysis of the feature
maps. The DSP's main task is formatting of the data. while the NET32K processor extracts all
the features. Its speed of computation is more than 100 billion multiply-accumulates per second
with operands that have one or two bits of resolution. Images with a size of Sl2xS 12 pixels are
processed at a rate of more than 10 frames per second. and 64 convolution kernels. each with
a size of 16x 16 pixels. can be scanned simultaneously over the image. Each such kernel IS
tuned to detect the presence of a feature. such as a line, an edge or a comer.
Address Block Location with a Neural Net System
....................................................................................
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Figure 1: Schematic of the whole NET32K system. Each of the dashed
boxes represents one 6U VME board. The
conununication paths.
aITOWS
show the
3. SEQUENCE OF ALGORITHMS
The final result of the address block location system is a box describing a tight bmmd around
the destination address, if the address is machine printed. Of handwritten addresses, only the
zip code is read, and hence, one has to find a tight boundary around the zip code. This
information is then passed along to reader modules of the address reading machine. There is no
a priori knowledge about the writing style. Therefore the system first has to discriminate
between handwritten and machine Plinted text. At the end of the address block location process,
additional algorithms are executed to improve the accuracy of the reader. An overview of the
sequence of algorithms used to solve these tasks is shown in Figure 2. The whole process is
divided into three major steps: Preprocessing, feature extraction. and high-level analysis based
on the feature information.
3.1. Preprocessing
To quickly get an idea about the complexity of the image, a coarse evaluation of its layout is
done. By sampling the density of the black pixels in various places of the image, one can see
already whether the image is clean or noisy and whether the text is lightly printed or is dark.
787
788
Oraf and Cosatto
The images are divided into four categories, depending on their darkness and the level of noise.
'This infonnation is used in the subsequent processing to guide the choice of the features. Only
about one percent of the pixels are taken into account for this analysis, therefore, it can be
executed quickly on the SPARC2 processor.
clean. light
Preprocessing
clean. dark
IF.
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Extract features
NET32K
8 Feature
maps
Extract text lines
Cluster lines into groups
- - - Classify groups of lines
MACHINE PRINT
Analyse group of lines
Determine level of noise
Clean with NET32K;
HANDWRITIEN
Cluster text segments into lines
Analyse group of lines
Segment lines to find ZIP
Determine slanVskew angle;
Figure 2: Schematic of the sequence of algorithms for finding the
position of the address blocks.
3.2. Feature Extraction
After the preprocessing, the image is sent to the NET32K board where simple geometrical
features, such as edges, corners and lines are extracted. Up to 16 different feature maps are
generated, where a pixel in one of the maps indicates the presence of a feature in this location.
Some of these feature maps are used by the host processor, for example, to decide whether text
is handwritten or machine printed. Other feature maps are combined and sent once more
through the NET32K processor in order to search for combinations of features representing
more complex features. Typically, the feature maps are thresholded, so that only one bit per
pixel is kept. More resolution of the computation results is available from the neural net chips.
but in this way the amount of data that has to be analyzed is minimal. and one bit of resolution
turned out to be sufficient.
Examples of kernels used for the detection of strokes and text lines are shown in Figure 3. In
the chip, usually four line detectors of increasing height plus eight stroke detectors of different
orientations are stored. Other detectors are tuned to edges and strokes of machine printed text.
The line detectors respond to any black line of the proper height. Due to the large width of 16
Address Block Location with a Neural Net System
pixels. a kernel stretches over one or even several characters. Hence a text line gives a response
similar to that produced by a continuous black line. When the threshold is set properly. a text
line in the original image produces a continuous line in the feature map. even across the gaps
between characters and across small empty spaces between words. For an interpretation of a
line feature map only the left and right end points of each connected component are stored. In
this way one obtains a compact representation of the lines' positions that are well suited for the
high-level analysis of the layout.
Kernel: Line detector
?
Image
t
the NET32K syste
IC::GUla
Feature
Kernel: Stroke detector
Feature map
Figure 3:Examples of convolution kernels and their results. The kernels' sizes
are 16x16 pixels, and their pixels' values are + 1, O. -1 . The upper part illustrates
the response of a line detector on a machine printed text line. The lower kernel
extracts strokes of a celtain orientation from handwritten text.
Handwritten lines are detected by a second technique, because they are more irregular in height
and the characters may be spaced apm1 widely. Detectors for strokes, of the type shown in the
lower half ofFigw-e 3. are well suited for sensing the presence of handwritten text. The feature
maps resulting form handwritten text tend to exhibit blobs of pixels along the text line. By
smearing such feature maps in horizontal direction the responses of individual strokes are
merged into lines that can then be used in the same way as described for the machine printed
lines.
Horizontal smearing of text lines. combined with connected component analysis is a well-known
789
790
Graf and Cosatto
technique, often applied in layout analysis, to find words and whole lines of text. But when
applied to the pixels of an image, such an approach works well only in clean images. As soon
as there is noise present, this technique produces ilTegular responses. The key to success in a
real world environment is robustness against noise. By extracting features first and then
analyzing the feature maps, we drastically reduce the influence of noise. Each of the convolution
kernels covers a range of 256 pixels and its response depends on several dozens of pixels inside
this area. If pixels in the image are corrupted by noise, this has only a minor effect on the result
of the convolution and, hence, the appearance of the feature map.
When the analysis is started, it is unknown, whether the address is machine printed or hand
written. In order to distinguish between the two writing styles, a simple one-layer classifier
looks at the results of four stroke detectors and of four line detectors. It can determine reliably
whether text is handwritten or machine printed. Additional useful information that can be
extracted easily from the feature maps, is the skew angle of handwritten text. People tend to
write with a skew anywhere from -45 degrees to almost +90 degrees. In order to improve the
accuracy of a reader, the text is first deskewed. The most time consuming part of this operation
is to determine the skew angle of the writing. The stroke detector with the maximum response
over a line is a good indicator of the skew angle of the text. We compared this simple technique
with several alternatives and found it to be as reliable as the best other algorithm and much
faster to compute.
3.3. High-level Analysis
The results of the feature extraction process are line segments, each one marked as handwritten
or machine printed. Only the left and right end points of such lines are stored. At this point,
there may still be line segments in this group that do not correspond to text, but rather to solid
black lines or to line drawings. Therefore each line segment is checked, to determine whether
the ratio of black and white pixels is that found typically in text.
Blocks of lines are identified by clustering the line segments into groups. Then each block is
analyzed, to see whether it can represent the destination address. For this purpose such features
as the number of lines in the block, its size, position, etc. are used. These features are entered
into a classifier that ranks each of the blocks. Certain conditions, such as a size that is too large,
or if there are too many text lines in the block, will lead to an attempt to split blocks. If no good
result is obtained, clustering is tried again with a changed distance metric, where the horizontal
and the vertical distances between lines are weighted differently.
If an address is machine printed, the whole address block is passed on to the reader, since not
only the zip code, but the whole address, including the city name, the street name and the name
of the recipient have to be read. A big problem for the reader present images of poor quality,
particularly those with background noise and texture. State-of-the-art readers handle machine
printed text reliably if the image quality is good, but they may fail totally if the text is buried in
noise. For that reason, an address block is cleaned before sending it to the reader. Feature
extraction with the NET32K board is used once more for this task, this time with detectors tuned
to find all the strokes of the machine printed text. Applying stroke detectors with the proper
width allows a good discrimination between the text and any noise. Even texture that consists
of lines can be rejected reliably, if the line thickness of the texture is not the same as that of the
text.
Address Block Location with a Neural Net System
.:
.
"3"" /"ksiQ \i~.\. Cal! [~
"
', '
~"S'~e".I ?
. ~..~
~
===t
,o;;;r;;;a.e;2 .
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.
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Figure 4: Example of an envelope image at various stages of the processing. Top: The
result of the clustering process to find the bounding box of the address. Bottom right: The
text lines within the address block are marked. Bottom left: Cuts in the text line with the
zip code and below that the result of the reader. (The zip code is actually the second
segment sent to the reader; the first one is the string 'USA').
If the address is handwritten, only the zip code is sent to the reader. In order to find the zip code,
an analysis of the internal stmcture of the address block has to be done, which starts with finding
the true text lines. Handwritten lines are often not straight, may be heavily skewed, and may
contain large gaps. Hence simple techniques, such as connected component analysis, do not
provide proper results. ClusteJing of the line segments obtained from the feature maps, provides
a reliable solution of this problem. Once the lines are found, each one is segmented into words
and some of them are selected as candidates for the zip code and are sent to the reader. Figure
4 shows an example of an envelope image as it progresses through the various processing steps.
The system has been tested extensively on overall more than 100,000 images. Most of these
tests were done in the assembled address reader, but during development of the system, large
791
792
Graf and Cosatto
tests were also done with the address location module alone. One of the problems for evaluating
the peIformance is the lack of an objective quality measure. When has an address been located
correctly? Cutting off a small part of the address may not be detrimental to the final
interpretation, while a bounding box that includes some additional text may slow the reader
down too much. or it may throw off the interpretation. Therefore, it is not always clear when a
bounding box, describing the address' location, is tight enough. Another important factor
affecting the accw-acy numbers is, how many candidate blocks one actually considers. For all
these reasons, accw-acy numbers given for address block location have to be taken with some
caution. The results mentioned here were obtained by judging the images by eye. If images are
clean and the address is surrounded by a white space larger than two line heights, the location
is found correctly in more than 98% of the cases. Often more than one text block is found and
of these the destination address is the first choice in 90% of the images, for a typical layout. If
the image is very noisy, which actually happens surprisingly often, a tight bound around the
address is found in 85% of the cases. These results were obtained with 5,000 images, chosen
from more than 100,000 images to represent as much variety as possible. Of these 5,000 images
more than 1,200 have a texture around the address, and often this texture is so dark that a
human has difficulties to make out each character.
4. CONCLUSION
Most of our algorithms described here consist of two parts: feature extraction implemented with
a convolution and interpretation, typically implemented with a small classifier. Surprisingly
many algorithms can be cast into such a fOimat. This common framework for algorithms has
the advantage of facilitating the implementation, in particular when algorithms are mapped into
hardware. Moreover, the feature extraction with large convolution kernels makes the system
robust against noise. This robustness is probably the biggest advantage of our approach. Most
existing automatic reading systems are very good as long as the images are clean, but they
deteriorate rapidly with decreasing image quality.
'The biggest drawback of convolutions is that they require a lot of computation. In fact, without
special purpose hardware, convolutions are often too slow. Our system relies on the NET32K
new-al net chips to obtain the necessary throughput. The NET32K system is, we believe, at the
moment the fastest board system for this type of computation. This speed is obtained by
systematically exploiting the fact that only a low resolution of the computation is required. This
allows to use analog computation inside the chip and hence much smaller circuits than would
be the case in an all-digital circuit.
References
United States Postal Service, (1992), Proc. Advanced Technology Conf., Vol. 3, Section on
address block location: pp. 1221 - 1310.
P.W. Palumbo, S.N. Srihari, J. Soh, R. Sridhar, V. Demjanenko, (1992), !'Postal Address Block
Location in Real Time", IEEE COMPUTER, Vol. 25n, pp. 34 - 42.
H.P. Oraf and D. Henderson, (1990), "A Reconfigurable CMOS Neural Network", Digest
IEEE Int. Solid State Circuits Conf. p. 144.
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marked:2 content:1 typical:1 discriminate:1 internal:1 people:1 tested:2 |
7,003 | 78 | 358
LEARNING REPRESENTATIONS BY RECIRCULATION
Geoffrey E. Hinton
Computer Science and Psychology Departments, University of Toronto,
Toronto M5S lA4, Canada
James L. McClelland
Psychology and Computer Science Departments, Carnegie-Mellon University,
Pittsburgh, PA 15213
ABSTRACT
We describe a new learning procedure for networks that contain groups of nonlinear units arranged in a closed loop. The aim of the learning is to discover codes
that allow the activity vectors in a "visible" group to be represented by activity
vectors in a "hidden" group. One way to test whether a code is an accurate
representation is to try to reconstruct the visible vector from the hidden vector. The
difference between the original and the reconstructed visible vectors is called the
reconstruction error, and the learning procedure aims to minimize this error. The
learning procedure has two passes. On the fust pass, the original visible vector is
passed around the loop, and on the second pass an average of the original vector and
the reconstructed vector is passed around the loop. The learning procedure changes
each weight by an amount proportional to the product of the "presynaptic" activity
and the difference in the post-synaptic activity on the two passes. This procedure is
much simpler to implement than methods like back-propagation. Simulations in
simple networks show that it usually converges rapidly on a good set of codes, and
analysis shows that in certain restricted cases it performs gradient descent in the
squared reconstruction error.
INTRODUCTION
Supervised gradient-descent learning procedures such as back-propagation 1
have been shown to construct interesting internal representations in "hidden" units
that are not part of the input or output of a connectionist network. One criticism of
back-propagation is that it requires a teacher to specify the desired output vectors. It
is possible to dispense with the teacher in the case of "encoder" networks 2 in which
the desired output vector is identical with the input vector (see Fig. 1). The purpose
of an encoder network is to learn good "codes" in the intermediate, hidden units. If
for, example, there are less hidden units than input units, an encoder network will
perform data-compression 3 . It is also possible to introduce other kinds of constraints
on the hidden units, so we can view an encoder network as a way of ensuring that the
input can be reconstructed from the activity in the hidden units whilst also making
nus research was supported by contract NOOOl4-86-K-00167 from the Office of Naval Research
and a grant from the Canadian National Science and Engineering Research Council. Geoffrey Hinton
is a fellow of the Canadian Institute for Advanced Research. We thank: Mike Franzini, Conrad
Galland and Geoffrey Goodhill for helpful discussions and help with the simulations.
? American Institute of Physics 1988
359
the hidden units satisfy some other constraint.
A second criticism of back-propagation is that it is neurally implausible (and
hard to implement in hardware) because it requires all the connections to be used
backwards and it requires the units to use different input-output functions for the
forward and backward passes. Recirculation is designed to overcome this second
criticism in the special case of encoder networks.
output units
I \
hidden units
/
r-.
input units
Fig. 1. A diagram of a three layer encoder network that learns good codes using
back-propagation. On the forward pass, activity flows from the input units in the
bottom layer to the output units in the top layer. On the backward pass, errorderivatives flow from the top layer to the bottom layer.
Instead of using a separate group of units for the input and output we use the
very same group of "visible" units, so the input vector is the initial state of this group
and the output vector is the state after information has passed around the loop. The
difference between the activity of a visible unit before and after sending activity
around the loop is the derivative of the squared reconstruction error. So, if the
visible units are linear, we can perfonn gradient descent in the squared error by
changing each of a visible unit's incoming weights by an amount proportional to the
product of this difference and the activity of the hidden unit from which the
connection emanates. So learning the weights from the hidden units to the output
units is simple. The harder problem is to learn the weights on connections coming
into hidden units because there is no direct specification of the desired states of these
units. Back-propagation solves this problem by back-propagating error-derivatives
from the output units to generate error-derivatives for the hidden units.
Recirculation solves the problem in a quite different way that is easier to implement
but much harder to analyse.
360
THE RECIRCULATION PROCEDURE
We introduce the recirculation procedure by considering a very simple
architecture in which there is just one group of hidden units. Each visible unit has a
directed connection to every hidden unit, and each hidden unit has a directed
connection to every visible unit. The total input received by a unit is
Xj = LYiWji - 9j
(1)
i
where Yi is the state of the i th unit, K'ji is the weight on the connection from the i th to
the Jib unit and 9j is the threshold of the Jh unit. The threshold tenn can be
eliminated by giving every unit an extra input connection whose activity level is
fIXed at 1. The weight on this special connection is the negative of the threshold, and
it can be learned in just the same way as the other weights. This method of
implementing thresholds will be assumed throughout the paper.
The functions relating inputs to outputs of visible and hidden units are smooth
monotonic functions with bounded derivatives. For hidden units we use the logistic
function:
y. = <1(x.) =
J
J
I
I +e-Xj
(2)
Other smooth monotonic functions would serve as well. For visible units, our
mathematical analysis focuses on the linear case in which the output equals the total
input, though in simulations we use the logistic function.
We have already given a verbal description of the learning rule for the hiddento-visible connections. The weight, Wij , from the Ih hidden unit to the itlr visible
unit is changed as follows:
f:t.wij = ?y/I) [Yi(O)-Yi(2)]
(3)
where Yi(O) is the state of the i th visible unit at time 0 and Yi(2) is its state at time 2
after activity has passed around the loop once. The rule for the visible-to-hidden
connections is identical:
(4)
where y/I) is the state of the lh hidden unit at time I (on the frrst pass around the
loop) and y/3) is its state at time 3 (on the second pass around the loop). Fig. 2
shows the network exploded in time.
In general, this rule for changing the visible-to-hidden connections does not
perfonn steepest descent in the squared reconstruction error, so it behaves differently
from back-propagation. This raises two issues: Under what conditions does it work,
and under what conditions does it approximate steepest descent?
361
time =3
time = 1
time =2
time =0
Fig. 2. A diagram showing the states of the visible and hidden units exploded in
time. The visible units are at the bottom and the hidden units are at the top. Time
goes from left to right.
CONDITIONS UNDER WHICH RECIRCULATION
APPROXIMATES GRADIENT DESCENT
For the simple architecture shown in Fig. 2, the recirculation learning procedure
changes the visible-to-hidden weights in the direction of steepest descent in the
squared reconstruction error provided the following conditions hold:
1. The visible units are linear.
2. The weights are symmetrical (i.e. wji=wij for all i,j).
3. The visible units have high regression.
"Regression" means that, after one pass around the loop, instead of setting the
activity of a visible unit, i, to be equal to its current total input, x i (2), as determined
by Eq 1, we set its activity to be
y;(2)
= AY;(O) + (I-A)x;(2)
(5)
where the regression, A, is close to 1. Using high regression ensures that the visible
units only change state slightly so that when the new visible vector is sent around the
loop again on the second pass, it has very similar effects to the first pass. In order to
make the learning rule for the hidden units as similar as possible to the rule for the
visible units, we also use regression in computing the activity of the hidden units on
the second pass
(6)
For a given input vector, the squared reconstruction error, E, is
For a hidden unit, j,
362
where
For a visible-to...hidden weight wj ;
dE,
dE
= Yj(1)Yi(O)-dwj ;
dYj(l)
So, using Eq 7 and the assumption that Wkj=wjk for all k,j
dE
dw??}l
=y/(l) y;(O) [LYk(2) Yk'(2) Wjk k
LYk(O) Yk'(2) Wjk]
k
The assumption that the visible units are linear (with a gradient of 1) means that
for all k, Yk'(2) = 1. So using Eq 1 we have
dE = y.'(l) y.(O)[x.(3)-x~1)]
dw ..
}
I
)
}
(8)
}l
Now, with sufficiently high regression, we can assume that the states of units
only change slightly with time so that
and
Yt(O) ::::: y;(2)
So by substituting in Eq 8 we get
dE
1
-aw::::: (1 _ A) y;(2) [y/3) ji
y/l)]
(9)
An interesting property of Eq 9 is that it does not contain a tenn for the gradient
of the input-output function of unit } so recirculation learning can be applied even
when unit} uses an unknown non-linearity. To do back-propagation it is necessary to
know the gradient of the non-linearity, but recirculation measures the gradient by
measuring the effect of a small difference in input, so the tenn y/3)-y/l) implicitly
contains the gradient.
363
A SIMULATION OF RECIRCULATION
From a biological standpoint, the synunetry requirement that wij=Wji is
unrealistic unless it can be shown that this synunetry of the weights can be learned.
To investigate what would happen if synunetry was not enforced (and if the visible
units used the same non-linearity as the hidden units), we applied the recirculation
learning procedure to a network with 4 visible units and 2 hidden units. The visible
vectors were 1000, 0100, 0010 and 0001, so the 2 hidden units had to learn 4
different codes to represent these four visible vectors. All the weights and biases in
the network were started at small random values uniformly distributed in the range
-0.5 to +0.5. We used regression in the hidden units, even though this is not strictly
necessary, but we ignored the teon 1/ (1 - A) in Eq 9.
Using an E of 20 and a A. of 0.75 for both the visible and the hidden units, the
network learned to produce a reconstruction error of less than 0.1 on every unit in an
average of 48 weight updates (with a maximum of 202 in 100 simulations). Each
weight update was perfonned after trying all four training cases and the change was
the sum of the four changes prescribed by Eq 3 or 4 as appropriate. The final
reconstruction error was measured using a regression of 0, even though high
regression was used during the learning. The learning speed is comparable with
back-propagation, though a precise comparison is hard because the optimal values of
E are different in the two cases. Also, the fact that we ignored the tenn 1/ (1- A.)
when modifying the visible-to-hidden weights means that recirculation tends to
change the visible-to-hidden weights more slowly than the hidden-to-visible weights,
and this would also help back -propagation.
It is not inunediately obvious why the recirculation learning procedure works
when the weights are not constrained to be synunetrical, so we compared the weight
changes prescribed by the recirculation procedure with the weight changes that
would cause steepest descent in the sum squared reconstruction error (i.e. the weight
changes prescribed by back-propagation). As expected, recirculation and backpropagation agree on the weight changes for the hidden-to-visible connections, even
though the gradient of the logistic function is not taken into account in weight
adjustments under recirculation. (Conrad Galland has observed that this agreement
is only slightly affected by using visible units that have the non-linear input-output
function shown in Eq 2 because at any stage of the learning, all the visible units tend
to have similar slopes for their input-output functions, so the non-linearity scales all
the weight changes by approximately the same amount.)
For the visible-to-hidden connections, recirculation initially prescribes weight
changes that are only randomly related to the direction of steepest descent, so these
changes do not help to improve the perfonnance of the system. As the learning
proceeds, however, these changes come to agree with the direction of steepest
descent. The crucial observation is that this agreement occurs after the hidden-tovisible weights have changed in such a way that they are approximately aligned
(symmetrical up to a constant factor) with the visible-to-hidden weights. So it
appears that changing the hidden-to-visible weights in the direction of steepest
descent creates the conditions that are necessary for the recirculation procedure to
cause changes in the visible-to-hidden weights that follow the direction of steepest
descent.
It is not hard to see why this happens if we start with random, zero-mean
364
visible-to-hidden weights. If the visible-to-hidden weight wji is positive, hidden unit
j will tend to have a higher than average activity level when the ith visible unit has a
higher than average activity. So Yj will tend to be higher than average when the
reconstructed value of Yi should be higher than average -- i.e. when the tenn
[Yi(O)-Yi(2)] in Eq 3 is positive. It will also be lower than average when this tenn is
negative. These relationships will be reversed if w ji is negative, so w ij will grow
faster when wJi is positive than it will when wji is negative. Smolensky4 presents a
mathematical analysis that shows why a similar learning procedure creates
symmetrical weights in a purely linear system. Williams 5 also analyses a related
learning rule for linear systems which he calls the "symmetric error correction"
procedure and he shows that it perfonns principle components analysis. In our
simulations of recirculation, the visible-to-hidden weights become aligned with the
corresponding hidden-to-visible weights, though the hidden-to-visible weights are
generally of larger magnitude.
A PICTURE OF RECIRCULATION
To gain more insight into the conditions under which recirculation learning
produces the appropriate changes in the visible-to-hidden weights, we introduce the
pictorial representation shown in Fig. 3. The initial visible vector, A, is mapped into
the reconstructed vector, C, so the error vector is AC. Using high regression, the
visible vector that is sent around the loop on the second pass is P, where the
difference vector AP is a small fraction of the error vector AC. If the regression is
sufficiently high and all the non-linearities in the system have bounded derivatives
and the weights have bounded magnitudes, the difference vectors AP, BQ, and CR
will be very small and we can assume that, to first order, the system behaves linearly
in these difference vectors. If, for example, we moved P so as to double the length
of AP we would also double the length of BQ and CR.
Fig. 3. A diagram showing some vectors (A, P) over the visible units, their
"hidden" images (B, Q) over the hidden units, and their "visible" images (C, R)
over the visible lUlits. The vectors B' and C' are the hidden and visible images of
A after the visible-to-hidden weights have been changed by the learning procedure.
365
Suppose we change the visible-to-hidden weights in the manner prescribed by
Eq 4, using a very smaIl value of ?. Let Q' be the hidden image of P (i.e. the image
of P in the hidden units) after the weight changes. To first order, Q' will lie between
B and Q on the line BQ. This follows from the observation that Eq 4 has the effect
of moving each y/3) towards y/l) by an amount proportional to their difference.
Since B is close to Q, a weight change that moves the hidden image of P from Q to
Q' will move the hidden image of A from B to B', where B' lies on the extension of
the line BQ as shown in Fig. 3. If the hidden-to-visible weights are not changed, the
visible image of A will move from C to C', where C' lies on the extension of the line
CR as shown in Fig. 3. So the visible-to-hidden weight changes will reduce the
squared reconstruction error provided the vector CR is approximately parallel to the
vector AP.
But why should we expect the vector CR to be aligned with the vector AP? In
general we should not, except when the visible-to-hidden and hidden-to-visible
weights are approximately aligned.
The learning in the hidden-to-visible
connections has a tendency to cause this alignment. In addition, it is easy to modify
the recirculation learning procedure so as to increase the tendency for the learning in
the hidden-to-visible connections to cause alignment. Eq 3 has the effect of moving
the visible image of A closer to A by an amount proportional to the magnitude of the
error vector AC. If we apply the same rule on the next pass around the loop, we
move the visible image of P closer to P by an amount proportional to the magnitude
of PRo If the vector CR is anti-aligned with the vector AP, the magnitude of AC will
exceed the magnitude of PR, so the result of these two movements will be to
improve the alignment between AP and CR. We have not yet tested this modified
procedure through simulations, however.
This is only an infonnal argument and much work remains to be done in
establishing the precise conditions under which the recirculation learning procedure
approximates steepest descent. The infonnal argument applies equally well to
systems that contain longer loops which have several groups of hidden units
arranged in series. At each stage in the loop, the same learning procedure can be
applied, and the weight changes will approximate gradient descent provided the
difference of the two visible vectors that are sent around the loop aligns with the
difference of their images. We have not yet done enough simulations to develop a
clear picture of the conditions under which the changes in the hidden-to-visible
weights produce the required alignment.
USING A HIERARCHY OF CLOSED LOOPS
Instead of using a single loop that contains many hidden layers in series, it is
possible to use a more modular system. Each module consists of one "visible" group
and one "hidden" group connected in a closed loop, but the visible group for one
module is actually composed of the hidden groups of several lower level modules, as
shown in Fig. 4. Since the same learning rule is used for both visible and hidden
units, there is no problem in applying it to systems in which some units are the
visible units of one module and the hidden units of another. Ballard6 has
experimented with back-propagation in this kind of system, and we have run some
simulations of recirculation using the architecture shown in Fig. 4. The network
366
learned to encode a set of vectors specified over the bottom layer. After learning,
each of the vectors became an attractor and the network was capable of completing a
partial vector, even though this involved passing information through several layers.
00
00
00
0000
0000
Fig 4. A network in which the hidden units of the bottom two modules are the
visible units of the top module.
CONCLUSION
We have described a simple learning procedure that is capable of fonning
representations in non-linear hidden units whose input-output functions have
bounded derivatives. The procedure is easy to implement in hardware, even if the
non-linearity is unknown. Given some strong assumptions, the procedure petforms
gradient descent in the reconstruction error. If the synunetry assumption is violated,
the learning procedure still works because the changes in the hidden-to-visible
weights produce symmetry. H the assumption about the linearity of the visible units
is violated, the procedure still works in the cases we have simulated. For the general
case of a loop with many non-linear stages, we have an informal picture of a
condition that must hold for the procedure to approximate gradient descent, but we
do not have a fonnal analysis, and we do not have sufficient experience with
simulations to give an empirical description of the general conditions under which
the learning procedure works.
REFERENCES
1. D. E. Rumelhart, G. E. Hinton and R.I. Williams, Nature 323, 533-536 (1986).
2. D. H. Ackley, G. E. Hinton and T. 1. Sejnowski, Cognitive Science 9,147-169
(1985).
3. G. Cottrell, 1. L. Elman and D. Zipser, Proc. Cognitive Science Society, Seattle,
WA (1987).
4. P. Smolensky, Technical Report CU-CS-355-87, University of Colorado at
Boulder (1986).
5. R.I. Williams, Technical Report 8501, Institute of Cognitive Science, University
ofCalifomia, San Diego (1985).
6. D. H. Ballard, Proc. American Association for Artificial Intelligence, Seattle, W A
(1987).
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7,004 | 780 | Dynamic Modulation of Neurons and Networks
Eve Marder
Center for Complex Systems
Brandeis University
Waltham, MA 02254 USA
Abstract
Biological neurons have a variety of intrinsic properties because of the
large number of voltage dependent currents that control their activity.
Neuromodulatory substances modify both the balance of conductances
that determine intrinsic properties and the strength of synapses. These
mechanisms alter circuit dynamics, and suggest that functional circuits
exist only in the modulatory environment in which they operate.
1 INTRODUCTION
Many studies of artificial neural networks employ model neurons and synapses that are
considerably simpler than their biological counterparts. A variety of motivations underly
the use of simple models for neurons and synapses in artificial neural networks. Here,
I discuss some of the properties of biological neurons and networks that are lost in overly
simplified models of neurons and synapses. A fundamental principle in biological nervous
systems is that neurons and networks operate over a wide range of time scales, and that
these are modified by neuromodulatory substances. The flexible, multiple time scales in
the nervous system allow smooth transitions between different modes of circuit operation.
2 NEURONS HAVE DIF'FERENT INTRINSIC PROPERTIES
Each neuron has complex dynamical properties that depend on the number and kind of
ion channels in its membrane. Ion channels have characteristic kinetics and voltage
511
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Marder
dependencies that depend on the sequence of amino acids of the protein. Ion channels
may open and close in several milliseconds; others may stay open for hundreds of
milliseconds or several seconds.
Some neurons are silent unless they receive synaptic inputs. Silent neurons can be
activated by depolarizing synaptic inputs, and many will fire on rebound from a
hyperpolarizing input (postinhibitory rebound). Some neurons are tonically active in the
absence of synaptic inputs, and synaptic inputs will increase or decrease their firing rate.
Some neurons display rhythmic bursts of action potentials. These bursting neurons can
display stable patterns of oscillatory activity, that respond to perturbing stimuli with
behavior characteristic of oscillators, in that their period can be stably reset and
entrained. Bursting neurons display a number of different voltage and time dependent
conductances that interact to produce slow membrane potential oscillations with rapid
action potentials riding on the depolarized phase. In a neuron such as R15 of Aplysia
(Adams and Levitan 1985) or the AB neuron of the stomatogastric ganglion (STG)
(Harris-Warrick and Flamm 1987), the time scale of the burst is in the second range, but
the individual action potentials are produced in the 5-10msec time scale.
Neurons can generate bursts by combining a variety of different conductances. The
particular balance of these conductances can have significant impact on the oscillator's
behavior (Epstein and Marder 1990; Kepler et aI1990; Skinner et aI1993), and therefore
the choice of oscillator model to use must be made with care (Somers and Kopell 1993).
Some neurons have a balance of conductances that give them bistable membrane
potentials, allowing to produce plateau potentials. Typically, such neurons have two
relatively stable states, a hyperpolarized silent state, and a sustained depolarized state in
which they fire action potentials. The transition between these two modes of activity can
be made with a short depolarizing or hyperpolarizing pulse (Fig. 1). Plateau potentials,
like "flip-flops" in electronics, are a "short-term memory" mechanism for neural circuits.
The intrinsic properties of neurons can be modified by sustained changes in membrane
potential. Because the intrinsic properties of neurons depend on the balance of
conductances that activate and inactivate in different membrane potential ranges and over
a variety of time scales, hyperpolarization or depolarization can switch a neuron between
modes of intrinsic activity (Llinas 1988; McCormick 1991; Leresche et aI1991).
An interesting "memory-like" effect is produced by the slow inactivation properties of
some K+ currents (McCormick 1991; Storm 1987). In cells with such currents a
sustained depolarization can "amplify" a synaptic input from subthreshold to
suprathreshold, as the sustained depolarization causes the K+ current to inactivate
(Marom and Abbott 1994; Turrigiano, Marder and Abbott in preparation). This is another
"short-term memory" mechanism that does not depend on changes in synaptic efficacy.
Dynamic Modulation of Neurons and Networks
A. CONTROL
\20mV
I04nA
1...,
Figure 1: Intracellular recording from the DG neuron ofthe crab STG. A: control saline,
a depolarizing current pulse elicits action potentials for its duration. B: In
SDRNFLRFamide, a short depolarization elicits a plateau potential that lasts until a short
hyperpolarizing current pulse terminates it. Modified from Weimann et al 1993.
2 INTRINSIC MEMBRANE PROPERTIES ARE MODULATED
Biological nervous systems use many substances as neurotransmitters and
neuromodulators. The effects of these substances include opening of rapid, relatively nonvoltage dependent ion channels, such as those mediating conventional rapid synaptic
potentials. Alternatively, modulatory substances can change the number or type of
voltage-dependent conductances displayed by a neuron, and in so doing dramatically
modify the intrinsic properties of a neuron. In Fig. 1, a peptide, SDRNFLRFamide
transforms the DG neuron of the crab STG from a state in which it fires only during a
depolarizing pulse to one in which it displays long-lasting plateau properties (Weimann
et al 1993). The salient feature here is that modulatory substances can elicit slow
membrane properties not otherwise expressed.
3 SYNAPTIC STRENGTH IS MODULATED
In most neural network models synaptic weights are modified by learning rules, but are
not dependent on the temporal pattern of presynaptic activity. In contrast, in many
biological synapses the amount of transmitter released depends on the frequency of firing
of the presynaptic neuron. Facilitation, the increase in the amplitude of the postsynaptic
current when' the presynaptic neuron is activated several times in quick succession is quite
common. Other synapses show depression. The same neuron may show facilitation at
some of its terminals while showing depression at others (Katz et al 1993). The
facilitation and depression properties of any given synapse can not be deduced on first
principles, but must be determined empirically.
Synaptic efficacy is often modified by modulatory substances. A dramatic example is seen
in the Aplysia gill withdrawal reflex, where serotonin significantly enhances the amplitude
of the monosynaptic connection from the sensory to motor neurons (Clark and Kandel
1993; Emptage and Carew 1993). The effects of modulatory substances can occur on
different branches on a neuron independently (Clark and Kandel 1993), and the same
modulatory substance may have different actions at different sites of the same neuron.
513
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Marder
Electrical synapses are also subject to neuromodulation (Dowling, 1989). For example,
in the retina dopamine reversibly uncouples horizonal cells.
Modulation of synaptic strength can be quite extreme; in some cases synaptic contacts
may be virtually invisible in some modulatory environments, while strong in others. The
implications of this for circuit ooeration will be discussed below.
Hormones
Neuromodulators
.DA
? ACh
.OA
LJ5-HT
?
Oct
II CCAP
II cCCK
?
?
AST
Buc
ID
.HA .LK
rnm GABA
?
Oct
IIlomTK
? APCH
cCCK
I'IlomTK
1'? 1
Proc
Myomod
?
?
RPCH
SOAN
1'1 TNRN
.
Sensory Transmitters
?
ACh }
~~~
Figure 2: Modulatory substances found in inputs to the STG. See Harris-Warrick et aI.,
1992 for details. Figure courtesy of P. Skiebe.
4 TRANS:MITTERS ARE COLOCALIZED IN NEURONS
The time course of a synaptic potential evoked by a neurotransmitter or modulator is a
characteristic property of the ion channels gated by the transmitter and/or the second
messenger system activated by the signalling molecule. Synaptic currents can be relatively
fast, such as the rapid action of ACh at the vertebrate skeletal neuromuscular junction
where the synaptic currents decay in several milliseconds. Alternatively, second
messenger activated synaptic events may have durations lasting hundreds of milliseconds,
seconds, or even minutes. Many neurons contain several differen.t neurotransmitters.
It is common to find a small molecule such as glutamate or GABA colocalized with an
amine such as serotonin or histamine and one or more neuropeptides. To describe the
synaptic actions of such neurons, it is necessary to determine for each- signalling molecule
how its release depends on the frequency and pattern of activity in the presynaptic
Dynamic Modulation of Neurons and Networks
terminal, and characterize its postsynaptic actions. This is important, because different
mixtures of cotransmitters, and consequently of postsynaptic action may occur with
different presynaptic patterns of activity.
5 NEURAL NETWORKS ARE MULTIPLY MODULATED
Neural networks are controlled by many modulatory inputs and substances. Figure 2
illustrates the patterns of modulatory control to the crustacean stomatogastric nervous
system, where the motor patterns produced by the only 30 neurons of the stomatogastric
ganglion are controlled by about 60 input fibers (Coleman et a11992) that contain at least
15 different substances, including a variety of amines, amino acids, and neuropeptides
(Marder and Weimann 1992; Harris-Warrick et aI1992). Each of these
modulatory substances produces characteristic and different effects on the motor patterns
of the STG (Figs. 3,4). This can be understood if one remembers that the intrinsic
membrane properties as well as the strengths of the synaptic connections within this
group of neurons are all subject to modulation. Because each cell has many conductances,
many of which are subject to modulation, and because of the large number of synaptic
connections, the modes of circuit operation are theoretically large.
6 CIRCUIT RECONFIGURA TION BY MODULATORY CONTROL
Figure 3 illustrates that modulatory substances can tune the operation of a single
functional circuit. However, neuromodulatory substances can also produce far more
extensive changes in the functional organization of neuronal networks. Recent work on
the STG demonstrates that sensory and modulatory neurons and substances can cause
neurons to switch between different functional circuits, so that the same neuron is part
of several different pattern generating circuits at different times (Hooper and Moulins
1989; Dickinson et al 1990; Weimann et al 1991; Meyrand et al 1991; Heinzel et al
1993).
In the example shown in Fig. 4, in control saline the LG neuron is firing in time with the
fast pyloric rhythm (the LP neuron is also firing in pyloric time), but there is no ongoing
gastric rhythm. When the gastric rhythm was activated by application of the peptide
SDRNFLRF NHz , the LG neuron fired in time with the gastric rhythm (Weimann et al
1993). These and other data lead us to conclude that it is the modulatory environment
that constructs the functional circuit that produces a given behavior (Meyrand et al
1991). Thus, by tuning intrinsic membrane properties and synaptic strengths,
neuromodulatory agents can recombine the same neurons into a variety of circuits,
capable of generating remarkably distinct outputs.
Acknowledgements
I thank Dr. Petra Skiebe for Fig 3 art work. Research was supported by NSI7813.
SIS
516
Marder
CONTROL
PILOCARPINE
SEROTONIN
Figure 3: Different forms of the pyloric rhythm different modulators. Each panel, the top
two traces: simulataneous intracellular recordings from LP and PD neurons of crab STG;
bottom trace: extracellular recording, Ivn nerve. Control, rhythmic pyloric activity
absent. Substances were bath applied, the pyloric patterns produced were different.
Modified from Marder and Weimann 1992.
dgn
Figure 4: Neurons switch between different pattem-genreating circuits. Left panel, the
gastric rhythm not active (monitored by DG neuron), LG neuron in time with the pyloric
rhythm (seen as activity in LP neuron). Right panel, gastric rhythm activated by
SDRNFLRFamide, monitored by the DG neuron bursts recorded on the dgn. LG now
fired in alternation with DG neuron. Pyloric time is seen as the interruptions in the
activity of the VD neuron. Modified from Marder and Weimann 1992.
Dynamic Modulation of Neurons and Networks
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| 780 |@word hyperpolarized:1 open:2 pulse:4 lowfrequency:1 dramatic:1 electronics:1 efficacy:2 current:13 si:1 must:2 evans:1 hyperpolarizing:3 physiol:3 underly:1 motor:7 tenn:1 nervous:7 signalling:2 coleman:2 short:5 kepler:2 simpler:1 burst:5 sustained:4 behavioral:2 theoretically:1 rapid:5 behavior:3 terminal:2 jm:2 vertebrate:1 monosynaptic:1 circuit:15 panel:3 kind:1 depolarization:4 selverston:1 nj:1 temporal:2 demonstrates:1 rm:2 control:9 inactivating:2 understood:1 local:1 modify:2 acad:1 switching:1 id:1 oxford:1 firing:4 modulation:8 bursting:5 evoked:1 dif:1 range:3 lost:1 lf:2 elicit:1 significantly:1 suggest:1 protein:1 amplify:1 close:1 ga:1 selection:1 ast:1 cybern:3 conventional:1 quick:1 center:1 courtesy:1 duration:3 independently:1 stomatogastric:8 rule:1 insight:1 facilitation:7 construction:1 dickinson:2 mammalian:1 bottom:1 role:1 electrical:2 remote:1 decrease:1 environment:3 pd:1 dynamic:6 depend:4 recombine:1 neurophysiol:1 neurotransmitter:3 fiber:1 distinct:1 fast:3 describe:1 activate:1 modulators:1 artificial:2 quite:2 otherwise:1 serotonin:4 sequence:1 rr:1 turrigiano:2 biophysical:1 borealis:2 reset:1 combining:1 bath:1 fired:2 neurone:1 potassium:1 ach:3 p:2 produce:5 generating:3 adam:2 oscillating:1 coupling:1 strong:1 waltham:1 petra:1 bistable:1 suprathreshold:1 govind:1 carew:2 repertoire:1 biological:7 kinetics:1 crab:7 exp:1 bj:1 released:1 relay:1 tonically:1 proc:2 geniculate:1 peptide:2 mit:1 modified:7 ck:1 inactivation:2 moulins:4 voltage:5 release:2 oflong:1 transmitter:3 contrast:1 stg:7 dependent:6 typically:1 flexible:1 warrick:5 art:1 integration:1 construct:1 rebound:2 alter:1 others:3 stimulus:1 employ:1 opening:1 retina:2 dg:5 individual:1 flamm:2 phase:1 fire:3 saline:2 ab:1 conductance:8 organization:1 multiply:1 mixture:1 extreme:1 activated:6 natl:1 tj:1 implication:1 capable:1 necessary:1 unless:1 re:1 modeling:1 wb:1 hundred:2 characterize:1 dependency:2 hooper:2 considerably:1 deduced:1 fundamental:1 stay:1 central:1 neuromodulators:2 recorded:1 slowly:2 dr:1 potential:14 mp:1 mv:1 depends:2 tion:1 doing:1 depolarizing:4 vivo:1 ir:1 acid:2 characteristic:4 succession:1 subthreshold:1 ofthe:1 identification:1 produced:4 calabrese:1 reversibly:1 comp:1 kopell:2 oscillatory:2 synapsis:8 plateau:4 messenger:2 synaptic:21 ed:2 frequency:4 storm:2 monitored:2 crustacean:1 car:1 electrophysiological:2 amplitude:2 nerve:1 marom:2 llinas:2 synapse:1 until:1 simulataneous:1 mode:5 stably:1 epstein:2 riding:1 usa:2 effect:6 contain:2 counterpart:1 postinhibitory:1 skinner:2 during:1 rhythm:8 rat:1 hippocampal:1 invisible:1 common:2 functional:6 hyperpolarization:1 perturbing:1 empirically:1 rl:1 discussed:1 katz:2 significant:1 dowling:2 cambridge:1 ai:2 neuromodulatory:4 tuning:1 fk:1 stable:2 recent:1 emptage:2 alternation:1 seen:3 care:1 gill:1 determine:2 period:1 ii:2 branch:2 multiple:6 smooth:1 hormone:1 long:2 controlled:2 impact:1 dopamine:2 ion:6 cell:5 receive:1 remarkably:1 neuromuscular:1 operate:2 depolarized:2 induced:1 recording:3 subject:3 virtually:1 entrained:1 eve:1 variety:6 switch:3 modulator:1 silent:3 absent:1 nusbaum:1 cause:2 weimann:11 action:10 depression:4 dramatically:1 modulatory:16 tune:1 transforms:1 amount:1 generate:1 dgn:2 exist:1 amine:2 sl:1 millisecond:4 overly:1 skeletal:1 group:1 salient:1 rnm:1 inactivate:2 threshold:1 abbott:4 ht:1 colocalized:2 respond:1 pattem:1 somers:2 oscillation:1 horizonal:1 display:4 activity:12 marder:19 strength:5 occur:2 r15:1 relatively:3 extracellular:1 heinzel:2 neur:2 gaba:2 membrane:11 terminates:1 postsynaptic:3 lp:3 lasting:2 discus:1 mechanism:5 neuromodulation:2 flip:1 junction:1 operation:3 differen:1 top:1 include:1 levitan:2 contact:1 pergamon:1 md:1 interruption:1 enhances:1 elicits:2 thank:1 sci:1 oa:1 vd:1 lateral:1 presynaptic:6 induction:1 balance:4 lg:4 mediating:1 trace:2 gated:1 allowing:1 mccormick:3 neuron:67 aplysia:5 displayed:1 flop:1 neurobiology:1 gastric:7 extensive:1 connection:3 trans:1 dynamical:1 pattern:14 below:1 pig:1 tb:1 including:1 memory:3 event:1 examination:1 glutamate:1 pilocarpine:1 pyloric:8 lk:1 shortterm:1 remembers:1 acknowledgement:1 synchronization:1 interesting:1 clark:3 generator:2 ivn:1 agent:1 principle:2 cancer:3 course:1 supported:1 last:1 thalamocortical:1 guinea:1 allow:1 wide:1 rhythmic:2 transition:2 ferent:1 sensory:5 made:2 simplified:1 programme:1 brandeis:1 far:1 claiborne:1 buc:1 active:2 conclude:1 alternatively:2 channel:5 mj:1 molecule:3 nature:1 sem:1 interact:1 complex:2 da:2 intracellular:2 neurosci:3 motivation:1 mediate:1 amino:2 neuronal:2 fig:5 site:1 je:1 slow:4 axon:1 msec:1 kandel:3 ib:1 kirk:1 minute:1 substance:17 showing:1 er:1 decay:1 evidence:1 fusion:1 intrinsic:12 illustrates:2 mill:1 ganglion:4 expressed:1 reflex:1 harris:5 ma:1 oct:2 conditional:2 bioi:4 ai1992:1 consequently:1 oscillator:5 absence:2 change:5 determined:1 withdrawal:1 modulated:3 dorsal:1 preparation:1 ongoing:1 |
7,005 | 781 | A Comparison of Dynamic Reposing and
Tangent Distance for Drug Activity
Prediction
Thomas G. Dietterich
Arris Pharmaceutical Corporation and Oregon State University
Corvallis, OR 97331-3202
Ajay N. Jain
Arris Pharmaceutical Corporation
385 Oyster Point Blvd., Suite 3
South San Francisco, CA 94080
Richard H. Lathrop and Tomas Lozano-Perez
Arris Pharmaceutical Corporation and MIT Artificial Intelligence Laboratory
545 Technology Square
Cambridge, MA 02139
Abstract
In drug activity prediction (as in handwritten character recognition), the features extracted to describe a training example depend
on the pose (location, orientation, etc.) of the example. In handwritten character recognition, one of the best techniques for addressing this problem is the tangent distance method of Simard,
LeCun and Denker (1993). Jain, et al. (1993a; 1993b) introduce a
new technique-dynamic reposing-that also addresses this problem. Dynamic reposing iteratively learns a neural network and then
reposes the examples in an effort to maximize the predicted output values. New models are trained and new poses computed until
models and poses converge. This paper compares dynamic reposing
to the tangent distance method on the task of predicting the biological activity of musk compounds. In a 20-fold cross-validation,
216
A Comparison of Dynamic Reposing and Tangent Distance for Drug Activity Prediction
dynamic reposing attains 91 % correct compared to 79% for the
tangent distance method, 75% for a neural network with standard
poses, and 75% for the nearest neighbor method.
1
INTRODUCTION
The task of drug activity prediction is to predict the activity of proposed drug
compounds by learning from the observed activity of previously-synthesized drug
compounds. Accurate drug activity prediction can save substantial time and money
by focusing the efforts of chemists and biologists on the synthesis and testing of
compounds whose predicted activity is high. If the requirements for highly active
binding can be displayed in three dimensions, chemists can work from such displays
to design new compounds having high predicted activity.
Drug molecules usually act by binding to localized sites on large receptor molecules
or large enyzme molecules. One reasonable way to represent drug molecules is
to capture the location of their surface in the (fixed) frame of reference of the
(hypothesized) binding site. By learning constraints on the allowed location of
the molecular surface (and important charged regions on the surface), a learning
algorithm can form a model of the binding site that can yield accurate predictions
and support drug design.
The training data for drug activity prediction consists of molecules (described by
their structures, i.e., bond graphs) and measured binding activities. There are two
complications that make it difficult to learn binding site models from such data.
First, the bond graph does not uniquely determine the shape of the molecule. The
bond graph can be viewed as specifying a (possibly cyclic) kinematic chain which
may have several internal degrees of freedom (i.e., rotatable bonds). The conformations that the graph can adopt, when it is embedded in 3-space, can be assigned
energies that depend on such intramolecular interactions as the Coulomb attraction,
the van der Waal's force, internal hydrogen bonds, and hydrophobic interactions.
Algorithms exist for searching through the space of conformations to find local
minima having low energy (these are called "conformers"). Even relatively rigid
molecules may have tens or even hundreds of low energy conformers. The training
data does not indicate which of these conformers is the "bioactive" one-that is,
the conformer that binds to the binding site and produces the observed binding
activity.
Second, even if the bioactive conformer were known, the features describing the
molecular surface-because they are measured in the frame of reference of the binding site-change as the molecule rotates and translates (rigidly) in space.
Hence, if we consider feature space, each training example (bond graph) induces a
family of 6-dimensional manifolds. Each manifold corresponds to one conformer as
it rotates and translates (6 degrees of freedom) in space. For a classification task,
a positive decision region for "active" molecules would be a region that intersects
at least one manifold of each active molecule and no manifolds of any inactive
molecules. Finding such a decision region is quite difficult, because the manifolds
are difficult to compute.
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Dietterich, Jain, Lathrop, and Lozano-Perez
A similar "feature manifold problem" arises in handwritten character recognition.
There, the training examples are labelled handwritten digits, the features are extracted by taking a digitized gray-scale picture, and the feature values depend on
the rotation, translation, and zoom of the camera with respect to the character.
=
1, ... , N be training examWe can formalize this situation as follows. Let Xi, i
ples (i.e., bond graphs or physical handwritten digits), and let I(Xi) be the label
associated with Xi (i.e., the measured activity of the molecule or the identity of the
handwritten digit). Suppose we extract n real-valued features V( Xi) to describe object Xi and then employ, for example, a multilayer sigmoid network to approximate
I(x) by j(x) g(V(x?. This is the ordinary supervised learning task.
=
However, the feature manifold problem arises when the extracted features depend
on the "pose" of the example. We will define the pose to be a vector P of parameters
that describe, for example, the rotation, translation, and conformation of a molecule
or the rotation, translation, scale, and line thickness of a handwritten digit. In this
case, the feature vector V(x,p) depends on both the example and the pose.
Within the handwritten character recognition community, several techniques have
been developed for dealing with the feature manifold problem. Three existing approaches are standardized poses, the tangent-prop method, and the tangent-distance
method. Jain et al. (1993a, 1993b) describe a new method-dynamic reposingthat applies supervised learning simultaneously to discover the "best" pose pi of
each training example Xi and also to learn an approximation to the unknown function I(x) as j(Xi) = g(V(Xi'p;?. In this paper, we briefly review each of these
methods and then compare the performance of standardized poses, tangent distance, and dynamic reposing to the problem of predicting the activity of musk
molecules.
2
2.1
FOUR APPROACHES TO THE FEATURE
MANIFOLD PROBLEM
STANDARDIZED POSES
The simplest approach is to select only one of the feature vectors V( Xi, Pi) for each
S(Xi), that computes a standard pose
example by constructing a function, Pi
for each object. Once Pi is chosen for each example, we have the usual supervised learning task-each training example has a unique feature vector, and we can
approximate 1 by j(x) g(V(x, S(x?).
=
=
The difficulty is that S can be very hard to design. In optical character recognition,
S typically works by computing some pose-invariant properties (e.g., principal axes
of a circumscribing ellipse) of Xi and then choosing Pi to translate, rotate, and scale
Xi to give these properties standard values. Errors committed by OCR algorithms
can often be traced to errors in the S function, so that characters are incorrectly
positioned for recognition.
In drug activity prediction, the standardizing function S must guess which conformer is the bioactive conformer. This is exceedingly difficult to do without additional information (e.g., 3-D atom coordinates of the molecule bound in the binding
A Comparison of Dynamic Reposing and Tangent Distance for Drug Activity Prediction
site as determined by x-ray crystallography). In addition, S must determine the
orientation of the bioactive conformers within the binding site. This is also quite
difficult-the bioactive conformers must be mutually aligned so that shared potential chemical interactions (e.g., hydrogen bond donors) are superimposed.
2.2
TANGENT PROPAGATION
The tangent-prop approach (Simard, Victorri, LeCun, & Denker, 1992) also employs a standardizing function S, but it augments the learning procedure with the
constraint that the output of the learned function g(V( x, p)) should be invariant
with respect to slight changes in the poses of the examples:
II\7
p
g(V(x,p))
Ip=S(x)
I = 0,
where II . II indicates Euclidean norm. This constraint is incorporated by using the
left-hand-side as a regularizer during backpropagation training.
Tangent-prop can be viewed as a way of focusing the learning algorithm on those
input features and hidden-unit features that are invariant with respect to slight
changes in pose. Without the tangent-prop constraint, the learning algorithm
may identify features that "accidentally" discriminate between classes. However,
tangent-prop still assumes that the standard poses are correct. This is not a safe
assumption in drug activity prediction.
2.3
TANGENT DISTANCE
The tangent-distance approach (Simard, LeCun & Denker, 1993) is a variant of the
nearest-neighbor algorithm that addresses the feature manifold problem . Ideally,
the best distance metric to employ for the nearest-neighbor algorithm with feature
manifolds is to compute the "manifold distance"-the point of nearest approach
between two manifolds:
This is very expensive to compute, however, because the manifolds can have highly
nonlinear shapes in feature space, so the manifold distance can have many local
mInIma.
The tangent distance is an approximation to the manifold distance. It is computed
by approximating the manifold by a tangent plane in the vicinity of the standard
poses. Let Ji be the Jacobian matrix defined by (Jdik = 8V(Xi,Pi)ij8(Pih, which
gives the plane tangent to the manifold of molecule Xi at pose Pi. The tangent
distance is defined as
=
=
S(xI) and P2
S(X2)' The column vectors a and b give the change
where PI
in the pose required to minimize the distance between the tangent planes approximating the manifolds. The values of a and b minimizing the right-hand side can
be computed fairly quickly via gradient descent (Simard, personal communication).
In practice, only poses close to S(xd and S(X2) are considered, but this provides
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Dietterich, Jain, Lathrop, and Lozano-Perez
more opportunity for objects belonging to the same class to adopt poses that make
them more similar to each other.
In experiments with handwritten digits, Simard, LeCun, and Denker (1993) found
that tangent distance gave the best performance of these three methods.
2.4
DYNAMIC REPOSING
All of the preceding methods can be viewed as attempts to make the final predicted
output j(x) invariant with respect to changes in pose. Standard poses do this by
not permitting poses to change . Tangent-prop adds a local invariance constraint.
Tangent distance enforces a somewhat less local invariance constraint.
In dynamic reposing, we make j invariant by defining it to be the maximum value
(taken over all poses p) of an auxiliary function g:
j(x)
= max
g(V(x,p)).
p
The function 9 will be the function learned by the neural network.
Before we consider how 9 is learned, let us first consider how it can be used to
predict the activity of a new molecule x'. To compute j(x'), we must find the pose
p'. that maximizes g(V(x',p'*?. We can do this by performing a gradient ascent
starting from the standard pose S(x) and moving in the direction of the gradient
of 9 with respect to the pose: \7plg(V(X',p'?.
This process has an important physical analog in drug activity prediction. If x' is
a new molecule and 9 is a learned model of the binding site, then by varying the
pose p' we are imitating the process by which the molecule chooses a low-energy
conformation and rotates and translates to "dock" with the binding site.
In handwritten character recognition, this would be the dual of a deformable template model: the template (g) is held fixed, while the example is deformed (by
rotation, translation, and scaling) to find the best fit to the template.
The function 9 is learned iteratively from a growing pool of feature vectors. Initially,
the pool contains only the feature vectors for the standard poses of the training examples (actually, we start with one standard pose of each low energy conformation
of each training example). In iteration j, we apply backpropagation to learn hypothesis gj from selected feature vectors drawn from the pool. For each molecule,
one feature vector is selected by performing a forward propagation (i.e., computing
9(V(Xi' Pi?)) of all feature vectors of that molecule and selecting the one giving the
highest predicted activity for that molecule.
After learning gj, we then compute for each conformer the pose
gj(V(Xi' p?:
?+1
Pi
P1+1 that maximizes
= argmax
gj(V(Xi'p?.
p
From the chemical perspective, we permit each of the molecules to "dock" to the
current model gj of the binding site.
The feature vectors V(Xi,Pi?+1 ) corresponding to these poses are added to the pool
of poses, and a new hypothesis gj+l is learned. This process iterates until the poses
A Comparison of Dynamic Reposing and Tangent Distance for Drug Activity Prediction
cease to change. Note that this algorithm is analogous to the EM procedure (Redner
& Walker, 1984) in that we accomplish the simultaneous optimization of 9 and the
poses {Pi} by conducting a series of separate optimizations of 9 (holding {Pi} fixed)
and {pd (holding 9 fixed).
We believe the power of dynamic reposing results from its ability to identify the
features that are critical for discriminating active from inactive molecules. In the
initial, standard poses, a learning algorithm is likely to find features that "accidentally" discriminate actives from inactives. However, during the reposing process,
inactive molecules will be able to reorient themselves to resemble active molecules
with respect to these features. In the next iteration, the learning algorithm is
therefore forced to choose better features for discrimination.
Moreover, during reposing, the active molecules are able to reorient themselves so
that they become more similar to each other with respect to the features judged
to be important in the previous iteration. In subsequent iterations, the learning
algorithm can "tighten" its criteria for recognizing active molecules.
In the initial, standard poses, the molecules are posed so that they resemble each
other along all features more-or-Iess equally. At convergence, the active molecules
have changed pose so that they only resemble each other along the features important for discrimination.
3
3.1
AN EXPERIMENTAL COMPARISON
MUSK ACTIVITY PREDICTION
We compared dynamic reposing with the tangent distance and standard pose methods on the task of musk odor prediction. The problem of musk odor prediction has
been the focus of many modeling efforts (e.g., Bersuker, et al., 1991; Fehr, et al.,
1989; Narvaez, Lavine & Jurs, 1986). Musk odor is a specific and clearly identifiable sensation, although the mechanisms underlying it are poorly understood.
Musk odor is determined almost entirely by steric (i.e., "molecular shape") effects
(Ohloff, 1986). The addition or deletion of a single methyl group can convert an
odorless compound into a strong musk. Musk molecules are similar in size and
composition to many kinds of drug molecules.
We studied a set of 102 diverse structures that were collected from published studies
(Narvaez, Lavine & Jurs, 1986; Bersuker, et al., 1991; Ohloff, 1986; Fehr, et al.,
1989). The data set contained 39 aromatic, oxygen-containing molecules with musk
odor and 63 homologs that lacked musk odor. Each molecule was conformationally searched to identify low energy conformations. The final data set contained
6,953 conformations of the 102 molecules (for full details of this data set, see Jain,
et al., 1993a). Each of these conformations was placed into a starting pose via a
hand-written S function. We then applied nearest neighbor with Euclidean distance, nearest neighbor with the tangent distance, a feed-forward network without
reposing, and a feed-forward network with the dynamic reposing method. For dynamic reposing, five iterations of reposing were sufficient for convergence. The time
required to compute the tangent distances far exceeds the computation times of
the other algorithms. To make the tangent distance computations feasible, we only
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Dietterich, Jain, Lathrop, and Lozano-Perez
Table 1: Results of 20-fold cross-validation on 102 musk molecules.
Percent Correct
Method
Nearest neighbor (Euclidean distance)
75
Neural network (standard poses)
75
Nearest neighbor (Tangent distance)
79
Neural network (dynamic reposing)
91
Table 2: Neural network cross-class predictions (percent correct)
N
Molecular class:
Standard poses
Dynamic reposing
85
100
76
90
74
85
57
71
computed the tangent distance for the 200 neighbors that were nearest in Euclidean
distance. Experiments with a subset of the molecules showed that this heuristic introduced no error on that subset.
Table 1 shows the results of a 20-fold cross-validation of all four methods. The
tangent distance method does show improvement with respect to a standard neural network approach (and with respect to the standard nearest neighbor method).
However, the dynamic reposing method outperforms the other two methods substantially.
An important test for drug activity prediction methods is to predict the activity
of molecules whose molecular structure (i.e., bond graph) is substantially different
from the molecules in the training set. A weakness of many existing methods for
drug activity prediction (Hansch & Fujita, 1964; Hansch, 1973) is that they rely on
the assumption that all molecules in the training and test data sets share a common
structural skeleton. Because our representation for molecules concerns itself only
with the surface of the molecule, we should not suffer from this problem. Table 2
shows four structural classes of molecules and the results of "class holdout" experiments in which all molecules of a given class were excluded from the training set
and then predicted. Cross-class predictions from standard poses are not particularly
good. However, with dynamic reposing, we obtain excellent cross-class predictions.
This demonstrates the ability of dynamic reposing to identify the critical discriminating features. Note that the accuracy of the predictions generally is determined
by the size of the training set (i.e., as more molecules are withheld, performance
drops). The exception to this is the right-most class, where the local geometry of
the oxygen atom is substantially different from the other three classes.
A Comparison of Dynamic Reposing and Tangent Distance for Drug Activity Prediction
4
CONCLUDING REMARKS
The "feature manifold problem" arises in many application tasks, including drug
activity prediction and handwritten character recognition. A new method, dynamic
reposing, exhibits performance superior to the best existing method, tangent distance, and to other standard methods on the problem of musk activity prediction.
In addition to producing more accurate predictions, dynamic reposing results in a
learned binding site model that can guide the design of new drug molecules. Jain,
et al., (1993a) shows a method for visualizing the learned model in the context of
a given molecule and demonstrates how the model can be applied to guide drug
design. Jain, et al., (1993b) compares the method to other state-of-the-art methods for drug activity prediction and shows that feed-forward networks with dynamic
reposing are substantially superior on two steroid binding tasks. The method is currently being applied at Arris Pharmaceutical Corporation to aid the development
of new pharmaceutical compounds.
Acknowledgements
Many people made contributions to this project. The authors thank Barr Bauer,
John Burns, David Chapman, Roger Critchlow, Brad Katz, Kimberle Koile, John
Park, Mike Ross, Teresa Webster, and George Whitesides for their efforts.
References
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7,006 | 782 | Optimal Unsupervised Motor Learning
Predicts the Internal Representation of
Barn Owl Head Movements
Terence D. Sanger
Jet Propulsion Laboratory
MS 303-310
4800 Oak Grove Drive
Pasadena, CA 91109
Abstract
(Masino and Knudsen 1990) showed some remarkable results which
suggest that head motion in the barn owl is controlled by distinct
circuits coding for the horizontal and vertical components of movement. This implies the existence of a set of orthogonal internal coordinates that are related to meaningful coordinates of the external
world. No coherent computational theory has yet been proposed
to explain this finding. I have proposed a simple model which provides a framework for a theory of low-level motor learning. I show
that the theory predicts the observed microstimulation results in
the barn owl. The model rests on the concept of "Optimal Unsupervised Motor Learning", which provides a set of criteria that
predict optimal internal representations. I describe two iterative
Neural Network algorithms which find the optimal solution and
demonstrate possible mechanisms for the development of internal
representations in animals.
1
INTRODUCTION
In the sensory domain, many algorithms for unsupervised learning have been proposed. These algorithms learn depending on statistical properties of the input
data, and often can be used to find useful "intermediate" sensory representations
614
Bam Owl Head Movements
u
p
y
z
Figure 1: Structure of Optimal Unsupervised Motor Learning. z is a reduced-order
internal representation between sensory data y and motor commands u. P is the
plant and G and N are adaptive sensory and motor networks. A desired value
of z produces a motor command u
N z resulting in a new intermediate value
GPNz.
z=
=
by extracting important features from the environment (Kohonen 1982, Sanger
1989, Linsker 1989, Becker 1992, for example). An extension of these ideas to the
domain of motor control has been proposed in (Sanger 1993). This work defined the
concept of "Optimal Unsupervised Motor Learning" as a method for determining
optimal internal representations for movement. These representations are intended
to model the important controllable components of the sensory environment, and
neural networks are capable of learning the computations necessary to gain control
of these components.
In order to use this theory as a model for biological systems, we need methods to
infer the form of biological internal representations so that these representations
can be compared to those predicted by the theory. Discrepancies between the
predictions and results may be due either to incorrect assumptions in the model, or
to constraints on biological systems which prevent them from achieving optimality.
In either case, such discrepancies can lead to improvements in the model and are
thus important for our understanding of the computations involved. On the other
hand, if the model succeeds in making qualitative predictions of biological responses,
then we can claim that the biological system possesses the optimality properties of
the model, although it is unlikely to perform its computations in exactly the same
manner.
2
BARN OWL EXPERIMENTS
A relevant set of experiments was performed by (Masino and Knudsen 1990) in
the barn owl. These experiments involved microstimulation of sites in the optic
tectum responsible for head movement. By studying the responses to stimulation
at different sites separated by short or long time intervals, it was possible to infer the
existence of distinct "channels" for head movement which could be made refractory
by prior stimulation. These channels were oriented in the horizontal and vertical
directions in external coordinates, despite the fact that the neck musculature of the
barn owl is sufficiently complex that such orientations appear unrelated to any set
615
616
Sanger
of natural motor coordinates. This result raises two related questions. First, why
are the two channels orthogonal with respect to external Cartesian coordinates, and
second, why are they oriented horizontally and vertically?
The theory of Optimal Unsupervised Motor Learning described below provides a
model which attempts to answer both questions. It automatically develops orthogonal internal coordinates since such coordinates can be used to minimize redundancy
in the internal representation and simplify computation of motor commands. The
selection of the internal coordinates will be based on the statistics of the components
of the sensory data which are controllable, so that if horizontal and vertical movements are distinguished in the environment then these components will determine
the orientation of intermediate channels. We can hypothesize that the horizontal
and vertical directions are distinguished in the owl by their relation to sensory information generated from physical properties of the environment such as gravity or
symmetry properties of the owl's head. In the simulation below, I show that reasonable assumptions on such symmetry properties are sufficient to guarantee horizontal
and vertical orientations of the intermediate coordinate system.
3
OPTIMAL UNSUPERVISED MOTOR LEARNING
Optimal Unsupervised Motor Learning (OUML) attempts to invert the dynamics of
an unknown plant while maintaining control of the most important modes (Sanger
1993). Figure 1 shows the general structure of the control loop, where the plant P
maps motor commands u into sensory outputs y = Pu, the adaptive sensory transformation G maps sensory data y into a reduced order intermediate representation
z
Gy, and the adaptive motor transformation N maps desired values of z into the
motor commands u N z which achieve them. Let z = G P N z be the value of the
intermediate variables after movement, and f) = P NGy be the resulting value of the
sensory variables. For any chosen value of z we want
z, so that we successfully
control the intermediate variables.
=
=
z=
In (Sanger 1993) it was proposed that we want to choose z to have lower dimensionality than y and to represent only the coordinates which are most important
for controlling the desired behavior. Thus, in general, f) =/; y and Ily - f)1I is the
performance error. OUML can then be described as
1. Minimize the movement error 1If) - yll
2. Subject to accurate control
z=
z.
These criteria lead to a choice of internal representation that maximizes the loop
gain through the plant.
Theorem 1:
(Sanger 1993) For any sensory mapping G there exists a motor
mapping N such t~at z = z, and [; _ E[lIy - f)1I] is mi1!.imized when G is chosen to
minimize E[lly - G-1Gyll]' where G-l is such that GG-l = I.
The function G is an arbitrary right inverse of G, and this function determines the
asymptotic values of the unobserved modes. In other words, since G in general is
Gy will not respond to all the modes in y so that
dimensionality-reducing, z
dissimilar states may project to identical intermediate control variables z. The
=
Barn Owl Head Movements
Plant 1
Linear
RBF
Polynomial
II
Motor
Linear
Linear
Polynomial
Sensory
Eigenvectors of E[yy'l ]
Eigenvectors of basis function outputs
Eigenvectors of basis function outputs
Figure 2: Special cases of Theorem 1. If the plant inverse is linear or can be
approximated using a sum of radial basis functions or a polynomial, then simple
closed-form solutions exist for the optimal sensory network and the motor network
only needs to be linear or polynomial.
a-
1 G is a projection operator that determines the resulting plant output
function
fJ for any desired value of y. Unsupervised motor learning is "optimal" when the
1 G is the best approximation to the statistical
projection surface determined by
density of desired values of y.
a-
Without detailed knowledge of the plant, it may be difficult to find the general
solution described by the theorem. Fortunately, there are several important special
cases in which simple closed-form solutions exist. These cases are summarized
in figure 2 and are determined by the class of functions to which the plant inverse
belongs. If the plant inverse can be approximated as a sum of radial basis functions,
then the motor network need only be linear and the optimal sensory network is given
by the eigenvectors of the autocorrelation matrix of the basis function outputs (as
in (Sanger 1991a)). If the plant inverse can be approximated as a polynomial over
a set of basis functions (as in (Sanger 1991b)), then the motor network needs to be
a polynomial, and again the optimal sensory network is given by the eigenvectors
of the autocorrelation matrix of the basis function outputs.
Since the model of the barn owl proposed below has a linear inverse we are interested
in the linear case, so we know that the mappings Nand G need only be linear and
that the optimal value of G is given by the eigenvectors of the autocorrelation matrix
of the plant outputs y. In fact, it can be shown that the optimal Nand G are given
by the matrices ofleft and right singular vectors of the plant inverse (Sanger 1993).
Although several algorithms for iterative computation of eigenvectors exist, until
recently there were no iterative algorithms for finding the left and right singular
vectors. I have developed two such algorithms, called the "Double Generalized
Hebbian Algorithm" (DGHA) and the "Orthogonal Asymmetric Encoder" (OAE).
(These algorithms are described in detail elsewhere in this volume.) DGHA is
described by:
!J..G
!J..NT
r(zyT - LT[zzT]G)
r(zu T - LT[zzT]N T )
while OAE is described by:
!J..G
!J..NT
r(zyT - LT[zzT]G)
r( Gy - LT[GGT]z)uT
where LT[ ] is an operator that sets the above diagonal elements of its matrix
argument to zero, y = Pu, z = Gy, z = NT u, and r is a learning rate constant.
617
618
Sanger
Movement Sensors
Neck Muscles
e
u
Sensory Transform
Motor Transform
y
N
z
Figure 3: Owl model, and simulation results. The "Sensory Transform" box shows
the orientation tuning of the learned internal representation.
4
SIMULATION
I use OUML to simulate the owl head movement experiments described in (Masino
and Knudsen 1990), and I predict the form of the internal motor representation. I
assume a simple model for the owl head using two sets of muscles which are not
aligned with either the horizontal or the vertical direction (see the upper left block
of figure 3). This model is an extreme oversimplification of the large number of
muscle groups present in the barn owl neck, but it will serve to illustrate the case
of muscles which do not distinguish the horizontal and vertical directions.
I assume that during learning the owl gives essentially random commands to the
muscles, but that the physics of head movement result in a slight predominance of
either vertical or horizontal motion. This assumption comes from the symmetry
properties of the owl head, for which it is reasonable to expect that the axes of
rotational symmetry lie in the coronal, sagittal, and transverse planes, and that
the moments of inertia about these axes are not equal. I model sensory receptors
using a set of 12 oriented directionally-tuned units, each with a half-bandwidth at
half-height of 15 degrees (see the upper right block of figure 3). Together, the Neck
Muscles and Movement Sensors (the two upper blocks of figure 3) form the model
of the plant which transforms motor commands u into sensory outputs y. Although
this plant is nonlinear, it can be shown to have an approximately linear inverse on
Barn Owl Head Movements
Desired Direction
Figure 4: Unsupervised Motor Learning successfully controls the owl head simulation.
its range.
The sensory units are connected through an adaptive linear network G to three
intermediate units which will become the internal coordinate system z. The three
intermediate units are then connected back to the motor outputs through a motor
network N so that desired sensory states can be mapped onto the motor commands
necessary to produce them. The sensory to intermediate and intermediate to motor
mappings were allowed to adapt to 1000 random head movements, with learning
controlled by DGHA.
5
RESULTS
After learning, the first intermediate unit responded to the existence of a motion,
and did not indicate its direction. The second and third units became broadly
tuned to orthogonal directions. Over many repeated learning sessions starting from
random initial conditions, it was found that the intermediate units were always
aligned with the horizontal and vertical axes and never with the diagonal motor
axes. The resulting orientation tuning from a typical session is shown in the lower
right box of figure 3.
Note that these units are much more broadly tuned than the movement sensors
(the half-bandwidth at half-height is 45 degrees). The orientation of the internal
channels is determined by the assumed symmetry properties of the owl head. This
information is available to the owl as sensory data, and OUML allows it to determine
the motor representation. The system has successfully inverted the plant, as shown
in figure 4.
(Masino and Knudsen 1990) investigated the intermediate representations in the
owl by taking advantage of the refractory period of the internal channels. It was
found that if two electrical stimuli which at long latency tended to move the owl's
head in directions located in adjacent quadrants were instead presented at short
latency, the second head movement would be aligned with either the horizontal or
vertical axis. Figure 5 shows the general form of the experimental results, which
are consistent with the hypothesis that there are four independent channels coding
619
620
Sanger
Move 2a
Move 2a
Move 1
Move 1
Move 2b
iliL Move 2b
Short Interval
Long Interval
Figure 5: Schematic description of the owl head movement experiment. At long
interstimulus intervals (lSI), moves 2a and 2b move up and to the right, but at
short lSI the rightward channel is refractory from move 1 and thus moves 2a and
2b only have an upward component.
---I
I
??
..
or
11
I
a.
"".
- - ..
...
'10
h.
0'
~"""'tfII.,
Figure 6: Movements align with the vertical axis as the lSI shortens. a. Owl
data (reprinted with permission from (Masino and Knudsen 1990?. h. Simulation
results.
the direction of head movement, and that the first movement makes either the
left, right, up, or down channels refractory. As the interstimulus interval (lSI) is
shortened, the alignment of the second movement with the horizontal or vertical
axis becomes more pronounced. This is shown in figure 6a for the barn owl and 6b
for the simulation. If we stimulate sites that move in many different directions, we
find that at short latency the second movement always aligns with the horizontal
or vertical axis, as shown in figure 7a for the owl and figure 7b for the simulation.
6
CONCLUSION
Optimal Unsupervised Motor Learning provides a model for adaptation in low-level
motor systems. It predicts the development of orthogonal intermediate representations whose orientation is determined by the statistics of the controllable components of the sensory environment . The existence of iterative neural algorithms for
both linear and nonlinear plants allows simulation of biological systems, and I have
Barn Owl Head Movements
....
?;
a.
I.ONG
"TEaVAL
I
~
-.. ..
.,
"
i
?~
SHORT
INTERVAL
-
.,--"--
h.
Figure 7: At long lSI, the second movement can occur in many directions, but
at short lSI will tend to align with the horizontal or vertical axis. a. Owl data
(reprinted with permission from (Masino and Knudsen 1990)). h. Simulation results.
shown that the optimal internal representation predicts the horizontal and vertical
alignment of the internal channels for barn owl head movement.
Acknowledgements
Thanks are due to Tom Masino for helpful discussions as well as for allowing reproduction of the figures from (Masino and Knudsen 1990). This report describes
research done within the -laboratory of Dr. Emilio Bizzi in the department of Brain
and Cognitive Sciences at MIT. The author was supported during this work by a
National Defense. Science and Engineering Graduate Fellowship, and by NIH grants
5R37 AR26710 and 5ROINS09343 to Dr. Bizzi.
References
Becker S., 1992, An Information-Theoretic Unsupervised Learning Algorithm for
Neural Networks, PhD thesis, Univ. Toronto Dept. Computer Science.
Kohonen T., 1982, Self-organized formation of topologically correct feature maps,
Biological Cybernetics, 43:59-69.
Linsker R., 1989, How to generate ordered maps by maximizing the mutual information between input and output signals, Neural Computation, 1:402-411.
Masino T ., Knudsen E. I., 1990, Horizontal and vertical components of head movement are controlled by distinct neural circuits in the barn owl, Nature, 345:434-437.
Sanger T. D., 1989, Optimal unsupervised learning in a single-layer linear feedforward neural network, Neural Networks, 2:459-473.
Sanger T. D., 1991a, Optimal hidden units for two-layer nonlinear feedforward
neural networks, International Journal of Pattern Recognition and Artificial Intelligence, 5(4):545-561, Also appears in C. H. Chen, ed., Neural Networks in Pattern
Recognition and Their Applications, World Scientific, 1991, pp. 43-59.
Sanger T. D., 1991b, A tree-structured adaptive network for function approximation
in high dimensional spaces, IEEE Trans. Neural Networks, 2(2):285-293.
Sanger T. D., 1993, Optimal unsupervised motor learning, IEEE Trans. Neural
Networks, in press.
621
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7,007 | 783 | A Hodgkin-Huxley Type Neuron Model
That Learns Slow Non-Spike Oscillation
Kenji Doya*
Allen I. Selverston
Department of Biology
University of California, San Diego
La Jolla, CA 92093-0357, USA
Peter F. Rowat
Abstract
A gradient descent algorithm for parameter estimation which is
similar to those used for continuous-time recurrent neural networks
was derived for Hodgkin-Huxley type neuron models. Using membrane potential trajectories as targets, the parameters (maximal
conductances, thresholds and slopes of activation curves, time constants) were successfully estimated. The algorithm was applied to
modeling slow non-spike oscillation of an identified neuron in the
lobster stomatogastric ganglion. A model with three ionic currents
was trained with experimental data. It revealed a novel role of
A-current for slow oscillation below -50 mY.
1
INTRODUCTION
Conductance-based neuron models, first formulated by Hodgkin and Huxley [10],
are commonly used for describing biophysical mechanisms underlying neuronal behavior. Since the days of Hodgkin and Huxley, tens of new ionic channels have
been identified [9]. Accordingly, recent H-H type models have tens of variables and
hundreds of parameters [1, 2]. Ideally, parameters of H-H type models are determined by voltage-clamp experiments on individual ionic currents. However, these
experiments are often very difficult or impossible to carry out. Consequently, many
parameters must be hand-tuned in computer simulations so that the model behavior
resembles that of the real neuron. However, a manual search in a high dimensional
*current address: The Salk Institute, CNL, P.O. Box 85800, San Diego, CA 92186-5800.
566
A Hodgkin-Huxley Type Neuron Model That Learns Slow Non-Spike Oscillation
I
Figure 1: A connectionist's view of the H-H neuron model.
parameter space is very unreliable. Moreover, even if a good match is found between the model and the real neuron, the validity of the parameters is questionable
because there are, in general, many possible settings that lead to apparently the
same behavior .
We propose an automatic parameter tuning algorithm for H-H type neuron models
[5]. Since a H-H type model is a network of sigmoid functions, multipliers, and
leaky integrators (Figure 1), we can tune its parameters in a manner similar to the
tuning of connection weights in continuous-time neural network models [6, 12]. By
training a model from many initial parameter points to match the experimental
data, we can systematically estimate a region in the parameter space, instead of a
single point.
We first test if the parameters of a spiking neuron model can be identified from the
membrane potential trajectories. Then we apply the learning algorithm to a model
of slow non-spike oscillation of an identified neuron in the lobster stomatogastric
ganglion [7]. The resulting model suggests a new role of A-current [3] for slow
oscillation in the membrane potential range below -50 m V.
2
STANDARD FORM OF IONIC CURRENTS
Historically, different forms of voltage dependency curves have been used to represent the kinetics of different ionic channels. However, in order to derive a simple,
efficient learning algorithm, we chose a unified form of voltage dependency curves
which is based on statistical physics of ionic channels [11] for all the ionic currents
in the model.
The dynamics of the membrane potential v is given by
Gil
=I
- LIj,
(1)
j
where G is the membrane capacitance and I is externally injected current. The j-th
ionic current Ij is the product of the maximum conductance 9j, activation variable
567
568
Doya, Selverston, and Rowat
aj, inactivation variable bj , and the difference of the membrane potential v from
the reversal potential Vrj. The exponents Pi and qj represent multiplicity of gating
elements in the ionic channels and are usually an integer between 0 and 4. Variables
aj and bj are assumed to obey the first order differential equation
(2)
Their steady states
ajoo
and bjoo are sigmoid functions of the membrane potential
xoo(v)
= 1+
1
e-~'"
()'
v-v",
(x=aj,bj ),
(3)
where Vx and Sx represent the threshold and slope of the steady state curve, respectively. The rate coefficients ka ? (v) and kb ? (v) have the voltage dependence
[11]
]]
k x (v ) -- -1 cosh sx( v - v x) ,
tx
2
where tx is the time constant.
3
(4)
ERROR GRADIENT CALCULUS
Our goal is to minimize the average error over a cycle with period T:
E=
~
iT ~(v(t)
- v*(t?2dt,
(5)
where v*(t) is the target membrane potential trajectory.
We first derive the gradient of E with respect to the model parameters ( ... , Oi, ... ) =
( ... , 9j, va], Saj' taj' ... ). In studies of recurrent neural networks, it has been shown
that teacher forcing is very important in training autonomous oscillation patterns [4,
6, 12, 13]. In H-H type models, teacher forcing drives the activation and inactivation
variables by the target membrane potential v*(t) instead of vet) as follows.
x = kx(v*(t?? (-x +xoo(v*(t?)
(x
= aj,bj ).
(6)
We use (6) in place of (2) during training.
The effect of a small change in a parameter Oi of a dynamical system
x = F(X; ... , Oi, ... ),
(7)
is evaluated by the variation equation
.
of
y = oX y
of
+ OOi'
(8)
which is an n-dimensional linear system with time-varying coefficients [6, 12]. In
general, this variation calculus requires O(n 2 ) arithmetics for each parameter. However, in the case of H-H model with teacher forcing, (8) reduces to a first or second
order linear system. For example, the effect of a small change in the maximum
conductance 9j on the membrane potential v is estimated by
(9)
A Hodgkin-Huxley Type Neuron Model That Learns Slow Non-Spike Oscillation
where GCt) = l:k 9kak(t)Pkbk(t)Qk is the total membrane conductance. Similarly,
the effect of the activation threshold va] is estimated by the equations
GiJ = -G(t)y - 9jpjaj(t)pj- 1 bj (t)Qj(v(t) - Vrj) Z,
Z
= -kaj(t) [z + 8;j {aj(t) + ajoo(t) -
2aj(t) aj oo(t)}] .
(10)
The solution yet) represents the perturbation in v at time t, namely 8;b~). The
error gradient is then given by
aE
1 fT
.. av(t)
OBi = T Jo (v(t) - v (t)) OBi dt.
4
(11)
PARAMETER UPDATE
Basically, we can use arbitrary gradient-based optimization algorithms, for example,
simple gradient descent or conjugate gradient descent. The particular algorithm we
used was a continuous-time version of gradient descent on normalized parameters.
Because the parameters of a H-H type model have different physical dimensions and
magnitudes, it is not appropriate to perform simple gradient descent on them. We
represent each parameter by the default value Oi and the deviation Bi as below.
(12)
Then we perform gradient descent on the normalized parameters
Bi .
Instead of updating the parameters in batches, i.e. after running the model for T
and integrating the error gradient by (11), we updated the parameters on-line using
the running average of the gradient as follows.
.
Ta.D. o; = -.D.o,
av(t) OBi
- v (t)) OBi oBi'
1..
+ T(v(t)
Bi = -?.D. o, ,
(13)
where Ta is the averaging time and ? is the learning rate. This on-line scheme was
less susceptible to 2T-periodic parameter oscillation than batch update scheme and
therefore we could use larger learning rates.
5
PARAMETER ESTIMATION OF A SPIKING MODEL
First, we tested if a model with random initial parameters can estimate the parameters of another moqel by training with its membrane potential trajectories.
The default parameters Bi of the model was set to match the original H-H model
[10] (Table 1). Its membrane potential trajectories at five different levels of current
injection (I = 0,15,30,45, and 60J..lA/cm 2 ) were used alternately as the target v*(t).
We ran 100 trials after initializing Bi randomly in [-0.5,+0.5]. In 83 cases, the error
became less than 1.3 m V rms after 100 cycles of training. Figure 2a is an example of the oscillation patterns of the trained model. The mean of the normalized
569
570
Doya, Selverston, and Rowat
Table 1: Parameters of the spiking neuron model. Subscripts L, Na and K specifies leak, sodium and potassium currents, respectively. Constants: C=1J.lF/cm 2 ,
vNa=55mV,
vK=-72mV, vL=-50mV, PNa=3,
QNa=l,
PK=4,
QK=PL=qL=O,
Llv=20mV, (=0.1, Ta = 5T.
after learning
mean
s.d.
-0.017
0.252
0.248
-0.002
0.006
0.033
-0.052
0.073
-0.103
0.154
0.012
0.202
0.140
-0.010
0.093
0.330
0.264
0.050
-0.021
0.136
-0.061
0.114
-0.073
0.168
()i
gL
gNa
VaNa
SaNa
taNa
VbNa
SbNa
tbNa
gK
VaK
SaK
taK
default
0.3
120.0
-36.0
0.1
0.5
-62.0
-0.09
12.0
40.0
-50.0
0.06
5.0
value iii
mS/cm
mS/cm 2
mV
l/mV
msec
mV
l/mV
msec
mS/cm 2
mV
l/mV
msec
taX
v[
saK
vaK
gK
a_No [
IbNa
sbNa
vbNa
b_Na[________
taNa
saNa
vaNa
a_K [-------.....-
gNa
gL
o
10
20
time (ms)
(a)
30
gL gNa vaNa saNa taNa vbNasbNa IbNa gK vaK saK taK
(b)
Figure 2: (a) The trajectory of the spiking neuron model at I = 30J.lA/cm 2 ? v:
membrane potential (-80 to +40 mY). a and b: activation and inactivation variables
(0 to 1). The dotted line in v shows the target trajectory v*(t). (b) Covariance
matrix of the normalized parameters Oi after learning. The black and white squares
represent negative and positive covariances, respectively.
A Hodgkin-Huxley Type Neuron Model That Learns Slow Non-Spike Oscillation
Table 2: Parameters of the DG cell model. Constants: C=1J.lF/cm 2 , vA=-80mV ,
VH= -lOmV, vL=-50mV, PA=3, qA=l, PH=l, QH=PL=qL=O, ~v=20mV, (=0.1,
Ta = 2T.
iJ?t
gL
gA
VaA
SaA
taA
VbA
SbA
tllA
gH
VaH
SaH
taH
0.01
50
-12
0.04
7.0
-62
-0.16
300
0.1
-70
-0.14
3000
tuned (}i
0.025
41.0
-11.1
0.022
7.0
-76
-0.19
292
0.039
-75 .1
-0.11
4400
v[
mS cm
mS/cm 2
mV
1/mV
msec
mV
1/mV
msec
mS/cm 2
mV
1/mV
msec
a~[
-
---
b~[
'_H[
I_L
~ ~
"""'"
I _A
10000
20000
30000
40000
50000
tlme(msl
Figure 3: Oscillation pattern of the DG cell model. v: membrane potential (-70 to
-50 mY). a and b: activation and inactivation variables (0 to 1) . I: ionic currents
(-1 to +1 pAlcm 2 ).
parameters iii were nearly zero (Table 1) , which implies that the original parameter values were successfully estimated by learning . The standard deviation of each
parameter indicates how critical its setting is to replicate the given oscillation patterns. From the covariance matrix of the parameters (Figure 2b), we can estimate
the distribution of the solution points in the parameter space .
6
MODELING SLOW NON-SPIKE OSCILLATION
Next we applied the algorithm to experimental data from the "DG cell" of the lobster stomatogastric ganglion [7]. An isolated DG cell oscillates endogenously with
the acetylcholine agonist pilocarpine and the sodium channel blocker TTX. The
oscillation period is 5 to 20 seconds and the membrane potential is approximately
between -70 and -50 m V. From voltage-clamp data from other stomatogastric neurons [8], we assumed that A-current (potassium current with inactivation) [3] and
H-current (hyperpolarization-activated slow inward current) are the principal active
currents in this voltage range. The default parameters for these currents were taken
from [2] (Table 2).
571
572
Doya, Selverston, and Rowat
ionic currents
./
2
,,r .... ....
....
1
o ~'
~
W
V
..- ../
~
~
~
"
If
-2
-60 -40
-20
v
0
20
40
(mV)
Figure 4: Current-voltage curves of the DG cell model. Outward current is positive.
Figure 3 is an example of the model behavior after learning for 700 cycles. The
actual output v of the model, which is shown in the solid curve, was very close
to the target output v*(t), which is shown in the dotted curve. The bottom three
traces show the ionic currents underlying this slow oscillation. Figure 4 shows the
steady state I-V curves of three currents. A-current has negative conductance in
the range from -70 to -40 m V. The resulting positive feedback on the membrane
potential destabilizes a quiescent state. If we rotate the I-V diagram 180 degrees, it
looks similar to the I-V diagram for the H-H model; the faster outward A-current
in our model takes the role of the fast inward sodium current in the H-H model and
the slower inward H-current takes the role of the outward potassium current.
7
DISCUSSION
The results indicate that the gradient descent algorithm is effective for estimating
the parameters of H-H type neuron models from membrane potential trajectories.
Recently, an automatic parameter search algorithm was proposed by Bhalla and
Bower [1]. They chose only the maximal conductances as free parameters and used
conjugate gradient descent . The error gradient was estimated by slightly changing
each of the parameters. In our approach, the error gradient was more efficiently derived by utilizing the variation equations. The use of teacher forcing and parameter
normalization was essential for the gradient descent to work.
In order for a neuron to be an endogenous oscillator, it is required that a fast positive feedback mechanism is balanced with a slower negative feedback mechanism.
The most popular example is the positive feedback by the sodium current and the
negative feedback by the potassium current in the H-H model. Another common
example is the inward calcium current counteracted by the calcium dependent outward potassium current. We found another possible combination of positive and
negative feedback with the help of the algorithm: the inactivation of the outward
A-current and the activation of the slow inward H-current .
A Hodgkin-Huxley Type Neuron Model That Learns Slow Non-Spike Oscillation
Acknowledgements
The authors thank Rob Elson and Thom Cleland for providing physiological data
from stomatogastric cells. This study was supported in part by ONR grant N0001491-J-1720.
References
[1] U. S. Bhalla and J. M. Bower. Exploring parameter space in detailed single
neuron models: Simulations of the mitral and granule cells of the olfactory
bulb. Journal of Neurophysiology, 69:1948-1965, 1993.
[2] F. Buchholtz, J. Golowasch, I. R. Epstein, and E. Marder. Mathematical model
of an identified stomatogastric ganglion neuron. Journal of Neurophysiology,
67:332-340, 1992.
[3] J. A. Connor, D. Walter, and R. McKown. Neural repetitive firing, modifications of the Hodgkin-Huxley axon suggested by experimental results from
crustacean axons. Biophysical Journal, 18:81-102, 1977.
[4] K. Doya. Bifurcations in the learning of recurrent neural networks. In Proceedings of 1992 IEEE International Symposium on Circuits and Systems, pages
6:2777-2780, San Diego, 1992.
[5] K. Doya and A. I. Selverston. A learning algorithm for Hodgkin-Huxley type
neuron models. In Proceedings of IJCNN'93, pages 1108-1111, Nagoya, Japan,
1993.
[6] K. Doya and S. Yoshizawa. Adaptive neural oscillator using continuous-time
back-propagation learning. Neural Networks, 2:375-386, 1989.
[7] R. C. Elson and A. I. Selverston. Mechanisms of gastric rhythm generation
in the isolated stomatogastric ganglion of spiny lobsters: Bursting pacemaker
potential, synaptic interactions, and muscarinic modulation. Journal of Neurophysiology, 68:890-907, 1992.
[8] J. Golowasch and E. Marder. Ionic currents of the lateral pyloric neuron of
stomatogastric ganglion of the crab. Journal of Neurophysiology, 67:318-331,
1992.
[9] B. Hille. Ionic Channels of Excitable Membranes. Sinauer, 1992.
[10] A. L. Hodgkin and A. F. Huxley. A quantitative description of membrane
currents and its application to conduction and excitation in nerve. Journal of
Physiology, 117:500-544, 1952.
[11] H. Lecar, G. Ehrenstein, and R. Latorre. Mechanism for channel gating in
excitable bilayers. Annals of the New York Academy of Sciences, 264:304-313,
1975.
[12] P. F. Rowat and A.I. Selverston. Learning algorithms for oscillatory networks
with gap junctions and membrane currents. Network, 2:17-41, 1991.
[13] R. J. Williams and D. Zipser. Gradient based learning algorithms for recurrent
connectionist networks. Technical Report NU-CCS-90-9, College of Computer
Science, Northeastern University, 1990.
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7,008 | 784 | Learning Mackey-Glass from 25
examples, Plus or Minus 2
Mark Plutowski? Garrison Cottrell? Halbert White??
Institute for Neural Computation
*Department of Computer Science and Engineering
**Department of Economics
University of California, San Diego
La J oHa, CA 92093
Abstract
We apply active exemplar selection (Plutowski &. White, 1991;
1993) to predicting a chaotic time series. Given a fixed set of examples, the method chooses a concise subset for training. Fitting
these exemplars results in the entire set being fit as well as desired. The algorithm incorporates a method for regulating network
complexity, automatically adding exempla.rs and hidden units as
needed. Fitting examples generated from the Mackey-Glass equation with fractal dimension 2.1 to an rmse of 0.01 required about 25
exemplars and 3 to 6 hidden units. The method requires an order
of magnitude fewer floating point operations than training on the
entire set of examples, is significantly cheaper than two contending exemplar selection techniques, and suggests a simpler active
selection technique that performs comparably.
1
Introduction
Plutowski &. White (1991; 1993), have developed a method of active selection of
training exemplars for network learning. Active selection uses information about
the state of the network when choosing new exemplars. The approach uses the statistical sampling criterion Integrated Squared Bias (ISB) to derive a greedy selection
method that picks the training example maximizing the decrement in this measure.
(ISB is a special case of the more familiar Integrated Mean Squared Error in the
case that noise variance is zero.) We refer to this method as A.ISB. The method
automatically regulates network complexity by growing the network as necessary
1135
1136
Plutowski, Cottrell, and White
to fit the selected exemplars, and terminates when the model fits the entire set of
available examples to the desired accuracy. Hence the method is a nonparametric
regression technique. In this paper we show that the method is practical by applying it to the Mackey-Glass time series prediction task. We compare AISB with
the method of training on all the examples. AIS8 consistently learns the time
series from a small subset of the available examples, finding solutions equivalent
to solutions obtained using all of the examples. The networks obtained by AISB
consistently perform better on test data for single step prediction, and do at least
as well at iterated prediction, but are trained at much lower cost .
Having demonstra.ted that this particular type of exemplar selection is worthwhile,
we compare AISE with three other exemplar selection methods which are easier
to code and cost less to compute. We compare the total cost of training, as well
as the size of the exemplar sets selected. One of the three contending methods was
suggested by the AISB algorithm, and is also an active selection technique, as its
calculation involves the network state. Among the four exemplar selection methods,
we find that the two active selection methods provide the greatest computational
savings and select the most concise training sets.
2
The Method
We are provided with a set of N "candidate" examples of the form (Zi, g(zd) .
Given g, we can denote this as x N . Let 1(?, w) denote the network function parameterized by weights w. For a particular subset of the examples denoted x n , let
Wn = Wn (zn) minimize
Let w? be the "best" set of weights, which minimizes
where IJ is the distribution over the inputs. Our objective is to select a subset zn
of zN such that n < N, while minimizing J(/(z, wn ) - I(z, w?))21J(dz). Thus,
we desire a subset representative of the whole set. We choose the zn C zN giving
weights Wn that minimize the Integrated Squared Bias (ISB):
(1)
We generate zn incrementally. Given a candidate example Zn+l, let zn+l =
(zn, Zn+l). Selecting Zl optimally with respect to (1) is straightforward. Then
given zn minimizing ISB(zn), we opt to select Zn+l E zN maximizing ISB(zn)ISB(xn+1). Note that using this property for Zn+1 will not necessarily deliver the
globally optimal solution. Nevertheless, this approach permits a computationally
feasible and attractive method for sequential selection of training examples.
Learning Mackey-Glass from 25 Examples, Plus or Minus 2
Choosing Zn+l to maximize this decrement directly is expensive. We use the following simple approximation (see Plutowski &- White, 1991) for justification): Given
zn, select Zn+l E argmaxzn+l~ISB(xn+llzn), where
N
6ISB(x n+llz n ) = 6W n+l'
L
V w!(Zi, wn)(g(zd - !(Zi, w n ?,
i=l
and
n
H(zn ,wn ) = LV w!(Zj, wn)Vw!(Zi, w n }' .
i=l
In practice we approximate H appropriately for the task at hand. Although we
arrive at this criterion by making use of approximations valid for large n, this criterion has an appealing interpretation as picking the single example having individual
error gradient most highly correlated with the average error gradient of the entire
set of examples. Learning with this example is therefore likely to be especially informative. The 6ISB criterion thus possesses heuristic appeal in training sets of
any size.
3
The Algorithm
Before presenting the algorithm we first explain certain implementation details. We
integrated the ~I SB criterion with a straightforward method for regulating network
complexity. We begin with a small network and an initial training set composed of
a single exemplar. When a new exemplar is added, if training stalls, we randomize
the network weights and restart training. After 5 stalls, we grow the network by
adding another unit to each hidden layer.
Before we can select a new exemplar, we require that the network fit the current
training set "sufficiently well." Let en(zm) measure the rmse (root mean squared
error) network fit over m arbitrary examples zm when trained on xn. Let Fn E ~+
denote the rmse fit we require over the current set of n exemplars before selecting
a new one. Let FN E ~+ denote the rmse fit desired over all N examples. (Our
goal is en(zN) < FN.) It typically suffices to set Fn = FN, that is, to train to a fit
over the exemplars which is at least as stringent as the fit desired over the entire
set (normalized for the number of exemplars.) However,. active selection sometimes
chooses a new exemplar "too close" to previously selected exemplars even when this
is the case. This is easy to detect, and in this case we reject the new exemplar and
continue with training.
We use an "exemplar spacing" parameter d to detect when a new exemplar is too
close to a previous selection. Two examples Xi and Xi are "close" in this sense if
they are within Euclidean distance d, and if additionally Ig(Zi) - g(xi)1 < FN. The
additional condition allows the new exemplar to be accepted even when it is close to
a previous selection in input space, provided it is sufficiently far away in the output
space. In our experiments, the input and output space are of the same scale, so we
set d FN. When a new selection is too close to a current exemplar, we reject the
=
1137
1138
Plutowski, Cottrell, and White
new selection, reduce Fn by 20%, and continue training, resetting Fn = FN when
a subsequent selection is appended to the current training set. We now outline the
algorithm:
Initialize:
? Specify user-set parameters: initial network size, the desired fit FN, the
exemplar spacing parameter, and the maximum number of restarts .
? Select the first training set, xl = {xd. Set n
1 and Fn
FN. Train the
1
network on xl until en{x ) 5 Fn.
=
=
While(en(xN) > FN) {
Select a new exemplar, Zn+l E x N , maximizing 6.ISB.
H (Zn+l is "too close" to any Z E zn) {
Reject Zn+l
Reduce Fn by 20%. }
Else {
Append Zn+l to zn.
Increment n.
Set Fn = FN . }
While(en(zn} > Fn) {
Train the network on the current training set zn,
restarting and growing as necessary. }}
4
The Problem
We generated the data from the Mackey-Glass equation (Mackey &, Glass, 1977),
with T
17, a = 0.2, and b 0.1. We integrated the equation using fourth order
Runge-Kutta with step size 0.1, and the history initialized to 0.5. We generated two
data sets. We iterated the equation for 100 time steps before beginning sampling;
this marks t = O. The next 1000 time steps comprise Data Set 1. We generated
Data Set 2 from the 2000 examples following t 5000.
=
=
=
We used the standard feed-forward network architecture with [0, 1] sigmoids and one
or two hidden layers. Denoting the time series as z(t}, the inputs were z(t), x(t 6}, z(t - 12), z(t - 18), and the desired output is z(t + 6) (Lapedes &, Farber, 1987).
We used conjugate gradient optimization for all of the training runs. The line search
routine typically required 5 to 7 passes through the data set for each downhill step,
and was restricted to use no more than 10.
Initially, the single hidden layer network has a single hidden unit, and the 2 hidden
layer network has 2 units per hidden layer. A unit is added to each hidden layer
when growing either architecture. All methods use the same growing procedure.
Thus, other exemplar selection techniques are implemented by modifying how the
next training set is obtained at the beginning of the outer while loop. The method
of using all the training examples uses only the inner while loop.
In preliminary experiments we evaluated sensitivity of 6.ISB to the calculation of
H. We compared two ways of estimating H, in terms of the number of exemplars
Learning Mackey-Glass from 25 Examples, Plus or Minus 2
selected and the total cost of training. The first approach uses the diagonal terms
of H (Plutowski &. White, 1993). The second approach replaces H with the identity
matrix. Evaluated over 10 separate runs, fitting 500 examples to an rmse of 0.01,
~ISB gave similar results for both approaches, in terms of total computation used
and the number of exemplars selected. Here, we used the second approach.
5
The Comparisons
We performed a number of experiments, each comparing the ~ISB algorithm with
competing training methods. The competing methods include the conventional
method of using all the examples, henceforth referred to as "the strawman," as well
as three other data selection techniques. In each comparison we denote the cost
as the total number of floating point multiplies (the number of adds and divides is
always proportional to this count).
For each comparison we ran two sets of experiments. The first compares the total
cost of the competing methods as the fit requirement is varied between 0.02, 0.015,
and 0.01, using the first 500 examples from Data Set 1. The second compares the
cost as the size of the "candidate" set (the set of available examples) is varied using
the first 500, 625, 750, 875, and 1000 examples of Data Set I, and a tolerance of
0.01. To ensure that each method is achieving a comparable fit over novel data,
we evaluated each network over a test set. The generalization tests also looked at
the iterated prediction error (IPE) over the candidate set and test set (Lapedes &.
Farber, 1987). Here we start the network on the first example from the set, and
feed the output back into the network to obtain predictions in multiples of 6 time
steps. Finally, for each of these we compare the final network sizes. Each data point
reported is an average of five runs. For brevity, we only report results from the two
hidden layer networks.
6
Comparison With Using All the Examples
We first compare ~ISB with the conventional method of using all the available
examples, which we will refer to as "the strawman." For this test, we used the first
500 examples of Data Set 1. For the two hidden layer architecture, each method
required 2 units per hidden layer for a fit of 0.02 and 0.015 rmse, and from 3 to
4 (typically 3) units per hidden layer for a fit of 0.01 rmse. While both methods
did quite well on the generalization tests, ~ISB clearly did better. Whereas the
strawman networks do slightly worse on the test set than on the candidate set,
networks trained by ~ISB tended to give test set fits close to the desired (training)
fit. This is partially due to the control flow of the algorithm, which often fits the
candidate set better than necessary. However, we also observed ~ISB networks
exhibited a test set fit better than the candidate set fit 7 times over these 15 training
runs. This never occurred over any of the strawman runs.
Overall, ~ISB networks performed at least as well as the strawman with respect to
IPE. Figure 1a shows the second half of Data Set 1, which is novel to this network,
plotted along with the iterated prediction of a ~ISB network to a fit of 0.01, giving
an IPE of 0.081 rmse, the median IPE observed for this set of five runs. Figure 1b
shows the iterated prediction over the first 500 time steps of Data Set 2, which is
1139
1140
Plutowski, Cottrell, and White
4500 time steps later than the training set. The IPE is 0.086 rmse, only slightly
worse than over the "nearer" test set. This fit required 22 exemplars. Generalization
tests were excellent for both methods, although t1ISB was again better overall.
t1ISB networks performed better on Data Set 2 than they did on the candidate
set 9 times out of the 25 runs; this never occurred for the strawman. These effects
demand closer study before using them to infer that data selection can introduce a
beneficial bias. However, they do indicate that the t1ISB networks performed at
least as well as the strawman, ensuring the validity of our cost comparisons.
Figure 1: Itell.ted prediction for a 2 hidden layer network trained to 0.01 rmae over the
first 500 time steps of Data Set 1. The dotted line gives the network prediction; the solid
line is the target time series. Figure la, on the left, is over the next (consecutive) 500 time
steps of Data Set 1, with IPE = 0.081 rmse. Figure Ib, on the right, is over the first 500
steps of Data Set 2, with IPE = 0.086 rmse. This network was typical, being the median
IPE of 5 runs.
Figure 2a shows the average total cost versus required fit FN for each method.
The strawman required 109, 115, and 4740 million multiplies for the respective
tolerances, whereas t1ISB required 8, 28, and 219 million multiplies, respectively.
The strawman is severely penalized by a tighter fit because growing the network
to fit requires expensive restarts using all of the examples. Figure 2b shows the
average total cost versus the candidate set sizes. One reason for the difference is
that t1ISB tended to select smaller networks. For candidate sets of size 500, 625,
750 and 875, each method typically required 3 units per hidden layer, occasionally
4. Given 1000 examples, the strawman selected networks larger than 3 hidden units
per layer over twice as often as t1ISB. t1ISB also never required more than 4
hidden units per layer, while the strawman sometimes required 6. This suggests
that the growing technique is more likely to fit the data with a smaller network
when exemplar selection is used.
Cost
Cost
7000
6000
5000
4000
3000
2000
1000
35000
30000
25000
20000
15000
10000
5000
0.02
Figure 2: Cost (in millions of multiplies) oftraining t1ISB, compared to the Strawman.
Figure 2a on the left gives total cost versus the desired fit, and Figure 2b on the right
gives total cost versus the number of ca.ndidate examples. Each point is the average of 5
runs; the error bars are equal in width to twice the standard deviation.
Learning Mackey-Glass from 25 Examples, Plus or Minus 2
7
Contending Data Selection Techniques
The results above clearly demonstrate that exemplar selection can cut the cost of
training dramatically. In what follows we compare ~ISB with three other exemplar
selection techniques. Each of these is easier to code and cheaper to compute, and
are considerably more challenging contenders than the strawman. In addition to
comparing the overall training cost we will also evaluate their data compression
ability by comparing the size of the exemplar sets each one selects. We proceed in
the same manner as with ~ISB, sequentially growing the training set as necessary,
until the candidate set fit is as desired.
Two of these contending techniques do not depend upon the state of the network,
and are therefore are not "Active Selection" methods. Random Selection selects an
exampk randomly from the candidate set, without replacement, and appends it to
the current exemplar set. Uniform Grid exploits the time series representation of
our data set to select training sets composed of exemplars evenly spaced at regular
intervals in time. Note that Uniform Grid does. not append a single exemplar to
the training set, rather it selects an entirely new set of exemplars each time the
training set is grown. Note further that this technique relies heavily upon the time
series representation. The problem of selecting exemplars uniformly spaced in the
4 dimensional input space would be much more difficult to compute.
The third method, "Maximum Error," was suggested by the ~ISB algorithm, and
is also an Active Selection technique, since it uses the network in selecting new
exemplars. Note that the error between the network and the desired value is a
component of the tiISB criterion. ~ISB need not select an exemplar for which
network error is maximum, due to the presence of terms involving the gradient
of the network function. In comparison, the Maximum Error method selects an
exemplar maximizing network error, ignoring gradient information entirely. It is
cheaper to compute, typically requiring an order of magnitude fewer multiplies in
overhead cost as compared to tiISB. This comparison will test, for this particular
learning task, whether the gradient information is worth its additional overhead.
7.1 Comparison with Random Selection
Random Selection fared the worst among the four contenders. However, it still
performed better overall than the strawman method. This is probably because the
cost due to growing is cheaper, since early on restarts are performed over small
training sets. As the network fit improves, the likelihood of randomly selecting
an informative exemplar decreases, and Random Selection typically reaches a point
where it adds exemplars in rapid succession, often doubling the size of the exemplar
set in order to attain a slightly better fit. Random Selection also had a very high
variance in cost and number of exemplars selected.
7.2 Comparison with Uniform Grid and Maximum Error
Uniform Grid and Maximum Error are comparable with tiISB in cost as well as
in the size of the selected exemplar sets. Overall, tiISB and Maximum Error
performed about the same, with Uniform Grid finishing respectably in third place.
Maximum Error was comparable to ~ISB in generalization also, doing better on
the test set than on the candidate set 10 times out of 40, whereas tiISB did so a
1141
1142
Plutowski, Cottrell, and White
total of 16 times . This occurred only 3 times out of 40 for Uniform Grid.
Figure 3a shows that Uniform Grid requires more exemplars at all three tolerances,
whereas ~ISB and Maximum Errorselect about the same number. Figure 3b shows
that Uniform Grid typically requires about twice as many exemplars as the other
two. Maximum Error and ~ISB selected about the same number of exemplars,
typically selecting about 25 exemplars, plus or minus two.
n
60
Uniforrr
50
50
40
40
30
30
20
10
Max Er
De ta
0.02
20
ISB
10
0.015
0.02
n
ri 1-- I
* ~
Delt . . ISB
Max Error
Uniforn
~
~
750
875
-t
!,
RInse
500
625
1000 N
Figure 3: Number of examples selected by three contending selection techniques: Uniform, ~ISB (diamonds) and Max Error (triangles.) Figure 3a on the left gives number of
examples selected versus the desired fit, and Figure 3b on the right is versus the number
of candidate examples. The two Active Selection techniques selected about 25 exemplars,
?2. Each point is the average of 5 runSi the error bars are equal in width to twice the
standard deviation . The datapoints for ~I SB and Max Error are shifted slightly in the
graph to make them easier to distinguish.
8
Conclusions
These results clearly demonstrate that exemplar selection can dramatically lower
the cost of training. This particular learning task also showed that Active Selection
methods are better overall than two contending exemplar selection techniques.
~I S B and Maximum Error consistently selected concise sets of exemplars, reducing
the total cost of training despite the overhead associated with exemplar selection.
This particular learning task did not provide a clear distinction between the two
Active Selection techniques. Maximum Error is more attractive on problems of this
scope even though we have not justified it analytically, as it performs about as well
as ~ISB but is easier to code and cheaper to compute.
Acknowledgements
This work was supported by NSF grant IRI 92-03532.
References
Lapedes, Alan, and Robert Farber. 1987. "Nonlinear signal processing using neural networks. Prediction and system modelling." Los Alamos technical report LA-UR-87-2662.
Mackey, M.C., and L. Glass. 1977. "Oscillation and chaos in physiological control systems." Science 197, 287.
Plutowski, Mark E., and Halbert White. 1991. "Active selection of training examples for
network learning in noiseless environments." Technical Report No. CS91-180, CSE Dept.,
UCSD, La Jolla, California.
Plutowski, Mark E., and Halbert White. 1993. "Selecting concise training sets from clean
data." To appear, IEEE Transactions on Neural Networks. 3, 1.
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7,009 | 786 | Solvable Models of Artificial Neural
Networks
Sumio Watanabe
Information and Communication R&D Center
Ricoh Co., Ltd.
3-2-3, Shin-Yokohama, Kohoku-ku, Yokohama, 222 Japan
[email protected]
Abstract
Solvable models of nonlinear learning machines are proposed, and
learning in artificial neural networks is studied based on the theory
of ordinary differential equations. A learning algorithm is constructed, by which the optimal parameter can be found without
any recursive procedure. The solvable models enable us to analyze
the reason why experimental results by the error backpropagation
often contradict the statistical learning theory.
1
INTRODUCTION
Recent studies have shown that learning in artificial neural networks can be understood as statistical parametric estimation using t.he maximum likelihood method
[1], and that their generalization abilities can be estimated using the statistical
asymptotic theory [2]. However, as is often reported, even when the number of
parameters is too large, the error for the test.ing sample is not so large as the theory
predicts. The reason for such inconsistency has not yet been clarified, because it is
difficult for the artificial neural network t.o find the global optimal parameter.
On the other hand, in order to analyze the nonlinear phenomena, exactly solvable
models have been playing a central role in mathematical physics, for example, the
K-dV equation, the Toda lattice, and some statistical models that satisfy the Yang-
423
424
Watanabe
Baxter equation[3].
This paper proposes the first solvable models in the nonlinear learning problem. We
consider simple three-layered neural networks, and show that the parameters from
the inputs to the hidden units determine the function space that is characterized
by a differential equation. This fact means that optimization of the parameters
is equivalent to optimization of the differential equation. Based on this property,
we construct a learning algorithm by which the optimal parameters can be found
without any recursive procedure. Experimental result using the proposed algorithm
shows that the maximum likelihood estimator is not always obtained by the error
backpropagation, and that the conventional statistical learning theory leaves much
to be improved.
2
The Basic Structure of Solvable Models
Let us consider a function fc,w( x) given by a simple neural network with 1 input
unit, H hidden units, and 1 output unit,
H
fc,w(x) =
L CiIPw;{X),
(I)
i=1
where both C = {Ci} and w = {Wi} are parameters to be optimized, IPw;{x) is the
output of the i-th hidden unit.
We assume that {IPi(X) = IPw, (x)} is a set of independent functions in C H-class.
The following theorem is the start point of this paper.
Theorem 1 The H -th order differential equation whose fundamental system of solution is {IPi( x)} and whose H -th order coefficient is 1 is uniquely given by
(Dwg)(x)
=(_l)H H!H+l(g,1P1,1P2,
.. ?,IPH) = 0,
lVH(IP1, IP2, .. ?,IPH)
(2)
where ltVH is the H -th order Wronskian,
IPH
( 1)
IPH
(2)
'PH
(H-l)
'PI
(H-l)
'P2
(H -1)
IPH
For proof, see [4]. From this theorem, we have the following corollary.
Corollary 1 Let g(x) be a C H-class function. Then the following conditions for
g(x) and w = {wd are equivalent.
(1) There exists a set
(2) (Dwg)(x) = O.
C
= {cd
such that g{x)
= E~l CjIPw;(x).
Solvable Models of Artificial Neural Networks
Example 1 Let us consider a case, !Pw;(x)
= exp(WiX).
H
L Ci exp(WiX)
g(x) =
i=l
is equivalent to {DH + P1D H- 1 + P2DH-2 + ... + PH }g(x)
and a set {Pi} is determined from {Wi} by the relation,
= 0, where D = (d/dx)
H
zH
+ Plz H- 1 + P2zH-2 + ... + PIl
= II(z - Wi)
('Vz E C).
i=l
Example 2
(RBF)
A function g(x) is given by radial basis functions,
11
g(x)
=L
Ci exp{ -(x - Wi)2},
i=l
if and only if e- z2 {DIl + P1DIl-l + P2DIl-2 + ... + PIl }(e Z2 g(x)) = 0, where a set
{Pi} is determined from {Wi} by the relation,
11
zll + Plz ll - 1 + P2zll-2
+ ... + PII = II(z -
2Wi) ('Vz E C).
i=l
Figure 1 shows a learning algorithm for the solvable models. When a target function
g( x) is given, let us consider the following function approximation problem.
11
g(x) =
L Ci!Pw;(X) + E(X).
(3)
i=l
Learning in the neural network is optimizing both {cd and {wd such that E( x) is
minimized for some error function. From the definition of D w , eq. (3) is equivalent
to (Dwg)(x) = (Dw?)(x), where the term (Dwg)(x) is independent of Cj. Therefore,
if we adopt IIDwEIl as the error function to be minimized, {wd is optimized by
minimizing IIDwgll, independently of {Cj}, where 111112 = J II(x)1 2dx. After IIDwgll
is minimized, we have (Dw.g)(x) ~ 0, where w* is the optimized parameter. From
the corollary 1, there exists a set {cn such that g(x) ~ L:ci!Pw~(x), where {en
can be found using the ordinary least square method.
3
Solvable Models
For a general function !Pw, the differential operator Dw does not always have such
a simple form as the above examples. In this section, we consider a linear operator
L such that the differential equation of L!pw has a simple form.
Definition A neural network L: Cj!PWi (x) is called solvable ifthere exist functions
a, b, and a linear operator L such that
(L!pwJ(x) = exp{a{wj)x + b(wi)).
The following theorem shows that the optimal parameter of the solvable models can
be found using the same algorithm as Figure 1.
425
426
Watanabe
H
g(X) =
L Ci ~ (x) +E(X)
i=l
equiv.
to
It is difficult
optimize wi
independently ?f ci
t
There exits C i s.t.
=L
H
g(x)
Dw g(x) = Dw E(X)
i
i=l
Ci
<P
.(x)
wi
I
Least Square Method
=L <<P
~
II Dwg II :minimited - - W: optimized
. -.-----1
.
eqmv.
q,* g(x)
0
I
c i : optimized
H
g(x)
i=l
.(x)
wi
Figure 1: St.ructure of Solvable Models
Theorem 2 For a solvable model of a neuml network, the following conditions are
equivalent when Wi "# Wj (i "# j).
= E:!:l Ci<t'w;(X).
that {DH + P1D H- 1 + P2DH-2 + ... + PH }(Lg)(x) =
(1) There exist both {cd and {wd such that g(x)
(2) There exists {Pi} such
O.
(3) For arbitmry Q > 0, we define a sequence {Yn} by Yn = (Lg)(nQ). Then, there
exists {qd such that Yn + qlYn-l + q2Yn-2 + ... + qHYn-H = o.
Note that IIDwLgl12 is a quadratic form for {pd, which is easily minimized by the
least square method. En IYn + qlYn-l + ... + QHYn_HI 2 is also a quadratic form for
{Qd?
Theorem 3 The sequences { wd, {pd, and {qd in the theorem 2 have the following
relations.
H
z H+ PIZ H-l+ P2 ZH-2+ ... + PH
IT(z - a(wi)) ('Vz E C),
i=l
H
zH
+ qlzH-l + q2zH-2 + ... + qH
=
IT(z - exp(a(Wi)Q)) ('Vz E C).
i=l
For proofs of the above theorems, see [5]. These theorems show that, if {Pi} or
Solvable Models of Artificial Neural Networks
{qd is optimized for a given function g( x), then {a( wd} can be found as a set of
solutions of the algebraic equation.
Suppose that a target function g( x) is given. Then, from the above theorems,
the globally optimal parameter w* = {wi} can be found by minimizing IIDwLgll
independently of {cd. Moreover, if the function a(w) is a one-to-one mapping, then
there exists w* uniquely without permutation of {wi}, if and only if the quadratic
form II{DH + P1 DH-1 + ... + PH }g1l2 is not degenerate[4]. (Remark that, if it is
degenerate, we can use another neural network with the smaller number of hidden
units.)
Example 3
A neural network without scaling
H
fb,c(X) =
L CiU(X + bi),
(4)
i=1
is solvable when (F u)( x) I- 0 (a.e.), where F denotes the Fourier transform. Define
(Fg)(x)/(Fu)(x), then, it follows that
a linear operator L by (Lg)(x)
=
H
(Lfb,c)(X)
=L
Ci
exp( -vCi bi x).
(5)
i=l
By the Theorem 2, the optimal {bd can be obtained by using the differential
sequential equation.
Example 4 (MLP)
01'
the
A three-layered perceptron
H
~
fb,c(X) = L
Ci
tan
-1
(
X
i=1
+
bi
a . ),
(6)
z
is solvable. Define a linear operator L by (Lg)( x) = x . (F g)( x), then, it follows
that
H
(Lfb,c)(X)
=L
Ci exp( -(a.i
+ yCi bdx + Q(ai, bd) (x
~ 0).
(7)
i=1
where Q( ai, bi ) is some function of ai and bj. Since the function tan -1 (x) is monotone increasing and bounded, we can expect that a neural network given by eq.
(6) has the same ability in the function approximation problem as the ordinary
three-layered perceptron using the sigmoid function, tanh{x).
Example 5 (Finite Wavelet Decomposition)
H
fb,c(X)
=L
Cju(
x
= (d/dx)n(1/(l + x 2 ?
),
(8)
a.j
i=l
is solvable when u(x)
+ bj
A finite wavelet decomposition
(n
~ 1). Define a lineal' operator
L by
(Lg)(x) = x- n . (Fg)(x) then, it follows that
H
(Lfb,c)(X)
=L
i=1
Ci
exp( -(a.j
+ vCi bi)x + P(a.j, bi?
(x ~ 0).
(9)
427
428
Watanabe
where f3(ai, bi) is some function of ai and bi. Note that O"(x) is an analyzing wavelet,
and that this example shows a method how to optimize parameters for the finite
wavelet decomposition.
4
Learning Algorithm
We construct a learning algorithm for solvable models, as shown in Figure 1-
< <Learning Algorithm> >
(0) A target function g(x) is given.
(1) {Ym} is calculated by Ym = (Lg)(mQ).
(2) {qi} is optimized by minimizing L:m IYm + Q1Ym-l + Q2Ym-2 + ... + QHYm_HI 2.
(3) {Zi} is calculated by solving zH + q1zH-1 + Q2zH-2 + ... + QH = 0.
(4) {wd is determined by a( wd = (l/Q) log Zi.
(5) {cd is optimized by minimizing L:j(g(Xj) - L:i Cj<;?w;(Xj?2.
Strictly speaking, g(x) should be given for arbitrary x. However, in the practical
applicat.ion, if the number of training samples is sufficiently large so that (Lg)( x)
can be almost precisely approximated, this algorithm is available. In the third
procedure, to solve the algebraic equation, t.he DKA method is applied, for example.
5
5.1
Experimental Results and Discussion
The backpropagation and the proposed method
For experiments, we used a probabilit.y density fUllction and a regression function
given by
Q(Ylx)
h(x)
1
((y - h(X?2)
exp -
J27r0"2
1
-3" tan
20"2
-1
X - 1/3
1
-1 X - 2/3
( 0.04 ) + 6" tan ( 0.02 )
where 0" = 0.2. One hundred input samples were set at the same interval in [0,1),
and output samples were taken from the above condit.ional distribution.
Table 1 shows the relation between the number of hidden units, training errors,
and regression errors. In the table, the t.raining errol' in the back propagation shows
the square error obtained after 100,000 training cycles. The traiuing error in the
proposed method shows the square errol' by the above algorithm. And the regression error shows the square error between the true regression curve h( x) and the
estimated curve.
Figure 2 shows the true and estimated regression lines: (0) the true regression
line and sanlple points, (1) the estimated regression line with 2 hidden units, by
the BP (the error backpropagation) after 100,000 training cycles, (2) the estimated
regression line with 12 hidden units, by the BP after 100,000 training cycles, (3) the
Solvable Models of Artificial Neural Networks
Table 1: Training errors and regression errors
Hidden
Units
2
4
6
8
10
12
Backpropagation
Training Regression
0.7698
4.1652
3.3464
0.4152
0.4227
3.3343
0.4189
3.3267
3.3284
0.4260
3.3170
0.4312
Proposed Method
Training Regression
4.0889
0.3301
3.8755
0.2653
3.5368
0.3730
3.2237
0.4297
3.2547
0.4413
3.1988
0.5810
estimated line with 2 hidden units by the proposed method, and (4) the estimated
line with 12 hidden units by the proposed method.
5.2
Discussion
When the number of hidden units was small, the training errors by the BP were
smaller, but the regression errors were larger. Vlhen the number of hidden units
was taken to be larger, the training error by the BP didn't decrease so much as the
proposed method, and the regression error didn't increase so mnch as the proposed
method.
By the error back propagation , parameters dichl 't reach the maximum likelihood
estimator, or they fell into local minima. However, when t.he number of hidden
units was large, the neural network wit.hout. t.he maximum likelihood estimator
attained the bett.er generalization. It seems that paramet.ers in the local minima
were closer to the true parameter than the maximum likelihood estimator.
Theoretically, in the case of the layered neural networks, the maximum likelihood
estimator may not be subject to asymptotically normal distribution because the
Fisher informat.ion matrix may be degenerate. This can be one reason why the
experimental results contradict the ordinary st.atistical theory. Adding such a problem, the above experimental results show that the local minimum causes a strange
problem. In order to construct the more precise learning t.heory for the backpropagation neural network, and to choose the better parameter for generalization, we
maybe need a method to analyze lea1'1ling and inference with a local minimum.
6
Conclusion
We have proposed solvable models of artificial neural networks, and studied their
learning structure. It has been shown by the experimental results that the proposed
method is useful in analysis of the neural network generalizat.ion problem.
429
430
Watanabe
........'..'
: ..
~--------.
'.
H : the number of hidden units
....
.'"
".'
...
'
..
t.he t.raining error
E"eg : the regression error
Etrain :
"0
..
(0) True Curve and Samples.
Sample error sum = 3.6874
.
.,
......
.
.
"0
e"
~
....... ~
..: ...........:......::::.. .
"0,
:
..: .... "... ".
e" e " '
..
'.
...
'
.' . '..
(1) BP after 100,000 cycles
H = 2, Etrain = 4.1652, E"eg = 0.7698
. . . ...
....." :
. .
,'.
..
..
'.'
.
.....
?
?
'
.
0"
(2) TIP aft.er 100,000 cycles
H = 12, E Ir?a;" = 3.3170, E"eg = 0.4312
.
,".
..
'
'
(3) Proposed Method
H = 2, Etrain = 4.0889, Ereg
= 0.3301
?
......
. .:'{:
..
(4) Proposed Met.hod
H = 12, E'm;" = 3.1988, Ereg = 0.5810
Figure 2: Experimental Results
References
[I] H. White. (1989) Learning in artificial neural networks: a statistical perspective.
Neural Computation, 1, 425-464.
[2] N.Murata, S.Yoshizawa, and S.-I.Amari.(1992) Learning Curves, Model Selection
and Complexity of Neural Networks. Adlla:nces in Neural Information Processing
Systems 5, San Mateo, Morgan Kaufman, pp.607-614.
[3] R. J. Baxter. (1982) Exactly Solved Models in Statistical Mechanics, Academic
Press.
[4] E. A. Coddington. (1955) Th.eory of ordinary differential equations, the McGrawHill Book Company, New York.
[5] S. Watanabe. (1993) Function approximation by neural networks and solution
spaces of differential equations. Submitted to Neural Networks.
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7,010 | 787 | Computational Elements of the Adaptive
Controller of the Human Arm
Reza Shadmehr and Ferdinando A. Mussa-Ivaldi
Dept . of Brain and Cognitive Sciences
M. I. T ., Cambridge , MA 02139
Email : [email protected] , sandro@ai .mit.edu
Abstract
We consider the problem of how the CNS learns to control dynamics of a mechanical system. By using a paradigm where a subject's
hand interacts with a virtual mechanical environment , we show
that learning control is via composition of a model of the imposed
dynamics. Some properties of the computational elements with
which the CNS composes this model are inferred through the generalization capabilities of the subject outside the training data.
1
Introduction
At about the age of three months, children become interested in tactile exploration
of objects around them. They attempt to reach for an object , but often fail to
properly control their arm and end up missing their target. In the ensuing weeks,
they rapidly improve and soon they can not only reach accurately, they can also
pick up the object and place it. Intriguingly, during this period of learning they
tend to perform rapid, flailing-like movements of their arm, as if trying to "excite"
the plant that they wish to control in order to build a model of its dynamics.
From a control perspective , having a model of the arm's skeletal dynamics seems
necessary because of the relatively low gain of the fast acting feedback system
in the spinal neuro-muscular controllers (Crago et al. 1976), and the long delays in
transmission of sensory information to the supra-spinal centers. Such a model could
be used by the CNS to predict the muscular forces that need to be produced in order
to move the arm along a desired trajectory. Yet, this model by itself is not sufficient
1077
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Shadmehr and Mussa-Ivaldi
for performing a contact task because most objects which our hand interacts with
change the arm's dynamics significantly. We are left with a situation in which we
need to be able to quickly acquire a model of an object's dynamics so that we can
incorporate it in the control system for the arm. How we learn to construct a model
of a dynamical system and how our brains represent the composed model are the
subjects of this research.
2
Learning Dynamics of a Mechanical System
To make the idea behind learning dynamics evident, consider the example of controlling a robotic arm. The arm may be seen as an inertially dominated mechanical
admitance, accepting force as input and producing a change in state as its output:
q = H(q)-l (F - C(q, q))
(1)
where q is the configuration of the robot, H is the inertia tensor, F is the input
force from some controllable source (e.g., motors), and C is the coriolis/centripetal
forces. In learning to control the arm, i.e., having it follow a certain state trajectory
or reach a final state, we form a model which has as its input the desired change in
the state of the arm and receive from its output a quantity representing the force
that should be produced by the actuators. Therefore, what needs to be learned is
a map from state and desired changes in state to force:
iJ(q, q, iid)
= if(q)qd + C(q, q)
(2)
Combine the above model with a simple PD feedback system,
F = iJ + if K(q - qd)
+ if B(q -
qd)
and the dynamics of the system in Eq. (1) can now be written in terms of a new
variable s = q - qd, i.e., the error in the trajectory. It is easy to see that if we have
if ~ Hand C ~ C, and if J( and B are positive definite, then s will be a decreasing
function of time, i.e., the system will be globally stable.
Learning dynamics means forming the map in Eq. (2). The computational elements
which we might use to do this may vary from simple memory cells that each have an
address in the state space (e.g., Albus 1975, Raibert & Wimberly 1984, Miller et al.
1987), to locally linear functions restricted to regions where we have data (Moore
& Atkeson 1994), to sigmoids (Gomi & Kawato 1990) and radial basis functions
which can broadly encode the state space (Botros & Atkeson 1991). Clearly, the
choice that we make in our computational elements will affect how the learned map
will generalize its behavior to regions of the state space outside of the training data.
Furthermore, since the task is to learn dynamics of a mechanical system (as opposed
to, for example, dynamics of a financial market), certain properties of mechanical
systems can be used to guide us in our choice for the computational elements. For
example, the map from states to forces for any mechanical system can be linearly
parameterized in terms of its mass properties (Slotine and Li, 1991). In an inertially
dominated system (like a multi-joint arm) these masses may be unknown, but the
fact that the dynamics can be linearized in terms of the unknowns makes the task
of learning control much simpler and orders of magnitude faster than using, for
example, an unstructured memory based approach.
Computational Elements of the Adaptive Controller of the Human Arm
J
ji
f
/i
t
I~
--.J
B
Lfl
1. sec ci
2.5
.
,~
.. -., ..."
. ,'
..... .
"C
c
~ Lfl!.'!
1.sec
o
Figure 1: Dynamics of a real 2 DOF robot was learned so to produce a desired trajectory.
A: Schematic of the robot. The desired trajectory is the quarter circle. Performance of a
PD controller is shown by the gray line, as well as in B, where joint trajectories are drawn:
the upper trace is the shoulder joint and the lower trace is the elbow joint. Desired joint
trajectory is solid line, actual trajectory is the gray line. C: Performance when the PD
controller is coupled with an adaptive model. D: Error in trajectory. Solid line is PD,
Gray line is PD+adaptation.
To illustrate this point, consider the task of learning to control a real robot arm.
Starting with the assumption that the plant has 2 degrees of freedom with rotational joints, inertial dynamics of Eq. (2) can be written as a product of a known
matrix-function of state-dependent geometric transformations Y, and an unknown
(but constant) vector a, representing the masses, center of masses, and link lengths:
D( q , q, qd) = Y (q, q, qd) a . The matrix Y serves the function of referring the unknown masses to their center of rotation and is a geometric transformation which
can be derived from our assumption regarding the structure of the robot. It is
these geometric transformations that can guide us in choosing the computational
elements for encoding the sensory data (q and q).
We used this approach to learn to control a real robot. The adaptation law was
derived from a Lyapunov criterion, as shown by Slotine and Li (1991):
~ = _yT (q, q, qd) (q - qd(t)
+q-
qd(t))
The system converged to a very low trajectory tracking error within only three periods of the movement (Fig. 1). This performance is achieved despite the fact that
our model of dynamics ignores frictional forces, noise and delay in the sensors, and
dynamics of the actuators. In contrast, using a sigmoid function as the basic com-
1079
1080
Shadmehr and Mussa-Ivaldi
putational element of the map and training via back-propagation led to comparable
levels of performance in over 4000 repetitions of the training data (Shadmehr 1990) .
The difference in performance of these two approaches was strictly due to the choice
of the computational elements with which the map of Eq. (2) was formed.
Now consider the task of a child learning dynamics of his arm, or that of an adult
picking up a hammer and pounding a nail. We can scarcely afford thousands of
practice trials before we have built an adequate model of dynamics. Our proposal
is that because dynamics of mechanical systems are distinctly structured, perhaps
our brains also use computational elements that are particularly suited for learning
dynamics of a motor task (as we did in learning to control the robot in Fig. 1). How
to determine the structure of these elements is the subject of the following sections.
3
A Virtual Mechanical Environment
To understand how humans represent learned dynamics of a motor task, we designed
a paradigm where subjects reached to a target while their hand interacted with a
virtual mechanical environment. This environment was a force field produced by a
manipulandum whose end-effector was grasped by the subject. The field of forces
depended only on the velocity of the hand, e.g., F = Bx, as shown in Fig. 2A,
and significantly changed the dynamics of the limb: When the robot's motors were
turned off (null field condition), movements were smooth, straight line trajectories
to the target (Fig. 2B). When coupled with the field however, the hand 's trajectory
was now significantly skewed from the straight line path (Fig. 2C).
It has been suggested that in making a reaching movement, the brain formulates
a kinematic plan describing a straight hand path along a smooth trajectory to the
target (Morasso 1981) . Initially we asked whether this plan was independent of the
dynamics of the moving limb. If so, as the subject practiced in the environment,
the hand path should converge to the straight line, smooth trajectory observed in
the null field . Indeed , with practice , trajectories in the force field did converge to
those in the null field. This was quantified by a measure of correlation which for all
eight subjects increased monotonically with practice time.
If the CNS adapted to the force field by composing a model of its dynamics, then
removal of the field at the onset of movement (un-be-known to the subject) should
lead to discrepancies between the actual field and the one predicted by the subject's
model, resulting in distorted trajectories which we call after-effects. The expected
dynamics of these after-effects can be predicted by a simple model of the upper
arm (Shadmehr and Mussa-Ivaldi 1994). Since the after-effects are a by-product
of the learning process, we expected that as subjects adapted to the field, their
performance in the null field would gradually degrade. We observed this gradual
growth of the after-effects, leading to grossly distorted trajectories in the null field
after subjects had adapted to the force field (Fig. 2D). This evidence suggested that
the CNS composed a model of the field and used this model to compensate for the
forces which it predicted the hand would encounter during a movement.
The information contained in the learned model is a map whose input is the state
and the desired change in state of the limb, and whose output is force (Eq. 2). How
is this map implemented by the CNS? Let us assume that the approximation is via
Computational Elements of the Adaptive Controller of the Human Arm
15o,------------.
I
0.5
~
~
~
>-g
0
.
r -0.5
A
-1
-0.5
0.5
Hand x-velocrty (rrV$)
B
-150 '--------,-1~00--=:-50-~-------:5::-:-0-1:-:"':OO--,J150
Displacement (mm)
Figure 2: A: The virtual mechanical environment as a force field. B: Trajectory of reaching movements (center-out) to 8 targets in a null field. C: Average?standard-deviation
of reaches to same targets when the field was on, before adaptation. D: After-affects of
adaptation, i.e., when moving in a null field but expecting the field.
a distributed set of computational elements (Poggio 1990). What are the properties
of these elements? An important property may be the spatial bandwidth, i.e_, the
size of the receptive field in the input space (the portion of the input space where
the element generates a significant output). This property greatly influences how
the eNS might interpolate between states which it has visited during training, and
whether it can generalize to regions beyond the boundary of the training data.
For example, in eye movements, it has been suggested that a model of dynamics of
the eye is stored in the cerebellum (Shidara et al. 1992). Cells which encode this
model (Purkinje cells) vary their firing rate as a linear function of the state of the
eye, and the sum of their outputs (firing rates) correlates well with the force that the
muscles need to produce to move the eye. Therefore, the model of eye's dynamics
is encoded via cells with very large receptive fields. On the other hand, cells which
take part in learning a visual hyperacuity task may have very small receptive fields
(Poggio et al. 1992), resulting in a situation where training in a localized region
does not lead to generalization.
In learning control of our limbs, one possibility for the computational elements is the
neural control circuits in the spinal cord (Mussa-Ivaldi 1992). Upon activation of
1081
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Shadmehr and Mussa-Ivaldi
Test workspace
Trai n ed Wo rkspace
A
-fo:::;;...-+---:>10 em
X
150
100
50
I
0
-50
-100
B
-150
-100
-50
50
100
150
Displacement (mm)
Figure 3: A: Schematic of subject's arm and
the force field was presented and the "test"
measured. B: After-effects at the test region.
field shown in Fig. 2A to the novel workspace.
at the test region.
c
-1
-0.5
o
0.5
Hand .-velocity (""s)
the trained region of the workspace where
region where the transferred effects were
C: A joint-based translation of the force
This is the field that the subject expected
one such circuit, muscles produce a time varying force field, i.e., forces which depend
on the state of the limb (position and velocity) and time (Mussa-Ivaldi et al. 1990).
Let us call the force function produced by one such motor element h(q, q, t) . It
turns out that as one changes the amount of activation to a motor element, the
output forces essentially scale . When two such motor elements are activated, the
resulting force field is a linear combination of the two individual fields (Bizzi et al.
1991): f = 2::;=1 Cdi(q, q, t).
Now consider the task of learning to move in the field shown in Fig. 2A . The
model that the eNS builds is a map from state of the limb to forces imposed by the
environment. Following the above scenario, the task is to find coefficients Ci for each
element such that the output field is a good approximation of the environmental
field. Unlike the computational elements of a visual task however, we may postulate
that the motor elements are characterized by their broad receptive fields . This is
because muscular force changes gradually as a function of the state of the limb
and therefore its output force is non zero for wide region of the state space. It
follows that if learning dynamics was accomplished through formation of a map
whose computational elements were these motor functions, then because of the large
spatial bandwidth of the elements the composed model should be able to generalize
to well beyond the region of the training data.
Computational Elements of the Adaptive Controller of the Human Arm
To test this, we limited the region of the input space for which training data was
provided and quantified the subject's ability to generalize to a region outside the
training set. Specifically, we limited the workspace where practice movements in the
force field took place and asked whether local exposure to the field led to after-effects
in other regions (Fig. 3A). We found that local training resulted in after-effects in
parts ofthe workspace where no exposure to the field had taken place (Fig. 3B) . This
indicated that the model composed by the CNS predicted specific forces well outside
the region in which it had been trained. The existence of this generalization showed
that the computational elements with which the internal model was implemented
had broad receptive fields.
The transferred after-effects (Fig. 3B) show that at the novel region of the
workspace, the subject's model of the environment predicted very different forces
than the one on which the subject had been trained on (compare with Fig. 2D).
This rejected the hypothesis that the composed model was a simple mapping (i.e.,
translation in variant) in a hand-based coordinate system, i.e., from states of the
arm to forces on the hand. The alternate hypothesis was that the composed model
related observed states of the arm to forces that needed to be produced by the
muscles and was translation invariant in a coordinate system based on the joints
and muscles. This would be the case, for example, if the computational elements
encoded the state of the arm linearly (analogous to Purkinje cells for the case of
eye movements) in joint space .
To test this idea, we translated the field in which the subject had practiced to the
novel region in a coordinate system defined based on the joint space of the subject's
arm, resulting in the field shown in Fig. 3C. We recorded the performance of the
subjects in this new field at the novel region of the workspace (after they had been
trained on field of Fig. 2A) and found that performance was near optimum at the
first exposure. This indicated that the geometric structure of the composed model
supported transfer of information in an intrinsic, e.g., joint based, coordinate system. This result is consistent with the hypothesis that the computational elements
involved in this learning task broadly encode the state space and represent their
input in a joint-based coordinate system and not a hand-based one.
4
Conclusions
In learning control of an inertially dominated mechanical system, knowledge of the
system's geometric constraints can direct us to choose our computational elements
such that learning is significantly faciliated. This was illustrated by an example
of a real robot arm: starting with no knowledge of its dynamics, a reasonable
model was learned within 3 periods of a movements (as opposed to thousands of
movements when the computational elements were chosen without regard to the
geometric properties). We argued that in learning to control the human arm, the
CNS might also make assumption regarding geometric properties of its links and
use specialized computational elements which facilitate learning of dynamics.
One possibility for these elements are the discrete neuronal circuits found in the
spinal cord. The function of these circuits can be mathematically formulated such
that a map representing inverse dynamics of the arm is formed via a combination
of the elements. Because these computational elements encode their input space
1083
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Shadmehr and Mussa-Ivaldi
broadly, i.e., has significant output for a wide region of the input space, we expected
that if subjects learned a dynamical process from localized training data, then the
formed model should generalize to novel regions of the state space. Indeed we found
that the subjects transferred the training information to novel regions of the state
space, and this transfer took place in a coordinate system similar to that of the
joints and muscles. We therefore suggest that the eNS learns control of the arm
through formation of a model whose computational elements broadly encode the
state space, and that these elements may be neuronal circuits of the spinal cord.
Acknowledgments: Financial support was provided in part by the NIH (AR26710) and
the ONR (N00014/90/J/1946). R . S. was supported by the McDonnell-Pew Center for
Cognitive Neurosciences and the Center for Biological and Computational Learning.
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control by Purkinje cells in the cerebellum. Nature 365:50-52 . Slotine JJE, Li W (1991)
Applied Nonlinear Control, Prentice Hall, Englewood Cliffs, New Jersey.
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7,011 | 788 | U sing Local Trajectory Optimizers To
Speed Up Global Optimization In
Dynamic Programming
Christopher G. Atkeson
Department of Brain and Cognitive Sciences and
the Artificial Intelligence Laboratory
Massachusetts Institute of Technology, NE43-771
545 Technology Square, Cambridge, MA 02139
617-253-0788, [email protected]
Abstract
Dynamic programming provides a methodology to develop planners
and controllers for nonlinear systems. However, general dynamic
programming is computationally intractable. We have developed
procedures that allow more complex planning and control problems
to be solved. We use second order local trajectory optimization
to generate locally optimal plans and local models of the value
function and its derivatives. We maintain global consistency of the
local models of the value function, guaranteeing that our locally
optimal plans are actually globally optimal, up to the resolution of
our search procedures.
Learning to do the right thing at each instant in situations that evolve over time is
difficult, as the future cost of actions chosen now may not be obvious immediately,
and may only become clear with time. Value functions are a representational tool
that makes the consequences of actions explicit. Value functions are difficult to
learn directly, but they can be built up from learned models of the dynamics of the
world and the cost function. This paper focuses on how fast optimizers that only
produce locally optimal answers can playa useful role in speeding up the process
of computing or learning a globally optimal value function.
Consider a system with dynamics Xk+l = f(xk, Uk) and a cost function L(Xk, Uk),
663
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Atkeson
where x is the state of the system and u is a vector of actions or controls. The subscript k serves as a time index, but will be dropped in the equations that follow. A
goal of reinforcement learning and optimal control is to find a policy that minimizes
the total cost, which is the sum of the costs for each time step. One approach to
doing this is to construct an optimal value function, V(x). The value of this value
function at a state x is the sum of all future costs, given that the system started in
state x and followed the optimal policy P(x) (chose optimal actions at each time
step as a function of the state). A local planner or controller can choose globally
optimal actions if it knew the future cost of each action. This cost is simply the
sum of the cost of taking the action right now and the future cost of the state that
the action leads to, which is given by the value function.
u* = arg min (L(x, u) + V(f(x, u?)
u
(1)
Value functions are difficult to learn. The environment does not provide training
examples that pair states with their optimal cost (x, V(x?. In fact, it seems that the
optimal policy depends on the optimal value function, which in turn depends on the
optimal policy. Algorithms to compute value functions typically iteratively refine
a candidate value function and/or a corresponding policy (dynamic programming).
These algorithms are usually expensive. We use local optimization to generate
locally optimal plans and local models of the value function and its derivatives. We
maintain global consistency of the local models of the value function, guaranteeing
that our locally optimal plans are actually globally optimal, up to the resolution of
our search procedures.
1
A SIMPLE EXAMPLE: A PENDULUM
In this paper we will present a simple example to make our ideas clear. Figure 1
shows a simulated set of locally optimal trajectories in phase space for a pendulum
being driven by a motor at the joint from the stable to the unstable equilibrium
position. S marks the start point, where the pendulum is hanging straight down,
and G marks the goal point, where the pendulum is inverted (pointing straight up).
The optimization criteria quadratically penalizes deviations from the goal point
and the magnitude of the torques applied. In the three locally optimal trajectories
shown the pendulum either swings directly up to the goal (1), moves initially away
from the goal and then swings up to the goal (2), or oscillates to pump itself and
then swing to the goal (3). In what follows we describe how to find these locally
optimal trajectories and also how to find the globally optimal trajectory.
2
LOCAL TRAJECTORY OPTIMIZATION
We base our local optimization process on dynamic programming within a tube
surrounding our current best estimate of a locally optimal trajectory (Dyer and
McReynolds 1970, Jacobson and Mayne 1970). We have a local quadratic model
of the cost to get to the goal (V) at each time step along the optimal trajectory
(assume a time step index k in everything below unless otherwise indicated):
Vex) ~ Vo
1 T
+ Vxx + 2x
Vxxx
(2)
Using Local Trajectory Optimizers to Speed Up Global Optimization
I
/"
?
e
/
/
~
1/// ~
II I
VI ~
"
\
\
Is
\
v'
\
Go
')
e
I~
Figure 1: Locally optimal trajectories for the pendulum swing up task.
A locally optim al policy can be computed using local models of the plant (in this
case local linear models) at each time step along the trajectory:
= f(x, u) ~ Ax + Bu + c
(3)
and local quadratic m odels of the one step cost at each time step along the trajectory:
1
1
L(x,u) ~ 2xT Qx+ 2uTRu+xTSu+tTu
(4)
Xk+l
At each point along the trajectory the optimal policy is given by:
u opt = -(R + BTVxxB)-1 x
(BTVxxAx + ST x + BTVxxc + VxB + t)
One can integrate the plant dynamics forward in time based on the above policy,
and then integrate the value functions and its first and second spatial derivatives
backwards in time to compute an improved value function, policy, and trajectory.
For a one step cost of the form:
1
T
L(x, u) ~ 2(x - Xd) Q(x - Xd)+
1
T
T
2(u - Ud) R(u - Ud) + (x - Xd) S(n - Ud)
the backward sweep takes the following form (in discrete time):
Zx = VxA + Q(x - Xd)
Zu = VxB + R(u - Ud)
Zxx = ATVxxA + Q
Zux = BTVxxA + S
Zuu = BTVxxB + R
(9)
K = Z;;: Zux
VXk _ 1 = Zx - ZuK
VXXk _ 1 = Zxx - ZxuK
(10)
(11)
(12)
(5)
(6)
(7)
(8)
665
666
Atkeson
3
STANDARD DYNAMIC PROGRAMMING
A typical implementation of dynamic programming in continuous state spaces discretizes the state space into cells, and assigns a fixed control action to each cell.
Larson's state increment dynamic programming (Larson 1968) is a good example
of this type of approach. In Figure 2A we see the trajectory segments produced by
applying the constant action in each cell, plotted on a phase space for the example
problem of swinging up a pendulum.
4
USING LOCAL TRAJECTORY OPTIMIZATION
WITH DP
We want to minimize the number of cells used in dynamic programming by making
the cells as large as possible. Combining local trajectory optimization with dynamic
programming allows us to greatly reduce the resolution of the grid on which we do
dynamic programming and still correctly estimate the cost to get to the goal from
different parts of the space. Figure 2A shows a dynamic programming approach
in which each cell contains a trajectory segment applied to the pendulum problem.
Figure 2B shows our approach, which creates a set of locally optimal trajectories
to the goal. By performing the local trajectory optimizations on a grid and forcing
adjacent trajectories to be consistent, this local optimization process becomes a
global optimization process. Forcing adjacent trajectories to be consistent means
requiring that all trajectories can be generated from a single underlying policy.
A trajectory can be made consistent with a neighbor by using the neighboring
trajectory as an initial trajectory in the local optimization process, or by using the
value function from the neighboring trajectory to generate the initial trajectory in
the local optimization process. Each grid element stores the trajectory that starts
at that point and achieves the lowest cost.
The trajectory segments in figure 2A match the trajectories in 2B. Figures 2C and
2D are low resolution versions of the same problem. Figure 2C shows that some
of the trajectory segments are no longer correct. In Figure 2D we see the locally
optimal trajectories to the goal are still consistent with the trajectories in Figure 2B.
Using locally optimal trajectories which go all the way to the goal as building blocks
for our dynamic programming algorithm allows us to avoid the problem of correctly
interpolating the cost to get to the goal function on a sparse grid. Instead, the cost
to get to the goal is measured directly on the optimal trajectory from each node to
the goal. We can use a much sparser grid and still converge.
5
ADAPTIVE GRIDS BASED ON CONSTANT COST
CONTOURS
We can limit the search by "growing" the volumes searched around the initial and
goal states by gradually increasing a cost threshold Cg ? We will only consider states
around the goal that have a cost less than Cg to get to the goal and states around
the initial state that have a cost less than C g to get from the initial state to that
state (Figure 3B). These two regions will increase in size as Cg is increased. We stop
Using Local Trajectory Optimizers to Speed Up Global Optimization
A
B
c
o
Figure 2: Different dynamic programming techniques (see text).
667
668
Atkeson
Figure 3: Volumes defined by a cost threshold.
increasing Cg as soon as the two regions come into contact. The optimal trajectory
has to be entirely within the union of these two regions, and has a cost of 2Cg .
Instead of having the initial conditions of the trajectories laid out on a grid over the
whole space, the initial conditions are laid out on a grid over the surface separating
the inside and the outside surfaces of the volumes described above. The resolution
of this grid is adaptively determined by checking whether the value function of one
trajectory correctly predicts the cost of a neighboring trajectory. If it does not,
additional grid points are added between the inconsistent trajectories.
During this global optimization we separate the state space into a volume around
the goal which has been completely solved and the rest of the state space, in which
no exploration or computation has been done. Each iteration of the algorithm
enlarges the completely solved volume by performing dynamic programming from
a surface of slightly increased cost to the current constant cost surface. When the
solved volume includes a known starting point or contacts a similar solved volume
with constant cost to get to the boundary from the starting point, a globally optimal
trajectory from the start to the goal has been found.
6
DP BASED ON APPROXIMATING CONSTANT
COST CONTOURS
Unfortunately, adaptive grids based on constant cost contours still suffer from the
curse of dimensionality, having only reduced the dimensionality of the problem by
1. We are currently exploring methods to approximate constant cost contours. For
example, constant cost contours can be approximated by growing "key" trajectories.
Using Local Trajectory Optimizers to Speed Up Global Optimization
;'
/
\
"
Figure 4: Approximate constant cost contours based on key trajectories
A version of this is illustrated in Figure 4. Here, trajectories were grown along the
"bottoms" of the value function "valleys". The location of a constant cost contour
can be estimated by using local quadratic models of the value function produced
by the process which optimizes the trajectory. These approximate representations
do not suffer from the curse of dimensionality. They require on the order of T D2,
where T is the length of time the trajectory requires to get to the goal, and D is
the dimensionality of the state space.
7
SUMMARY
Dynamic programming provides a methodology to plan trajectories and design controllers and estimators for nonlinear systems. However, general dynamic programming is computationally intractable. We have developed procedures that allow more
complex planning problems to be solved. We have modified the State Increment
Dynamic Programming approach of Larson (1968) in several ways:
1. In State Increment DP, a constant action is integrated to form a trajectory
segment from the center of a cell to its boundary. We use second order local
trajectory optimization (Differential Dynamic Programming) to generate an
optimal trajectory and form an optimal policy in a tube surrounding the
optimal trajectory within a cell. The trajectory segment and local policy
are globally optimal, up to the resolution of the representation of the value
function on the boundary of the cell.
2. We use the optimal policy within each cell to guide the local trajectory
optimization to form a globally optimal trajectory from the center of each
669
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Atkeson
cell all the way to the goal. This helps us avoid the accumulation of interpolation errors as one moves from cell to cell in the state space, and avoid
limitations caused by limited resolution of the representation of the value
function over the state space.
3. The second order trajectory optimization provides us with estimates of
the value function and its first and second spatial derivatives along each
trajectory. This provides a natural guide for adaptive grid approaches.
4. During the global optimization we separate the state space into a volume
around the goal which has been completely solved and the rest of the state
space, in which no exploration or computation has been done. The surface separating these volumes is a surface of constant cost, with respect to
achieving the goal.
5. Each iteration of the algorithm enlarges the completely solved volume by
performing dynamic programming from a surface of slightly increased cost
to the current constant cost surface.
6. When the solved volume includes a known starting point or contacts a
similar solved volume with constant cost to get to the boundary from the
starting point, a globally optimal trajectory from the start to the goal has
been found. No optimal trajectory will ever leave the solved volumes. This
would require the trajectory to increase rather than decrease its cost to get
to the goal as it progressed.
7. The surfaces of constant cost can be approximated by a representation that
avoids the curse of dimensionality.
8. The true test of this approach lies ahead: Can it produce reasonable solutions to complex problems?
Acknowledgenlents
Support was provided under Air Force Office of Scientific Research grant AFOSR89-0500, by the Siemens Corporation, and by the ATR Human Information Processing Research Laboratories. Support for CGA was provided by a National Science
Foundation Presidential Young Investigator Award.
References
Bellman, R., (1957) Dynamic Programming, Princeton University Press, Princeton,
NJ.
Bertsekas, D.P., (1987) Dynamic Programming: Deterministic and Stochastic Models, Prentice-Hall, Englewood Cliffs, NJ.
Dyer, P. and S.R. McReynolds, (1970) The Computation and Theory of Optimal
Control, Academic Press, New York, NY.
Jacobson, D.H. and D.Q. Mayne, (1970) Differential Dynamic Programming, Elsevier, New York, NY.
Larson, R.E., (1968) State Increment Dynamic Programming, Elsevier, New York,
NY.
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7,012 | 789 | Assessing the Quality of Learned Local Models
Stefan Schaal
Christopher G. Atkeson
Department of Brain and Cognitive Sciences & The Artifical Intelligence Laboratory
Massachusetts Institute of Technology
545 Technology Square, Cambridge, MA 02139
email: [email protected], [email protected]
Abstract
An approach is presented to learning high dimensional functions in the case
where the learning algorithm can affect the generation of new data. A local
modeling algorithm, locally weighted regression, is used to represent the learned
function. Architectural parameters of the approach, such as distance metrics, are
also localized and become a function of the query point instead of being global.
Statistical tests are given for when a local model is good enough and sampling
should be moved to a new area. Our methods explicitly deal with the case where
prediction accuracy requirements exist during exploration: By gradually shifting
a "center of exploration" and controlling the speed of the shift with local prediction accuracy, a goal-directed exploration of state space takes place along the
fringes of the current data support until the task goal is achieved. We illustrate
this approach with simulation results and results from a real robot learning a
complex juggling task.
1
INTRODUCTION
Every learning algorithm faces the problem of sparse data if the task to be learned is sufficiently nonlinear and high dimensional. Generalization from a limited number of data
points in such spaces will usually be strongly biased. If, however, the learning algorithm
has the ability to affect the creation of new experiences, the need for such bias can be reduced. This raises the questions of (1) how to sample data the most efficient, and (2) how
to assess the quality of the sampled data with respect to the task to be learned. To address
these questions, we represent the task to be learned with local linear models. Instead of
constraining the number of linear models as in other approaches, infinitely many local
models are permitted. This corresponds to modeling the task with the help of (hyper-)
tangent planes at every query point instead of representing it in a piecewise linear fashion. The algorithm applied for this purpose, locally weighted regression (LWR), stems
from nonparametric regression analysis (Cleveland, 1979, Muller, 1988, Hardie 1990,
Hastie&Tibshirani, 1991). In Section 2, we will briefly outline LWR. Section 3 discusses
160
Assessing the Quality of Learned Local Models
several statistical tools for assessing the quality of a learned linear LWR model, how to
optimize the architectural parameters of LWR, and also how to detect outliers in the data.
In contrast to previous work, all of these statistical methods are local, i.e., they depend on
the data in the proximity of the current query point and not on all the sampled data. A
simple exploration algorithm, the shifting setpoint algorithm (SSA), is used in Section 4
to demonstrate how the properties of L WR can be exploited for learning control. The
SSA explicitly controls prediction accuracy during learning and samples data with the
help of optimal control techniques. Simulation results illustrate that this method work
well in high dimensional spaces. As a final example, the methods are applied to a real
robot learning a complex juggling task in Section 5.
2
LOCALLY WEIGHTED REGRESSION
Locally linear models constitute a good compromise between locally constant models
such as nearest neighbors or moving average and locally higher order models; the former
tend to introduce too much bias while the latter require fitting many parameters which is
computationally expensive and needs a lot of data. The algorithm which we explore here,
locally weighted regression (LWR) (Atkeson, 1992, Moore, 1991, Schaal&Atkeson,
1994), is closely related to versions suggested by Cleveland et al. (1979, 1988) and
Farmer&Siderowich (1987). A LWR model is trained by simply storing every experience as an input/output pair in memory. If an output Y, is to be generated from a given
input x" the
. it is computed by fitting a (hyper-) tangent plane at x , by means of weighted regressIOn:
(1)
where X is an mx(n+ 1) matrix of inputs to the regression, y the vector of corresponding
outputs, P(x,) the vector of regression parameters, and W the diagonal mxm matrix of
weights. The requested Y,results from evaluating the tangent plane at x ,i.e., Y = x~p.
The elements of W give points which are close to the current query poi~t x, a l~ger influence than those which are far away. They are determined by a Gaussian kernel:
w;(x,) = exp( (x; - x,lD(x,)(x; - x,) / 2k(x,)2)
(2)
w; is the weight 'for the i rh data point (xj,Yj) in memory given query point x . The matrix D(x,) weights the contribution of the individual input dimensions, and the factor
k(x,) determines how local the regression will be. D and k are architectural parameters
of LWR and can be adjusted to optimize the fit of the local model. In the following we
will just focus on optimizing k, assuming that D normalizes the inputs and needs no further adjustment; note that, with some additional complexity, our methods would also hold
for locally tuning D.
3
ASSESSING THE LOCAL FIT
In order to measure the goodness of the local model, several tests have been suggested.
The most widely accepted one is leave-one-out cross validation (CV) which calculates the
prediction error of every point in memory after recalculating (1) without this point
(Wahba&Wold 1975, Maron&Moore 1994). As an alternative measure, Cleveland et al.
(1988) suggested Mallow's Cp-test, originally developed as a way to select covariates in
linear regression analysis (Mallow, 1966). Hastie&Tibshirani (1991) showed that CV and
the Cp-test are closely related for certain classes of analyses. Hastie&Tibshirani (1991)
161
162
Schaal and Atkeson
also presented pointwise standard-error bands to assess the confidence in a fitted value
which correspond to confidence bands in the case of an unbiased fit All these tests are
essentially global by requiring statistical analysis over the entire range of data in memory. Such a global analysis is computationally costly, and it may also not give an adequate measure at the current query site Xq: the behavior of the function to be approximated may differ significantly in different places, and an averaging over all these behaviors is unlikely to be representative for all query sites (Fan&Gijbels, 1992).
It is possible to convert some of the above measures to be local. Global cross validation
has a relative in linear regression analysis, the PRESS residual error (e.g., Myers, 1990),
here formulated as a mean squared local cross validation error:
n is the number of data points in memory contributing with a weight Wj greater than
some small constant (e.g., Wi> 0.01) to the regression, and p is the dimensionality of ~.
The PRESS statistic performs leave-one-out cross validation computationally very efficient by not requiring the recalculation of ~ (Eq.(1)) for every excluded point.
Analogously, prediction intervals from linear regression analysis (e.g., Myers, 1990) can
be transformed to be a local measure too:
1'1 = x;~ ? (a/2,11'-p'
where
S2
S~1 + x: (XTWTWXfl Xq
(4)
is an estimate of the variance at x'I:
S2(X )
= (X~ - ytWTW(X~ - y)
n' - p'
q
(5)
and (a/2,,.'-' isStudent'st-valueof n'-p' degrees of freedom fora l00(I-a)% prediction bound. The direct interpretation of (4) as prediction bounds is only possible if y is
an unbiased estimate, which is usually hard to determine.
'I
Finally, the PRESS statistic can also be used for local outlier detection. For this PUIJJOse it
is reformulated as a standardized individual PRESS residual:
eiC,..,..
, .. (x q )=
~
S
T T T
-1
1- w?x.
(X W wx) X.W.
I
I
I
I
(6)
This measure has zero mean and unit variance. If it exceeds a certain threshold for a point
Xi' the point can be called an outlier.
An important ingredient to forming the measures (3)-(6) lies in the definition of n' and
p' as given in (3). Imagine that the weighting function (2) is not Gaussian but rather a
function that clips data points whose distance from the current query point exceeds a certain threshold and that the remaining r data points all contribute with unit weight. This
reduced data regression coincides correctly with a r -data regression since n' = r . In the
case of the soft-weighting (2). the definition of n' ensures the proper definition of the
moments of the data. However, the definition of p', i.e., the degrees of freedom of the regression, is somewhat arbitrary since it is unclear how many degrees of freedom have ac-
Assessing the Quality of Learned Local Models
tually been used. Defining p' as in (3) guarantees that p' < n' and renders all results
more pessimistic when only a small number of data points contribute to the regression.
The statistical tests (3) and (4) can not only be
(a) :
used as a diagnostic tool, but they can also
1.5
1\.
serve
to optimize the architectural parameters
.
:,
ofLWR. This results in a function fitting tech.,J;
,..,
nique which is called supersmoothing in statis0.5
tics (Hastie&Tibshirani, 1991). Fan&Gijbels
(1992) investigated a method for this purpose
that required estimation of the second deriva.0.S.o+.2~+-+-0~"0.""2c..,.....,....,..O'?..;..;........~0T-.8~"0.~e~-i--r'~1.2 tive of the function to be approximated and the
data density distribution. These two measures
are
not trivially obtained in high dimensions
(b)
1.5
and we would like to avoid using them. Figure
1 shows fits of noisy data from the function
,,..
y = x- sin\2n:x 3 ) COS(2n:x3) exp(x4) with
0.5
95% prediction intervals around the fitted values. In Figure la, global one-leave-out cross
validation was applied to optimize k (cf.
.o.5.+0.2~""""~"0.-'2~""'0.?""""""~0f-.8~"0.8~""""""".....-'r-...-112 Eq.(2?. In the left part of the graph the fit
x ??>
starts to follow noise. Such behavior is to be
expected since the global optimization of k
(c)
1.5
also took into account the quickly changing
- _. _. predcton int.rv.
regions on the right side of the graph and thus
" nai., data
chose a rather small k. In Figure 1b mini0.5
mization of the local one-leave-out cross validation error was applied to fit the data, and in
o
Figure 1c prediction intervals were mini.0.5.+0.2..,......,.-.,.....,~...,0.2-.-,......,.,..0.? ...,.....,....~0r-.8,....,.....,r-r0.8~...,...,-..,......,.-.,....,1.2 mized. These two fits cope nicely with both
J(-->
the high frequency and the low frequency reFigure 1: Optimizing the LWR fit using: (a) gions of the data and recover the true function
global cross validation; (b) local cross valida- rather well. The extrapolation properties of lotion; (c) local prediction intervals.
cal cross validation are the most appropriate
given that the we know the true function.
Interestingly, at the right end of Figure 1c, the minimization of the prediction intervals
suddenly detects that global regression has a lower prediction interval than local regression and jumps into the global mode by making k rather large. In both local methods
there is always a competition between local and global regression. But sudden jumps take
place only when the prediction interval is so large that the data is not trustworthy anyway.
2
A
A
To some extend, the statistical tests (3)-(6) implicitly measure the data density at the current query point and are thus sensitive towards little data support, characterized by a
small n'. This property is desirable as a diagnostic tool, particularly if the data sampling
process can be directed towards such regions. However, if a fixed data set is to be analyzed which has rather sparse and noisy data in several regions, a fit of the data with local
optimization methods may result in too jagged an approximation since the local fitting
mistakes the noise in such regions as high frequency portion of the data. Global methods
avoid this effect by biasing the function fitting in such unfavorable areas with knowledge
from other data regions and will produce better results if this bias is appropriate.
163
164
Schaal and Atkeson
4
THE SHIFTING SETPOINT EXPLORATION ALGORITHM
In this section we want to give an example of how LWR and its statistical tools can be
used for goal directed data sampling in learning control. If the task to be learned is high
dimensional it is not possible to leave data collection to random exploration; on the one
hand this would take too much time. and on the other hand it may cause the system to enter unsafe or costly regions of operation. We want to develop an exploration algorithm
which explicitly avoids with such problems. The shifting setpoint algorithm (SSA) attempts to decompose the control problem into two separate control tasks on different time
scales. At the fast time scale. it acts as a nonlinear regulator by trying to keep the controlled system at some chosen setpoints in order to increase the data density at these setpoints. On a slower time scale. the setpoints are shifted by controlling local prediction accuracy to accomplish a desired goal. In this way the SSA builds a narrow tube of data
support in which it knows the world. This data can be used by more sophisticated control
algorithms for planning or further exploration.
The algorithm is graphically illustrated in the example of a mountain car in Figure 2. The
task of the car is to drive at a given constant horizontal speed xdesired from the left to the
right of Figure 2a. xduired need not be met precisely; the car should also minimize its fuel
consumption. Initially. the car knows nothing about the world and cannot look ahead. but
it has noisy feedback of its position and velocity. Commands. which correspond to the
thrust F of the motor. can be generated at 5Hz. The mountain car starts at its start point
with one arbitrary initial action for the first time step; then it brakes and starts all over
again. assuming the system can be reset somehow. The discrete one step dynamics of the
car are modeled by an LWR forward model:
x...,xt
= f(Xc..,.,.elll. F ).
where
x = (x.xl
(7)
After a few trials~ the SSA searches the data in memory for the point (x;u"elll.F,x~?xt)resl
whose outcome x lI?xt can be predicted with the smallest local prediction interval. This
best point is declared the setpoint of this stage:
T )T = (T
FAT)T
T F S ,XS,OIl'
( XS,ill'
XC~IIl' 'X llm bltSl
(8)
and its local linear model results from a corresponding LWR lookup:
A
XS,OIll
(9)
= f(xS,u.,F s ):::: AxS;1I + BFs + C
Based on this liDear model. an optimal LQ controller (e.g., Dyer&McReynolds. 1970) can
be constructed. This results in a control law of the form:
(10)
After these calculations. the mountain car learned one controlled action for the first time
step, However. since the initial action was chosen arbitrarily, XS,OIII will be significantly
away from the desired speed Xdesir?d. A reduction of this error is achieved as follows,
First, the SSA repeats one step actions with the LQ controller until suffjcient data is collected to reduce the prediction intervals ofLWR lookups for (x~,ill,Fs) (Eq.(9)) below a
certain threshold. Then it shifts the setpoint towards the goal according to the procedure:
1) calculate the error of the predicted output state:
err S o,d = xde .
2) take the derivfltive of the error with respect to the comm'and Fs
for (XIill.FS) (cf. (9)):
d -
Xs
III
sr;om a LWR lookup
Assessing the Quality of Learned Local Models
aerr S,OI" = aerr S,Old aXS,OMI = _ aXS,Old = _ B
aFs
aXSpld aFs
aFs
and calculate a correction Ms from solving: -BMs = a errs old ; a
E [0,1] determines how much of the error should be compensated for in one step.
3) update Fs: Fs = Fs - Ms and calculate the new X SOM1 with LWR (Eq.(9?.
4) assess the fit for the updated setpoint with prediction intervals. If the quality is above
a certain threshold, continue with I), otherwise terminate shifting.
Figure 2: The mountain car: (a) landscape across which the car has to drive at constant velocity
of 0.8 mIs, (b) contour plot of data density in phase space as generated by using multistage
SSA, (c) contour plot of data density in position-action space, (d) 2-dimensional mountain car
0.1
10
?
2D
Polltlon E" ... [III)
30
40
10
[J Ylloclty EITOf ["'")
In this way, the output state of the setpoint
shifts towards the goal until the data support
falls below a threshold. Now the mountain
car perfonns several new trials with the new
setpoint and the correspondingly updated
LQ controller. After the quality of fit statistics rise above a threshold, the setpoint can
be shifted again. As soon as the first stage's
setpoint reduces the error Xdesj~d - Xs old sufficiently, a new stage is created and the
mountain car tries to move one step further in its world. The entire procedure is repeated
for each new stage until the car knows how to move across the landscape. Figure 2b and
Figure 2c show the thin band of data which the algorithm collected in state space and position-action space, These two pictures together form a narrow tube of knowledge in the
input space of the forward model.
Figure 3: Mean prediction error of local models
165
166
Schaal and Atkeson
The example of the mountain car can easily be scaled up to arbitrarily high dimensions by
making the mountain a multivariate function. We tried versions up to a 5-dimensional
mountain corresponding to a 9\15 ~ 9\10 forward model; Figure 2d shows the 2-dimensional version. The results of learning had the same quality as in the ID example. Figure
3 shows the prediction errors of the local models after learning for the ID. 2D ?...? and 5D
mountain car. To obtain these errors. the car was started at random positions within its
data support from where it drove along the desired trajectory. The difference between the
predicted next state and the actual outcome at each time step was averaged. Position errors stayed within 2-4 cm on the 10m long landscape. and velocity errors within 0.020.05 m/s. The dimensionality of the problem did not affect the outcome significantly.
5
ROBOT JUGGLING
To test our algorithms in a real world experiment. we implemented them on a juggling
robot. The juggling task to be performed.
devil sticking. is illustrated in Figure 4a. For
the robot. devil sticking was slightly simpli(a)
fied by attaching the devil stick to a boom.
as illustrated in Figure 4b. The task state was
encoded as a 5-dimensional state vector.
taken at the moment when the devilstick hit
one of the hand sticks; the throw action was
parameterized as 5-dimensional action vector. This resulted in a 9\10 ~ 9\5 discrete
forward model of the task. Initially the robot
was given default actions for the left-hand
and right-hand throws; the quality of these
throws. however. was far away from achieving steady juggling. The robot started with
no initial experiences and tried to build con(b)
trollers to perform continuous juggling. The
goal
states for the SSA developed automati(;j
cally from the requirement that the left hand
~,~~--------------------~
~1OIIJ
had to learn to throw the devilstick to a place
where the right hand had sufficient data support to control the devilstick. and vice versa.
Figure 4c shows a typical learning curve for
this task. It took about 40 trials before the
21
3,
51
4'
"
Trial Number
left and the right hand learned to throw the
(C)
devilstick such that both hands were able to
Figure 4: (a) illustration of devilsticking, (b) a cooperate. Then. performance quickly went
devils ticking robot, (c) learning curve of robot up to long runs up to 1200 consecutive hits.
Humans usually need about one week of one
hour practicing per day before they achieve decent juggling performance. In comparison
to this. the learning algorithm performed very well. However. it has to be pointed out that
the learned controllers were only local and could not cope with larger perturbations. A detailed description of this experiment can be found in Schaal&Atkeson (1994).
Assessing the Quality of Learned Local Models
CONCLUSIONS
One of the advantages of memory-based nonparametric learning methods lies in the least
commitment strategy which is associated with them. Since all data is kept in memory, a
lookup can be optimized with respect to the architectural parameters. Parametric approaches do not have this ability if they discard their training data; if they retain it, they
essentially become memory-based. The origin of nonparametric modeling in traditional
statistics provides many established statistical methods to inspect the quality of what has
been learned by the system. Such statistics formed the backbone of the SSA exploration
algorithm. So far we have only examined some of the most obvious statistical tools which
directly relate to regression analysis. Many other methods from other statistical frameworks may be suitable as well and will be explored by our future work.
Acknowledgements
Support was provided by the Air Force Office of Scientific Research, by the Siemens
Corporation, the German Scholarship Foundation and the Alexander von Humboldt
Foundation to Stefan Schaal, and a National Science Foundation Presidential Young
Investigator Award to Christopher G. Atkeson. We thank Gideon Stein for implementing
the first version of LWR on a DSP board, and Gerrie van Zyl for building the devil
sticking robot and implementing the first version of learning of devil sticking.
References
Atkeson, C.G. (1992), "Memory-Based Approaches to Approximating Continuous Functions", in:
Casdagli, M.; Eubank, S. (eds.): Nonlinear Modeling and Forecasting. Redwood City, CA: Addison Wesley (1992).
Cleveland, W.S., Devlin, S.l, Grosse, E. (1988), "Regression by Local Fitting: Methods, Properties, and Computational Algorithms". Journal of &onometrics 37,87 -114, North-Holland (1988).
Cleveland, W.S. (1979), "Robust Locally-Weighted Regression and Smoothing Scatterplots".
Journal of the American Statistical Association ,no.74, pp.829-836 (1979).
Dyer, P., McReynolds, S.R. (1970), The Computation and Theory of Optima I Comrol, New York:
Academic Press (1970).
Fan, J., Gijbels, I. (1992), "Variable Bandwidth And Local Linear Regression Smoothers", The
Annals of Statistics, vol.20, no.4, pp.2008-2036 (1992).
Farmer, J.D., Sidorowich, J.I (1987), "Predicting Chaotic Dynamics", Kelso, IA.S., Mandell, AJ.,
Shies inger, M.F., (eds.):Dynamic Patterns in Complex Systems, World Scientific Press (1987).
HardIe, W. (1991), Smoothing Techniques with Implementation in S, New York, NY: Springer.
Hastie, T.l; Tibshirani, R.J. (1991), Generalized Additive Models, Chapman and Hall.
Mallows, C.L. (1966), "Choosing a Subset Regression", unpublished paper presented at the annual
meeting of the American Statistical Association, Los Angles (1966).
Maron, 0., Moore, A.W. (1994), "Hoeffding Races: Accelerating Model Selection Search for
Classification and Function Approximation", in: Cowan, J. , Tesauro, G., and Alspector, 1. (eds.)
Advances in Neural Information Processing Systems 6, Morgan Kaufmann (1994).
Muller, H.-G. (1988), Nonparametric Regression Analysis of Longitudinal Data, Lecture Notes in
Statistics Series, vo1.46, Berlin: Springer (1988).
Myers, R.H. (1990), Classical And Modern Regression With Applications, PWS-KENT (1990).
Schaal, S., Atkeson, C.G. (1994), "Robot Juggling: An Implementation of Memory-based
Learning", to appear in: Control Systems Magazine, Feb. (1994).
Wahba, G., Wold, S. (1975), "A Completely Automatic French Curve: Fitting Spline Functions By
Cross-Validation", Communications in Statistics, 4(1) (1975).
167
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7,013 | 79 | 840
LEARNING IN NETWORKS OF
NONDETERMINISTIC ADAPTIVE LOGIC ELEMENTS
Richard C. Windecker*
AT&T Bell Laboratories, Middletown, NJ 07748
ABSTRACT
This paper presents a model of nondeterministic adaptive automata that are
constructed from simpler nondeterministic adaptive information processing
elements. The first half of the paper describes the model. The second half discusses
some of its significant adaptive properties using computer simulation examples.
Chief among these properties is that network aggregates of the model elements can
adapt appropriately when a single reinforcement channel provides the same positive
or negative reinforcement signal to all adaptive elements of the network at the same
This holds for multiple-input, multiple-output, multiple-layered,
time.
combinational and sequential networks. It also holds when some network elements
are "hidden" in that their outputs are not directly seen by the external
environment.
INTRODUCTION
There are two primary motivations for studying models of adaptive automata
constructed from simple parts. First, they let us learn things about real biological
systems whose properties are difficult to study directly: We form a hypothesis
about such systems, embody it in a model, and then see if the model has reasonable
learning and behavioral properties. In the present work, the hypothesis being tested
is: that much of an animal's behavior as determined by its nervous system is
intrinsically nondeterministic; that learning consists of incremental changes in the
probabilities governing the animal's behavior; and that this is a consequence of the
animal's nervous system consisting of an aggregate of information processing
elements some of which are individually nondeterministic and adaptive. The second
motivation for studying models of this type is to find ways of building machines
that can learn to do (artificially) intelligent and practical things. This approach has
the potential of complementing the currently more developed approach of
programming intelligence into machines.
We do not assert that there is necessarily a one-to-one correspondence
between real physiological neurons and the postulated model information processing
elements. Thus, the model may be loosely termed a "neural network model," but is
more accurately described as a model of adaptive automata constructed from simple
adaptive parts.
* The main ideas in this paper were conceived and initially developed while the
author was at the University of Chiang Mai, Thailand (1972-73). The ideas were
developed further and put in a form consistent with existing switching and
automata theory during the next four years. For two of those years, the author
was at the University of Guelph, Ontario, supported of National Research
Council of Canada Grant #A6983.
? American Institute of Physics 1988
841
It almost certainly has to be a property of any acceptable model of animal
learning that a single reinforcement channel providing reinforcement to all the
adaptive elements in a network (or subnetwork) can effectively cause that network
to adapt appropriately. Otherwise, methods of providing separate, specific
reinforcement to all adaptive elements in the network must be postulated. Clearly,
the environment reinforces an animal as a whole and the same reinforcement
mechanism can cause the animal to adapt to many types of situation. Thus, the
reinforcement system is non-specific to particular adaptive elements and particular
behaviors. The model presented here has this property.
The model described here is a close cousin to the family of models recently
described by Barto and coworkers 1-4. The most significant difference are: 1) In
the present model, we define the timing discipline for networks of elements more
explicitly and completely. This particular timing discipline makes the present
model consistent with a nondeterministic extension of switching and automata
theory previously described 0. 2) In the present model, the reinforcement algorithm
that adjusts the weights is kept very simple. With this algorithm, positive and
negative reinforcement have symmetric and opposite effects on the weights. This
ensures that the logical signals are symmetric opposites of each other. (Even small
differences in the reinforcement algorithm can make both subtle as well as profound
differences in the behavior of the model.) We also allow, null, or zero,
reinforcemen t.
As in the family of models described by Barto, networks constructed within
the present model can get "stuck" at a sUboptimal behavior during learning and
therefore not arrive at the optimal adapted state. The complexity of the Barto
reinforcement algorithm is designed partly to overcome this tendency. In the
present work, we emphasize the use of training strategies when we wish to ensure
that the network arrives at an optimal state. (In nature, it seems likely that getting
"stuck" at suboptimal behavior is common.) In all networks studied so far, it has
been easy to find strategies that prevent the network from getting stuck.
The chief contributions of the present work are: 1) The establishment of a
close connection between these types of models and ordinary, nonadaptive,
switching and automata theory 0. This makes the wealth of knowledge in this area,
especially network synthesis and analysis methods, readily applicable to the study
of adaptive networks. 2) The experimental demonstration that sequential
("recurrent") nondeterministic adaptive networks can adapt appropriately. Such
networks can learn to produce outputs that depend on the recent sequence of past
inputs, not just the current inputs. 3) The demonstration that the use of training
strategies can not only prevent a network from getting stuck, but may also result in
more rapid learning. Thus, such strategies may be able to compensate, or even
more than compensate, for reduced complexity in the model itself.
References 2-4 and 6 provide a comprehensive background and guide to the
literature on both deterministic and nondeterministic adaptive automata including
those constructed from simple parts and those not.
THE MODEL ADAPTIVE ELEMENT
The model adaptive element postulated in this work is a nondeterministic,
adaptive generalization of threshold logic 7. Thus, we call these elements
Nondeterministic Adaptive Threshold-logic gates (NATs). The output chosen by a
NAT at any given time is not a function of its inputs. Rather, it is chosen by a
stochastic process according to certain probabilities. It is these probabilities that
are a function of the inputs.
A NAT is like an ordinary logic gate in that it accepts logical inputs that are
two-valued and produces a logical output that is two-valued. We let these values be
842
+ 1 and -1. A NAT also has a timing input channel and a reinforcement input
channel. The NAT operates on a three-part cycle: 1) Logical input signals are
changed and remain constant. 2) A timing signal is received and the NAT selects a
new output based on the inputs at that moment. The new output remains
constant. 3) A reinforcement signal is received and the weights are incremented
according to certain rules.
Let N be the number of logical input channels, let Xi represent the ith input
signal, and let z be the output. The NAT has within it N+ 1 "weights,"
wo, WI! ... , WN. The weights are confined to integer values. For a given set of
inputs, the gate calculates the quantity W:
Then the probability that output z =
+ 1 is chosen is:
w _--=-=-
P(z = +1) -
Je
1
.j2;u
-
00
2q2
W/v2q
dx = _1_
..;;
J
e-(l d~
(2)
- 00
where ~ = xjV2u. (An equivalent formulation is to let the NAT generate a
random number, Wq, according to the normal distribution with mean zero and
variance u 2 . Then if W > - Wq, the gate selects the output z = + 1. If
W < - Wq, the gate selects output z = -1. If W = - Wq, the gate selects output
-1 or + 1 with equal probability.)
Reinforcement signals, R, may have one of three values: + 1, -1, and 0
representing positive, negative, and no reinforcement, respectively. If + 1
reinforcement is received, each weight is incremented by one in the direction that
makes the current output, z, more likely to occur in the future when the same
inputs are applied; if -1 reinforcement is received, each weight is incremented in
the direction that makes the current output less likely; if 0 reinforcement is
received, the weights are not changed. These rules may be summarized: ~wo = zR
and ~Wj = xjzR for i > o.
NATs operate in discrete time because if the NAT can choose output + 1 or
-1, depending on a stochastic process, it has to be told when to select a new
output. It cannot "run freely," or it could be constantly changing output. Nor can
it change output only when its inputs change because it may need to select a new
output even when they do not change.
The normal distribution is used for heuristic reasons. If a real neuron (or an
aggregate of neurons) uses a stochastic process to produce nondeterministic
behavior, it is likely that process can be described by the normal distribution. In
any case, the exact relationship between P{z = + 1) and W is not critical. What is
important is that P(z = + 1) be monotonically increasing in W, go to 0 and 1
asymptotically as W goes to - 00 and + 00, respectively, and equal 0.5 at W = O.
The parameter u is adjustable. We use 10 in the computer simulation
experiments described below. Experimentally, values near 10 work reasonably well
for networks of NATs having few inputs. Note that as u goes to zero, the behavior
of a NAT approximates that of an ordinary deterministic ada pt,ive threshold logic
gate with the difference that the output for the case W = 0 is not arbitrary: The
NAT will select output +1 or -1 with equal probability.
Note that for all values of W, the probabilit,ies are greater than zero that
either + 1 or -1 will be chosen, although for large values of W (relative to u) for all
843
practical purposes, the behavior is deterministic. There are many values of the
weights that cause the NAT to approximate the behavior of a deterministic
threshold logic gate. ~or the same reasons that deterministic threshold logic gates
cannot realize all 22 functions of N variables 7, so a NAT cannot learn to
approximate any deterministic function; only the threshold logic functions.
Note also that when the weights are near zero, a NAT adapts most rapidly
when both positive and negative reinforcement are used in approximately equal
amounts. As the NAT becomes more likely to produce the appropriate behavior,
the opportunity to use negative reinforcement decreases while the opportunity to
use positive reinforcement increases. This means that a NAT cannot learn to
(nearly) always select a certain output if negative reinforcement alone is used.
Thus, positive reinforcement has an important role in this model. (In most
deterministic models, positive reinforcement is not useful.)
Note further that there is no hysteresis in NAT learning. For a given
configuration of inputs, a + 1 output followed by a + 1 reinforcement has exactly the
same effect on all the weights as a -1 output followed by a -1 reinforcement. So
the order of such events has no effect on the final values of the weights.
Finally, if only negative reinforcement is applied to a NAT, independent of
output, for a particular combination of inputs, the weights will change in the
direction that makes W tend toward zero and once there, follow a random walk
centered on zero. (The further W is from zero, the more likely its next step will be
toward zero.) If all possible input combinations are applied with more or less equal
probability, all the weights will tend toward zero and then follow random walks
centered on zero. In this case, the NAT will select + 1 or -1 with more or less
equal probability without regard to its inputs.
NETWORKS
NATs may be connected together in networks (NAT-nets). The inputs to a
NAT in such a network can be selected from among: 1) the set of inputs to the
entire network, 2) the set of outputs from other NATs in the network, and 3) its
own output. The outputs of the network may be chosen from among: 1) the inputs
to the network as a whole, and 2) the outputs of the various NATs in the network .
Following Ref. 5, we impose a timing discipline on a NAT-net. The network is
organized into layers such that each NAT belongs to one layer. Letting L be the
number of layers, the network operates as follows: 1) All NATs in a given layer
receive timing signals at the same time and select a new output at the same time.
2) Timing signals are received by the different layers, in sequence, from 1 to L. 3)
Inputs to the network as a whole are levels that may change only before Layer 1
receives its timing signal. Similarly, outputs from the network as a whole are
available to the environment only after Layer L has received its timing signal.
Reinforcement to the network as a whole is accepted only after outputs are made
available to the environment. The same reinforcement signal is distributed to all
NATs in the network at the same time.
With these rules, NAT-nets operate through a sequence of timing cycles. In
each cycle: 1) Network inputs are changed. 2) Layers 1 through L select new
outputs, in sequence. 3) Network outputs are made available to the environment.
4) Reinforcement is received from the environment. We call each such cycle a
"trial" and a sequence of such trials is a "session."
This model is very general. If, for each gate, inputs are selected only from
among the inputs to the network as a whole and from the outputs of gates in layers
preceding it in the timing cycle, then the network is combinational. In this case, the
probability of the network producing a given output configuration is a function of
the inputs at the start of the timing cycle. If at least one NAT has one input from a
844
NAT in the same layer or from a subsequent layer in the timing cycle, then the
network is sequential. In this case, the network may have "internal states" that
allow it to remember information from one cycle to the next. Thus, the
probabilities governing its choice of outputs may depend on inputs in previous
cycles. So sequential NAT-nets may have short-term memory embodied in internal
states and long-term memory embodied in the weights. In Ref. 5, we showed that
sequential networks can be constructed by adding feedback paths to combinational
networks and any sequential network can be put in this standard form.
In information-theoretic terms: 1) A NAT-net with no inputs and some
outputs is an "information source." 2) A NAT-net with both inputs and outputs is
an information "channel." 3) A combinational NAT-net is "memory-less" while a
sequential NAT-net has memory. In this context, note that a NAT-net may operate
in an environment that is either deterministic or nondeterministic. Both the logical
and the reinforcement inputs can be selected by stochastic processes. Note also
that nondeterministic and deterministic elements as well as adaptive and
nonadaptive elements can be combined in one network. (It may be that the
decision-making parts of an animal's nervous system are nondeterministic and
adaptive while the information transmitting parts (sensory data-gathering and the
motor output parts) are deterministic and nonadaptive.)
One capability that combinational NAT-nets possess is that of "pattern
recognizers." A network having many inputs and one or a few outputs can
"recognize" a small subset of the potential input patterns by producing a particular
output pattern with high probability when a member of the recognized subset
appears and a different output pattern otherwise. In practice, the number of
possible input patterns may be so large that we cannot present them all for training
purposes and must be content to train the network to recognize one subset by
distinguishing it (with different output pattern) from another subset. In this case,
if a pattern is subsequently presented to the network that has not been in one of
the training sets, the probabilities governing its output may approach one or zero,
but may well be closer to 0.5. The exact values will depend on the details of the
training period. If the new pattern is similar to those in one of the training sets, the
NAT-net will often have a high probability of producing the same output as for that
set. This associative property is the analog of the well known associative property
in deterministic models. If the network lacks sufficient complexity for the
separation we wish to make, then it cannot be trained. For example, a single Ninput NAT cannot be trained to recognize any arbitrary set of input patterns by
selecting the + 1 output when one of them is presented and -1 otherwise. It can
only be trained to make separations that correspond to threshold functions.
A combinational NAT-net can also produce patterns. By analogy with a
pattern recognizer, a NAT-net with none or a few inputs and a larger number of
outputs can learn for each input pattern to produce a particular subset of the
possible output patterns. Since the mapping may be few-to-many, instead of
many-to-few, the goal of training in this case mayor may not be to have the
network approximate deterministic behavior. Clearly, the distinction between
pattern recognizers and pattern prod ucers is somewhat arbitrary: in general, NATnets are pattern transducers that map subsets of input patterns into subsets of
output patterns. A sequential network can "recognize" patterns in the timesequence of network inputs and produce patterns in the time-sequence of outputs.
SIMULATION EXPERIMENTS
In this Section, we discuss computer simulation results for three types of
multiple-element networks. For two of these types, certain strategies are used to
train the networks. In general, these strategies have two parts that alternate, as
845
needed. The first part is a general scheme for providing network inputs and
reinforcement that tends to train all elements in the network in the desired
direction. The second part is substituted temporarily when it becomes apparent
that the network is getting stuck in some suboptimal behavior. It is focussed on
getting the network unstuck. The strategies used here are intuitive. In general,
there appear to be many strategies that will lead the network to the desired
behavior. While we have made some attempt to find strategies that are reasonably
efficient, it is very unlikely that the ones used are optimal. Finally, these strategies
have been tested in hundreds of training sessions. Although they worked in all such
sessions, there may be some (depending on the sequence of random numbers
generated) in which they would not work .
In describing the networks simulated, Figs. 1-3, we use the diagramatic
conventions defined in Ref. 5: We put all NATs in the same layer in a vertical line,
with the various layers arranged from left to right in their order in the timing cycle.
Inputs to the entire network corne in from the left; outputs go out to the right.
Because the timing cycle is fixed, we omit the timing inputs in these figures. For
similar reasons, we also omit the reinforcement inputs.
In the simulations described here, the weights in the NATs start at zero
making the network outputs completely random in the sense that on any given
trial, all outputs are equally likely to occur, independent of past or present inputs.
As learning proceeds, some or all the weights become large, so that the NAT-net's
selection of outputs is strongly influenced by some or all of its inputs and internal
connections. (Note that if the weights do not start at zero, they can be driven close
to zero by using negative reinforcement.) In general, the optimum behavior toward
which the network adapts is deterministic. However, because the probabilities are
never identically equal to zero or one, we apply an arbitrary criterion and say that a
NAT-net has learned the appropriate behavior when that criterion is satisfied. In
real biological systems, we cannot know the weights or the exact probabilities
governing the behavior of the individual adaptive elements. Therefore, it is
appropriate to use a criterion based on observable behavior. For example, the
criterion might be that the network selects the correct response (and continues to
receive appropriate reinforcement) 25 times in a row .
Note that NAT-nets can adapt appropriately when the environment is not
deliberately trying to make the them behave in a particular way. For example, the
environment may provide inputs according to some (not necessarily deterministic)
pattern and there may be some independent mechanism that determines whether
the NAT-net is responding appropriately or not and provides the reinforcement
accordingly. One paradigm for this situation is a game in which the NAT-net and
the environment are players. The reinforcement scheme is simple: if, according to
the rules of the game, the NAT-net wins a play (= trial) of the game, reinforcement
is + 1 , if it loses, -1.
For a NAT-net to adapt appropriately in this situation, the game must consist
of a series of similar plays. If the game is competitive, the best strategy a given
player has depends on how much information he has about the opponent and vice
versa. If a player assumes that his opponent is all-knowing, then his best strategy is
to minimize his maximum loss and this often means playing at random, or a least
according to certain probabilities. If a player knows a lot about how his opponent
plays, his best strategy may be to maximize gain. This often means playing
according to some deterministic strategy.
The example networks described here are special cases of three types: pattern
producing (combinational multiple-output) networks, pattern recogmzmg
(combinational multiple-input, multiple-layered, few-output) networks, and game
playing (sequential) networks. The associative properties of NATs and NAT-nets
846
are not emphasized here because they are analogous to the well known associative
properties of other related models.
A Class of Simple Pattern Producing Networks
A simple class of pattern producing
networks consists of the single-layer type
shown in Fig. 1. Each of M NATs in such a
o~- z,
network has no inputs, only an output. As a
consequence, each has only one weight, Woo
Z2
The network is a simple, adaptive, information
source.
O~- ~3
Consider first the case in which the
??
network contains only one NAT and we wish to
train it to always produce a simple "pattern,"
???
+ 1. We give positive reinforcement when it
selects + 1 and negative reinforcement
Z18
otherwise. If Wo starts at 0, it will quickly
gr.ow large making the probability of selecting
+ 1 approach unity. The criterion we use for
Fig. 1. A Simple Pattern
deciding that the network is trained is that it
Producing Network
produce a string of 25 correct outputs. Table I
shows that in 100 sessions, this one-NAT network selected + 1 output for the next
25 trials starting, on average, at trial 13.
Next consider a network with two NATs. They can produce four different
output patterns. If both weights are 0, they will produce each of the patterns with
equal probability. But they can be trained to produce one pattern (nearly) all the
time. If we wish to train this subnetwork to produce the pattern (in vector
notation) [+1 +1], one strategy is to give no
reinforcement if it produces patterns [-1 +1] or
M Min Ave Max
[+1 -1), give it positive reinforcement if it
1
1
13
26
produces [+1 +1] and negative reinforcement if
2
8
25
43
it produces [-1 -1]. Table I shows that in 100
4
18
35
60
sessions, this network learned to produce the
8
44
70
109
desired pattern (by producing a string of 25
16
215
49
115
correct outputs) in about 25 trials. Because we
initially gave reinforcement only about 50% of
the time, it took longer to train two NATS Table I. Training Times For
than one.
Networks Per Fig. 1.
Next, consider the 16-NAT network in
Fig. 1. Now there are 216 possible patterns the network can produce. When all the
weights are zero, each has probability 2- 16 of being produced. An ineffective
strategy for training this network is to provide positive reinforcement when the
desired pattern is produced, negative reinforcement when its opposite is produced,
and zero reinforcement otherwise. A better strategy is to focus on one output of the
network at a time, training each NAT separately (as above) to have a high
probability of producing the desired output. Once all are trained to a relatively
high level, the network as a whole has a reasonable chance of producing exactly the
correct output. Now we can provide positive reinforcement when it does and no
reinforcement otherwise. With this two-stage hybrid strategy, the network will
soon meet the training criterion. The time it takes to train a network of M
elements with a strategy of this type is roughly proportional to M, not 2(M - 1), as
for the first strategy.
...
0--.'
...
???
?
0--..
?
?
847
A still more efficient strategy is to alternate between a general substrategy
and a substrategy focussed on keeping the network from getting "stuck ." One
effective general substrategy is to give positive reinforcement when more than half
of the NATs select the desired output, negative reinforcement when less than half
select the desired output, and no reinforcement when exactly half select the desired
output. This substrategy starts out with approximately equal amounts of positive
and negative reinforcement being applied. Soon, the network selects more than half
of the outputs correctly more and more of the time. Unfortunately, there will tend
to be a minority subset with low probability of selecting the correct output. At this
stage, we must recognize this subset and switch to a substrategy that focuses on the
elements of this subset following the strategy for one or two elements, above. When
all NATs have a sufficiently high probability of selecting the desired output,
training can conclude with the first substrategy.
The strategies used to obtain the results for M = 4,8, and 16 in Table I were
slightly more complicated variants of this two-part strategy. In all of them, a
running average was kept of the number of right responses given by each NAT.
Letting OJ be the "correct" output for Zj, the running average after the tt" trial,
Aj( t), is:
Aj(t) = BAj(t - 1)
+
(3)
CjZj(t)
where B is a fraction generally in the range 0.75 to 0.9. If Aj(t) for a particular i
gets too far below the combined average for all i, then training focuses on the it"
element until its average improves. The significance of the results given in Table I
is not the details of the strategies used, nor how close the training times may be to
the optimum. Rather, it is the demonstration that training strategies exist such
that the training time grows significantly more slowly than in proportion to M.
A Simple Pattern Recognizing Network
As mentioned above, there are fewer
threshold logic functions of N variables (for
N > 1) than the total possible functions.
x, -~))~---......p)oo-- Z
For N = 2, there are 14. The remining two
X2 _-0lil_1:;,._ ___
are the "exclusive or" (XOR) and its
complement. Multi-layered networks are
needed to realize these functions, and an Fig. 2. A Two-Element Network
important test of any adaptive network
That Learns XOR
model is its ability to learn XOR. The
network in Fig. 2 is one of the simplest networks capable of learning this function.
Table II gives the results of 100 training sessions with this network. The strategy
used to obtain these
Ave
Max
Min
Function
Network
results again had two
106
57
18
parts. The general part
OR
Fig. 2
1992
681
218
XOR
Fig. 2
consisted of supplying
-700
-3500
-14,300
each of the four possible
XOR
Ref. 2
2232
input patterns to the
XOR
Ref. 8
network
in
rotation,
Table II. Training Times For The
glvmg
appropriate
Network In Fig. 2.
reinforcement each trial.
The second part involved
keeping a running average (similar to Eq. (3)) of the responses of the network by
input combination. When the average for one combination fell significantly behind
?.
848
the average for all, training was focused on just that combination until performance
improved. The criterion used for deciding when training was complete was a
sequence of 50 correct responses (for all input patterns together).
For comparison, Table II shows results for the same network trained to realize
the normal OR function. Also shown for comparison are numbers taken from Refs.
2 and 8 for the equivalent network in those different models. These are
nondeterministic and deterministic models, respectively. The numbers from Ref. 2
are not exactly comparable with the present results for several reasons. These
include: 1) The criterion for judging when the task was learned was not the same;
2) In Ref. 2, the "wrong" reinforcement was deliberately applied 10% of the time to
test learning in this situation; 3) Neither model was optimized for the particular
task at hand. Nonetheless, if these (and other) differences were taken into account,
it is likely that the NAT-net would have learned the XOR function significantly
faster.
The significance of the present results is that they suggest that the use of a
training strategy can not only prevent a network from getting stuck, but may also
facilitate more rapid learning. Thus, such strategies can compensate, or more than
compensate, for reduced complexity in the reinforcement algorithm.
A Simple Game-Playing Network
Here, we consider NAT-nets in the context of the game of "matching
pennies." In this game, each player has a stack of pennies. At each play of the
game, each player places one of his pennies, heads up or heads down, but covered, in
front of him. Each player uncovers his penny at the same time. If they match,
player A adds both to his stack, otherwise, player B takes both.
Game theory says that the strategy of each player that minimizes his
maximum loss is to play heads and tails at random. Then A cannot predict B's
behavior and at best can win 50% of the time and likewise for B with respect to A.
This is a conservative strategy on the part of each player because each assumes that
the other has (or can derive through a sequence of plays), and can use, information
about the other player's strategy. Here, we make the different assumption that: 1)
Player B does not play at random, 2) Player B has no information about A's
strategy, and 3) Player B is incapable of inferring any information about A through
a sequence of plays and in any event is incapable of changing its strategy. Then, if
A has no information about B's pattern of playing at the start of the game, A's best
course of action is to try to infer a non-random pattern in B's playing through a
sequence of plays and subsequently take advantage of that knowledge to win more
often than 50% of the time. An adaptive NAT-net, as A, can adapt appropriately
in situations of this type. For example, suppose a single NAT of the type in Fig. 1
plays A, where + 1 output means heads, -1 output means tails. A third agent
supplies reinforcement + 1 if the NAT wins a play, -1 otherwise. Suppose B plays
heads with 0.55 probability and tails with 0.45 probability. Then A will learn over
time to play heads 100% of the time and thereby maximize its total winnings by
winning 55% of the time.
A more complicated situation is the following. Suppose B repeats its own
move two plays ago 80% of the time, and plays the opposite 20% of the time. A
NAT-net with the potential to adapt to this strategy and win 80% of the time is
shown in Fig. 3. This is a sequential network shown in the standard form of a
combinational network (in the dotted rectangle) plus a feedback path. The input to
the network at time tis B's play at t - 1. The output is A's move. The top NAT
selects its output at time t based partly on the bottom NAT's output at time
t - 1. The bottom NAT selects its output at t - 1 based on its input at that time
which is B's output at t - 2. Thus, the network as a whole can learn to select its
849
output based on B's play two time increments past. Simulation of 100 sessions
resulted in the network learning to do this
98 times. On average, it took 468 plays
(Min 20, max 4137) to reach the point at
which the network repeated B's move two
H i - - - -.... Z
x----~
times past on the next 50 plays. For two
sessions the network got stuck (for an
unknown number of plays greater than
25,000) playing the opposite of B's last
move or always playing tails. {The first
two-part strategy found that trains the
network to repeat B's output two time
increments past without getting stuck (not
Fig. 3. A Sequential Gamein the game-playing context) took an
Playing Network
average of 260 trials (Min 25, Max 1943) to
meet the training criterion.)
The significance of these results is that a sequential NAT-net can learn to
produce appropriate behavior. Note that hidden NATs contributed to appropriate
behavior for both this network and the one that learned XOR, above.
CONCLUDING REMARKS
The examples above have been kept simple in order to make them readily
understandable. They are not exhaustive in the sense of covering all possible types
of situations in which NAT-nets can adapt appropriately. Nor are they definitive in
the sense of proving generally and in what situations NAT-nets can adapt
appropriately. Rather, they are illustrative in the sense of demonstrating a variety
of significant adaptive abilities. They provide an existence proof that NAT-nets can
adapt appropriately and relatively easily in a wide variety of situations.
The fact that nondeterministic models can learn when the same reinforcement
is applied to all adaptive elements, while deterministic models generally cannot,
supports the hypothesis that animal nervous systems may be (partly)
nondeterministic. Experimental characterization of how animal learning does, or
does not get "stuck," as a function of learning environment or training strategy,
would be a useful test of the ideas presented here.
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1.
2.
3.
4.
5.
6.
7.
8.
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Barto, A. G., Human Neurobiology, 4, 229-256, 1985.
Barto, A. G., R. S. Sutton, and C. W. Anderson, IEEE Transactions on
Systems, Man, and Cybernetics, SMC-13, No.5, 834-846, 1983.
Barto, A. G., and P. Anandan, IEEE Transactions on Systems, Man, and
Cybernetics, SMC-15, No.3, 360-375, 1985.
Windecker, R. C., Information Sciences, 16, 185-234 (1978).
Rumelhart, D. E., and J. L. McClelland, Parallel Distributed Processing, MIT
Press, Cambridge, 1986.
Muroga, S., Threshold Logic And Its Applications, Wiley-Interscience, New
York, 1971.
Rumelhart, D. E., G. E. Hinton, and R. J. Williams, Chapter 8 in Ref. 6.
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7,014 | 790 | A Connectionist Model of the Owl's
Sound Localization System
D alliel J. Rosen?
Department of Psychology
Stanford University
Stanford, CA 94305
David E. Rumelhart
Department of Psychology
Stanford University
Stanford, CA 94305
Eric. I. Knudsen
Department of Neurobiology
Stanford University
Stanford, CA 94305
Abstract
,,"'e do not have a good understanding of how theoretical principles
of learning are realized in neural systems. To address this problem
we built a computational model of development in the owl's sound
localization system. The structure of the model is drawn from
known experimental data while the learning principles come from
recent work in the field of brain style computation. The model
accounts for numerous properties of the owl's sound localization
system, makes specific and testable predictions for future experiments, and provides a theory of the developmental process.
1
INTRODUCTION
The barn owl, Tyto Alba, has a remarkable ability to localize sounds in space. In
complete darkness it catches mice with nearly flawless precision. The owl depends
upon this skill for survival, for it is a nocturnal hunter who uses audition to guide
?Current address: Keck Center for Integrative Neuroscience, UCSF, 513 Parnassus
Ave., San Francisco, CA 94143-0444.
606
A Connectionist Model of the Owl's Sound Localization System
its search for prey (Payne, 1970; Knudsen, Blasdel and Konishi, 1979). Central to
the owl's localization system are the precise auditory maps of space found in the
owl's optic tectum and in the external nucleus of the inferior colliculus (lex).
The development of these sensory maps poses a difficult problem for the nervous
system, for their accuracy depends upon changing relationships between the animal
and its environment. The owl encodes information about the location of a sound
source by the phase and amplitude differences with which the sound reaches the
owl's two ears. Yet these differences change dramatically as the animal matures
and its head grows. The genome cannot "know" in advance precisely how the
animal's head will develop - many environmental factors affect this process - so it
cannot encode the precise development of the auditory system. Rather, the genome
must design the auditory system to adapt to its environment, letting it learn the
precise interpretation of auditory cues appropriate for its head and ears.
In order to understand the nature of this developmental process, we built a connectionist model of the owl's sound localization system, using both theoretical principles
of learning and knowledge of owl neurophysiology and neuroanatomy.
2
THE ESSENTIAL SYSTEM TO BE MODELED
The owl calculates the horizontal component of a sound source location by measuring the interaural time difference (lTD) of a sound as it reaches the two ears
(Knudsen and Konishi, 1979). It computes the vertical component of the signal by
determining the interaurallevel difference (ILD) of that same sound (Knudsen and
Konishi, 1979). The animal processes these signals through numerous sub-cortical
nuclei to form ordered auditory maps of space in both the ICx and the optic tectum.
Figure 1 shows a diagram of this neural circuit.
Neurons in both the ICx and the optic tectum are spatially tuned to auditory
stimuli. Cells in these nuclei respond to sound signals originating from a restricted
region of space in relation to the owl (Knudsen, 1984). Neurons in the ICx respond
exclusively to auditory signals. Cells in the optic tectum, on the other hand, encode
both audito!y and visual sensory maps, and drive the motor system to orient to the
location of an auditory or visual signal.
Researchers study the owl's development by systematically altering the animal's
sensory experience, usually in one of two ways. They may fit the animal with a
sound attenuating earplug, altering its auditory experience, or they may fit the owl
with displacing prisms, altering its visual experience.
Disturbance of either auditory or visual cues, during a period when the owl is developing to maturity, causes neural and behavioral changes that bring the auditory
map of space back into alignment with the visua.l map, and/or tune the auditory system to be sensitive to the appropriate range of binaural sound signals. The earplug
induced changes take place at the level of the VLVp, where ILD is first computed
(Mogdans and Knudsen, 1992). The visually induced adjustment of the auditory
maps of space seems to take place at the level of the ICx (Brainard and Knudsen,
1993b). The ability of the owl to adjust to altered sensory signals diminishes over
time, and is greatly restricted in adulthood (Knudsen and Knudsen, 1990).
607
608
Rosen, Rumelhart, and Knudsen
OVERVIEW of the BARN OWL'.
SOUND LOCALIZATION SYSTEM
(
~~dIC
(
NUCLBJS
NUCLBJS
MAGNOCEWJLAAIS
MAGNOCB.LULAAIS
T"'*'a
L"".
TIn*'II
Figure 1: A chart describing the flow of auditory information in the owl's sound
localization system. For simplicity, only the connections leading to the one of the
bilateral optic tecta are shown. Nuclei labeled with an asterisk (*) are included in
the model. Nuclei that process ILD and/or lTD information are so labeled.
3
THE NETWORK MODEL
The model has two major components: a network architecture based on the neurobiology of the owl's localization system, as shown in Figure 1, and a learning rule
derived from computational learning theory. The elements of the model are standard connectionist units whose output activations are sigmoidal functions of their
weighted inputs. The learning rule we use to train the model is not standard. In
the following section we describe how and why we derived this rule.
3.1
DEFINING THE GOAL OF THE NETWORK
The goal of the network, and presumably the owl, is to accurately map sound signals
to sound source locations. The network must discover a model of the world which
best captures the relationship between sound signals and sound source locations.
Recent work in connectionist learning theory has shown us ways to design networks
that search for the model that best fits the data at hand (Buntine and Weigend,
1991; MacKay, 1992; Rumelhart, Durbin, Golden and Chauvin, in press). In this
section we apply such an analysis to the localization network.
A Connectionist Model of the Owl's Sound Localization System
Table 1: A table showing the mathematical terms used in the analysis.
I
TERM
M
1J
P(MI1J)
<
X,Y>i
xi
Yi
Yi
Yij
Wij
7Jj
:F(7Jj)
C
3.2
I MEANING
The Model
The Data
Probability of the Model given the Data
The set of i input/target training pairs
The input vector for training trial i
The target vector for training trial i
The output vector for training trial i
The value of output unit j on training trial i
The weight from unit j to unit i
The netinput to unit j
The activation function of unit j evaluated at its netinput
The term to be maximized by the network
DERIVING THE FUNCTION TO BE MAXIMIZED
The network should maximize the probability of the model given the data. Using
Bayes' rule we write this probability as:
P(MI1J) = P(1JIM)P(M)
P(1J)
.
Here M represents the model (the units, weights and associated biases) and D
represents the data. We define the data as a set of ordered pairs, [< soundsignal, location - signal >d, which represent the cues and targets normally used to
train a connectionist network. In the owl's case the cues are the auditory signals,
and the target information is provided by the visual system. (Table 1 lists the
mathematical terms we use in this section.)
We simplify this equation by taking the natural logarithm of each side giving:
In P(MI1J) = In P(1JIM)
+ InP(M) -In P(1J).
Since the natural logarithm is a monotonic transformation, if the network maximizes
the second equation it will also maximize the first.
The final term in the equation, In P(1J), represents the probability of the ordered
pairs the network observes. Regardless of which model the network settles upon,
this term remains the same - the data are a constant during training. Therefore we
can ignore it when choosing a model.
The second term in the equation, In P(M), represents the probability of the model.
This is the prior term in Bayesian analysis and is our estimation of how likely it
is that a particular model is true, regardless of the data. 'Ve will discuss it below.
For now we will concentrate on maximizing In P(1JIM).
609
610
Rosen, Rumelhart, and Knudsen
3.3
ASSUMPTIONS ABOUT THE NETWORK'S ENVIRONMENT
We assume that the training data - pairs of stylized auditory and visual signals are independent of one another and re-write the previous term as:
=
InP(VIM)
L:lnP?
i,Y>i 1M),
i
The i subscript denotes the particular data, or training, pair. We further expand
this term to:
In P(VIM)
= Lin P(ih Iii 1\ M) + L: In P(Xi).
i
i
We ignore the last term, since the sound signals are not dependent on the model.
vVe are left, then, with the task of maximizing Li In P(Ui Iii 1\ M). It is important
to note that Yi represents a visual signal, not a localization decision. The network
attempts to predict its visual experience given its auditory experience. It does not
predict the probability of making an accurate localization decision. If we assume
that visual signals provide the target values for the network, then this analysis shows
that the auditory map will always follow the visual map, regardless of whether this
leads to accurate localization behavior or not. Our assumption is supported by
experiments showing that, in the owl, vision does guide the formation of auditory
spatial maps (Knudsen and Knudsen, 1985; Knudsen, 1988).
Next, we must clarify the relationship between the inputs, Xi and the targets, ih.
\Ve know that the real world is probabilistic - that for a given input there exists
some distribution of possible target values. We need to estimate the shape of this
distribution. In this case we assume that the target values are binomially distributed
- that given a particular sound signal, the visual system did or did not detect a
sound source at each point in owl-centered space.
Having made this assumption, we can clarify our interpretation of the network
output array, Y~. Each element, Yij, of this vector represents the activity of output
unit j on training trial i. We assume that the output activation of each of these
units represents the expected value of its corresponding target, Yij. In this case
the expected value is the mean of a binomial distribution. So the value of each
output unit Yij represents the probability that a sound signal originated from that
particular location. vVe now write the probability of the data given the model as:
P(yilxi 1\ M) =
II yft (1 -
Yij )l-Yi j
.
j
Taking the natural log of the probability and summing over all data pairs we get:
C=
L L: Yij In Yij + (1 i
Yij) In( 1 - Yij)
j
where C is the term we want to maximize. This is the standard cross-entropy term.
3.4
DERIVING THE LEARNING RULE
Having defined our goal we derive a learning rule appropriate to achieving that goal.
To determine this rule we compute :~ where 7}j is the net input to a unit. (In these
A Connectionist Model of the Owl's Sound Localization System
equations we have dropped the i subscript, which denotes the particular training
trial, since this analysis is identical for all trials.) We write this as:
where aF( '1]j) is the derivative of a unit's activation function evaluated at its net
input.
Next we choose an appropriate activation function for the output units. The logistic
1_,,"), is a good choice for two reasons. First, it is bounded by
function, F('1]j)
l+e ,
zero and one. This makes sense since we assume that the probability that a sound
signal originated at anyone point in space is bounded by zero and one. Second,
when we compute the derivative of the logistic function we get the following result:
=(
aF('1]j)
= F('1]j)(I- F('1]j)) = 1/j(1- 1/j).
This term is the variance of a binomial distribution and when we return to the
derivative of our cost function, we see that this variance term is canceled by the
denominator. The final derivative we use to compute the weight changes at the
output units is therefore:
ac
~ <X
u'1]j
(Yj
-
~
)
Yj .
The weights to other units in the network are updated according to the standard
backpropagation learning algorithm.
3.5
SPECIFYING MODEL PRIORS
There are two types of priors in this model. First is the architectural one. We design
a fixed network architecture, described in the previous section, based upon our
knowledge of the nuclei involved in the owl's localization system. This is equivalent
to setting the prior probability of this architecture to 1, and all others to O.
We also use a weight elimination prior. This and similar priors may be interpreted
as ways to reduce the complexity of a network (\Veigend, Huberman and Rumelhart,
1990). The network, therefore, maximizes an expression which is a function of both
its error and complexity.
3.6
TRAINING
We train the model by presenting it with input to the core of the inferior colli cui us
(ICc), which encodes interaural phase and time differences (IPD/ITD), and the
angular nuclei, which encode sound level. The outputs of the network are then
compared to target values, presumed to come from the visual system. The weights
are adjusted in order to minimize this difference. \Ve mimic plug training by varying
the average difference between the two angular input values. We mimic prism
training by systematically changing the target values associated with an input.
611
612
Rosen, Rumelhart, and Knudsen
Figure 2: The activity level of lex units in response to a particular auditory input immediately after simulated prism training was begun (left), midway through
training (middle) and after training was completed (right).
4
RESULTS and DISCUSSION
The trained network localizes accurately, shows appropriate auditory tuning curves
in each of the modeled nuclei, and responds appropriately to manipulations that
mimic experiments such as blocking inhibition at the level of the lex. The network
also shows appropriate responses to changing average binaural intensity at the level
of the VLVp, the lateral shell and the lex.
Furthermore, the network exhibits many properties found in the developing owl..
The model appropriately adjusts its auditory localization behavior in simulated
earplug experiments and this plasticity takes place at the level of the VLVp. As
earplug simulations are begun progressively later in training, the network's ability
to adapt to plug training gradually diminishes, following a time course of plasticity
qualitatively similar to the sensitive and critical periods described in the owl.
The network adapts appropriately in simulated prism studies and the changes in
response to these simulations primarily take place along the lateral shell to lex
connections. As with the plug studies, the network's ability to adapt to prisms
diminishes over time. However, unlike the mature owl, a highly trained network
retains the ability to adapt in a simulated prism experiment.
We also discovered that the principally derived learning rule better models intermediate stages of prism adjustment than does a standard back-propagation network.
Brainard and Knudsen (1993a) report observing two peaks of activity across the tectum in response to an auditory stimulus during prism training - one corresponding
to the pre-training response and one corresponding to the newly learned response.
Over time the pre-trained response diminishes while the newly learned one grows.
As shown in Figure 2, the network exhibits this same pattern of learning. Networks
we trained under a standard back-propagation learning algorithm do not. Such a
A Connectionist Model of the Owl's Sound Localization System
result lends support to the idea that the owl's localization system is computing a
function similar to the one the network was designed to learn.
In addition to accounting for known data, the network predicts results of experiments it was not designed to mimic. Specifically, the network accurately predicted
that removal of the animal's facial ruff, which causes ILD to vary with azimuth
instead of elevation, would have no effect on the animal's response to varying ILD.
The network accomplishes the goals for which it was designed. It accounts for much,
though not all, of the developmental data, it makes testable predictions for future
experiments, and since we derived the learning rule in a principled fashion, the
network provides us with a specific theory of the owl's sound localization system.
References
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Knudsen, E. (1984). Auditory properties of space-tuned units in owl's optic tectum.
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7,015 | 791 | Supervised Learning with Growing Cell
Structures
Bernd Fritzke
Institut fiir Neuroinformatik
Ruhr-U niversitat Bochum
Germany
Abstract
We present a new incremental radial basis function network suitable for classification and regression problems. Center positions
are continuously updated through soft competitive learning. The
width of the radial basis functions is derived from the distance
to topological neighbors. During the training the observed error
is accumulated locally and used to determine where to insert the
next unit. This leads (in case of classification problems) to the
placement of units near class borders rather than near frequency
peaks as is done by most existing methods. The resulting networks
need few training epochs and seem to generalize very well. This is
demonstrated by examples.
1
INTRODUCTION
Feed-forward networks of localized (e.g., Gaussian) units are an interesting alternative to the more frequently used networks of global (e.g., sigmoidal) units. It
has been shown that with localized units one hidden layer suffices in principle to
approximate any continuous function, whereas with sigmoidal units two layers are
necessary.
In the following we are considering radial basis function networks similar to those
proposed by Moody & Darken (1989) or Poggio & Girosi (1990). Such networks
consist of one layer L of Gaussian units. Each unit eEL has an associated vector
We E R n indicating the position of the Gaussian in input vector space and a standard
255
256
Fritzke
deviation
by
Uc.
For a given input datum
eE R n the activation of unit c is described
D c ('"C) -_ exp (_
lie -
wcll2)
2?
Uc
(1)
On top of the layer L of Gaussian units there are m single layer percepirons.
Thereby, m is the output dimensionality of the problem which is given by a number
of input/output pairsl (e, () E (Rn x Rm). Each of the single layer perceptrons
computes a weighted sum of the activations in L:
Oi(e)
= L Wij Dj (0
iE{1, ... ,m}
(2)
jEL
With Wij we denote the weighted connection from local unit j to output unit i.
Training of a single layer perceptron to minimize square error is a very well understood problem which can be solved incrementally by the delta rule or directly by
linear algebra techniques (Moore-Penrose inverse). Therefore, the only (but severe)
difficulty when using radial basis function networks is choosing the number of local
units and their respective parameters, namely center position wand width u.
One extreme approach is to use one unit per data points and to position the units
directly at the data points. If one chooses the width of the Gaussians sufficiently
small it is possible to construct a network which correctly classifies the training
data, no matter how complicated the task is (Fritzke, 1994). However, the network
size is very large and might even be infinite in the case of a continuous stream of
non-repeating stochastic input data. Moreover, such a network can be expected to
generalize poorly.
Moody & Darken (1989), in contrast, propose to use a fixed number of local units
(which is usually considerably smaller than the total number of data points). These
units are first distributed by an unsupervised clustering method (e.g., k-means).
Thereafter, the weights to the output units are determined by gradient descent.
Although good results are reported for this method it is rather easy to come up
with examples where it would not perform well: k-means positions the units based
on the density of the training data, specifically near density peaks. However, to approximate the optimal Bayesian a posteriori classifier it would be better to position
units near class borders. Class borders, however, often lie in regions with a particularly low data density. Therefore, all methods based on k-means-like unsupervised
placement of the Gaussians are in danger to perform poorly with a fixed number of
units or - similarly undesirable - to need a huge number of units to achieve decent
performance.
From this one can conclude that - in the case of radial basis function networks
- it is essential to use the class labels not only for the training of the connection
weights but also for the placement of the local units. Doing this forms the core of
the method proposed below.
IThroughout this article we assume a classification problem and use the corresponding
terminology. However, the described method is suitable for regression problems as well.
Supervised Learning with Growing Cell Structures
2
SUPERVISED GROWING CELL STRUCTURES
In the following we present an incremental radial basis function network which
is able to simultaneously determine a suitable number of local units, their center
positions and widths as well as the connection weights to the output units. The
basic idea is a very simple one :
O. Start with a very small radial basis function network.
1. Train the current network with some I/O-pairs from the training data.
2. Use the observed accumulated error to determine where in input vector
space to insert new units.
3. If network does not perform well enough goto 1.
One should note that during the training phase (Step 1.) error is accumulated
over several data items and this accumulated error is used to determine where to
insert new units (Step 2.). This is different from the approach of Platt (1991) where
insertions are based on single poorly mapped patterns. In both cases, however, the
goal is to position new units in regions where the current network does not perform
well rather than in regions where many data items stem from.
In our model the center positions of new units are interpolated from the positions
of existing units. Specifically, after some adaptation steps we determine the unit
q which has accumulated the maximum error and insert a new unit in between q
and one of its neighbors in input vector space. The interpolation procedure makes
it necessary to allow the center positions of existing units to change. Otherwise, all
new units would be restricted to the convex hull of the centers of the initial network.
We do not necessarily insert a new unit in between q and its nearest neighbor.
Rather we like to choose one of the units with adjacent Voronoi regions 2 . In the
two-dimensional case these are the direct neighbors of q in the Delaunay triangulation (Delaunay-neighbors) induced by all center positions. In higher-dimensional
spaces there exists an equivalent based on hypertetrahedrons which, however, is
very hard to compute. For this reason, we arrange our units in a certain topological
structure (see below) which has the property that if two units are direct neighbors
in that structure they are mostly Delaunay-neighbors. By this we get with very
little computational effort an approximate subset of the Delaunay-neighbors which
seems to be sufficient for practical purposes.
2.1
NETWORK STRUCTURE
The structure of our network is very similar to standard radial basis function networks. The only difference is that we arrange the local units in a k-dimensional
1, triangles
topological structure consisting of connected simplices 3 (lines for k
=
2The Voronoi region of a unit c denotes the part of the input vector space which consists
of points for which c is the nearest unit.
3 A historical reason for this specific approach is the fact that the model was developed
from an unsupervised network (see Fritzke, 1993) where the k-dimensional neighborhood
was needed to reduce dimensionality. We currently investigate an alternative (and more
257
258
Fritzke
=
=
for k
2, tetrahedrons for k
3 and hypertetrahedrons for larger k). This arrangement is done to facilitate the interpolation and adaptation steps described
below. The initial network consists of one k-dimensional simplex (k + 1 local units
fully connected with each other). The neighborhood connections are not weighted
and do not directly influence the behavior of the network. They are, however, used
to determine the width of the Gaussian functions associated with the units. Let
for each Gaussian unit c denote Ne the set of direct topological neighbors in the
topological structure. Then the width of c is defined as
(je
= (1/INe l)
L: Ilwe - wdl12
(3)
dEN c
which is the mean distance to the topological neighbors. If topological neighbors
have similar center positions (which will be ensured by the way adaptation and
insertion is done) then this leads to a covering ofthe input vector space with partially
overlapping Gaussian functions.
2.2
ADAPTATION
It was mentioned above that several adaptation steps are done before a new unit is
inserted. One single adaptation step is done as follows (see fig. 1):
? Chose an I/O-pair (e,(),e E Rn,( E Rm) from the training data.
e (the so-called best-matching unit).
Move the centers of s and its direct topological neighbors towards e.
? determine the unit s closest to
?
dWe
= en (e -
eb and en are small constants with eb
> > en.
we)
for all c E N~
? Compute for each local unit eEL the activation De(e)
? Compute for each output unit i the activation Oi
(see eqn. 1)
(see eqn. 2)
? Compute the square error by
m
SE = L:?(i - Oi)2
i=l
? Accumulate error at best-matching unit s:
derrs = SE
? Make Delta-rule step for the weights (a denotes the learning rate):
iE{1, ... ,m},jEL
Since together with the best-matching unit always its direct topological neighbors
are adapted, neighboring units tend to have similar center positions. This property can be used to determine suitable center positions for new units as will be
demonstrated in the following.
Supervised Learning with Growing Cell Structures
a) Before ...
b) during, and ...
c) ... after adaptation
Figure 1: One adaptation step. The center positions of the current network are
shown and the change caused by a single input signal. The observed error SE for
this pattern is added to the local error variable of the best-matching unit.
f
a) Before ...
b) ... and after insertion
Figure 2: Insertion of a new unit. The dotted lines indicate the Voronoi fields.
The unit q has accumulated the most error and, therefore, a new unit is inserted
between q and one of its direct neighbors.
2.3
INSERTION OF NEW UNITS
After a constant number A of adaptation steps a new unit is inserted. For this
purpose the unit q with maximum accumulated error is determined. Obviously,
q lies in a region of the input vector space where many misclassifications occur.
One possible reason for this is that the gradient descent procedure is unable to find
suitable weights for the current network. This again might be caused by the coarse
resolution at this region of the input vector space: if data items from different
classes are covered by the same local unit and activate this unit to about the same
degree then it might be the case that their vectors of local unit activations are
nearly identical which makes it hard for the following single layer perceptrons to
distinguish among them. Moreover, even if the activation vectors are sufficiently
different they still might be not linearly separable.
accurate) approximation of the Delaunay triangulation which is based on the "Neural-Gas"
method proposed by Martinetz & Schulten (1991).
259
260
Fritzke
o
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o
o
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a) two spiral problem: 194 points in two
classes
b) decision regions for CascadeCorrelation (reprinted with permission
from Fahlman & Lebiere, 1990)
Figure 3: Two spiral problem and learning results of a constructive network.
The insertion of a new local unit near q is likely to improve the situation: This unit
will probablY be activated to a different degree by the data items in this region and
will, therefore, make the problem easier for the single layer perceptrons.
What exactly are we doing? We choose one of the direct topological neighbors of
q, say a unit f (see also fig. 2). Currently this is the neighbor with the maximum
accumulated error. Other choices, however, have shown good results as well, e.g., the
neighbor with the most distant center position or even a randomly picked neighbor.
We insert a new unit r in between q and f and initialize its center by
(4)
We connect the new unit with q and f and with all common neighbors of q and f.
The original connection between q and f is removed. By this we get a structure of
k-dimensional simplices again. The new unit gets weights to the output units which
are interpolated from the weights of its neighbors. The same is done for the initial
error variable which is linearly interpolated from the variables of the neighbors of r.
After the interpolation all the weights of r and its neighbors and the error variables
of these units are multiplied by a factor INrl/(INrl + 1)1. This is done to disturb
the output of the network as less as possible 4 ? However, the by far most important
4The redistribution of the error variable is again a relict from the unsupervised version
(Fritzke, 1993). There we count signals rather than accumulate error. An elaborate
scheme for redistributing the signal counters is necessary to get good local estimates of the
probability density. For the supervised version this redistribution is harder to justify since
the insertion of a new unit in general makes previous error information void. However,
even though there is still some room for simplification, the described scheme does work
very well already in its present form.
Supervised Learning with Growing Cell Structures
o
a) final network with 145 cells
b) decision regions
Figure 4: Performance of the Growing Cell Structures on the two spiral benchmark.
decision seems to be to insert the new unit near the unit with maximum error. The
weights and the error variables adjust quickly after some learning steps.
2.4
SIMULATION RESULTS
Simulations with the two spiral problem (fig. 3a) have been performed. This classification benchmark has been widely used before so that results for comparison
are readily available.Figure 3b) shows the result of another constructive algorithm.
The data consist of 194 points arranged on two interlaced spirals in the plane. Each
spiral corresponds to one class. Due to the high nonlinearity of the task it is particular difficult for networks consisting of global units (e.g., multi-layer perceptrons).
However, the varying density of data points (which is higher in the center of the
spirals) makes it also a challenge for networks of local units.
As for most learning problems the interesting aspect is not learning the training
examples but rather the performance on new data which is often denoted as generalization. Baum & Lang (1991) defined a test set of 576 points for this problem
consisting of three equidistant test points between each pair of adjacent same-class
training points. They reported for their best network 29 errors on the test set in
the mean.
In figure 4 a typical network generated by our method can be seen as well as the
corresponding decision regions. No errors on the test set of Baum and Lang are
made. Table 1 shows the necessary training cycles for several algorithms. The new
growing network uses far less cycles than the other networks.
Other experiments have been performed with a vowel recognition problem (Fritzke,
1993). In all simulations we obtained significantly better generalization results
than Robinson (1989) who in his thesis investigated the performance of several
connectionist and conventional algorithms on the same problem. The necessary
261
262
Fritzke
Table 1: Training epochs necessary for the two spiral problem
network model
Backpropagation
Cross Entropy BP
Cascade-Correlation
Growmg Cell Structures
epochs
20000
10000
1700
180
test error
yes
yes
yes
no
reported in
Lang & Witbrock (1989)
Lang & Witbrock (1989)
Fahlman & Lebiere (1990)
Fntzke ( 1993)
number of training cycles for our method was lower by a factor of about 37 than
the numbers reported by Robinson (1993, personal communication).
REFERENCES
Baum, E. B. & K. E. Lang [1991]' "Constructing hidden units using examples and queries,"
in Advances in Neural Information Processing Systems 3, R.P. Lippmann, J.E.
Moody & D.S. Touretzky, eds., Morgan Kaufmann Publishers, San Mateo, 904910.
Fahlman, S. E. & C. Lebiere [1990], "The Cascade-Correlation Learning Architecture,"
in Advances in Neural Information Processing Systems 2, D.S. Touretzky, ed.,
Morgan Kaufmann Publishers, San Mateo, 524-532.
Fritzke, B. [1993], "Growing Cell Structures - a self-organizing network for unsupervised
and supervised learning," International Computer Science Institute, TR-93-026,
Berkeley.
Fritzke, B. [1994], "Making hard problems linearly separable - incremental radial basis
function approaches," (submitted to ICANN'94: International Conference on Artificial Neural Networks), Sorrento, Italy.
Lang, K. J. & M. J. Witbrock [1989], "Learning to tell two spirals apart," in Proceedings
of the 1988 Connectionist Models Summer School, D. Touretzky, G. Hinton & T .
Sejnowski, eds., Morgan Kaufmann, San Mateo, 52-59.
Martinetz, T. M. & K. J. Schulten [1991]' "A "neural-gas" network learns topologies," in
Artificial Neural Networks, T. Kohonen, K. Makisara, O. Simula & J. Kangas,
eds., North-Holland, Amsterdam, 397-402.
Moody, J. & C. Darken [1989], "Learning with Localized Receptive Fields," in Proceedings
of the 1988 Connectionist Models Summer School, D. Touretzky, G. Hinton & T.
Sejnowski, eds., Morgan Kaufmann, San Mateo, 133-143.
Platt, J. C. [1991], "A Resource-Allocating Network for Function Interpolation," Neural
Computation 3, 213-225.
Poggio, T. & F. Girosi [1990], "Regularization Algorithms for Learning That Are Equivalent to Multilayer Networks," Science 247, 978-982.
Robinson, A. J. [1989], "Dynamic Error Propagation Networks," Cambridge University,
PhD Thesis, Cambridge.
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7,016 | 792 | Figure of Merit Training for Detection and
Spotting
Eric I. Chang and Richard P. Lippmann
MIT Lincoln Laboratory
Lexington, MA 02173-0073, USA
Abstract
Spotting tasks require detection of target patterns from a background of
richly varied non-target inputs. The performance measure of interest for
these tasks, called the figure of merit (FOM), is the detection rate for
target patterns when the false alarm rate is in an acceptable range. A
new approach to training spotters is presented which computes the FOM
gradient for each input pattern and then directly maximizes the FOM
using b ackpropagati on. This eliminates the need for thresholds during
training. It also uses network resources to model Bayesian a posteriori
probability functions accurately only for patterns which have a
significant effect on the detection accuracy over the false alarm rate of
interest. FOM training increased detection accuracy by 5 percentage
points for a hybrid radial basis function (RBF) - hidden Markov model
(HMM) wordspotter on the credit-card speech corpus.
1 INTRODUCTION
Spotting tasks require accurate detection of target patterns from a background of richly varied non-target inputs. Examples include keyword spotting from continuous acoustic input,
spotting cars in satellite images, detecting faults in complex systems over a wide range of
operating conditions, detecting earthquakes from continuous seismic signals, and finding
printed text on images which contain complex graphics. These problems share three common characteristics. First, the number of instances of target patterns is unknown. Second,
patterns from background, non-target, classes are varied and often difficult to model accurately. Third, the performance measure of interest, called the figure of merit (FOM), is the
detection rate for target patterns when the false alarm rate is over a specified range.
Neural network classifiers are often used for detection problems by training on target and
background classes, optionally normalizing target outputs using the background output,
1019
1020
Chang and Lippmann
PUTATIVE HITS
nA
uS
I NORMALIZATION AND THRESHOLDING I
jl
A
Us
~~ SACKG ROUND
CLASSIFIER
t
INPUT PATTERN
Figure 1. Block diagram of a spotting system.
and thresholding the resulting score to generate putative hits, as shown in Figure 1. Putative
hits in this figure are input patterns which generate normalized scores above a threshold.
We have developed a hybrid radial basis function (RBF) - hidden Markov model (HMM)
keyword spotter. This wordspotter was evaluated using the NIST credit card speech database as in (Rohlicek, 1993, Zeppenfeld, 1993) using the same train/evaluation split of the
training conversations as was used in (Zeppenfeld, 1993). The system spots 20 target keywords, includes one general filler class, and uses a Viterbi decoding backtrace as described
in (Lippmann, 1993) to backpropagate errors over a sequence of input speech frames. The
performance of this spotting system and its improved versions is analyzed by plotting detection versus false alarm rate curves as shown in Figure 2. These curves are generated by
adjusting the classifier output threshold to allow few or many putative hits. Wordspotter putative hits used to generate Figure 2 correspond to speech frames when the difference between the cumulative log Viterbi scores in output HMM nodes of word and filler models is
above a threshold. The FOM for this wordspotter is defined as the average keyword detection rate when the false alarm rate ranges from 1 to 10 false alarms per keyword per hour.
The 69.7% figure of merit for this system means that 69.7% of keyword occurrences are
detected on the average while generating from 20 to 200 false alarms per hour of input
speech.
2 PROBLEMS WITH BACKPROPAGATION TRAINING
Neural network classifiers used for spotting tasks can be trained using conventional backpropagation procedures with 1 of N desired outputs and a squared error cost function. This
approach to training does not maximize the FOM because it attempts to estimate Bayesian
a posteriori probability functions accurately for all inputs even if a particular input has little
effect on detection accuracy at false alarm rates of interest. Excessive network resources
may be allocated to modeling the distribution of common background inputs dissimilar
from targets and of high-scoring target inputs which are easily detected. This problem can
be addressed by training only when network outputs are above thresholds. This approach is
problematic because it is difficult to set the threshold for different keywords, because using
fixed target values of 1.0 and 0.0 requires careful normalization of network output scores to
prevent saturation and maintain backpropagation effectiveness, and because the gradient
calculated from a fixed target value does not reflect the actual impact on the FOM.
Figure of Merit Training for Detection and Spotting
100
A SPLIT OF CREDIT-CARD
TRAINING DATA
z 90
(J)
80
0 70
W
t- 60
w
c 50
tO 40
w
0
i=
????????? :::.:./.::.:: ..!!!..
/'-:,/,
/f
a:: 30
a::
0
0
0~
20
10
0
-
~.~ ......... u::.~ ...:::..
","I'I .. ~.~I'iI.I:
l- -
FOM BACK-PROP (FOM: 69.7%)
./
EMBEDDED REESTIMATION (FOM: 64.5%)
ISOLATED WORD TRAIN (FOM: 62.5%)
0
2
4
6
8
10
FALSE ALARMS PER KW PER HR
Figure 2. Detection vs. false alarm rate curve for a 20-word hybrid wordspotter.
Figure 3 shows the gradient of true hits and false alarms when target values are set to be 1.0
for true hits and 0.0 for false alarms, the output unit is sigmoidal, and the threshold for a
putative hit is set to roughly 0.6. The gradient is the derivative of the squared error cost with
respect to the input of the sigmodal output unit. As can be seen, low-scoring hits or false
alarms that may affect the FOM are ignored, the gradient is discontinuous at the threshold,
the gradient does not fall to zero fast enough at high values, and the relative sizes of the hit
and false alarm gradients do not reflect the true effect of a hit or false alarm on the FOM.
3 FIGURE OF MERIT TRAINING
A new approach to training a spotter system called "figure of merit training" is to directly
compute the FOM and its derivative. This derivative is the change in FOM over the change
in the output score of a putative hit and can be used instead of the derivative of a squarederror or other cost function during training. Since the FOM is calculated by sorting true hits
and false alarms separately for each target class and forming detection versus false alarm
curves, these measures and their derivatives can not be computed analytically. Instead, the
FOM and its derivative are computed using fast sort routines. These routines insert a new
0.2
r--------------------,
THRESHOLD
!zw
?Ci
a:
HIT GRADIENT
L.......
0
I-----------f------==-'-'""!l
.0.2
L....................L..............~_'_'_~............L....................J'_'_'_~.................J.......................L.<_.'-'--'-J.......o............
<!)
GRADIENT
o
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0,8
0.9
OUTPUT VALUE
Figure 3. The gradient for a sigmoid output unit when the target value for true hits is set to
1.0 and the target value for false alarms is set to 0.0.
1021
1022
Chang and Lippmann
putative hit into an already sorted list and calculate the change in the FOM caused by that
insertion. The running putative hit list used to compute the FOM is updated after every new
putative hit is observed and it must contain all putative hits observed during the most recent
past training cycle through all training patterns. The gradient estimate is smoothed over
nearby putative hit scores to account for the quantized nature of detection versus false alarm
rate curves.
Figure 4 shows plots of linearly scaled gradients for the 20-word hybrid wordspotter. Each
value on the curve represents the smoothed change in the FOM that occurs when a single
hit or false alarm with the specified normalized log output score is inserted into the current
putative hit list. Gradients are positive for putative hits corresponding to true hits and negative for false alarms. They also fall off to zero for putative hits with extremely high or low
scores. Shapes of these curves vary across words. The relative importance of a hit or false
alarm, the normalized output score which results in high gradient values, and the shape of
the gradient curve varies. Use of a squared error or other cost function with sigmoid output
nodes would not generate this variety of gradients or automatically identify the range of putative hit scores where gradients should be high. Application ofFOM training requires only
the gradients shown in these curves with no supplementary thresholds. Patterns with low
and high inputs will have a minimal effect during training without using thresholds because
they produce gradients near zero.
Different keywords have dramatically different gradients. For example, credit-card is long
and the detection rate is high. The overall FOM thus doesn't change much if more true hits
are found. A high scoring false alarm, however, decreases the FOM drastically. There is thus
a large negative gradient for false alarms for credit-card. The keywords account and check
are usually short in duration and thus more difficult to detect, thus any increase in number
of true hits strongly increases the overall FOM. On the other hand, since in this database,
the words account and check occur much less frequently than credit-card, a high scoring
false alarm for the words account and check has less impact on the overall FOM. The gradient for false alarms for these words is thus correspondingly smaller. Comparing the
curves in Figure 3 with the fixed prototypical curve in Figure 4 demonstrates the dramatic
differences in gradients that occur when the gradient is calculated to maximize the FOM
directly instead of using a threshold with sigmoid output nodes.
"ACCOUNT"
"CHECK'
"CREDIT-CARD'
0.3 r - - - - - - - ,
HIT
o
FA
~
w
is -03
?
ffi
-D.6
-0. 9 ~--'--.L..-L--'--.L...-L---'
-100 0
100 200 300 -100
0
100
200
300 -100
0
100
200
300
PUTATIVE HIT SCORE
Figure 4. Figure of merit gradients computed for true hits (HIT) and false alarms (FA)
with scores ranging from -100 to 300 for the keywords account, check, and
credit-card.
Figure of Merit Training for Detection and Spotting
FaM training is a general technique that can applied to any "spotting" task where a set of
targets must be discriminated from background inputs. FaM training was successfully tested using the hybrid radial basis function (RBF) - hidden Markov model (HMM) keyword
spotter described in (Lippmann, 1993).
4 IMPLEMENTATION OF FOM TRAINING
FaM training is applied to our high-performance HMM wordspotter after forward-backward training is complete. Word models in the HMM wordspotter are first used to spot on
training conversations. The FaM gradient of each putative hit is calculated when this hit is
inserted into the putative hit list. The speech segment corresponding to a putative hit is excised from the conversation speech file and the corresponding keyword model is used to
match each frame with a particular state in the model using a Viterbi backtrace (shown in
Figure 5.) The gradient is then used to adjust the location of each Gaussian component in a
node as in RBF classifiers (Lippmann, 1993) and also the state weight of each state. The
state weight is a penalty added for each frame assigned to a state. The weight for each individual state is adjusted according to how important each state is to the detection of the
keyword. For example, many false alarms for the word card are words that sound like part
of the keyword such as hard or far. The first few states of the card model represent the sound
/kJ and false alarms stay in these front states only a short time. If the state weight of the first
few states of the card model is large, then a true hit has a larger score than false alarms.
The putative hit score which is used to detect peaks representing putative hits is generated
according to
Stota I =
In this equation,
Stotal
Sk eywor d -
is the putative hit score,
(EQ 1)
SJrll
r er .
Skeyword
is the log Viterbi score in the
RAW KEYWORD SCORE
??
? ? ?
? ? ? ? ?
RADIAL
BASIS
FUNCTIO
NODES
V~TERBI
ALIGNMENT
Figure 5. State weights and center updates are applied to the state that is matched to each
frame in a Viterbi backtrace.
1023
1024
Chang and Lippmann
last node of a specific keyword model computed using the Viterbi algorithm from the beginning of the conversation to the frame where the putative hit ended, and SIi/Ie r is the log
Viterbi score in the last node of the filler model. The filler score is used to normalize the
keyword score and approximate a posterior probability. The keyword score is calculated using a modified form of the Viterbi algorithm
a./ (t + 1) = max(a /. (t) + a /,../ , a./- 1 (t) + a./-1, /.) + d /. (t, x) + W /.?
(EQ2)
This equation is identical to the normal Viterbi recursion for left-to-right linear word models after initialization, except the extra state score wi is added. In this equation, a i (t) is
the log Viterbi score in node i at time t, a j . is the log of the transition probability from
node i to node j ,and d j (t, x) is the log lik~lihood distance score for node i for the input
feature vector x at time t .
Word scores are computed and a peak-picking algorithm looks for maxima above a low
threshold. After a peak representing a putative hit is detected, frames of a putative hit are
aligned with the states in the keyword model using the Viterbi backtrace and both the means
of Gaussians in each state and state weights of the keyword model are modified. State
weights are modified according to
(EQ 3)
In this equation, Wj (t) is the state weight in node i at time t, gradient is the FOM
gradient for the putative hit, llstate is the stepsize for state weight adaptation, and
duration is the number of frames aligned to node i . If a true hit occurs, and the gradient
is positive, the state weight is increased in proportion to the number of frames assigned to
a state. If a false alarm occurs, the state weight is reduced in proportion to the number of
frames assigned to a state. The state weight will thus be strongly positive if there are many
more frames for a true hit that for a false alarm. It will be strongly negative if there are more
frames for a false alarm than for a true hit. High state weight values should thus improve
discrimination between true hits and false alarms.
The center of the Gaussian components within each node, which are similar to Gaussians
in radial basis function networks, are modified according to
m .. (t+ 1)
V
= miJ. (t)
x.(t) -m .. (t)
+gradientxllcenterX J
/J
a /J..
(EQ4)
In this equation, m j . (t) is the j th component of the mean vector for a Gaussian hidden
node in HMM state 1at time t, gradient is the FOM gradient, llcenter is the stepsize
for moving Gaussian centers, x? (t) is the value of the j th component of the input feature
vector at time t, and a j . is the'standard deviation of the j th component of the Gaussian
hidden node in HMM st~te i .
For each true hit, the centers of Gaussian hidden nodes in a state move toward the observation vectors of frames assigned to a particular state. For a false alarm, the centers move
away from the observation vectors that are assigned to a particular state. Over time, the centers move closer to the true hit observation vectors and further away from false alarm observation vectors.
Figure of Merit Training for Detection and Spotting
0.95 , - - - - - - - - - - - - - - - - - - - - - . ,
0.9
FEMALE TRAIN
0.85
0.8 \--_,,-r
0.75
FOM
0.7
0.65
0.6
L---~~~~------~
MALE TEST
0.55
0.5
L..-_.....J-_ _.l.....-_--'--_ _.J...-_---'-_ _...L...-_--'-_-----'
o
20
40
60
80
100
120
NUMBER OF CONVERSATIONS
140
160
Figure 6. Change in FOM vs. the number of conversations that the models have been
trained with. There were 25 male training conversations and 23 female training
conversations.
5 EXPERIMENTAL RESULTS
Experiments were performed using a HMM wordspotter that was trained using maximum
likelihood algorithm. More complicated models were created for words which occur frequently in the training set. The word models for card and credit-card were increased to four
mixtures per state. The models for cash, charge, check, credit, dollar, interest, money,
month, and visa were increased to two mixtures per state. All other word models had one
mixture per state. The number of states per keyword is roughly 1.5 times the number of phonemes in each keyword. Covariance matrices were diagonal and variances were estimated
separately for all states. All systems were trained on the first 50 talkers in the credit card
training corpus and evaluated using the last 20 talkers.
An initial set of models was trained during 16 passes through the training data using wholeword training and Viterbi alignment on only the excised words from the training conversations. This training provided a FOM of 62.5% on the 20 evaluation talkers. Embedded forward-backward reestimation training was then performed where models of keywords and
fillers are linked together and trained jointly on conversations which were split up into sentence-length fragments. This second stage ofHMM training increased the FOM by two percentage points to 64.5%. The detection rate curves of these systems are shown in Figure 2.
FOM training was then performed for six passes through the training data. On each pass,
conversations were presented in a new random order. The change in FOM for the training
set and the evaluation set is shown in Figure 6. The FOM on the training data for both male
and female talkers increased by more than 10 percentage points after roughly 50 conversations had been presented. The FOM on the evaluation data increased by 5.2 percentage
points to 69.7% after three passes through the training data, but then decreased with further
training. This result suggests that the extra structure learned during the final three training
passes is overfitting the training data and providing poor performance on the evaluation set.
Figure 7 shows the spectrograms of high scoring true hits and false alarms for the word card
generated by our wordspotter. All false alarms shown are actually the occurrences of the
word car. The spectrograms of the true hits and the false alarms are very similar and the
actual excised speech segments are difficult even for humans to distinguish.
1025
1026
Chang and Lippmann
A) True hits for card
Figure 7. Spectrograms of high scoring true hit and false alarm for the word card.
6 SUMMARY
Detection of target signals embedded in a noisy background is a common and difficult problem distinct from the task of classification. The evaluation metric of a spotting system,
called Figure of Merit (FOM), is also different from the classification accuracy used to evaluate classification systems. FOM training uses a gradient which directly reflects a putative
hit's impact on the FOM to modify the parameters of the spotting system. FOM training
does not require careful adjustment of thresholds and target values and has been applied to
improve a wordspotter's FOM from 64.5% to 69.7% on the credit card database. POM
training can also be applied to other spotting tasks such as arrhythmia detection and address
block location.
ACKNOWLEDGEMENT
This work was sponsored by the Advanced Research Projects Agency. The views expressed are those
of the authors and do not reflect the official policy or position of the U.S. Government. Portions of
this work used the HTK Toolkit developed by Dr. Steve Young of Cambridge University.
BIBLIOGRAPHY
R. Lippmann & E. Singer. (1993) Hybrid HMM/Neural-NetworkApproaches to Wordspotting. In ICASSP '93, volume I, pages 565-568.
J. Rohlicek et. al. (1993) Phonetic and Language Modeling for Wordspotting. In ICASSP '93, volume
II, pages 459-462.
T. Zeppenfeld, R. Houghton & A. Waibel. (1993) Improving the MS-TDNN for Word Spotting. In
ICASSP '93, volume II, pages 475-478.
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7,017 | 793 | Directional Hearing by the Mauthner
System
.Audrey L. Gusik
Department of Psychology
University of Colorado
Boulder, Co. 80309
Robert c. Eaton
E. P. O. Biology
University of Colorado
Boulder, Co. 80309
Abstract
We provide a computational description of the function of the Mauthner system. This is the brainstem circuit which initiates faststart escapes in teleost fish in response to sounds. Our simulations, using back propagation in a realistically constrained feedforward network, have generated hypotheses which are directly interpretable in terms of the activity of the auditory nerve fibers, the
principle cells of the system and their associated inhibitory neurons.
1
1.1
INTRODUCTION
THE M.AUTHNER SYSTEM
Much is known about the brainstem system that controls fast-start escapes in teleost
fish. The most prominent feature of this network is the pair of large Mauthner
cells whose axons cross the midline and descend down the spinal cord to synapse
on primary motoneurons. The Mauthner system also includes inhibitory neurons,
the PHP cells, which have a unique and intense field effect inhibition at the spikeinitiating zone of the Mauthner cells (Faber and Korn, 1978). The Mauthner system
is part of the full brainstem escape network which also includes two pairs of cells
homologous to the Mauthner cell and other populations of reticulospinal neurons.
With this network fish initiate escapes only from appropriate stimuli, turn away
from the offending stimulus, and do so very rapidly with a latency around 15 msec
in goldfish. The Mauthner cells play an important role in these functions. Only one
574
Directional Hearing by the Mauthner System
fires thus controlling the direction of the initial turn, and it fires very quickly (4-5
msec). They also have high thresholds due to instrinsic membrane properties and
the inhibitory inlluence of the PHP cells. (For reviews, see Eaton, et al, 1991 and
Faber and Korn, 1978.)
Acoustic stimuli are thought to be sufficient to trigger the response (Blader, 1981),
both Mauthner cells and PHP cells receive innervation from primary auditory fibers
(Faber and Korn, 1978). In addition, the Mauthner cells have been shown physiologically to be very sensitive to acoustic pressure (Canfield and Eaton, 1990).
1.2
LOCALIZING SOUNDS UNDERWATER
In contrast to terrestrial vertebrates, there are several reasons for supposing that
fish do not use time of arrival or intensity differences between the two ears to localize
sounds: underwater sound travels over four times as fast as in air; the fish body
provides no acoustic shadow; and fish use a single transducer to sense pressure which
is conveyed equally to the two ears. Sound pressure is transduced into vibrations
by the swim bladder which, in goldfish, is mechanically linked to the inner ear.
Fish are sensitive to an additional component of the acoustic wave, the particle
motion. Any particle ofthe medium taking part in the propagation of a longitudenal
wave will oscillate about an equilibrium point along the axis of propagation. Fish
have roughly the same density as water, and will experience these oscillations. The
motion is detected by the bending of sensory hairs on auditory receptor cells by
the otolith, an inertial mass suspended above the hair cells. This component of the
sound will provide the axis of propagation, but there is a 180 degree ambiguity.
Both pressure and particle motion are sensed by hair cells of the inner ear. In
goldfish these signals may be nearly segregated. The linkage with the swim bladder
impinges primarily on a boney chamber containing two of the endorgans of the inner
ear: the saccule and the lagena. The utricle is a third endorgan also thought to
mediate some acoustic function, without such direct input from the 3wimbladder.
Using both of these components fish can localize sounds. According to the phase
model (Schuijf, 1981) fish analyze the phase difference between the pressure component of the sound and the particle displacement component to calculate distance
and direction. When pressure is increasing, particles will be pushed in the direction of sound propagation, and when pressure is decreasing particles will be pulled
back. There will be a phase lag between pressure and particle motion which varies
with frequency and distance from the sound source. This, and the separation of the
pressure from the displacement signals in the ear of some species pose the greatest
problems for theories of sound localization in fish.
The acoustically triggered escape in goldfish is a uniquely tractable problem in
underwater sound localization. First, there is the fairly good segregation of pressure
from particle motion at the sensory level. Second I the escape is very rapid. The
decision to turn left or right is equivalent to the firing of one or the other Mauthner
cell, and this happens within about 4 msec. With transmission delay, this decision
relies only on the initial 2 msec or so of the stimulus. For most salient frequencies,
the phase lag will not introduce uncertainty: both the first and second derivatives
of particle position and acoustic pressure will be either positive or negative.
575
576
Guzik and Eaton
1.3
THE XNOR MODEL
A
Active
pressure
input
Active
displacement
input
left
Mauthner
output
Right
Mauthner
output
p+
Ol
On
Ofr
p+
DR
Off
On
p-
OL
orr
On
p-
DR
On
Off
B
Left sound
source
OR---a
p+ - - - - - - - . .
1)---_ _..;:Jo.._ _ _
DL
P+
--..,----p.
1 ) - - - - DR
p-
OL---a
. . inhibitory
0- excitatory
No response
Figure 1 Truth table and minimal network for the XNOR model.
Given the above simplification of the problem, we can see that each Mauthner
cell must perform a logical operation (Guzik and Eaton, 1993j Eaton et al, 1994).
The left Mauthner cell should fire when sounds are located on the left, and this
occurs when either pressure is increasing and particle motion is from the left or
when pressure is decreasing and particle motion is from the right. We can call
displacement from the left positive for the left Mauthner cell, and immediately we
Directional Hearing by the Mauthner System
have the logical operator exclusive-nor (or XNOR). The right Mauthner cell must
solve the same problem with a redefinition of right displacement as positive. The
conditions for this logic gate are shown in figure 1A for both Mauthner cells. This
analysis simplifies our task of understanding the computational role of individual
elements in the system. For example, a minimal network could appear as in figure
lB.
In this model PHP units perform a logical sub-task of the XNOR as AND gates.
This model requires at least two functional classes of PHP units on each side of
the brain. These PHP units will be activated for the combinations of pressure
and displacement that indicate a sound coming from the wrong direction for the
Mauthner cell on that side. Both Mauthner cells are activated by sufficient changes
in pressure in either direction, high or low, and will be gated by the PHP cells. This
minimal model emerged from explorations of the system using the connectionist
paradigm, and inspired us to extend our efforts to a more realistic context.
2
THE NETWORK
We used a connectionist model to explore candidate solutions to the left/right discrimination problem that include the populations of neurons known to exist and
include a distributed input resembling the sort available from the hair cells of the
inner ear. We were interested in generating a number of alternative solutions to be
better prepared to interpret physiological recordings from live goldfish, and to look
for variations of, or alternatives to, the XNOR model.
2.1
THE .ARCHITECTURE
As shown in figure 2, there are four layers in the connectionist model. The input
layer consists of four pools of hair cell units. These represent the sensory neurons
of the inner ear. There are two pools on each side: the saccule and the utricle.
Treating only the horizontal plane, we have ignored the lagena in this model. The
saccule is the organ of pressure sensation and the utricle is treated as the organ
of particle motion. Each pool contains 16 hair cell units maximally responsive for
displacements of their sensory hairs in one particular direction. They are activated
as the eosine of the difference between their preferred direction and the stimulus
dellection. All other units use sigmoidal activation functions.
The next layer consists of units representing the auditory fibers of the VIIIth nerve.
Each pool receives inputs from only one pool of hair cell units, as nerve fibers have
not been seen to innervate more than one endorgan. There are 10 units per fiber
pool.
The fiber units provide input to both the inhibitory PHP units, and to the Mauthner
units. There are four pools of PHP units, two on each side of the fish. One set
on each side represents the collateral PHP eells, and the other set represents the
commissural PHP cells (Faber and Korn, 1978). Both types receive inputs from the
auditory fibers. The collaterals project only to the Mauthner cell on the same side.
The commissurals project to both Mauthner cells. There are five units per PHP
pool.
577
578
Guzik and Eaton
The Mauthner cell units receive inputs from saecular and utricular fibers on their
same side only, as well as inputs from a single collateral PHP population and both
commissural PHP populations.
Left Saccule Left Utricle Right Utricle Right Saccule
Hair Cells
Auditory Nerve
Fiber Pools
PHPs
Left Mauthner
Right Mautlll1er
Figure 2 The architecture.
Weights from the PHP units are all constrained to be negative, while all others are
constrained to be positive. The weights are implemented using the function below,
positive or negative depending on the polarity of the weight.
f(w) = 1/2 (w
+ In cosh(w) + In 2)
The function asymptotes to zero for negative values, and to the identity function for
values above 2. This function vastly improved learning compared with the simpler,
but highly nonlinear exponential function used in earlier versions of the model.
2.2
TRAINING
We used a total of 240 training examples. We began with a set of 24 directions for
particle motion, evenly distributed around 360 degrees. These each appeared twice,
once with increasing pressure and once with decreasing pressure, making a base set
of 48 examples. Pressure was introduced as a deflection across saccular hair cells of
either 0 degrees for low pressure, or 180 degrees for high pressure. These should be
thought of as reflecting the expansion or compression of the swim bladder. Targets
for the Mauthner cells were either 0 or 1 depending upon the conditions as described
in the XNOR model, in figure lA.
Directional Hearing by the Mauthner System
Next by randomly perturbing the activations of the hair cells for these 48 patterns,
we generated 144 noisy examples. These were randomly increased or decreased up
to 10%. An additional 48 examples were generated by dividing the hair cell adivity
by two to represent sub-threshold stimuli. These last 48 targets were set to zero.
The network was trained in batch mode with backpropagation to minimize a crossentropy error measure, using conjugate gradient search. Unassisted backpropagation was unsuccessful at finding solutions.
For the eight solutions discussed here, two parameters were varied at the inputs. In
some solutions the utride was stimulated with a vedor sum of the displacement and
the pressure components, or a "mixed" input. In some solutions the hair cells in the
utride are not distributed uniformly, but in a gaussian manner with the mean tuning
of 45 degrees to the right or left, in the two ears respedively. This approximates
the actual distribution of hair cells in the goldfish utride (Platt, 1977).
3
RESULTS
Analyzing the activation of the hidden units as a fundion of input pattern we found
activity consistent with known physiology, nothing inconsistent with our knowledge
of the system, and some predidions to be evaluated during intracellular recordings
from PHP cells and auditory afFerents.
First, many PHP cells were found exhibiting a logical fUndion, which is consistent
with our minimal model described above. These tended to projed only to one
Mauthner cell unit, which suggests that primarily the collateral PHP cells will
demonstrate logical properties. Most logical PHP units were NAND gates with
very large weights to one Mauthner cell. An example is a unit which is on for all
stimuli except those having displacements anywhere on the left when pressure is
high.
Second, saccular fibers tended to be either sensitive to high or low pressure, consistent with recordings of Furukawa and Ishii (1967). In addition there were a dass
which looked like threshold fibers, highly active for all supra-threshold stimuli, and
inactive for all sub-threshold stimuli. There were some fibers with no obvious seledivity, as well.
Third, utricular fibers often demonstrate sensitivity for displacements exclusively
from one side ofthe fish, consistent with our minimal model. Right and left utricular
fibers have not yet been demonstrated in the real system.
Utricular fibers also demonstrated more coarsely tuned, less interpretable receptive
fields. All solutions that included a mixed input to the utrieie, for example, produced fibers that seemed to be "not 180 degree" ,or "not 0 degree", countering the
pressure vedors. We interpret these fibers as doing dean-up given the absence of
negative weights at that layer.
Fourth, sub-threshold behavior of units is not always consistent with their suprathreshold behavior. At sub-threshold levels of stimulation the adivity of units may
not refted their computational role in the behavior. Thus, intracellular recordings
should explore stimulus ranges known to elicit the behavior.
579
580
Guzik and Eaton
Fifth, Mauthner units usually receive very strong inputs from pressure fibers. This
is consistent with physiological recordings which suggest that the Mauthner cells
in goldfish are more sensitive to sound pressure than displacement (Canfield and
Eaton, 1990).
Sixth, Mauthner cells always acquired rdatively equal high negative biases. This is
consistent with the known low input resistance of the real Mauthner eells, giving
them a high threshold (Faber and Korn, 1978).
Seventh, PHP cells that maintain substantial bilateral connections tend to be tonically active. These contribute additional negative bias to the Mauthner cells. The
relative sizes of the connections are often assymetric. This suggests that the commissural PHP cells serve primarily to regulate Mauthner threshold, ensure behavioral
response only to intense stimuli, consistent with Faber and Korn (1978). These cells
could only contribute to a partial solution of the XNOR problem.
Eighth, all solutions consistently used logic gate PHP units for only 50% to 75%
of the training examples. Probably distributed solutions relying on the direct connections of auditory nerve fibers to Mauthner cells were more easily learned, and
logic gate units only developed to handle the unsolved eases. Cases solved without
logic gate units were solved by assymetric projections to the Mauthner cells of one
polarity of pressure and one class of direction fibers, left or right.
Curiously, most of these eases involved a preferential projection from high pressure
fibers to the Mauthner units, along with directional fibers encoding displacements
from each Mauthner unit's positive direction. This means the logic gate units
tended to handle the low pressure eases. This may be a result of the presence of
the assymetric distributions of utricular hair cells in 6 out of the 8 solutions.
4
CONCLUSIONS
\Ve have generated predictions for the behavior of neurons in the Mauthner system
under different conditions of acoustic stimulation. The predictions generated with
our connectionist model are consistent with our interpretation of the phase model
for underwater sound localization in fishes as a logical operator. The results are also
consistent with previously described properties of the Mauthner system. Though
perhaps based on the characteristics more of the training procedure, our solutions
suggest that we may find a mixed solution in the fish. Direct projections to the
Mauthner cells from the auditory nerve perhaps handle many of the commonly
encountered acoustic threats. The results of Blaxter (1981) support the idea that
fish do escape from stimuli regardless of the polarity of the initial pressure change.
Without significant nonlinear processing at the Mauthner cell itsdf, or more complex processing in the auditory fibers, direct connections could not handle all of
these eases. These possibilities deserve exploration.
We propose different computational roles for the two classes of inhibitory PHP
neurons. We expect only unilaterally-projecting PHP cells to demonstrate some
logical function of pressure and particle motion. We believe that some elements of
the Mauthner system must be found to demonstrate such minimal logical functions
if the phase modd is an explanation for left-right discrimination by the Mauthner
system.
Directional Hearing by the Mauthner System
We are currently preparing to deliver controlled acoustic stimuli to goldfish during
acute intracellular recording procedures from the PHP neurons, the afferent fibers
and the Mauthner cells. Our insights from this model will greatly assist us in
designing the stimulus regimen, and in interpreting our experimental results. Plans
for future computational work are of a dynamic model that will include the results of
these physiological investigations, as well as a more realistic version of the Mauthner
cell .
.Acknowledgements
We are grateful for the technical assistance of members of the Boulder Connectionist
Research Group, especially Don Mathis for help in debugging and optimizing the
original code. We thank P.L. Edds-Walton for crucial discussions. This work was
supported by a grant to RCE from the National Institutes of Health (ROI NS22621).
References
Blader, J.H.S., J.A.B. Gray, and E.J. Denton (1981) Sound and startle responses
in herring shoals. J. Mar. BioI. Assoc. UK, 61: 851-869
Canfield, J.G. and R.C. Eaton (1990) Swimbladder acoustic pressure transduction
intiates Mauthner-mediated escape. Nature, 3~7: 760-762
Eaton, R.C., J.G. Canfield and A.L. Guzik (1994) Left-right discrimination of sound
onset by the Mauthner system. Brain Behav. Evol., in pre66
Eaton, R.C., R. DiDomenico and J. Nissanov (1991) Role of the Mauthner cell in
sensorimotor integration by the brain stem escape network. Brain Behav. Evol.,
37: 272-285
Faber, D.S. and H. Korn (1978) Electrophysiology of the Mauthner cell: Basic
properties, synaptic mechanisms and associated networks. In Neurobiology of the
Mauthner Cell, D.S. Faber and H. Korn (eds) , Raven Press, NY, pp. 47-131
Fay, R.R.(1984) The goldfish ear codes the axis of acoustic particle motion in three
dimensions. Science, 225: 951-954
Furukawa, T. and Y. Ishii (1967) Effects of static bending of sensory hairs on sound
reception in the goldfish. Japanese J. Physiol., 17: 572-588
Guzik, A.L. and R.C. Eaton (1993) The XNOR model for directional hearing by
the Mauthner system. Soc. Neurosci. Abstr.
PIaU, C. (1977) Hair cell distribution and orientation in goldfish otolith organs. J.
Compo Neurol., 172: 283-298
Schuijf, A. (1981) Models of acoustic localization. In Hearing and Sound Communication in Fishes, W.N. Tavolga, A.N . Popper and R.R. Fay (eds.), Springer, New
York,. pp. 267-310
581
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7,018 | 794 | Recognition-based Segmentation of
On-line Cursive Handwriting
Nicholas S. Flann
Department of Computer Science
Utah State University
Logan, UT 84322-4205
flannGnick.cs.usu.edu
Abstract
This paper introduces a new recognition-based segmentation approach to recognizing on-line cursive handwriting from a database
of 10,000 English words. The original input stream of z, y pen coordinates is encoded as a sequence of uniform stroke descriptions that
are processed by six feed-forward neural-networks, each designed
to recognize letters of different sizes. Words are then recognized by
performing best-first search over the space of all possible segmentations. Results demonstrate that the method is effective at both
writer dependent recognition (1.7% to 15.5% error rate) and writer
independent recognition (5.2% to 31.1% error rate).
1
Introduction
With the advent of pen-based computers, the problem of automatically recognizing
handwriting from the motions of a pen has gained much significance. Progress has
been made in reading disjoint block letters [Weissman et. ai, 93]. However, cursive
writing is much quicker and natural for humans, but poses a significant challenge to
pattern recognition systems because of its variability, ambiguity and need to both
segment and recognize the individual letters. Recent techniques employing selforganizing networks are described in [Morasso et. ai, 93] and [Schomaker, 1993].
This paper presents an alternative approach based on feed-forward networks.
On-line handwriting consists of writing with a pen on a touch-terminal or digitizing
777
778
Flann
(a)
(b)
(c)
(d)
(e)
Figure 1: The five principal stages of preprocessing: (a) The original data, z, Y
values sampled every 10mS. (b) The slant is normalized through a shear transformation; (c) Stroke boundaries are determined at points where y velocity equals 0 or
pen-up or pen-down events occur; (d) Delayed strokes are reordered and associated
with corresponding strokes of the same letters; (e) Each stroke is resampled in space
to correspond to exactly 8 points. Note pen-down strokes are shown as thick lines,
pen-up strokes as thin lines.
Recognition-Based Segmentation of On-Line Cursive Handwriting
tablet. The device produces a regular stream of z, y coordinates, describing the
positions of the pen while writing. Hence the problem of recognizing on-line cursively written words is one of mapping a variable length sequence of z, y coordinates
to a variable length sequence of letters. Developing a system that can accurately
perform this mapping faces two principal problems: First, because handwriting is
done with little regularity in speed, there is unavoidable variability in input size.
Second, because no pen-up events or spatial gaps signal the end of one letter and the
beginning of the next, the system must perform both segmentation and recognition.
This second problem necessitates the development of a recognition-based segmentation approach. In [Schenkel et al., 93] one such approach is described for connected
block letter recognition where the system learns to recognize segmentation points.
In this paper an alternative method is presented that first performs exhaustive
recognition then searches the space of possible segmentations. The remainder of
the paper describes the method in more detail and presents results that demonstrate its effectiveness at recognizing a variety of cursive handwriting styles.
2
Methodology
The recognition system consists of three subsystems: (a) the preprocessor that maps
the initial stream of z, y coordinates to a stream of stroke descriptions; (b) the letter
classifier that learns to recognize individual letters of different size; and (c) the word
finder that performs recognition-based segmentation over the output of the letter
classifier to identify the most likely word written.
2.1
Preprocessing
The preprocessing stage follows steps outlined in [Guerfali & Plamondon, 93] and
is illustrated in Figure 1. First the original data is smoothed by passing it through
a low-pass filter, then reslanted to make the major stroke directions vertical. This
is achieved by computing the mean angle of all the individual lines then applying
a shear transformation to remove it. Second, the strokes boundaries are identified
as points when if = 0 or when the pen is picked up or put down. Zero y velocity
was chosen rather than minimum absolute velocity [Morasso et. ai, 93] since it was
found to be more robust. Third, delayed strokes such as those that dot an i or cross
a t are reordered to be associated with their corresponding letter. Here the delayed
stroke is placed to immediately follow the closest down stroke and linked into the
stroke sequence by straight line pen-up strokes. Fourth, each stroke is resampled in
the space domain (using linear interpolation) so as to represent it as exactly eight
z, y coordinates. Finally the new stream of z, y coordinates is converted to a stream
of 14 feature values.
Eight of these features are similar to those used in [Weissman et. ai, 93], and represent the angular acceleration (as the sin and cos of the angle), the angular velocity
of the line (as the sin and cos of the angle), the z, y coordinates (z has a linear
ramp removed), and first differential ox,Oy. One feature denotes whether the pen
was down or up when the line was drawn. The remaining features encode more
abstract information about the stroke.
779
780
Flann
?
32
Figure 2: The pyramid-style architecture of the network used to recognize 2 stroke
letters. The input size is 32 x 14; 32 is from the 4 input strokes (each represented by
8 resampled points), two central strokes from the letter and the 2 context strokes,
one each side; 14 is from the number of features employed to represent each point.
Not all the receptive fields are shown. The first hidden layer consists of 7 fields,
4 over each stroke and 3 more spanning the stroke boundaries. The next hidden
layer consists of 5 fields, each spanning 3 x 20 inputs. The output is a 32 bit
error-correcting code.
J.) ~"I v~c.'fJcr/ "~lI"")c' (/" .p/ ~'l q\) /.h.l/ .....')"/\1\.1 Jt?z./I' l'-V..c..A.U,I A {jAAVv ....... t A ~...J)'l~.n",l1v..-t..>...,--ZUv ..... U.,.,,,( .lI\
..,. ..n...d t...rt '( f,l.-v tV 'i> r' 1/"J1. tt I'-' V (,fJ 1\./11 \....-"\ ~ r.r S)y' U Iv' hV (..;
.-Y.w .M/l.JYV.JJ. ~ At.. ~ fA. "'"'I.t.N. ~ .I..L.r.,.. U. f" I' ry \{\J?'\J)1 LA ~
r"Yi.IW11. . .
\..0.m
~ W ~.-;,(...vy..p/v~\.6\~ J..v
bY)U> _~.bA ~ u...Yv:.)~
rn ~ ~d~ AlAt
)AA. \.Oe!;\IVY' M1~~ /\.$\ t.W f1- ~~,
Figure 3: Examples of the class "other" for stroke sizes 1 though 6. Each letter is
a random fragment of a word, such that it is not an alphabetic letter.
Recognition-Based Segmentation of On-Line Cursive Handwriting
2.2
Letter Recognition
The letter classifier consists of six separate pyramid-style neural-networks, each
with an architecture suitable for recognizing a letter of one through six strokes.
A neural network designed to recognize letters of size j strokes encodes a mapping from a sequence of j + 2 stroke descriptions to a 32 bit error-correcting code
[Dietterich & Bakiri, 91]. Experiments have shown this use of a context window
improves performance, since the allograph of the current letter is dependent on the
allographs of the previous and following letters. The network architecture for stroke
size two is illustrated in Figure 2. The architecture is similar to a time-delayed
neural-network [Lang & Waibel, 90] in that the hierarchical structure enables different levels of abstract features to be learned. However, the individual receptive
fields are not shared as in a TDNN, since translational variance is not problem and
the sequence of data is important.
The networks are trained using 80% of the raw data collected. This set is further
divided into a training and a verification set. All training and verification data is
preprocessed and hand segmented, via a graphical interface, into letter samples.
These are then sorted according to size and assembled into distinct training and
verification sets. It is often the case that the same letter will appear in multiple
size files due to variability in writing and different contexts (such as when an 0 is
followed by a 9 it is at least a 3 stroke allograph, while an 0 followed by an 1 is
usually only a two stroke allograph). Included in these letter samples are samples
of a new letter class "other," illustrated in Figure 3. Experiments demonstrated
that use of an "other" class tightens decision boundaries and thus prevents spurious
fragments-of which there are many during performance-from being recognized as
real letters. Each network is trained using back-propagation until correctness on
the verification set is maximized, usually requiring less than 100 epochs.
2.3
Word Interpreter
To identify the correct word, the word interpreter explores the space of all possible
segmentations of the input stroke sequence. First, the input sequence is partitioned
into all possible fragments of size one through six, then the appropriately sized
network is used to classify each fragment. An example of this process is illustrated
as a matrix in Figure 4(a).
The word interpreter then performs a search of this matrix to identify candidate
words. Figure 4(b) and Figure 4(c) illustrates two sets of candidate words found
for the example in Figure 4(a). Candidates in this search process are generated
according to the following constraints:
? A legal segmentation point of the input stream is one where no two adjacent fragments overlap or leave a gap. To impose this constraint the i'th
fragment of size j may be extended by all of the i + j fragments, if they
exist.
? A legal candidate letter sequence must be a subsequence of a word in the
provided lexicon of expected English words.
781
782
Flann
DktioJliU)' Siu-107.a!:l
UiL-tiollary Siz .. - (J
1?AAE
1)ARE
2)ARE
2)ARf
3)ARf
&)QAf
S)ORf
Figure 4: (a) The matrix of fragments and their classifications that is generated by
applying the letter recognizers to a sample of the word are. The original handwriting
sample, following preprocessing, is given at the top of the matrix. The bottom row
of the matrix corresponds to all fragments of size one (with zero overlap), the second
row to all fragments of size two (with an overlap of one stroke) etc. The column
of letters in each fragment box represents the letter classifications generated by
the neural network of appropriate size. The higher the letter in the column, the
more confident the classification. Those fragments with no high scoring letter were
recognized as examples of the class "other." (b) The first five candidates found by
the word recognizer employing no lexicon. The first column is the word recognized,
the second column is the score for that word, the third is the sequence of fragments
and their classifications. (c) The first five candidates found by the word recognizer
employing a lexicon of 10748 words.
Recognition-Based Segmentation of On-Line Cursive Handwriting
In a forward search, a candidate of size n consists of: (a) a legal sequence of fragments It, 12, .. . , In that form a prefix of the input stroke sequence, (b) a sequence
of letters It, 12 , ? ?? , In that form a prefix of an English word from the given dictionary and (c) a score s for this candidate, defined as the average letter recognition
error:
E?-l 6(1., Ii)
8
==---:.,;...;.,,;.~
=
n
where 6(/i, Ii) is the hamming distance between letter Is's code and the actual code
produced by the neural network when given Ii as input. This scoring function is
the same as employed in [Edelman et. ai, 90].
The best word candidate is one that conforms with the constraints and has the
lowest score. Although this is a reasonable scoring function, it is easy to show
that it is not admissible when used as an evaluation function in forward search.
With a forward search, problems arise when the prefix of the correct word is poorly
recognized. To help combat this problem without greatly increasing the size of the
search space, both forward and backward search is performed.
Search is initiated by first generating all one letter and one fragment prefix and suffix
candidates. Then at each step in the search, the candidate with the lowest score is
expanded by considering the cross product of all legal letter extensions (according to
the lexicon) with all legal fragment extensions (according to the fragment-sequence
constraints) . The list of candidates is maintained as a heap for efficiency. The search
process terminates when the best candidate satisfies: (1) the letter sequence is a
complete word in the lexicon and (2) the fragment sequence uses all the available
input strokes.
The result of this bi-directional search process is illustrated in Figure 4(a)(b), where
the five best candidates found are given for no lexicon and a large lexicon. The use
of a 10,748 word lexicon eliminates meaningless fragment sequences, such as cvre,
which is a reasonable segmentation, but not in the lexicon. The first two candidates
are the same fragment sequence, found by the two search directions. The third
candidate with a 10,748 word dictionary illustrates an alternative segmentation of
the correct word. This candidate was identified by a backward search, but not a
forward search, due to the poor recognition of the first fragment.
3
Evaluation
To evaluate the system, 10 writers have provided samples of approximately 100
words picked by a random process, biased to better represent uncommon letters.
Two kinds of experiments were performed. First, to test the ability of the system to
learn a variety of writing styles, the system was tested and trained on distinct sets
of samples from the same writer. This experiment was repeated 10 times, once for
each writer. The error rate varied between 1.7% and 15.5%, with a mean of 6.2%,
when employing a database of 10,748 English words. The second experiments tested
the ability of the system to recognize handwriting of a writer not represented in the
training set. Here the set of 10 samples were split into two sets, the training set
of 9 writers with the remaining 1 writer being the test set. The error rate was
understandably higher, varying between 5.2% and 31.1%, with a mean of 10.8%,
when employing a database of 10,748 English words.
783
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4
Summary
This paper has presented a recognition-based segmentation approach for on-line
cursive handwriting. The method is very flexible because segmentation is performed
following exhaustive recognition. Hence, we expect the method to be successful
with more natural unconstrained writing, which can include mixed block, cursive
and disjoint letters, diverse orderings of delayed strokes, overwrites and erasures.
Acknowledgements
This work was supported by a Utah State University Faculty Grant. Thanks to
Balaji Allamapatti, Rebecca Rude and Prashanth G Bilagi for code development.
References
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[Morasso et. ai, 93] Morasso P., Barberis, S. Pagliano S. & Vergano, D. (1993).
Recognition experiments of cursive dynamic handwriting with selforganizing networks. Pattern Recognition, Vol. 26, No.3, pp. 451-460.
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(1993). Recognition-based segmentation of on-line hand-printed words. In
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[Schomaker, 1993] Schomaker L. (1993). Using stroke or character based selforganizing maps in the recognition of on-line connected cursive script. Pattern Recognition, Vol. 26. No.3., pp. 442-450.
[Srihari & Bozinovic, 87] Srihari S. N. & Bozinovic R. M. (1987). A multi-level
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[Weissman et. ai, 93] Weissman H., Schenkel M., Guyon I., Nohl C. & Henderson
D. (1993). Recognition-based segmentation of on-line run-on hand printed
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7,019 | 795 | Analysis of Short Term Memories for Neural
Networks
Jose C. Principe, Hui-H. Hsu and Jyh-Ming Kuo
Computational NeuroEngineering Laboratory
Department of Electrical Engineering
University of Florida, CSE 447
Gainesville, FL 32611
[email protected]
Abstract
Short term memory is indispensable for the processing of time
varying information with artificial neural networks. In this paper a
model for linear memories is presented, and ways to include
memories in connectionist topologies are discussed. A comparison
is drawn among different memory types, with indication of what is
the salient characteristic of each memory model.
1
INTRODUCTION
An adaptive system that has to interact with the external world is faced with the
problem of coping with the time varying nature of real world signals. Time varying
signals, natural or man made, carry information in their time structure. The problem
is then one of devising methods and topologies (in the case of interest here, neural
topologies) that explore information along time.This problem can be appropriately
called temporal pattern recognition, as opposed to the more traditional case of static
pattern recognition. In static pattern recognition an input is represented by a point in
a space with dimensionality given by the number of signal features, while in temporal
pattern recognition the inputs are sequence of features. These sequence of features
can also be thought as a point but in a vector space of increasing dimensionality.
Fortunately the recent history of the input signal is the one that bears more
information to the decision making, so the effective dimensionality is finite but very
large and unspecified a priori. How to find the appropriate window of input data
1011
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Principe, Hsu, and Kuo
(memory depth) for a given application is a difficult problem. Likewise, how to
combine the information in this time window to better meet the processing goal is
also nontrivial. Since we are interested in adaptive systems, the goal is to let the
system find these quantities adaptively using the output error information.
These abstract ideas can be framed more quantitatively in a geometric setting (vector
space). Assume that the input is a vector [u(l), ... u(n), .... ] of growing size. The
adaptive processor (a neural network in our case) has a fixed size to represent this
information, which we assign to its state vector [x1(n), .... xN(n)] of size N. The
usefulness of xk(n) depends on how well it spans the growing input space (defined by
the vector u(n?, and how well it spans the decision space which is normally
associated with the minimization of the mean square error (Figure 1). Therefore, in
principle, the procedure can be divided into a representational and a mapping
problem.
The most general solution to this problem is to consider a nonlinear projection
manifold which can be modified to meet both requirements. In terms of neural
topologies, this translates to a full recurrent system, where the weights are adapted
such that the error criterion is minimized. Experience has shown that this is a rather
difficult proposition. Instead, neural network researchers have worked with a wealth
of methods that in some way constrain the neural topology.
Projection space
Nonlinear mapping
error
~
Optimal
Decision space
Figure 1. Projection ofu(n) and the error for the task. (for simplicity we
are representing only linear manifolds)
The solution that we have been studying is also constrained. We consider a linear
manifold as the projection space, which we call the memory space. The projection of
u(n) in this space is subsequently mapped by means of a feedforward neural network
(multilayer perceptron) to a vector in decision space that minimizes the error
criterion. This model gives rise to the focused topologies. The advantage of this
constrained model is that it allows an analytical study of the memory structures, since
they become linear filters. It is important to stress that the choice of the projection
space is crucial for the ultimate performance of the system, because if the projected
version of u(n) in the memory space discards valuable information about u(n), then
Analysis of Short Term Memories for Neural Networks
the nonlinear mapping will always produce sub-optimal results.
2
Projection in the memory space
If the projection space is linear, then the representational problem can be studied with
linear system concepts. The projected vector u(n) becomes Yn
N
Yn =
L w0n-k
(1)
k=l
where xn are the memory traces. Notice that in this equation the coefficients wk are
independent of time, and their number fixed to N. What is the most general linear
structure that implements this projection operation? It is the generalizedfeedfonvard
structure [Principe et aI, 1992] (Figure 2), which in connectionist circles has been
called the time lagged recursive network [Back and Tsoi, 1992]. One can show that
the defining relation for generalized feedforward structures is
gk (n) = g (n) ? gk-l (n)
k';? 1
where ? represents the convolution operation, and go (n) = (5 (n) . This relation
means that the next state vector is constructed from the previous state vector by
convolution with the same function g(n), yet unspecified. Different choices of g(n)
will provide different choices for the projection space axes. When we apply the input
u(n) to this structure, the axes of the projection space become xk(n), the convolution
of u(n) with the tap signals. The projection is obtained by linearly weighting the tap
signals according to equation (1).
Figure 2. The generalizedfeedfonvard structure
We define a memory structure as a linear system whose generating kernel g(n) is
causal g (n) = 0 fo r n < 0 and normalized, i.e.
00
L Ig(n)1
= 1
n=O
We define memory depth D as the modified center of mass (first moment in time) of
the last memory tap.
00
D =
L ngk(n)
n=O
And we define the memory resolution R as the number of taps by unit time, which
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Principe, Hsu, and Kuo
becomes liD. The purpose of the memory structure is to transform the search for an
unconstrained number of coefficients (as necessary if we worked directly with u(n?
into one of seeking a fixed number of coefficients in a space with time varying axis.
3
Review of connectionist memory structures
The gamma memory [deVries and Principe, 1992] contains as special cases the
context unit [Jordan, 1986] and the tap delay line as used in TDNN [Waibel et aI,
1989]. However, the gamma memory is also a special case of the generalized
feedforward filters where g (n) = Jl (1 - Jl) n which leads to the gamma functions as
the tap signals. Figure 3, adapted from [deVries and Principe, 1993], shows the most
common connectionist memory structures and its characteristics.
As can be seen when k=l, the gamma memory defaults to the context unit, and when
Jl=1 the gamma memory becomes the tap delay line. In vector spaces the context unit
represents a line, and by changing 11 we are finding the best projection of u(n) on this
line. This representation is appropriate when one wants long memories but low
resolution.
Likewise, in the tap delay line, we are projecting u(n) in a memory space that is
uniquely determined by the input signal, i.e. once the input signal u(n) is set, the axes
become u(n-k) and the only degree of freedom is the memory order K. This memory
structure has the highest resolution but lacks versatility, since one can only improve
the input signal representation by increasing the order of the memory. In this respect,
the simple context unit is better (or any memory with a recursive parameter), since
the neural system can adapt the parameter 11 to project the input signal for better
performance.
We recently proved that the gamma memory structure in continuous time represents
a memory space that is rigid [Principe et aI, 1994] . When minimizing the output mean
square error, the distance between the input signal and the projection space
decreases. The recursive parameter in the feedforward structures changes the span of
the memory space with respect to the input signal u(n) (which can be visualized as
some type of complex rotation). In terms of time domain analysis, the recursive
parameter is finding the length of the time window (the memory depth) containing
the relevant information to decrease the output mean square error. The recursive
parameter Jl can be adapted by gradient descent learning [deVries and Principe,
1992], but the adaptation becomes nonlinear and multiple minima exists.Notice that
the memory structure is stable for O<Jl<2.
The gamma memory when utilized as a linear adaptive filter extends Widrow's
ADALINE [de Vries et aI, 1992], and results in a more parsimonious filter for echo
cancellation [Palkar and Principe, 1994]. Preliminary results with the gamma
memory in speech also showed that the performance of word spotters improve when
11 is different from one (i.e. when it is not the tap delay line). In a signal such as
speech where time warping is a problem, there is no need to use the full resolution
provided by the tap delay line. It is more important to trade depth by resolution.
Analysis of Short Term Memories for Neural Networks
4
Other Memory Structures
There are other memory structures that fit our definition. Back and Tsoi proposed a
lattice structure that fits our definition of generalized feedforward structure.
Essentially this system orthogonalizes the input, uncorrelating the axis of the vector
space (or the signals at the taps of the memory). This method is known to provide the
best speed of adaptation because gradient descent becomes Newton's method (after
the lattice parameters converge). The problem is that it becomes more computational
demanding (more parameters to adapt, and more calculations to perform).
Tape delay line
u(tJ
-0
Delay operator: Z-l
Memory resolution: 1.
memory depth: K
Context Unit
$
z nnmnin
yet)
1--1--+--.
Delay operator:
Memory depth: 1/J,l
1-
Memory resolution: J,l
z-(1-J,l)
Gamma memory
G(z)
Delay operator:
z-
(1- J,l)
Memory depth: klJ,l
Memory resolution: J,l
Figure 3. Connectionist memory structures
Laguerre memories
A set of basis intimately related to the gamma functions is the Laguerre bases. The
1015
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Principe, Hsu, and Kuo
Laguerre bases is an orthogonal span of the gamma space [Silva, 1994], which means
that the information provided by both memories is the same. The advantage of the
Laguerre is that the signals at the taps (the basis) are less correlated and so the
adaptation speed becomes faster for values of Jl close to 0 or 2 [Silva, 1994] (the
condition number of the matrix created by the tap signals is bounded). Notice that the
Laguerre memory is still very easy to compute (a lowpass filter followed by a cascade
of first order all pass filters).
aguerre memory
z domain
Delay operator:
Z-
z-
(1 - Jl)
z-
(1 - Jl)
-1
(1- Jl)
Gamma II memories.
The Gamma memory has a multiple pole that can be adaptively moved along the real
Z domain axis, i.e. the Gamma memory can only implement lowpass (0< Jl <1) or
highpass (1 <Jl <2) transfer functions. We experimentally observed that in nonlinear
prediction of chaotic time series the recursive parameter sometimes adapts to values
less than one. The highpass creates an extra ability to match the prediction by
alternating the signs of the samples in the gamma memory (the impulse response for
1< Jl <2 is alternating in sign). But with a single real parameter the adaptation is
unable to move the poles to complex values. Two conditions come to mind that
require a memory structure with complex poles. First, the information relevant for
the signal processing task appears in periodic bursts, and second, the input signal is
corrupted by periodic noise. A memory structure with adaptive complex poles can
successfully cope with these two conditions by selecting in time the intervals where
the information is concentrated (or the windows that do not provide any information
for the task). Figure 3 shows one possible implementation for the Gamma II kernel.
Notice that for stability, the parameter u must obey the condition Jl (1 +~) < 2 and
o <Jl <2. Complex poles are obtained for u> O. These parameters can be adapted by
gradient descent [Silva et aI, 1992]. In terms of versatility, the Gamma II has a pair
of free complex poles, the Gamma I has a pole restricted to the real line in the Z
domain, and the tap delay line has the pole set at the origin of the Z domain (z=O). A
multilayer perceptron equipped with an input memory layer with the Gamma II
memory structure implements a nonlinear mapping on an ARMA model of the input
signal.
5
How to use Memory structures in Connectionist networks.
Although we have presented this theory with the focused architectures (which
Analysis of Short Term Memories for Neural Networks
corresponds to a nonlinear moving average model (NMAX?, the memory structures
can be placed anywhere in the neural topology. Any nonlinear processing element can
feed one of these memory kernels as an extension of [Wan, 1990]. If the memory
structures are used to store traces of the output of the net, we obtain a nonlinear
autoregressive model (NARX). If they are used both at the input and output, they
represent a nonlinear ARMAX model shown very powerful for system identification
tasks. When the memory layer is placed in the hidden layers, there is no
corresponding linear model.
Gamma II
Delay operator: _Jl_[z_-_<l_-_Jl)_]_
[z - (l - Jl)] 2+ ~Jl2
One must realize that these types of memory structures are recursive (except the tap
delay line), so their training will involve gradients that depend on time. In the focused
topologies the network weights can still be trained with static backpropagation, but
the recursive parameter must be trained with real time recurrent learning (RTRL) or
backpropagation through time (BPTT). When memory structures are scattered
through out the topology, training can be easily accomplished with backpropagation
through time, provided a systematic way is utilized to decompose the global
dynamics in local dynamics as suggested in [Lefebvre and Principe, 1993].
6
Conclusions
The goal of this paper is to present a set of memory structures and show their
relationship. The newly introduced Gamma II is the most general of the memories
reviewed. By adaptively changing the two parameters u,Jl the memory can create
complex poles at any location in the unit circle. This is probably the most general
memory mechanism that needs to be considered. With it one can model poles and
zeros of the system that created the signal (if it accepts the linear model).
In this paper we addressed the general problem of extracting patterns in time. We
have been studying this problem by pre-wiring the additive neural model, and
decomposing it in a linear part -the memory space- that is dedicated to the storage of
past values of the input (output or internal states), and in a nonlinear part which is
static. The memory space accepts local recursion, which creates a powerful
representational structure and where stability can be easily enforced (test in a single
parameter). Recursive memories have the tremendous advantage of being able to
trade memory depth by resolution. In vector spaces this means changing the relative
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Principe, Hsu, and Kuo
position between the projection space and the input signal. However, the problem of
finding the best resolution is still open (this means adaptively finding k, the memory
order). Likewise ways to adaptively find the optimal value of the memory depth need
improvements since the gradient procedures used up to now may be trapped in local
minima. It is still necessary to modify the definition of memory depth such that it
applies to both of these new memory structures. The method is to define it as the
center of mass of the envelope of the last kernel.
Acknowledgments:This work was partially supported by NSF grant ECS #920878.
7
Iteferences
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response neural networks," Connectionist Models, Proc. of the 1990 Summer School,
pp.131-137, 1990.
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7,020 | 796 | Robust Parameter Estimation And
Model Selection For Neural Network
Regression
Yong Liu
Department of Physics
Institute for Brain and Neural Systems
Box 1843, Brown University
Providence, RI 02912
yong~cns.brown.edu
Abstract
In this paper, it is shown that the conventional back-propagation
(BPP) algorithm for neural network regression is robust to leverages (data with :n corrupted), but not to outliers (data with y
corrupted). A robust model is to model the error as a mixture of
normal distribution. The influence function for this mixture model
is calculated and the condition for the model to be robust to outliers
is given. EM algorithm [5] is used to estimate the parameter. The
usefulness of model selection criteria is also discussed. Illustrative
simulations are performed.
1
Introduction
In neural network research, the back-propagation (BPP) algorithm is the most
popular algorithm. In the regression problem y = 7](:n, w) + ?, in which 7](:n, 8)
denote a neural network with weight 8, the algorithm is equivalent to modeling
the error as identically independently normally distributed (i.i.d.), and using the
maximum likelihood method to estimate the parameter [13]. Howerer, the training
data set may contain surprising data points either due to errors in y space (outliers)
when the response vectors ys of these data points are far away from the underlying
function surface, or due to errors in :n space (leverages), when the the feature vectors
192
Robust Parameter Estimation and Model Selection for Neural Network Regression
xs of these data points are far away from the mass of the feature vectors of the rest
of the data points. These abnormal data points may be able to cause the parameter
estimation biased towards them. A robust algorithm or robust model is the one
that overcome the influence of the abnormal data points.
A lot of work has been done in linear robust regression [8, 6, 3]. In neural network.
it is generally believed that the role of sigmoidal function of the basic computing
unit in the neural net has some significance in the robustness of the neural net
to outliers and leverages. In this article, we investigate this more thoroughly. It
turns out the conventional normal model (BPP algorithm) is robust to leverages
due to sigmoidal property of the neurons, but not to outliers (section 2). From the
Bayesian point of view [2], modeling the error as a mixture of normal distributions
with different variances, with flat prior distribution on the variances, is more robust.
The influence function for this mixture model is calculated and condition for the
model to be robust to outliers is given (section 3.1). An efficient algorithm for
parameter estimation in this situation is the EM algorithm [5] (section 3.2). In
section 3.3, we discuss a choice of prior and its properties. In order to choose
among different probability models or different forms of priors, and neural nets
with different architecture, we discuss the model selection criteria in section 4.
Illustrative simulations on the choice of prior, or the t distribution model, and the
normal distribution model are given. Model selection statistics, is used to choose
the degree of freedom oft distribution, different neural network, and choose between
a t model and a normal model (section 4 and 5).
2
Issue Of Robustness In Normal Model For Neural Net
Regression
One way to think of the outliers and leverages is to regard them as a data perturbation on the data distribution of the good data. Remember that a estimated
parameter T = T(F) is an implicit function of the underlying data distribution F.
To evaluate the influence of T by this distribution perturbation, we use the influence
function [6] of estimator T at point z = (x, y) with data distribution F, which is
defined as
T((1 - t)F + t~z) - T(F)
IF(T, z, F ) --1?Imt -+ 0+ ----.:'-'------'------'-----'----'(1 )
t
in which ~:r: has mass 1 at
This definition is equivalent to a definition of derivative with respect to F except what we are dealing now is the derivative of functional.
This definition gives the amount of change in the estimator T with respect to a distribution perturbation t~z at point z = (x, y). For a robust estimation of the
parameter, we expect the estimated parameter does not change significantly with
respect to a data perturbation. In another word, the influence function is bounded
for a robust estimation.
x. 1
Denote the conditional probability model of y given x as i.i.d. f(ylx,8) with parameter 8. If the error function is the negative log-likelihood plus or not plus a penalty
term, then a general property of the influence function of the estimated parameter
B, is IF(B, (Xi, Yi), F) ex \71l1ogf(ydxi, B) (for proof, see [11]). Denote the neural
lThe probability density of the distribution D.", is 6(y - 2:).
193
194
Liu
net, with h hidden units and the dimension of the output being one (d y = 1), as
h
17(:z:,8) =
L akO"( Wk:Z: + tk)
(2)
k=l
in which O"(:z:) is the sigmoidal function or 1/(1 + exp(:z:)) and 8 = {ak, Wk, td.
For a normal model, f(yl:z:, 8, 0") = JV(Yj 17(:Z:, 8), 0") in which .N'(y; c, 0") denotes dy variate normal distribution with mean c and covariance matrix 0"2 I. Straightforward
calculation yield (d y = 1)
IF(8, (:Z:i' Yi), F)
<X
(O"(Wi:z:+ti))hXl
)
(
(y - 17(X, 8))
(O,:O"',(w:x + t~)x )
aiO" (WiX + td
hx 1
(3)
Since y with a large value makes the influence function unbounded, thus the normal
model or the back-propagation algorithm for regression is not robust to outliers.
Since 0"' (wx +t) tends to be zero for x that is far away from the projection wx +i =
0, the influence function is bounded for a abnormal x, or the normal model for
regression is robust to leverages. This analysis can be easily extented to a neural
net with multiple hidden layers and multiple outputs. Since the neural net model
is robust to leverages, we shall concentrate on the discussion of robustness with
respect to outliers afterwards.
3
3.1
Robust Probability Model And Parameter Estimation
Mixture Model
One method for the robust estimation is by the Bayesian analysis [2J. Since our
goal is to overcome the influence of outliers in the data set, we model the error as
a mixture of normal distributions, or,
f(yl:z:,8,0")
=
J
f(ylx,8,q,0")7r(q)dq
(4)
with f(ylx, 8, q, 0") = N'(y; 17(x, 8), 0"2 /q) and the prior distribution on q is denoted
as 7r(q). Intuitively, a mixture of different normal distributions with different qs, or
different variances, somehow conveys the idea that a data point is generated from
a normal distribution with large variance, which can be considered to be outliers,
or from that with small variance, which can be considered to be good data. This
requires 7r(q) to be flat to accommodate the abnormal data points. A case of
extreme non-flat prior is to choose 7r(q) = 6(q - 1), which will make f(ylx, 8, 0") to
be a normal distribution model. This model has been discussed in previous section
and it is not robust to outliers.
Calculation yields (d y = 1) the influence function as
~
IF(8, (x, y), F)
(
<X
(y - 17(x, 8)) w
(O"(ti\X+ti))hXl
)
( a:a',( wix + t~)x )
ai a (Wi X + td
hx 1
(5)
Robust Parameter Estimation and Model Selection for Neural Network Regression
in which
(6)
where expectation is taken with respect to the posterior distribution of q, or
7r(qly , x , 0- , 8) = f(ylx,9,q,u)'1r(q)
For the influence function to be bounded for a y
f(ylx,9,u)
with large value, (y - 7](x, 8))w must be bounded. This is the condition on 7r(q)
when the distribution f(ylx, 8, 0-) is robust to outliers. It can be noticed that the
mixture model is robust to leverages for the same reason as in the case of the normal
distribution model.
3.2
Algorithm For Parameter Estimation
An efficient parameter estimation method for the model in equation 4 is the EM
algorithm [5]. In EM algorithm, a portion ofthe parameter is regarded as the missing observations. During the estimation, these missing observations are estimated
through previous estimated parameter of the model. Afterwards, these estimated
missing observations are combined with the real observations to estimate the parameter of the model. In our mixture model, we shall regard {qi, i = 1, ... n} as the
missing observations. Denote w = {Xi, Yi, i = 1, ... n} as the training data set.
It is a straight forward calculation for the EM algorithm (see Liu, 1993b) once one
w~it~ ~o~n
the full probability f( {Yi, qdl{xd, 0-, 8). The algorithm is equivalent to
mmimIzmg
n
L w~S-l)(Yi -
(7)
7](Xi' 8))2
i=l
and estimating
0-
at the s step by (0-2)(S)
= ~ l:~l W~!-l)(Yi -
7](Xi' 8(5?))2.
=
If we use f(ylx, 8, cr) oc exp( -p(IY-7]( x, 8) 110-)) and denote 1/J(z)
p' (z), calculation
yield, w
E [qly, x, 0-, 8]
""~z) Iz=IY-71(X,9)I/u' This has exact the same choice of
=
wr
=
weight S - 1 ) as in the iterative reweighted regression algorithm [7]. What we have
here, different from the work of Holland et al., is that we link the EM algorithm
with the iterative reweighted algorithm, and also extend the algorithm to a much
more general situation. The weighting Wi provides a measure of the goodness of a
data point. Equation 7 estimates the parameters based on the portion of the data
that are good. A penalization term on 8 can also be included in equation 7. 2
3.3
Choice Of Prior
There are a lot choices of prior distribution 7r(q) (for discussion, see [11]). We
only discuss the choice IIq '" X~, i.e., a chi distribution with II degree of freedom.
By intergrating equation 4 f(Ylx 8 0-) = r'( v +dl/)/2) (1 + (Y-71(x,9?2)-(Il+dl/)/2.
,
"
r(V/2)(q2V'1r)ctl//2
vu 2
It is a dy variate t distribution with II degree of freedom, mean 0 and covariance
matrix cr 2 I. Calculation yields, E [q ly, x, 0-, 8]
= v + (Y-~tx~9?)27u2
The t distribution
prior on 8 can be 1r(8) ex: e- a (A ,9)/(2cr 2 ), which yields a additional penalization term
0:( A, 8) in equation 7, in which A denotes a tunning parameters of the penalization.
2A
195
196
Liu
becomes a normal distribution as 1.1 goes to infinity. For finite 1.1, it has heavier tail
than the normal distribution and thus is appropriate for regression with to outliers.
Actually the condition for robustness, (y - 1J(x,8))w being bounded for a y with
large value, is satisfied. The weighting w ex: 1/{1 + [Y -1J(x,8)f /a 2 } balances the
influence of the ys with large values, achieving robustness with respect to outliers
for the t distribution.
4
Model Selection Criteria
The meaning of model is in a broad sense. It can be the degree of penalization, or
a probability model, or a neural net architecture, or the combination of the above.
A lot of work has been done in model selection [1, 17, 15, 4, 13, 14, 10, 12] . The
choice of a model is based on its prediction ability. A natural choice is the expected
negative log-likelihood. This is equivalent to using the Kullback-Leibler measure [9]
for model selection, or -E [logf(ylx, model)] + E [log f(ylx, true model)]. This has
problem if the model can not be normalized as in the case of a improper prior.
Equation 7 implies that we can use
Tm(w) = -
1
n
,,"",*
Lt Wi (Yi - 1J(Xi' (Ld)
A
2
(8)
neff i=l
as the cross-validation [16] assessment of model m, in which neff = Ei wi, wi is
the convergence limit of w~s), or equation 6, and 8_ i is the estimator of 8 with
ith data deleted from the full data set. The successfulness of the cross-validation
method depends on a robust parameter estimation. The cross-validation method is
to calculate the average prediction error on a data based on the rest of the data in
the training data set. In the presence of outliers, predicting an outlier based on the
rest of the data, is simply not meaningful in the evaluation of the model. Equation
8 takes consideration of the outliers. Using result from [10], we can show [11] with
penalization term 0:(>',8),
Tm(w)
1 ~ *
-L
t w i (Yi-1J(Xi,8)) 2
~
(9)
A
+
neff
i=1
~
t
eff
wirigJ
i=1
[2:
wi (gigJ - ri(i) + 'VI) 'VI)O:(>',
8)]-1 rigi
(10)
i
in which gi = 'V1)1J(xi,8), (i = 'V1)'V~1J(xi,8) and ri = Yi -1J(xi,8). Thus if the
models in comparison contains a improper prior, the above model selection statistics
can be used.
If the models in comparison have close forms of f(ylx, 8, u), the average negative
log-likelihood can be used as the model selection criteria. In Liu's work [10], an
approximation form for the unbiased estimation of expected negative log-likelihood
was provided. If we use the negative log-likelihood plus a penalty term 0:(>.,8) as
the parameter optimization criteria, the model selection statistics is
1~
A
Tm(w) = - - Ltlogf(Yilxi,8_i)
n .
~=1
~
1~
1
-- Ltlogf(Yilxi,8) + -Tr(C
n
i=1
A
n
-1
D)
(11)
Robust Parameter Estimation and Model Selection for Neural Network Regression
which
C
E~=l V' (1 log f(Yi lXi, B)V'~ log f(Yi lXi, B)
and D = - E~=l V'(1V'pogf(Yilxi,B) + V'(1\7~a().,8). The optimal model is the
one that minimizes this statistics. If the true underlying distribution is the normal
distribution model and there is no penalization terms, it is easy to prove C -+ D as
n goes to infinite. Then the statistics becomes AIC [1].
10
o
S
o
-1.5
~--
o
o
o
00
o
__~______~~__~______~____~______~____~
o
1
2
3
4
5
6
7
Figure 1: BPP fit to data set with leverages, and comparison with BPP fit to the
data set without the leverages. An one hidden layer neural net with 4 hidden units,
is fitted to a data set with 10 leverage, which are on the right side of X = 3.5, by
using the conventional BPP method. The main body of the data (90 data points)
was generated from Y = sin(x) + ? with ?
.V(?j 0, a = 0.2). It can be noticed
that the fit on the part of good data points was not dramatically influenced by the
leverages. This verified our theoretical result about the robustness of a neural net
with respect to leverages
'"V
5
Illustrative Simulations
For the results shown in figure 2 and 3, the training data set contains 93 data point
from Y == sin( x) + ? and seven Y values (outliers) randomly generated from region [1,
2), in which ? '" .:\:'( ?j 0, a = 0.2). The neural net we use is of the form in equation
2. Denote h as the number of hidden units in the neural net. The caption of each
figure (1, 2, 3) explains the usefulness of the parameter estimation algorithm and
the model selection.
Acknowledgements
The author thanks Leon N Cooper, M. P. Perrone. The author also thanks his wife
Congo This research was supported by grants from NSF, ONR and ARO.
References
[1] H. Akaike. Information theory and an extension of the maximum likelihood
197
198
Liu
1.1
1
.--.--~--~~--~--~--~~--~--~--~~--~~
0.9
0.8
0.7
0.6
0.5
Tm statistics
MSE on the test set (x 10- 1 )
OA'=-----'-__-'--__..L----l_ _---'-_ _---L--_L-----..l_ _---L-_ _..l.--_L-..---'-_ _-'---..d
(3,3)(2,3)( 4,3)(2,4)(3,4)(5,3)(3,5)( 1,3)(3,7)( n,3X n,4X n,5X n, 7)
models (/.I, h), n stands for normal distribution model (BPP fit)
Figure 2: Model selection statistics Tm for fits to data set with outliers, tests on
a independent data set with 1000 data points from y = sin(:z:) + ?, where ? '"
JV(f.; 0, U 0.2). it can be seen that Tm statistics is in consistent with the error on
the test data set. The Tm statistics favors t model with small /.I than for the normal
distribution models.
=
2
0
0
0
1.5
00
0
0
1
0.5
Y
0
0
t3 model with outliers
BPP fit with outliers
PP fit without outliers
-0.5
-1
-1.5
0
1
2
4
3
5
6
7
:z:
Figure 3: Fits to data set with outliers, and comparison with BPP fit to the data
set without the outliers. The best fit in the four BPP fits (h = 3), according to Tm
statistics, was influenced by the outliers, tending to shift upwards. Although the
distribution is not a t distribution at all, the best fit by the EM algorithm under
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the BPP fit, actually is almost the same as the BPP fit (h = 3) to the training data
set without the outliers. This is due to the fact that a t distribution has a heavy
tail to accommodate the outliers
Robust Parameter Estimation and Model Selection for Neural Network Regression
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and Akaike's criterion. J. Roy. Stat. Soc., Ser. B, 39(1):44-47, 1977.
199
| 796 |@word nd:1 simulation:3 covariance:2 qly:2 tr:1 accommodate:2 ld:1 liu:9 contains:2 surprising:1 must:1 wx:2 cook:1 ith:1 stahel:1 characterization:1 detecting:1 math:1 provides:1 sigmoidal:3 unbounded:1 symposium:1 prove:1 huber:1 expected:2 weisberg:1 brain:1 chi:1 td:3 becomes:2 provided:1 estimating:3 underlying:3 bounded:5 mass:2 what:2 minimizes:1 remember:1 ti:3 xd:1 ser:3 schwartz:1 normally:1 unit:4 ly:1 grant:1 tends:1 limit:1 ak:1 plus:3 equivalence:1 yj:1 vu:1 empirical:1 significantly:1 projection:1 word:1 close:1 selection:19 influence:15 equivalent:4 conventional:3 missing:4 straightforward:1 go:2 independently:1 estimator:4 q:1 regarded:1 tunning:1 his:1 exact:1 caption:1 akaike:2 roy:3 role:1 calculate:1 region:1 improper:2 dempster:1 easily:1 tx:1 effective:1 ability:1 statistic:13 gi:1 favor:1 think:1 noisy:1 laird:1 net:12 aro:1 convergence:1 tk:1 stat:6 soc:3 implies:1 concentrate:1 correct:1 eff:1 explains:1 hx:2 generalization:2 extension:1 considered:2 normal:20 exp:2 yilxi:3 estimation:18 ctl:1 cr:3 publication:2 likelihood:8 sense:1 hidden:5 issue:1 among:1 denoted:1 smoothing:2 mackay:1 once:1 validatory:1 broad:1 spline:1 randomly:1 cns:1 technometrics:1 freedom:3 investigate:1 evaluation:1 numer:1 mixture:9 extreme:1 incomplete:1 theoretical:1 fitted:1 modeling:2 giles:1 aio:1 goodness:1 usefulness:2 wix:2 providence:1 corrupted:2 combined:1 thoroughly:1 thanks:2 density:1 international:1 physic:1 yl:2 iy:2 moody:2 thesis:1 satisfied:1 choose:4 derivative:2 imt:1 wk:2 depends:1 vi:2 performed:1 view:1 lot:3 portion:2 il:1 square:1 variance:5 kaufmann:2 yield:5 ofthe:1 t3:1 hxl:2 bayesian:4 straight:1 submitted:2 influenced:2 definition:3 petrov:1 pp:1 conveys:1 proof:1 popular:1 bpp:12 actually:2 back:3 response:1 sufficiency:1 done:2 box:1 implicit:1 ei:1 nonlinear:1 assessment:2 propagation:3 somehow:1 brown:2 contain:1 true:2 normalized:1 unbiased:2 regularization:1 leibler:2 iteratively:1 reweighted:3 sin:3 during:1 illustrative:3 oc:1 criterion:7 generalized:1 stone:2 upwards:1 meaning:1 consideration:1 neff:3 tending:1 functional:1 discussed:2 extend:1 tail:2 ai:1 surface:1 posterior:1 commun:1 onr:1 yi:11 seen:1 morgan:2 additional:1 ii:2 afterwards:2 multiple:2 full:2 calculation:5 believed:1 cross:7 y:2 qi:1 prediction:3 regression:16 basic:1 expectation:1 biased:1 rest:3 cowan:1 logf:1 leverage:13 presence:1 identically:1 easy:1 variate:2 fit:14 architecture:2 wahba:1 idea:1 tm:9 shift:1 heavier:1 penalty:2 york:1 cause:1 dramatically:1 generally:1 ylx:12 amount:1 rousseeuw:1 nsf:1 estimated:6 wr:1 shall:2 iz:1 four:1 achieving:1 deleted:1 jv:2 verified:1 v1:2 wife:1 almost:1 decision:1 dy:2 ydxi:1 abnormal:4 layer:2 aic:1 infinity:1 ri:3 flat:3 yong:2 leon:1 jackknife:1 department:1 influential:1 according:2 combination:1 perrone:1 craven:1 em:8 wi:7 outlier:28 intuitively:1 taken:1 equation:9 turn:1 discus:3 iiq:1 away:3 appropriate:1 robustness:6 lxi:2 denotes:2 noticed:2 link:1 oa:1 seven:1 lthe:1 reason:1 berger:1 balance:1 negative:5 neuron:1 observation:5 finite:1 situation:2 perturbation:4 hanson:2 california:1 able:1 oft:1 natural:1 hampel:1 predicting:1 technology:1 prior:11 acknowledgement:1 asymptotic:2 expect:1 penalization:6 validation:6 degree:5 consistent:1 article:1 dq:1 principle:1 editor:3 rubin:1 heavy:1 supported:1 l_:2 side:1 institute:2 distributed:1 regard:2 overcome:2 calculated:2 dimension:2 stand:1 forward:1 author:2 adaptive:1 far:3 lippmann:1 kullback:2 dealing:1 xi:9 iterative:2 robust:29 mse:1 significance:1 main:1 body:1 cooper:1 wiley:2 weighting:2 x:1 dl:2 phd:1 lt:1 simply:1 welsch:1 u2:1 holland:2 conditional:1 goal:1 ann:2 towards:1 change:2 springerverlag:1 included:1 infinite:1 except:1 meaningful:1 evaluate:1 ex:3 |
7,021 | 797 | Learning in Computer Vision and Image
Understanding
Hayit Greenspan
Department of Electrical Engineering
California Institute of Technology, 116-81
Pasadena, CA 91125
There is an increasing interest in the area of Learning in Computer Vision and
Image Understanding, both from researchers in the learning community and from
researchers involved with the computer vision world. The field is characterized by
a shift away from the classical, purely model-based, computer vision techniques,
towards data-driven learning paradigms for solving real-world vision problems.
Using learning in segmentation or recognition tasks has several advantages over
classical model-based techniques. These include adaptivity to noise and changing
environments, as well as in many cases, a simplified system generation procedure.
Yet, learning from examples introduces a new challenge - getting a representative
data set of examples from which to learn. Applications of learning systems to practical problems have shown that the performance of the system is often critically
dependent on both the size and quality of the training set. Federico Girosi of
MIT suggested the use of prior information as a general method for synthesizing many training examples from few exemplars. Prototypical transformations are
used for general 3D object recognition. Face-recognition was presented as a particular example. Dean Pomerleau of Carnegie Mellon addressed the training
data problem as well, within the context of ALVINN, a neural network vision system which drives an autonomous van without human intervention. Some general
problems emerge, such as getting sufficient training data for the more unexpected
scenes including passing cars and intersections. Several techniques for exploiting
prior geometric knowledge during training and testing of the neural-network, were
presented. A somewhat different perspective was presented by Bartlett Mel of
Caltech. Bartlett introduced a 3D object recognition approach based on concepts
from the human visual system. Here the assumption is that a large database of examples exists, with varying viewing angles and distances, as is available to human
observers as they manipulate and inspect common objects.
A different issue of interest was using learning schemes in general recognition frameworks which can handle several different vision problems. Hayit Greenspan of
Caltech suggested combining unsupervised and supervised learning approaches
within a multiresolution image representation space, for texture and shape recognition. It was suggested that shifting the input pixel representation to a more robust
representation (using a pyramid filtering approach) in combination with learning
1182
Learning in Computer Vision and Image Understanding
schemes can combine the advantages of both approaches. Jonathan Marshall of
Univ. of North Carolina concentrated on unsupervised learning and proposed
that a common set of unsupervised learning rules might provide a basis for communication between different visual modules (such as stereopsis, motion perception,
depth and so forth).
The role of unsupervised learning in vision tasks, and its combination with supervised learning, was an issue of discussion. The question arose on how much unsupervised learning is actually unsupervised. Some a-priori knowledge, or bias, is always
present (e.g., the metric chosen for the task). Eric Saund of Xerox introduced
the window registration problem in unsupervised learning of visual features. He
argued that there is a strong dependence on the window placement as slight shifts
in the window placement can represent confounding assignments of image data to
the input units of the classifying network. Chris Williams of Toronto introduced
the use of unsupervised learning for classifying objects. Given a set of images, each
of which contains one instance of a small but unknown set of objects imaged from
a random viewpoint, unsupervised learning is used to discover the object classes.
Data is grouped into objects via a mixture model which is trained with the EM
algorithm.
Real-world computer vision applications in which learning can playa major role,
and the challenges involved, was an additional theme in the workshop. Yann Le
Cun of AT&T described a handwritten word recognizer system of multiple modules, as an example of a large scale vision system. Yann suggested that increasing
the role of learning in all modules allows one to minimize the amount of hand-built
heuristics and improves the robustness and generality of the system. Challenges
include training large learning machines which are composed of multiple, heterogeneous modules, and what the modules should contain. Padhraic Smyth of JPL
introduced the challenges for vision and learning in the context of large scientific
image databases. In this domain there is often a large amount of data which typically has no ground truth labeling. In addition, natural objects can be much more
difficult to deal with than man made objects. Learning can be valuable here, as a
low-cost solution and sometimes the only solution (with model-based schemes being
impractical). The task of face recognition was addressed by Joachim Buhmann of
Bonn. Elastic matching was introduced for translation, rotation and scale invariant
recognition. Methods to combine unsupervised and supervised data clustering with
elastic matching to learn a discriminant metric and enhance saliency of prototypes
were discussed. Related issues from a recent AAAI forum on Machine Learning in
Computer Vision, were presented by Rich Zemel of the Salk Institute.
In Conclusion
The vision world is very diverse with each different task introducing a whole spectrum of challenges and open issues. Currently, many of the approaches are very
application dependent. It is clear that much effort still needs to be put in the
definition of the underlying themes of the field as combined across the different
application domains. There was general agreement at the workshop that the issues
brought up should be pursued further and discussed at future follow-up workshops.
Special thanks to Padhraic Smyth, Tommy Poggio, and Rama Chellappa for their
contribution to the organization of the workshop.
1183
| 797 |@word concept:1 contain:1 classical:2 forum:1 question:1 open:1 imaged:1 carolina:1 deal:1 human:3 dependence:1 viewing:1 during:1 distance:1 argued:1 mel:1 contains:1 chris:1 discriminant:1 motion:1 image:7 ground:1 yet:1 common:2 rotation:1 difficult:1 major:1 shape:1 girosi:1 synthesizing:1 recognizer:1 discussed:2 he:1 slight:1 pursued:1 pomerleau:1 unknown:1 currently:1 mellon:1 inspect:1 grouped:1 mit:1 brought:1 communication:1 toronto:1 always:1 arose:1 greenspan:2 varying:1 community:1 playa:1 introduced:5 recent:1 confounding:1 perspective:1 combine:2 joachim:1 driven:1 tommy:1 california:1 dependent:2 caltech:2 suggested:4 additional:1 typically:1 somewhat:1 perception:1 pasadena:1 challenge:5 window:3 paradigm:1 increasing:2 built:1 discover:1 underlying:1 pixel:1 issue:5 multiple:2 including:1 shifting:1 what:1 priori:1 natural:1 characterized:1 buhmann:1 special:1 field:2 scheme:3 transformation:1 manipulate:1 impractical:1 technology:1 unsupervised:10 heterogeneous:1 vision:14 metric:2 future:1 represent:1 sometimes:1 unit:1 few:1 intervention:1 pyramid:1 prior:2 composed:1 addition:1 understanding:3 geometric:1 engineering:1 addressed:2 adaptivity:1 generation:1 prototypical:1 filtering:1 organization:1 interest:2 might:1 sufficient:1 introduces:1 mixture:1 viewpoint:1 classifying:2 translation:1 practical:1 testing:1 bias:1 poggio:1 prototype:1 institute:2 procedure:1 face:2 shift:2 emerge:1 area:1 van:1 bartlett:2 depth:1 matching:2 world:4 word:1 instance:1 effort:1 rich:1 made:1 marshall:1 simplified:1 passing:1 assignment:1 put:1 context:2 cost:1 introducing:1 clear:1 dean:1 amount:2 williams:1 concentrated:1 stereopsis:1 spectrum:1 combined:1 rule:1 thanks:1 learn:2 diverse:1 handle:1 carnegie:1 autonomous:1 enhance:1 ca:1 robust:1 elastic:2 alvinn:1 smyth:2 aaai:1 padhraic:2 domain:2 changing:1 agreement:1 registration:1 recognition:8 whole:1 noise:1 database:2 role:3 module:5 angle:1 electrical:1 representative:1 north:1 salk:1 yann:2 hayit:2 theme:2 saund:1 valuable:1 observer:1 environment:1 trained:1 contribution:1 solving:1 minimize:1 placement:2 purely:1 eric:1 scene:1 basis:1 saliency:1 jpl:1 exists:1 bonn:1 workshop:4 handwritten:1 critically:1 texture:1 univ:1 researcher:2 drive:1 chellappa:1 department:1 xerox:1 zemel:1 labeling:1 combination:2 intersection:1 across:1 definition:1 heuristic:1 em:1 visual:3 cun:1 unexpected:1 involved:2 federico:1 invariant:1 truth:1 advantage:2 knowledge:2 car:1 improves:1 segmentation:1 towards:1 actually:1 man:1 combining:1 available:1 supervised:3 follow:1 multiresolution:1 away:1 forth:1 generality:1 getting:2 exploiting:1 robustness:1 hand:1 clustering:1 include:2 object:9 rama:1 jonathan:1 quality:1 exemplar:1 scientific:1 strong:1 |
7,022 | 798 | Autoencoders, Minimum Description Length
and Helmholtz Free Energy
Geoffrey E. Hinton
Department of Computer Science
University of Toronto
6 King's College Road
Toronto M5S lA4, Canada
Richard S. Zemel
Computational Neuroscience Laboratory
The Salk Institute
10010 North Torrey Pines Road
La Jolla, CA 92037
Abstract
An autoencoder network uses a set of recognition weights to convert an
input vector into a code vector. It then uses a set of generative weights to
convert the code vector into an approximate reconstruction of the input
vector. We derive an objective function for training autoencoders based
on the Minimum Description Length (MDL) principle. The aim is to
minimize the information required to describe both the code vector and
the reconstruction error. We show that this information is minimized
by choosing code vectors stochastically according to a Boltzmann distribution, where the generative weights define the energy of each possible
code vector given the input vector. Unfortunately, if the code vectors
use distributed representations, it is exponentially expensive to compute
this Boltzmann distribution because it involves all possible code vectors.
We show that the recognition weights of an autoencoder can be used to
compute an approximation to the Boltzmann distribution and that this approximation gives an upper bound on the description length. Even when
this bound is poor, it can be used as a Lyapunov function for learning
both the generative and the recognition weights. We demonstrate that
this approach can be used to learn factorial codes.
1 INTRODUCTION
Many of the unsupervised learning algorithms that have been suggested for neural networks
can be seen as variations on two basic methods: Principal Components Analysis (PCA)
3
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Hinton and Zemel
and Vector Quantization (VQ) which is also called clustering or competitive learning.
Both of these algorithms can be implemented simply within the autoencoder framework
(Baldi and Hornik, 1989; Hinton, 1989) which suggests that this framework may also
include other algorithms that combine aspects of both. VQ is powerful because it uses
a very non-linear mapping from the input vector to the code but weak because the code
is a purely local representation. Conversely, PCA is weak because the mapping is linear
but powerful because the code is a distributed, factorial representation. We describe a
new objective function for training autoencoders that allows them to discover non-linear,
factorial representations.
2 THE MINIMUM DESCRIPfION LENGTH APPROACH
One method of deriving a cost function for the activities of the hidden units in an autoencoder
is to apply the Minimum Description Length (MDL) principle (Rissanen 1989). We imagine
a communication game in which a sender observes an ensemble of training vectors and must
then communicate these vectors to a receiver. For our purposes, the sender can wait until
all of the input vectors have been observed before communicating any of them - an online
method is not required. Assuming that the components of the vectors are finely quantized
we can ask how many bits must be communicated to allow the receiver to reconstruct the
input vectors perfectly. Perhaps the simplest method of communicating the vectors would
be to send each component of each vector separately. Even this simple method requires
some further specification before we can count the number of bits required. To send the
value, Xi,c, of component i of input vector c we must encode this value as a bit string. If
the sender and the receiver have already agreed on a probability distribution that assigns
a probability p( x) to each possible quantized value, x, Shannon's coding theorem implies
that x can be communicated at a cost that is bounded below by -log p( x) bits. Moreover,
by using block coding techniques we can get arbitrarily close to this bound so we shall
treat it as the true cost. For coding real values to within a quantization width of t it is
often convenient to assume a Gaussian probability distribution with mean zero and standard
deviation (1'. Provided that (1' is large compared with t, the cost of coding the value x is then
-logt + 0.5 log 21r(1'2 + x 2 /2(1'2.
This simple method of communicating the trainjng vectors is generally very wasteful. If
the components of a vector are correlated it is generally more efficient to convert the input
vector into some other representation before communicating it. The essence of the MDL
principle is that the best model of the data is the one that minimizes the total number of
bits required to communicate it, including the bits required to describe the coding scheme.
For an autoencoder it is convenient to divide the total description length into three terms.
An input vector is communicated to the receiver by sending the activities of the hidden
units and the residual differences between the true input vector and the one that can be
reconstructed from the hidden activities. There is a code cost for the hidden activities and a
reconstruction cost for the residual differences. In addition there is a one-time model cost
for communicating the weights that are required to convert the hidden activities into the
output of the net. This model cost is generally very important within the MDL framework,
but in this paper we will ignore it. In effect, we are considering the limit in which there is
so much data that this limited model cost is negligible.
PCA can be viewed as a special case of MDL in which we ignore the model cost and we limit
the code cost by only using m hidden units. The question of how many bits are required
Autoencoders, Minimum Description Length, and Helmhotz Free Energy
to code each hidden unit activity is also ignored. Thus the only remaining term is the
reconstruction cost. Assuming that the residual differences are encoded using a zero-mean
Gaussian with the same predetermined variance for each component, the reconstruction
cost is minimized by minimizing the squared differences.
Similarly, VQ is a version of MDL in which we limit the code cost to at most log m bits by
using only m winner-lake-all hidden units, we ignore the model cost, and we minimize the
reconstruction cost.
In standard VQ we assume that each input vector is converted into a specific code. Surprisingly, it is more efficient to choose the codes stochastically so that the very same input
vector is sometimes communicated using one code and sometimes using another. This type
of "stochastic VQ" is exactly equivalent to maximizing the log probability of the data under
a mixture of Gaussians model. Each code of the VQ then corresponds to the mean of a
Gaussian and the probability of picking the code is the posterior probability of the input
vector under that Gaussian. Since this derivation of the mixture of Gaussians model is
crucial to the new techniques described later, we shall describe it in some detail.
2.1 The "bits-back" argument
The description length of an input vector using a particular code is the sum of the code cost
and reconstruction cost. We define this to be the energy of the code, for reasons that will
become clear later. Given an input vector, we define the energy of a code to be the sum
of the code cost and the reconstruction cost. If the prior probability of code i is 1f'i and its
squared reconstruction error is the energy of the code is
d;
Ei
= -log 1f'i -
k log t
k
d2
+ "2 log 21f'0'2 + 20'2
(1)
where k is the dimensionality of the input vector, 0'2 is the variance of the fixed Gaussian
used for encoding the reconstruction errors and t is the quantization width.
Now consider the following situation: We have fitted a VQ to some training data and, for a
particular input vector, two of the codes are equally good in the sense that they have equal
energies. In a standard VQ we would gain no advantage from the fact that there are two
equally good codes. However, the fact that we have a choice of two codes should be worth
something. It does not matter which code we use so if we are vague about the choice of
code we should be able to save one bit when communicating the code.
To make this argument precise consider the following communication game: The sender
is already communicating a large number of random bits to the receiver, and we want to
compute the additional cost of communicating some input vectors. For each input vector
we have a number of alternative codes h1 ... hi ... h m and each code has an energy, Ei. In a
standard VQ we would pick the code, j, with the lowest energy. But suppose we pick code
i with a probability Pi that depends on Ei. Our expected cost then appears to be higher
since we sometimes pick codes that do not have the minimum value of E.
< Cost >= LPiEi
(2)
i
where < ... > is used to denote an expected value. However, the sender can use her
freedom of choice in stochastically picking codes to communicate some of the random
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Hinton and Zemel
bits that need to be communicated anyway. It is easy to see how random bits can be used
to stochastically choose a code, but it is less obvious how these bits can be recovered by
the receiver, because he is only sent the chosen code and does not know the probability
distribution from which it was picked. This distribution depends on the particular input
vector that is being communicated. To recover the random bits, the receiver waits until all
of the training vectors have been communicated losslessly and then runs exactly the same
learning algorithm as the sender used. This allows the receiver to recover the recognition
weights that are used to convert input vectors into codes, even though the only weights that
are explicitly communicated from the sender to the receiver are the generative weights that
convert codes into approximate reconstructions of the input. After learning the recognition
weights, the receiver can reconstruct the probability distribution from which each code was
stochastically picked because the input vector has already been communicated. Since he
also knows which code was chosen, he can figure out the random bits that were used to do
the picking. The expected number of random bits required to pick a code stochastically is
simply the entropy of the probability distribution over codes
H=-
LPi logpi
(3)
So, allowing for the fact that these random bits have been successfully communicated, the
true expected combined cost is
(4)
Note that F has exactly the form of Helmholtz free energy. It can be shown that the
probability distribution which minimizes F is
e- E ;
Pi
=
Lj e-Ej
(5)
This is exactly the posterior probability distribution obtained when fitting a mixture of
Gaussians to an input vector.
The idea that a stochastic choice of codes is more efficient than just choosing the code with
the smallest value of E is an example of the concept of stochastic complexity (Rissanen,
1989) and can also be derived in other ways. The concept of stochastic complexity is
unnecessarily complicated if we are only interested in fitting a mixture of Gaussians.
Instead of thinking in terms of a stochastically chosen code plus a reconstruction error,
we can simply use Shannon's coding theorem directly by assuming that we code the input
vectors using the mixture of Gaussians probability distribution. However, when we start
using more complicated coding schemes in which the input is reconstructed from the
activities of several different hidden units, the formulation in terms of F is much easier
to work with because it liberates us from the constraint that the probability distribution
over codes must be the optimal one. There is generally no efficient way of computing the
optimal distribution, but it is nevertheless possible to use F with a suboptimal distribution
as a Lyapunov function for learning (Neal and Hinton, 1993). In MDL terms we are simply
using a suboptimal coding scheme in order to make the computation tractable.
One particular class of suboptimal distributions is very attractive for computational reasons.
In a factorial distribution the probability distribution over m d alternatives factors into d
independent distributions over m alternatives. Because they can be represented compactly,
Autoencoders, Minimum Description Length, and Helmhotz Free Energy
factorial distributions can be computed conveniently by a non-stochastic feed-forward
recognition network.
3 FACTORIAL STOCHASTIC VECTOR QUANTIZATION
Instead of coding the input vector by a single, stochastically chosen hidden unit, we could
use several different pools of hidden units and stochastically pick one unit in each pool.
All of the selected units within this distributed representation are then used to reconstruct
the input. This amounts to using several different VQs which cooperate to reconstruct the
input. Each VQ can be viewed as a dimension and the chosen unit within the VQ is the
value on that dimension. The number of possible distributed codes is m d where d is the
number of VQs and m is the number of units within a VQ. The weights from the hidden
units to the output units determine what output is produced by each possible distributed
code. Once these weights are fixed, they determine the reconstruction error that would be
caused by using a particular distributed code. If the prior probabilities of each code are also
fixed, Eq. 5 defines the optimal probability distribution over distributed codes, where the
index i now ranges over the m d possible codes.
Computing the correct distribution requires an amount of work that is exponential in d, so
we restrict ourselves to the suboptimal distributions that can be factored into d independent
distributions, one for each VQ. The fact that the correct distribution is not really factorial
will not lead to major problems as it does in mean field approximations of Boltzmann
machines (Galland, 1993). It will simply lead to an overestimate of the description length
but this overestimate can still be used as a bound when learning the weights. Also the excess
bits caused by the non-independence will force the generative weights towards values that
cause the correct distribution to be approximately factorial.
3.1 Computing the Expected Reconstruction Error
To perform gradient descent in the description length given in Eq. 4, it is necessary to
compute, for each training example, the derivative of the expected reconstruction cost with
respect to the activation probability of each hidden unit. An obvious way to approximate
this derivative is to use Monte Carlo simulations in which we stochastically pick one hidden
unit in each pool. This way of computing derivatives is faithful to the underlying stochastic
model, but it is inevitably either slow or inaccurate. Fortunately, it can be replaced by a
fast exact method when the output units are linear and there is a squared error measure for
the reconstruction. Given the probability, hi, of picking hidden unit i in VQ v, we can
compute the expected reconstructed output Yj for output unit j on a given training case
(6)
where bj is the bias of unit j and wji is the generative weight from ito j in VQ v. We
can also compute the variance in the reconstructed output caused by the stochastic choices
within the VQs. Under the assumption that the stochastic choices within different VQs are
independent, the variances contributed by the different VQs can simply be added.
(7)
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Hinton and Zemel
The expected squared reconstruction error for each output unit is Vi + (Yj - dj )2 where dj is
the desired output. So if the reconstruction error is coded assuming a zero-mean Gaussian
distribution the expected reconstruction cost can be computed exactlyl. It is therefore
straightforward to compute the derivatives, with respect to any weight in the network, of all
the terms in Eq. 4.
4 AN EXAMPLE OF FACTORIAL VECTOR QUANTIZATION
Zemel (1993) presents several different data sets for which factorial vector quantization
(FVQ) produces efficient encodings. We briefly describe one of those examples. The data
set consists of 200 images of simple curves as shown in figure 1. A network containing 4
VQs, each containing 6 hidden units, is trained on this data set. After training, the final
outgoing weights for the hidden units are as shown in figure 2. Each VQ has learned
to represent the height of the spline segment that connects a pair of control points. By
chaining these four segments together the image can be reconstructed fairly accurately. For
new images generated in the same way, the description length is approximately 18 bits for
the reconstruction cost and 7 bits for the code. By contrast, a stochastic vector quantizer
with 24 hidden units in a single competing group has a reconstruction cost of 36 bits and
a code cost of 4 bits. A set of 4 separate stochastic VQs each of which is trained on a
different 8x3 vertical slice of the image also does slightly worse than the factorial VQ (by
5 bits) because it cannot smoothly blend the separate segments of the curve together. A
purely linear network with 24 hidden units that performs a version of principal components
analysis has a slightly lower reconstruction cost but a much higher code cost.
Fixed x Positions
Random
y
------->
Positions
Figure 1: Each image in the spline dataset is generated by fitting a spline to 5 control
points with randomly chosen y-positions. An image is formed by blurring the spline with
a Gaussian. The intensity of each pixel is indicated by the area of white in the display. The
resulting images are 8x12 pixels, but have only 5 underlying degrees of freedom.
1 Each VQ contributes non-Gaussian noise and the combined noise is also non-Gaussian. But
since its variance is known, the expected cost of coding the reconstruction error using a Gaussian
prior can be computed exactly. The fact that this prior is not ideal simply means that the computed
reconstruction cost is an upper bound on the cost using a better prior.
Autoencoders, Minimum Description Length, and Helmhotz Free Energy
-':' ". I 1""-:
??"" :.>
: :] .':':.
"
"~--~
..
Figure 2: The outgoing weights of the hidden units for a network containing 4 VQs with 6
units in each, trained on the spline dataset. Each 8x 12 weight block corresponds to a single
unit, and each row of these blocks corresponds to one VQ.
5 DISCUSSION
A natural approach to unsupervised learning is to use a generative model that defines a
probability distribution over observable vectors. The obvious maximum likelihood learning
procedure is then to adjust the parameters of the model so as to maximize the sum of the
log probabilities of a set of observed vectors. This approach works very well for generative
models, such as a mixture of Gaussians, in which it is tractable to compute the expectations
that are required for the application of the EM algorithm. It can also be applied to the wider
class of models in which it is tractable to compute the derivatives of the log probability of
the data with respect to each model parameter. However. for non-linear models that use
distributed codes it is usually intractable to compute these derivatives since they require that
we integrate over all of the exponentially many codes that could have been used to generate
each particular observed vector.
The MDL principle suggest a way of making learning tractable in these more complicated
generative models. The optimal way to code an observed vector is to use the correct
posterior probability distribution over codes given the current model parameters. However,
we are free to use a suboptimal probability distribution that is easier to compute. The
description length using this suboptimal method can still be used as a Lyapunov function
for learning the model parameters because it is an upper bound on the optimal description
length. The excess description length caused by using the wrong distribution has the form
of a Kullback-Liebler distance and acts as a penalty term that encourages the recognition
weights to approximate the correct distribution as well as possible.
There is an interesting relationship to statistical physics. Given an input vector, each
possible code acts like an alternative configuration of a physical system. The function
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10
Hinton and Zemel
E defined in Eq. 1 is the energy of this configuration. The function F in Eq. 4 is
the Helmholtz free energy which is minimized by the thermal equilibrium or Boltzmann
distribution. The probability assigned to each code at this minimum is exactly its posterior
probability given the parameters of the generative model. The difficulty of performing
maximum likelihood learning corresponds to the difficulty of computing properties of the
equilibrium distribution. Learning is much more tractable if we use the non-equilibrium
Helmholtz free energy as a Lyapunov function (Neal and Hinton, 1993). We can then use
the recognition weights of an autoencoder to compute some non-equilibrium distribution.
The derivatives of F encourage the recognition weights to approximate the equilibrium
distribution as well as they can, but we do not need to reach the equilibrium distribution
before adjusting the generative weights that define the energy function of the analogous
physical system.
In this paper we have shown that an autoencoder network can learn factorial codes by using
non-equilibrium Helmholtz free energy as an objective function. In related work (Zemel
and Hinton 1994) we apply the same approach to learning population codes. We anticipate
that the general approach described here will be useful for a wide variety of complicated
generative models. It may even be relevant for gradient descent learning in situations where
the model is so complicated that it is seldom feasible to consider more than one or two of
the innumerable ways in which the model could generate each observation.
Acknowledgements
This research was supported by grants from the Ontario Information Technology Research
Center, the Institute for Robotics and Intelligent Systems, and NSERC. Geoffrey Hinton
is the Noranda Fellow of the Canadian Institute for Advanced Research. We thank Peter
Dayan, Yann Le Cun, Radford Neal and Chris Williams for helpful discussions.
References
Baldi, P. and Hornik, K. (1989) Neural networks and principal components analysis: Learning from examples without local minima. Neural Networks, 2, 53-58.
Galland, C. C. (1993) The limitations of deterministic Boltzmann machine learning. Network, 4, 355-379.
Hinton, G. E. (1989) Connectionist learning procedures. Artificial Intelligence, 40, 185234.
Neal, R., and Hinton. G. E. (1993) A new view of the EM algorithm that justifies incremental
and other variants. Manuscript available/rom the authors.
Rissanen.1. ( 1989) Stochastic Complexity in Statistical Inquiry. World Scientific Publishing Co .? Singapore.
Zemel. R. S. (1993) A Minimum Description Length Framework/or Unsupervised Learning.
PhD. Thesis. Department of Computer Science, University of Toronto.
Zemel, R. S. and Hinton. G. E. (1994) Developing Population Codes by Minimizing
Description Length. In I. Cowan, G. Tesauro. and I. Alspector (Eds.), Advances in Neural
In/ormation Processing Systems 6, San Mateo, CA: Morgan Kaufmann.
| 798 |@word briefly:1 version:2 d2:1 simulation:1 pick:6 configuration:2 recovered:1 current:1 activation:1 must:4 predetermined:1 generative:12 selected:1 intelligence:1 quantizer:1 quantized:2 toronto:3 height:1 become:1 consists:1 combine:1 fitting:3 baldi:2 expected:10 alspector:1 considering:1 provided:1 discover:1 bounded:1 moreover:1 underlying:2 lowest:1 what:1 string:1 minimizes:2 fellow:1 act:2 exactly:6 wrong:1 control:2 unit:28 grant:1 overestimate:2 before:4 negligible:1 local:2 treat:1 limit:3 encoding:2 approximately:2 plus:1 mateo:1 suggests:1 conversely:1 co:1 limited:1 range:1 faithful:1 yj:2 block:3 communicated:10 x3:1 procedure:2 area:1 convenient:2 road:2 wait:2 suggest:1 get:1 cannot:1 close:1 equivalent:1 deterministic:1 logpi:1 center:1 maximizing:1 send:2 straightforward:1 williams:1 assigns:1 communicating:8 factored:1 deriving:1 population:2 anyway:1 variation:1 analogous:1 imagine:1 suppose:1 exact:1 us:3 helmholtz:5 recognition:9 expensive:1 observed:4 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determine:2 maximize:1 equally:2 coded:1 variant:1 basic:1 expectation:1 represent:1 sometimes:3 lpi:1 robotics:1 addition:1 want:1 separately:1 crucial:1 finely:1 sent:1 cowan:1 ideal:1 canadian:1 easy:1 variety:1 independence:1 perfectly:1 suboptimal:6 restrict:1 competing:1 idea:1 vqs:8 pca:3 penalty:1 peter:1 cause:1 ignored:1 useful:1 generally:4 clear:1 factorial:12 amount:2 simplest:1 generate:2 singapore:1 neuroscience:1 shall:2 group:1 four:1 rissanen:3 nevertheless:1 wasteful:1 convert:6 sum:3 run:1 powerful:2 communicate:3 yann:1 lake:1 bit:24 bound:6 hi:2 display:1 activity:7 constraint:1 aspect:1 argument:2 performing:1 x12:1 department:2 developing:1 according:1 poor:1 logt:1 slightly:2 em:2 cun:1 making:1 vq:19 count:1 know:2 tractable:5 sending:1 available:1 gaussians:6 apply:2 save:1 alternative:4 galland:2 clustering:1 include:1 remaining:1 publishing:1 objective:3 already:3 question:1 added:1 blend:1 losslessly:1 gradient:2 distance:1 separate:2 thank:1 chris:1 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simply:7 sender:7 conveniently:1 nserc:1 radford:1 corresponds:4 viewed:2 king:1 towards:1 feasible:1 principal:3 called:1 total:2 la:1 shannon:2 college:1 outgoing:2 correlated:1 |
7,023 | 799 | Catastrophic interference in
connectionist networks: Can it be
predicted, can it be prevented?
Robert M. French
Computer Science Department
Willamette University
Salem, Oregon 97301
[email protected]
1
OVERVIEW
Catastrophic forgetting occurs when connectionist networks learn new information,
and by so doing, forget all previously learned information. This workshop focused
primarily on the causes of catastrophic interference, the techniques that have been
developed to reduce it, the effect of these techniques on the networks' ability to generalize, and the degree to which prediction of catastrophic forgetting is possible. The
speakers were Robert French, Phil Hetherington (Psychology Department, McGill
University, [email protected]), and Stephan Lewandowsky (Psychology
Department, University of Oklahoma, [email protected]).
2
PROTOTYPE BIASING AND FORCED SEPARATION
OF HIDDEN-LAYER REPRESENTATIONS
French indicated that catastrophic forgetting is at its worst when high representation overlap at the hidden layer combines with significant teacher-output error.
He showed that techniques to reduce this overlap tended to decrease catastrophic
forgetting. Activation sharpening, a technique that produces representations having a few highly active nodes and many low-activation nodes, was shown to be
effective because it reduced representation overlap. However, this technique was
ineffective for large data sets because creating localized representations reduced the
number of possible hidden-layer representations. Hidden layer representations that
were more distributed but still highly separated were needed. French introduced
prototype biasing, a technique that uses a separate network to learn a prototype
for each teacher pattern. Hidden-layer representations of new items are made to
resemble their prototypes. Each representation is also "separated" from the representation of the previously encountered pattern according to the difference between
the respective teachers. This technique produced hidden-layer representations that
1176
Catastrophic Interference in Connectionist Networks
were both distributed and well separated. The result was a significant decrease in
catastrophic forgetting.
3
ELIMINATING CATASTROPHIC INTERFERENCE
BY PRETRAINING
Hetherington presented a technique that consisted of prior training of the network on
a large body of items of the same type as the new items in the sequential learning
task. Hetherington measured the degree of actual forgetting, as did all of the
authors, by the method of savings, i.e., by determining how long the network takes
to relearn the original data set that has been "erased" by learning the new data.
He showed that when networks are given the benefit of relevant prior knowledge,
the representations of the new items are constrained naturally and interference may
be virtually eliminated. The previously encoded knowledge causes new items to be
encoded in more orthogonal manner (i.e., with less mutual overlap) than in a naive
(Le., non-pretrained) network. The resulting decrease in representation overlap
produced the virtual elimination of catastrophic forgetting.
Hetherington also presented another technique that substantially reduced catastrophic interference in the sequential learning task. Learning of new items takes
place in a windowed, or overlapping fashion. In other words, as new items are
learned the network continues learning on the most recently presented items.
4
THE RELATIONSHIP BETWEEN INTERFERENCE
AND GENERALIZATION
Lewandowsky examined the hypothesis that generalization is compromised in networks that had been "manipulated" to decrease catastrophic interference by creating semi-distributed (i.e., only partially overlapping) representations at the hidden
layer. He gave a theoretical analysis of the relationship between interference and
generalization and then presented results from several different simulations using
semi-distributed representations. His conclusions were that semi-distributed representations can significantly reduce catastrophic interference in backpropagation
networks without diminishing their generalization abilities. This was only true,
however, for techniques (e.g., activation sharpening) that reduced interference by
creating a more robust final weight pattern but that did not change the activation
surfaces of the hidden units. On the other hand, in models where interference is
reduced by eliminating overlap between receptive fields of static hidden units (i.e.,
by altering their response surface), generalization abilities are impaired.
In addition, Lewandowsky presented a technique that relied on orthogonalizing the
input vectors to a standard backpropagation network by converting standard asymmetric input vectors (each node at 0 or 1) to symmetric input vectors (each input
node at -lor 1). This technique was also found to significantly reduce catastrophic
interference.
1177
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7,024 | 8 | 242
THE SIGMOID NONLINEARITY IN PREPYRIFORM CORTEX
Frank H. Eeckman
University of California, Berkeley, CA 94720
ABSlRACT
We report a study ?on the relationship between EEG amplitude values and unit
spike output in the prepyriform cortex of awake and motivated rats. This relationship
takes the form of a sigmoid curve, that describes normalized pulse-output for
normalized wave input. The curve is fitted using nonlinear regression and is
described by its slope and maximum value.
Measurements were made for both excitatory and inhibitory neurons in the cortex.
These neurons are known to form a monosynaptic negative feedback loop. Both
classes of cells can be described by the same parameters.
The sigmoid curve is asymmetric in that the region of maximal slope is displaced
toward the excitatory side. The data are compatible with Freeman's model of
prepyriform burst generation. Other analogies with existing neural nets are being
discussed, and the implications for signal processing are reviewed. In particular the
relationship of sigmoid slope to efficiency of neural computation is examined.
INTRODUCTION
The olfactory cortex of mammals generates repeated nearly sinusoidal bursts of
electrical activity (EEG) in the 30 to 60 Hz. range 1. These bursts ride on top of a
slower ( 1 to 2 Hz.), high amplitude wave related to respiration. Each burst begins
shortly after inspiration and terminates during expiration. They are generated locally
in the cortex. Similar bursts occur in the olfactory bulb (OB) and there is a high
degree of correlation between the activity in the two structures!'
The two main cell types in the olfactory cortex are the superficial pyramidal cell
(type A), an excitatory neuron receiving direct input from the OB, and the cortical
granule cell (type B), an inhibitory interneuron. These cell groups are
monosynaptically connected in a negative feedback loop2.
Superficial pyramidal cells are mutually excitatory3, 4, 5 as well as being
excitatory to the granule cells. The granule cells are inhibitory to the pyramidal cells
as well as to each other3, 4, 6.
In this paper we focus on the analysis of amplitude dependent properties: How is
the output of a cellmass (pulses) related to the synaptic potentials (ie. waves)? The
concurrent recording of multi-unit spikes and EEG allows us to study these
phenomena in the olfactory cortex.
The anatomy of the olfactory system has been extensively studied beginning with
the work of S. Ramon y Cajal 7. The regular geometry and the simple three-layered
architecture makes these structures ideally suitable for EEG recording 4, 8. The EEG
generators in the various olfactory regions have been identified and their synaptic
connectivities have been extensively studied9, 10,5,4, 11,6.
The EEG is the scalar sum of synaptic currents in the underlying cortex. It can
be recorded using low impedance ? .5 Mohm) cortical or depth electrodes. Multiunit signals are recorded in the appropriate cell layers using high impedance (> .5
Mohm) electrodes and appropriate high pass filtering.
Here we derive a function that relates waves (EEG) to pulses in the olfactory
cortex of the rat. This function has a sigmoidal shape. The derivative of this curve
? American Institute of Physics 1988
243
gives us the gain curve for wave-to-pulse conversion. This is the forward gain for
neurons embedded in the cortical cellmass. The product of the forward gain values of
both sets of neurons (excitatory and inhibitory) gives us the feedback gain values.
These ultimately determine the dynamics of the system under study.
MATERIALS AND METI-IODS
A total of twenty-nine rats were entered in this study. In each rat a linear array of
6 100 micron stainless steel electrodes was chronically implanted in the prepyriform
(olfactory) cortex. The tips of the electrodes were electrolytically sharpened to
produce a tip impedance on the order of .5 to 1 megaohm. The electrodes were
implanted laterally in the midcortex, using stereotaxic coordinates. Their position was
verified electrophysiologically using a stimulating electrode in the olfactory tract.
This procedure has been described earlier by Freeman 12. At the end of the recording
session a small iron deposit was made to help in histological verification. Every
electrode position was verified in this manner.
Each rat was recorded from over a two week period following implantation. All
animals were awake and attentive. No stimulation (electrical or olfactory) was used.
The background environment for recording was the animal's home cage placed in the
same room during all sessions.
For the present study two channels of data were recorded concurrently. Channel
1 carried the EEG signal, filtered between 10 and 300 Hz. and digitized at 1 ms
intervals. Channel 2 carried standard pulses 5 V, 1.2 ms wide, that were obtained by
passing the multi-unit signal (filtered between 300 Hz. and 3kHz.) through a
window discriminator.
These two time-series were stored on disk for off-line processing using a PerkinElmer 3220 computer. All routines were written in FORTRAN. They were tested on
data files containing standard sine-wave and pulse signals.
DATA PROCESSING
The procedures for obtaining a two-dimensional conditional pulse probability
table have been described earlier 4. This table gives us the probability of occurrence
of a spike conditional on both time and normalized EEG amplitude value.
By counting the number of pulses at a fixed time-delay, where the EEG is
maximal in amplitude, and plotting them versus the normalized EEG amplitudes, one
obtains a sigmoidal function: The Pulse probability Sigmoid Curve (PSC) 13, 14.
This function is normalized by dividing it by the average pulse level in the record. It
is smoothed by passing it through a digital 1: 1: 1 filter and fitted by nonlinear
regression.
The equations are:
Q = Qmax ( 1- exp [ - ( ev - 1) I Qmax ]) for v> - uO
Q = -1
for v < - uO
(1 )
where uO is the steady state voltage, and Q = (p-PO)/pO.
and Qmax =(Pmax-PO)/pO.
PO is the background pulse count, Pmax is the maximal pulse count.
These equations rely on one parameter only. The derivation and justification for
these equations were discussed in an earlier paper by Freeman 13.
244
RESULTS
Data were obtained from all animals. They express normalized pulse counts, a
dimensionless value as a function of normalized EEG values, expressed as a Z-score
(ie. ranging from - 3 sd. to + 3 sd., with mean of 0.0). The true mean for the EEG
after filtering is very close to 0.0 m V and the distribution of amplitude values is very
nearly Gaussian.
The recording convention was such that high EEG-values (ie. > 0.0 to + 3.0 sd.)
corresponded to surface-negative waves. These in turn occur with activity at the
apical dendrites of the cells of interest. Low EEG values (ie. from - 3.0 sd. to < 0.0)
corresponded to surface-positive voltage values, representing inhibition of the cells.
The data were smoothed and fitted with equation (1). This yielded a Qrnax value
for every data file. There were on average 5 data files per animal. Of these 5, an
average of 3.7 per animal could be fitted succesfully with our technique. In 25 % of
the traces, each representing a different electrode pair, no correlations between spikes
and the EEG were found.
Besides Qmax we also calculated Q' the maximum derivative of the PSC,
representing the maximal gain.
There were 108 traces in all. In the first 61 cases the Qrnax value described the
wave-to-pulse conversion for a class of cells whose maximum firing probability is in
phase with the EEG. These cells were labelled type A cells 2. These traces
correspond to the excitatory pyramidal cells. The mean for Qmax in that group was
14.6, with a standard deviation of 1.84. The range was 10.5 to 17.8.
In the remaining 47 traces the Qmax described the wave-to-pulse conversion for
class B cells. Class B is a label for those cells whose maximal firing probability lags
the EEG maximum by approximately 1/4 cycle. The mean for Qrnax in that group
was 14.3, with a standard deviation of 2.05. The range in this group was 11.0 to
18.8.
The overall mean for Qmax was 14.4 with a standard deviation of 1.94. There is
no difference in Qmax between both groups as measured by the Student t-test. The
nonparametric Wilcoxon rank-sum test also found no difference between the groups
( p =0.558 for the t-test; p = 0.729 for the Wilcoxon).
Assuming that the two groups have Qmax values that are normally distributed (in
group A, mean = 14.6, median = 14.6; in group B, mean = 14.3, median = 14.1),
and that they have equal variances ( st. deviation group A is 1.84; st. deviation group
B is 2.05) but different means, we estimated the power of the t-test to detect that
difference in means.
A difference of 3 points between the Qmax's of the respective groups was
considered to be physiologically significant. Given these assumptions the power of
the t-test to detect a 3 point difference was greater than .999 at the alpha .05 level for
a two sided test. We thus feel reasonably confident that there is no difference
between the Qmax values of both groups.
The first derivative of the PSC gives us the gain for wave-to-pulse conversion4.
The maximum value for this first derivative was labelled Q'. The location at which
the maximum Q' occurs was labelled Vmax . Vmax is expressed in units of standard
deviation of EEG amplitudes.
The mean for Q' in group A was 5.7, with a standard deviation of .67, in group B
it was 5.6 with standard deviation of .73. Since Q' depends on Qmax, the same
statistics apply to both: there was no significant difference between the two groups
for slope maxima.
245
Figure 1. Distribution of Qmax values
group A
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The mean for Vmax was at 2.15 sd. +/- .307. In every case Vmax was on the
excitatory side from 0.00, ie. at a positive value of EEG Z-scores. All values were
greater than 1.00. A similar phenomenon has been reported in the olfactory bulb 4,
14, 15.
Figure 2. Examples of sigmoid fits.
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B cell
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246
COMPARISON WITH DATA FROM TIIE OB
Previously we derived Qrnax values for the mitral cell population in the olfactory
bulb14. The mitral cells are the output neurons of the bulb and their axons form the
lateral olfactory tract (LOT). The LOT is the main input to the pyramidal cells (type
A) in the cortex.
For awake and motivated rats (N = 10) the mean Qmax value was 6.34 and the
standard deviation was 1.46. The range was 4.41- 9.53. For anesthetized animals
(N= 8) the mean was 2.36 and the standard deviation was 0.89. The range was 1.153.62. There was a significant difference between anesthetized and awake animals.
Furthermore there is a significant difference between the Qmax value for cortical cells
and the Qmaxvalue for bulbar cells (non - overlapping distributions).
DISCUSSION
An important characteristic of a feedback loop is its feedback gain. There is ample
evidence for the existence of feedback at all levels in the nervous system. Moreover
specific feedback loops between populations of neurons have been described and
analyzed in the olfactory bulb and the prepyriform cortex 3, 9, 4.
A monosynaptic negative feedback loop has been shown to exist in the PPC,
between the pyramidal cells and inhibitory cells, called granule cells 3, 2, 6, 16.
Time series analysis of concurrent pulse and EEG recordings agrees with this idea.
The pyramidal cells are in the forward limb of the loop: they excite the granule
cells. They are also mutually excitatory 2,4,16. The granule cells are in the feedback
limb: they inhibit the pyramidal cells. Evidence for mutual inhibition (granule to
granule) in the PPC also exists 17, 6.
The analysis of cell firings versus EEG amplitude at selected time-lags allows one
to derive a function (the PSC) that relates synaptic potentials to output in a neural
feedback system. The first derivative of this curve gives an estimate of the forward
gain at that stage of the loop. The procedure has been applied to various structures in
the olfactory system 4, 13, 15, 14. The olfactory system lends itself well to this type
of analysis due to its geometry, topology and well known anatomy.
Examination of the experimental gain curves shows that the maximal gain is
displaced to the excitatory side. This means that not only will the cells become
activated by excitatory input, but their mutual interaction strength will increase. The
result is an oscillatory burst of high frequency ( 30- 60 Hz.) activity. This is the
mechanism behind bursting in the olfactory EEG 4, 13.
In comparison with the data from the olfactory bulb one notices that there is a
significant difference in the slope and the maximum of the PSC. In cortex the values
are substantially higher, however the Vmax is similar. C. Gray 15 found a mean
value of 2.14 +/- 0.41 for V max in the olfactory bulb of the rabbit (N= 6). Our value
in the present study is 2.15 +/- .31. The difference is not statistically significant.
There are important aspects of nonlinear coupling of the sigmoid type that are of
interest in cortical functioning. A sigmoid interaction between groups of elements
("neurons") is a prominent feature in many artificial neural nets. S. Grossberg has
extensively studied the many desirable properties of sigmoids in these networks.
Sigmoids can be used to contrast-enhance certain features in the stimulus. Together
with a thresholding operation a sigmoid rule can effectively quench noise. Sigmoids
can also provide for a built in gain control mechanism 18, 19.
247
Changing sigmoid slopes have been investigated by J. Hopfield. In his network
changing the slope of the sigmoid interaction between the elements affects the
number of attractors that the system can go to 20. We have previously remarked
upon the similarities between this and the change in sigmoid slope between waking
and anesthetized animals 14. Here we present a system with a steep slope (the PPC)
in series with a system with a shallow slope (the DB).
Present investigations into similarities between the olfactory bulb and Hopfield
networks have been reported 21, 22. Similarities between the cortex and Hopfieldlike networks have also been proposed 23.
Spatial amplitude patterns of EEG that correlate with significant odors exist in the
bulb 24. A transmission of "wave-packets" from the bulb to the cortex is known to
occur 25. It has been shown through cofrequency and phase analysis that the bulb
can drive the cortex 25, 26. It thus seeems likely that spatial patterns may also exist
in the cortex. A steeper sigmoid, if the analogy with neural networks is correct,
would allow the cortex to further classify input patterns coming from the olfactory
bulb.
In this view the bulb could form an initial classifier as well as a scratch-pad
memory for olfactory events. The cortex could then be the second classifier, as well
as the more permanent memory.
These are at present speculations that may turn out to be premature. They
nevertheless are important in guiding experiments as well as in modelling.
Theoretical studies will have to inform us of the likelihood of this kind of processing.
REFERENCES
1 S.L. Bressler and W.J. Freeman, Electroencephalogr. Clin. Neurophysiol. ~: 19
(1980).
.
2 W.J. Freeman, J. Neurophysiol. ll: 1 (1968).
3 W.J. Freeman, Exptl. Neurol. .lO.: 525 (1964).
4 W.J. Freeman, Mass Action in the Nervous System. (Academic Press, N.Y.,
1975), Chapter 3.
5 L.B. Haberly and G.M. Shepherd, Neurophys.~: 789 (1973).
6 L.B. Haberly and J.M. Bower, J. Neurophysiol. ll: 90 (1984).
7 S. Ramon y Cajal, Histologie du Systeme Nerveux de l'Homme et des Vertebres.
( Ed. Maloine, Paris, 1911) .
8 W.J. Freeman, BioI. Cybernetics . .3..5.: 21 (1979).
9 W. Rall and G.M. Shepherd, J. Neurophysiol.ll: 884 (1968).
10 G.M. Shepherd, Physiol. Rev. 5l: 864 (1972).
11 L.B. Haberly and J.L. Price, J. Compo Neurol. .l18.; 711 (1978).
12 W.J. Freeman, Exptl. Neurol. ~: 70 (1962).
13 W.J. Freeman, BioI. Cybernetics.ll: 237 (1979).
14 F.H. Eeckman and W.J. Freeman, AlP Proc. ill: 135 (1986).
15 C.M. Gray, Ph.D. thesis, Baylor College of Medicine (Houston,1986)
16 L.B. Haberly, Chemical Senses, .ll!: 219 (1985).
17 M. Satou et aI., J. Neurophysiol. ~: 1157 (1982).
18 S. Grossberg, Studies in Applied Mathematics, Vol LII, 3 (MIT Press, 1973)
p 213.
19 S. Grossberg, SIAM-AMS Proc. U: 107 (1981).
20 J.J Hopfield, Proc. Natl. Acad. Sci. USA 8.1: 3088 (1984).
21 W.A. Baird, Physica 2.m: 150 (1986).
22 W.A. Baird, AlP Proceedings ill: 29 (1986).
23 M. Wilson and J. Bower, Neurosci. Abstr. 387,10 (1987).
248
24 K.A. Grajski and W.J. Freeman, AlP Proc.lS.l: 188 (1986).
25 S.L. Bressler, Brain Res. ~: 285 (1986).
26 S.L. Bressler, Brain Res.~: 294 (1986).
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7,025 | 80 | 9
Stochastic Learning Networks and their Electronic Implementation
Joshua Alspector*. Robert B. Allen. Victor Hut. and Srinagesh Satyanarayanat
Bell Communications Research. Morristown. NJ 01960
We describe a family of learning algorithms that operate on a recurrent, symmetrically
connected. neuromorphic network that. like the Boltzmann machine, settles in the
presence of noise. These networks learn by modifying synaptic connection strengths on
the basis of correlations seen locally by each synapse. We describe a version of the
supervised learning algorithm for a network with analog activation functions. We also
demonstrate unsupervised competitive learning with this approach. where weight
saturation and decay play an important role. and describe preliminary experiments in
reinforcement learning. where noise is used in the search procedure. We identify the
above described phenomena as elements that can unify learning techniques at a physical
microscopic level.
These algorithms were chosen for ease of implementation in vlsi. We have designed a
CMOS test chip in 2 micron rules that can speed up the learning about a millionfold
over an equivalent simulation on a VAX lln80. The speedup is due to parallel analog
computation for snmming and multiplying weights and activations. and the use of
physical processes for generating random noise. The components of the test chip are a
noise amplifier. a neuron amplifier. and a 300 transistor adaptive synapse. each of which
is separately testable. These components are also integrated into a 6 neuron and 15
synapse network. Finally. we point out techniques for reducing the area of the
electronic correlational synapse both in technology and design and show how the
algorithms we study can be implemented naturally in electronic systems.
1. INTRODUCTION
Ibere has been significant progress. in recent years. in modeling brain function as the collective
behavior of highly interconnected networks of simple model neurons. This paper focuses on the
issue of learning in these networks especially with regard to their implementation in an electronic
system. Learning phenomena that have been studied include associative memoryllJ. supervised
leaming by error correction(2) and by stochastic search(3). competitive learning(4 ) lS) reinforcement
leamingI6 ). and other forms of unsupervised leaming(7). From the point of view of neural
plausibility as well as electronic implementation. we particularly like learning algorithms that
change synaptic connection strengths asynchronously and are based only on information
available locally at the synapse. This is illustrated in Fig. 1. where a model synapse uses only the
correlations of the neurons it connects and perhaps some weak global evaluation signal not
specific to individual neurons to decide how to adjust its conductance.
?
t
Address for correspondence: J. Alspector, BeU Communications ReselllCh, 2E-378, 435 South St., Morristown, Nl
07960 / (201) 8294342/ [email protected]
Pennanent address: University of California, Belkeley, EE Department, Cory HaU, Belkeley, CA 94720
* PennllDeDt address: Columbia University, EE Department, S.W. Mudd Bldg., New Yolk, NY 10027
@ American Institute of Physics 1988
10
S,
I
C.=<s
'1
i
's
j
>
S,
J
<r>
Hebb-type learning rule:
If
global scalar
evaluation
signal
C ij Increases,
(perhaps in the presence of r )
Increment W ij
Fig. 1. A local correlational synapse.
We believe that a stochastic search procedure is most compatible with this viewpoint. Statistical
procedures based on noise form the communication pathways by which global optimization can
take place based only on the interaction of neurons. Search is a necessary part of any learning
procedure as the network attempts to find a connection strength matrix that solves a particular
problem. Some learning procedures attack the search directly by gradient following through error
(orrection[8J (9J but electronic implementation requires specifying which neurons are input,
tudden and output in advanC'e and nece!;sitates global control of the error correction[2J procedure
m a way that requires specific connectivity and ~ynch!'Ony at the neural Jevel. There is also the
question of how such procedures would work with unsupervised methods and whether they might
get stuck in local minima. Stochastic processes can also do gradient foUowing but they are better
at avoiding minima, are compatible with asynchronous updates and local weight adjustments,
and, as we show in this paper, can generalize well to less supervifM!d learning.
The phenomena we studied are 1) analog activation, 2) noise, 3) semi-local Hebbian synaptic
modification, and 4) weight decay and saturation. These techniques were applied to problems in
supervised, unsupervised, and reinforcement learning. The goal of the study was to see if these
diverse learning styles can be unified at the microscopic level with a small set of physically
plausible and electronically implementable phenomena. The hope is to point the way for
powerful electronic learning systems in the future by elucidating the conditions and the types of
circuits that may be necessary. It may also be true that the conditions for electronic learning may
11
have some bearing on the general principles of biologicalleaming.
2. WCAL LEAltNlNG AND STOCHASl'IC SEARCH
2.1 Supervised Learning in Recurrent Networks with Analog Activations
We have previously shown! 10] how the supervised learning procedure of the Boltzmann
machine(3) can be implemented in an electronic system. This system works on a recurrent,
symmetrically connected network which can be characterized as settling to a minimum in its
Liapunov function(l]!II). While this architecture may stretch our criterion of neural plausibility, it
does provide for stability and analyzability. The feedback connectivity provides a way for a
supervised learning procedure to propagate information back through the network as the
stochastic search proceeds. More plausible would be a randomly connected network where
symmetry is a statistical approximation and inhibition damps oscillations, but symmetry is more
efficient and weD matched to our choice of learning rule and search procedure.
We have extended our electronic model of the Boltzmann machine to include analog activations.
Fig. 2 shows the model of the neuron we used and its tanh or sigmoid transfer function. The net
input consists of the usual weighted sum of activations from other neurons but, in the case of
Boltzmann machine learning, these are added to a noise signal chosen from a variety of
distributions so that the neuron performs the physical computation:
activation =1 (neti FI (EwijSj+noise ):::tanh(gain*neti)
Instead of counting the number of on-on and off-off cooccurrences of neurons which a synapse
connects, the correlation rule now defines the value of a cooccurrence as:
Cij=/i*/i
where Ii is the activation of neuron i which is a real value from -1 to 1. Note that this rule
effectively counts both on-on and off-off cooccurrences in the high gain limit. In this limit, for
Gaussian noise, the cumulative probability distribution for the neuron to have activation +1 (on)
is close to sigmoidal. The effect of noise "jitter" is illustrated at the bottom of the figure. The
weight change rule is still:
if Cij+ > Cij- then increment Wij .... else decrement
where the plus phase clamps the output neurons in their desired states while the minus phase
allows them to run free.
As? mentioned, we have studied a variety of noise distributions other than those based on the
Boltzmann distribution. The 2-2-1 XOR problem was selected as a test case since it has been
shown! 10] to be easily caught in local minima. The gain was manipulated in conditions with no
noise or with noise sampled from one of three distributions. The Gaussian distribution is closest
to true electronic thermal noise such as used in our implementation, but we also considered a
cut-off uniform distribution and a Cauchy distribution with long noise tails for comparison. The
inset to Fig. 3 shows a histogram of samples from the noise distributions used. The noise was
multiplied by the temperature to 'jitter' the transfer function. Hence. the jitter decreased as the
annealing schedule proceeded.
12
1;.
Vnol se
1;.
f.(r.
J
I
W II
II
vout or
+ noise)
1;.
Vln+
or r.
1;.
Vnol sl
WIJI J
+ noise = ne~
high IIIln
tr8nl'.. function
wUh noll. 'line"
Fig. 2. Electronic analog neuron.
Fig. 3 shows average performance across 100 runs for the last 100 patterns of 2000 training
pattern presentations. It can be seen that reducing the gain from a sharp step can improve
learning in a small region of gain, even without noise. There seems to be an optimal gain level.
However, the addition of noise for any distribution can substantially improve learning at all levels
of gain.
1
~
-----
-
~
0.9
Gaussian
Unifona
Cauchy
HO Hoise
tlCLI
~
u~
c
0.8
0
......-, .'.' ....u __. . . , ..
.,.j
~
~
8.0
0.7
&:
0.6-
...
0.5
10
-3
.,
10
-2
-1
10
Inverse Gain
Fig. 3. Proportion correct vs. inverse gain.
1
10
1
13
2.2 Stochastic Competitive Learning
We have studied how competitive leaming(4J[~) can be accomplished with stochastic local units.
Mter the presentation of the input pattern. the network is annealed and the weight is increased
between the winning cluster unit and the input units which are on. As shown in Fig. 4 this
approach was applied to the dipole problem of Rumelhart and Zipser. A 4x4 pixel array input
layer connects to a 2 unit competitive layer with recurrent inhibitory connections that are not
adjusted. The inhibitory connections provide the competition by means of a winner-lake-all
process as the network settles. The input patterns are dipoles - only two input units are turned
OIl at each pattern presentatiOll and they must be physically adjacent. either vertically or
horizontally. In this way, the network learns about the connectedness of the space and eventually
divides it into two equal spatial regions with each of the cluster units responding only to dipoles
from one of the halves. Rumelhart and Zipser renormalized the weights after each pattern and
picked the winning unit as the one with the highest activation. Instead of explicit nonnalization
of the weights. we include a decay term proportional to the weight. The weights between the
input layer and cluster layer are incremented for on-on correlations, but here there are no
alternating phases so that even this gross synchrony is not necessary. Indeed. if small time
constants are introduced to the weight updates. no external timing should be needed.
winner-lake-all
cluster layer
input/ayer
Pig. 4. Competitive learning network for the dipole problem.
Fig. S shows the results of several runs. A 1 at the po~ition of an input unit means that unit 1 of
the cluster layer has the larger weight leading to it from that position. A + between two units
means the dipole from these two units excites unit 1. A 0 and - means that unit 0 is the winner in
the complementary case. Note that adjacent l's should always have a + between them since both
weights to unit 1 are stronger. H, however, there is a 1 next to a 0, then there is a tension in the
dipole and a competition for dominance in the cluster layer. We define a figure of merit called
"surface tension" which is the number of such dipoles in dispute. The smaller the number, the
14
better. Note in Runs A and B, the number is reduced to 4, the minimum possible value, after
2000 pattern presentations. The space is divided vertically and horizontally, respectively. Run C
bas adopted a less favorable diagonal division with a surface tension of 6.
Number of dipole pattern presentations
2000
1400
0
200
800
0-0-0-0
1+0-0+1
+ + + +
1+1+1+1
+ +
1+1-0-0
+
0-0-0-0
1+1+1+1
+ + +
1+1+1-0
+ +
1-0-0-0
1+1+1+1
+ + + +
1+1+1+1
+ - + 0-0-0-0
1+1+1+1
+ + + +
1+1+1+1
- +
0-0-0-0
0-0-0-0
0-0-0-0
0-0-0-0
0-0-0+1
+ +
0-0-1+1
- + +
0-0-1+1
- + +
0-0+1+1
-0-0-1+1
-- +
- + +
-0-0-1+1
0-0-0-1
-++
-0-0-1+1
- + +
-0-0-1+1
--++
0-0+1+1
0-0-0-0
RUn A
0-0-0-0
0-0-0-0
0-0-0-0
---
0-0-0-0
0-0-0-0
-0-0-0+1
--+
-1-0-1+1
--+
0-0-0-0
+ - + +
1+0+1+1
0-0-0-0
Run B
0-0-0-0
0-0-0-0
Run C
0-0-0-0
0- 0-0-0
-
--
-
-
-
- -
+ +
0-0+1+1
-+++
0-1+1+1
-++ +
0+1+1+1
+ + +
0+1+1+1
+ + +
0-0-0-0
-
1+1+1+1
+ + + +
0+1+1+1
+ +
0-0-0-0
0-0-0-0
0-0-0-0
0-0-0-0
0+1+1+1
0-1+1+1
-
--
-
-
0-0-1+1
1+1+1+1
+ + +
0-0+1+1
+ +
0-0-0-1
- - +
0-0-0-1
-
--
Fig. 5. Results of competitive learning runs on the dipole problem.
Table 1 sbows the result of several competitive algorithms compared when averaged over 100
such runs. The deterministic algorithm of Rumelhart and Zipser gives an average surface tension
of 4.6 while the stochastic procedure is almost as good. Note that noise is essential in belping the
competitive layer settle. Without noise the surface tension is 9.8, sbowing that the winner-takeall procedure is not working properly.
Competitive learning algorithm
"surface tension"
Stochastic net with decay
- anneal: T=3H T=1.0
- no anneal: 70 @ T=1.0
4.8
Stochastic net with renonnallzation
5.6
Deterministic, winner-take-all
(Rumelhart & Zipser)
4.6
9.8
Table 1. Performance of competitive learning algorithms across 1()() runs.
We also tried a procedure where, instead of decay, weights were renormalized. The model is that
each neuron can support a maximum amount of weight leading into it. Biologically, this might
be the area that other neurons can form synapses on, so that one synapse cannot increase its
strength except at the expense of some of the others. Electronically, this can be implemented as
15
current emanating from a fixed clUTent source per neuron. As shown in Table 1, this works
nearly as well as decay. Moreover, preliminary results show that renormalization is especiaUy
effective when more then two cluster units are employed.
Both of the stochastic algorithms, which can be implemented in an electronic synapse in nearly
the same way as the supervised learning algorithm, divide the space just as the deterministic
normalization procedure14J does. This suggests that our chip can do both styles of learning,
supervised if one includes both phases and unsupervised if only the procedure of the minus phase
is used.
1.3 Reiolorcelfteot Learning
We have tried several approaches to reinforcement learning using the synaptic model of Fig. 1
where the evaluation signal is a scalar value available globally that represents how well the
system performed on each trial. We applied this model to an xor problem with only one output
unit. The reinforcement was r = 1 for the correct output and r = -1 otherwise. To the network,
this was similar to supervised learning since for a single unit, the output state is fully specified by
a scalar value. A major difference, however, is that we do not clamp the output unit in the
desired state in order to compare plus and minus phases. This feature of supervised learning has
the effect of adjusting weights to follow a gradient to the desired state. In the reinforcement
learning described here, there is no plus phase. This has a satisfying aspect in that no overall
synchrony is necessary to compare phases, but is also much slower at converging to a solution
because the network has to search the solution space without the guidance of a teacher clamping
the output units. This situation becomes much worse when there is more than one output unit. In
that case, the probability of reinforcement goes down exponentially with the number of outputs.
To test multiple outputs, we chose the simple replication problem whereby the output simply has
to replicate the input. We chose the number of bidden units equal to the input (or output).
10 the absence of a teacher to clamp the outputs, the network has to find the answer by chance,
guided only by a "critic" which rates its effort as "better" or "worse". This means the units must
somehow search the space. We use the same stochastic units as in the supervised or unsupervised
techniques, but now it is important to have the noise or the annealing temperature set to a proper
level. If it is too high, the reinforcement received is random rather than directed by the weights
in the network. If it is too low, the available states searched become too smaU and the probability
of finding the right solution decreases. We tuned our annealing schedule by looking at a
volatility measure defined at each neuron which is simply the fraction of the time the neuron
activation is above zero. We then adjust the final anneal temperature so that this number is
neither 0 or 1 (noise too low) nor 0.5 (noise too high). We used both a fixed annealing schedule
for all neurons and a unit-specific schedule where the noise was proportional to the sum of weight
magnitudes into the unit. A characteristic of reinforcement learning is that the percent correct
initially increases but then decreases and often oscillates widely. To avoid this, we added a factor
of (I - <r ? multiplying the final temperature. This helped to stabilize the learning.
In keeping with our simple model of the synapse, we chose a weight adjustment technique that
consisted of correlating the states of the connected neurons with the global reinforcement signal.
Each synapse measured the quantity R =rs;sj for each pattern presented. If R >0, then ~';j is
incremented and it is decremented if R <0. We later refined this procedure by insisting that the
reinforcement be greater than a recent average so that R =(r-<,. > hi Sj. This type of procedure
16
appears in previous work in a number of fonns.(12] (13) For r =?l only, this "excess
reinforcement" is the same as our previous algorithm but differs if we make a comparison
between short term and long tenn averages or use a graded reinforcement such as the negative of
the sum squared error. Following a suggestion by G. Hinton, we also investigated a more
complex technique whereby each synapse must store a time average of three quantities: <r>,
<SiSj>, and <rsiSj>. The definition now is R =<rsiSj>-<r><SjSj> and the rule is the same as
before. Statistically, this is the same as "excess reinforcement" if the latter is averaged over
trials. For the results reported below the values were collected across 10 pattern presentations. A
variation. which employed a continuous moving average, gave similar results.
Table 2 summarizes the perfonnance on the xor and the replication task of these reinforcement
learning techniques. As the table shows a variety of increasingly sophisticated weight adjustment
rules were explored; nevertheless we were unable to obtain good results with the techniques
described for more than S output units. In the third column, a small threshold had to be exceeded
prior to weight adjustment. In the fourth column, unit-specific temperatures dependent on the
sum of weights, were employed. The last column in the table refers to frequency dependent
learning where we trained on a single pattern until the network produced a correct answer and
then moved on to another pattern. This final procedure is one of several possible techniques
related to 'shaping' in operant learning theory in which difficult patterns are presented more often
to the network.
network
xor
24-1
2-2-1
-
eplication
2-2-2
3-3-3
444
S-S-S
6-6-6
t=1
time-averaged
+?=0.1
+T-I:W
+freq
(0.60) 0.64
(0.58) 0.57
(0.70) 0.88
(0.69) 0.74
(0.76) 0.88
(0.96) 1.00
(0.92)0.99
(0.85) 1.00
(0.98) 1.00
(0.78) 0.88
(0.94)0.94
(0.15) 0.21
(0.46) 0.46
(0.31) 0.33
(0.91) 0.97
(0.31) 0.62
(0.87) 0.99
(0.37)0.37
(0.97) 1.00
(0.97) 1.00
(0.75) 1.00
(0.13) 0.87
(0.02) 0.03
-
-
-
-
-
-
-
Table 2. Proportion correct performance of reinforcement learning
after (2K) and 10K patterns.
Our experiments. while incomplete, hint that reinforcement learning can also be implemented by
the same type of local-global synapse that characterize the other learning paradigms. Noise is
also necessary here for the random search procedure.
2... Sanunary of Study of hDdameatai Learning Par...eters
In summary, we see that the use of noise and our model of a local correlational synapse with a
DOn-specific global evaluation signal are two important features in all the learning paradigms.
Graded activation is somewhat less important. Weight decay seems to be quite important
although saturation can substitute for it in unsupervised learning. Most interesting from our point
of view is that all these phenomena are electronically implementable and therefore physically
17
plausible. Hopefully this means they are also related to true neural phenomena and therefore
provide a basis for unifying the various approaches of learning at a microscopic level.
3. ELECTRONIC IMPLEMENTATION
3.1 The Supervised LearDiog Chip
We have completed the design of the chip previously proposed.(IO] Its physical style of
computation speeds up learning a millionfold over a computer simulation. Fig. 6 shows a block
diagram of the neuron. It is a double differential amplifier. One branch forms a sum of the inputs
from the differential outputs of aU other neurons with connections to it. The other adds noise
from the noise amplifier. This first stage has low gain to preserve dynamic range at the summing
nodes. The second stage has high gain and converts to a single ended output. This is fed to a
switching arrangement whereby either this output state or some externally applied desired state is
fed into the final set of inverter stages which provide for more gain and guaranteed digital
complementarity .
Sdlslrld
Fig. 6. Block diagram of neuron.
The noise amplifier is shown schematically in Fig. 7. Thermal noise, with an nns level of tens of
microvolts, from the channel of an FET is fed into a 3 stage amplifier. Each stage provides a
potential gain of 100 over the noise bandwidth. Low pass feedback in each stage stabilizes the
DC output as well as controls gain and bandwidth by means of an externally controlled variable
resistance for tuning the annealing cycle.
Fig. 8 shows a block diagram of the synapse. The weight is stored in 5 flip-flops as a sign and
magnitude binary number. These flip-flops control the conductance from the outputs of neuron i
to the inputs of neuron j and vice-versa as shown in the figure. The conductance of the FETs are
in the ratio 1:2:4:8 to correspond to the value of the binary number while the sign bit determines
whether the true or complementary lines connect. The flip-flops are arranged in a counter which
is controUed by the correlation logic. If the plus phase correlations are greater than the minus
phase, then the counter is incremented by a single unit If less, it is decremented.
18
Vcontrol
I
I
l
I
>--.._V._.nOISI
Fig. 7. Block diagram of noise amplifier.
Sj
or
I
Sj
or
I
nior~
" T---~'-~__~--~
up.
.r----...
correlation
"ncrement
logic
sgn
down.
o
& set
i------lhnl
logic
WI)
or
JI
2
phase
3
Fig. 8. Block diagram of synapse.
Fig. 9 sbows the layout of a test chip. A 6 neuron, 15 synapse network may be seen in the lower
left comer. Eacb neuron bas attacbed to it a noise amplifier to assure that the noise is
uncorrelated. The network occupies an area about 2.5 mm on a side in 2 micron design rules.
Eacb 300 transistor synapse occupies 400 by 600 microns. In contrast, a biological synapse
occupies only about one square micron. The real miracle of biological learning is in the synapse
wbere plasticity operates on a molecular level, not in the neuron. We can't bope to compete using
transistors, bowevc:r small, especially in the digital domain. Aside from this small network, the
rest of the chip is occupied with test structures of the various components.
3.1 Analog Synapse
Analog circuit tecbni~ues can reduce the size of the synapse and increase its functionality.
Several recent papers( 4] II~I have shown how to make a voltage controlled resistor in MOS
technology. The voltage controlling the conductance representing the synaptic weight can be
obtained by an analog charge integrator from the correlated activation of the neurons which the
synapse in question connects. A charge integrator with a "leaky capacitor" bas a time constant
19
which can be used to make comparisons as a continuous time average over the last several trials.
thereby' adding temporal information. One can envision this time constant as being adaptive as
well. The charge integrator directly implements the analog Hebb-typel 16] correlation rules of
section 2.
~.~ ~,~~~' ~ .. ~i~ 'i~ ~ ~~ilf'~~
.'
?
~.,
??
' /., ~ "'" )A , ..?'<""~
:~";" ..
? ./ . '
. \ ' :": :" . _ .
??????????
*??
:i .
c?.. ..?
~.*
If. ., ? iii ? -
I I .?
~ii.:' ...
??.???? ???????????
., ? ? ? ?
'
~
.
_ ?
??
. .. . . .
~.~
It ill ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?
~.I ':;;::dU:;.;;;.UEEi
......... .
? 1!~'.'
-Jot- :
~ II.
~ ~" ,
i nr-':,~"?"";??;.
:i ii r. ? . '. ,:
:-"fO.,a'l.~"~;"
.......
..?.?.....
...?',......
.....
. . . . ..
.? ??".?
? . ' ;~-.
...:t~"1I ?
???
,;.:"
1IIi1.
~.... ...... ,? ?~I'
! : '.
.. ,.... " ~
.
:i ..................
JjI!
II~.~
'
' :" :::'.l-
,.;,:,...
s ?Ii,?iI'. .
?
?
.. .......
:::, ':.
Fig. 9. Chip layout.
3.3 Tecbnologicalbnprovemeots for Flectronic Neural Networks
It is still necessary to store the voltage which controls the analog conductance and we propose the
EPROMll7] or EEPROM device for this. Such a device can hold the value of the weight in the
same way that flip-flops do in the digital implementation of the synapse(lOJ. The process which
creates this device has two polysilicon layers which are useful for making high valued
capacitances in analog circuitry. In addition. the second polysilicon layer could be used to make
CCD devices for charge storage and transport. Coupled with the charge storage on a floating
gate(l8], this forms a compact. low power representation for weight values that apyroach
biological values. Another useful addition would be a high valued stable resistive layerl l9 . One
20
could thereby avoid space-wasting long-channel MOSFETs which are currently the only
rea~ble way to achieve high resistance in MOS technology. Lastly, the addition of a diffusion
step or two creates a Bi-CMOS process which adds high quality bipolar transistors useful in
analog design. Furthermore, one gets the logarithmic dependence of voltage on current in bipolar
technology in a natural, robust way, that is not subject to the variations inherent in using
MOSFETs in the subthreshold region. This is especially useful in compressing the dynamic
range in sensory processing[20J?
4. CONCLUSION
We have shown how a simple adaptive synapse which measures correlations can account for a
variety of learning styles in stochastic networks. By embellishing the standard CMOS process
and using analog design techniques. a technology suitable for implementing such a synapse
electronically can be developed. Noise is an important element in our formulation of learning. It
can help a network settle, interpolate between discrete values of conductance during learning. and
search a large solution space. Weight decay ("forgetting") and saturation are also important for
stability. These phenomena not only unify diverse learning styles but are electronically
implementabfe.
ACKNOWLEDGMENT:
This work has been influenced by many researchers. We would especially like to thank Andy
Barto and Geoffrey Hinton for valuable discussions on reinforcement learning, Yannis Tsividis
for contributing many ideas in analog circuit design, and Joel Gannett for timely releases of his
vlsi verification software.
21
References
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Corrf~r~nu.
edited by P. Losleben
| 80 |@word proceeded:1 trial:3 version:1 proportion:2 seems:2 stronger:1 replicate:1 simulation:2 propagate:1 tried:2 r:1 thereby:2 minus:4 solid:1 noll:1 electronics:1 tuned:1 envision:1 current:2 com:1 activation:13 si:1 must:3 plasticity:1 designed:1 update:2 v:1 aside:1 half:1 selected:1 tenn:1 device:5 liapunov:1 discovering:1 sys:3 short:1 dissertation:1 record:1 provides:2 node:1 attack:1 analyzability:1 sigmoidal:1 become:1 differential:2 replication:2 consists:1 resistive:1 pathway:1 dispute:1 forgetting:1 indeed:1 behavior:1 alspector:2 nor:1 brain:2 integrator:3 globally:1 becomes:1 matched:1 moreover:1 circuit:4 mass:1 sivilotti:1 substantially:1 developed:1 unified:1 finding:1 nj:1 ended:1 wasting:1 temporal:2 morristown:2 charge:6 bipolar:2 oscillates:1 control:5 unit:28 before:1 local:8 vertically:2 timing:1 limit:2 io:1 switching:3 acad:1 sutton:2 semiconductor:1 mead:1 punish:1 connectedness:1 might:2 plus:4 chose:3 au:1 studied:4 doctoral:1 specifying:1 suggests:1 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7,026 | 800 | Foraging in an Uncertain Environment Using
Predictive Hebbian Learning
P. Read Montague: Peter Dayan, and Terrence J. Sejnowski
Computational Neurobiology Lab, The Salk Institute,
100 ION. Torrey Pines Rd,
La Jolla, CA, 92037, USA
read~bohr.bcm.tmc.edu
Abstract
Survival is enhanced by an ability to predict the availability of food,
the likelihood of predators, and the presence of mates. We present a
concrete model that uses diffuse neurotransmitter systems to implement
a predictive version of a Hebb learning rule embedded in a neural architecture based on anatomical and physiological studies on bees. The
model captured the strategies seen in the behavior of bees and a number of
other animals when foraging in an uncertain environment. The predictive
model suggests a unified way in which neuromodulatory influences can
be used to bias actions and control synaptic plasticity.
Successful predictions enhance adaptive behavior by allowing organisms to prepare for future actions, rewards, or punishments. Moreover, it is possible to improve upon behavioral
choices if the consequences of executing different actions can be reliably predicted. Although classical and instrumental conditioning results from the psychological literature [1]
demonstrate that the vertebrate brain is capable of reliable prediction, how these predictions
are computed in brains is not yet known.
The brains of vertebrates and invertebrates possess small nuclei which project axons
throughout large expanses of target tissue and deliver various neurotransmitters such as
dopamine, norepinephrine, and acetylcholine [4]. The activity in these systems may report
on reinforcing stimuli in the world or may reflect an expectation of future reward [5, 6,7,8].
*Division of Neuroscience, Baylor College of Medicine, Houston, TX 77030
598
Foraging in an Uncertain Environment Using Predictive Hebbian Learning
A particularly striking example is that of the honeybee. Honeybees can be conditioned to
a sensory stimulus such as a color, visual pattern, or an odorant when the sensory stimulus
is paired with application of sucrose to the antennae or proboscis. An identified neuron,
VUMmxl, projects widely throughout the entire bee brain, becomes active in response
to sucrose, and its firing can substitute for the unconditioned odor stimulus in classical
conditioning experiments [8]. Similar diffusely projecting neurons in the bee brain may
substitute for reward when paired with a visual stimulus.
In this paper, we suggest a role for diffuse neurotransmitter systems in learning and behavior
that is analogous to the function we previously postulated for them in developmental selforganization[3, 2]. Specifically, we: (i) identify a neural substrate/architecture which is
known to exist in both vertebrates and invertebrates and which delivers information to
widespread regions of the brain; (ii) describe an algorithm that is both mathematically
sound and biologically feasible; and (iii) show that a version of this local algorithm, in the
context of the neural architecture, reproduces the foraging and decision behavior observed
in bumble bees and a number of other animals.
Our premise is that the predictive relationships between sensory stimuli and rewards are
constructed through these diffuse systems and are used to shape both ongoing behavior and
reward-dependent synaptic plasticity. We illustrate this using a simple example from the
ethological literature for which constraints are available at a number of different levels.
A Foraging Problem
Real and colleagues [9, 10] performed a series of experiments on bumble bees foraging on
artificial flowers whose colors, blue and yellow, predicted of the delivery of nectar. They
examined how bees respond to the mean and variability of this reward delivery in a foraging
version of a stochastic two-armed bandit problem [11]. All the blue flowers contained 2\-1l
of nectar, of the yellow flowers contained 6 \-1l, and the remaining j of the yellow flowers
contained no nectar at all. In practice, 85% of the bees' visits were to the constant yield
blue flowers despite the equivalent mean return from the more variable yellow flowers.
When the contingencies for reward were reversed, the bees switched their preference for
flower color within 1 to 3 visits to flowers. They further demonstrated that the bees could be
induced to visit the variable and constant flowers with equal frequency if the mean reward
from the variable flower type was made sufficiently high.
l
This experimental finding shows that bumble bees, like honeybees, can learn to associate
color with reward. Further, color and odor learning in honeybees has approximately the
same time course as the shift in preference descri bed above for the bumble bees [12]. It also
indicates that under the conditions of a foraging task, bees prefer less variable rewards and
compute the reward availability in the short term. This is a behavioral strategy utilized by
a variety of animals under similar conditions for reward [9, 10, 13] suggesting a common
set of constraints in the underlying neural substrate.
The Model
Fig. 1 shows a diagram of the model architecture, which is based on the considerations
above about diffuse systems. Sensory input drives the units 'B' and 'Y' representing blue
and yellow flowers. These neurons (outputs x~ and
respectively at time t) project
xi
599
600
Montague, Dayan, and Sejnowski
Action selection
Motor
systems
Lateral
inhibition
Figure 1: Neural architecture showing how predictions about future expected reinforcement can be made in the brain using a diffuse neurotransmitter system [3, 2]. In
the context of bee foraging [9], sensory input drives the units Band Y representing blue and
yellow flowers. These units project to a reinforcement neuron P through a set of variable
weights (filled circles w B and w Y) and to an action selection system. Unit S provides input
to n and fires while the bee sips the nectar. R projects its output rt through a fixed weight
to P. The variable weights onto P implement predictions about future reward rt (see text)
and P's output is sensitive to temporal changes in its input. The output projections of P, bt
(lines with arrows), influence learning and also the selection of actions such as steering in
flight and landing, as in equation 5 (see text). Modulated lateral inhibition (dark circle) in
the action selection layer symbolizes this. Before encountering a flower and its nectar, the
output of P will reflect the temporal difference only between the sensory inputs Band Y.
During an encounter with a flower and nectar, the prediction error bt is determined by the
output of B or Y and R, and learning occurs at connections w B and w Y. These strengths
are modified according to the correlation between presynaptic activity and the prediction
error bt produced by neuron P as in equation 3 (see text). Learning is restricted to visits to
flowers [14].
through excitatory connection weights both to a diffusely projecting neuron P (weights
w B and w Y) and to other processing stages which control the selection of actions such as
steering in flight and landing. P receives additional input rt through unchangeable wei~hts.
In the absence of nectar (rt = 0), the net input to P becomes V t W t ?Xt = w~x~ +w t x~.
=
The first assumption in the construction of this model is that learning (adjustment of
weights) is contingent upon approaching and landing on a flower. This assumption is
supported specifically by data from learning in the honeybee: color learning for flowers is
restricted to the final few seconds prior to landing on the flower and experiencing the nectar
[14].
This fact suggests a simple model in which the strengths of variable connections
adjusted according to a presynaptic correlational rule:
Wt
are
(1 )
where oc is the learning rate [15]. There are two problems with this formulation: (i) learning
would only occur about contingencies in the presence of a reinforcing stimulus (rt =/: 0);
Foraging in an Uncertain Environment Using Predictive Hebbian Learning
A
B
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:::::s
o
100.0
'----~---~----'
0.0
5.0
10.0
Nectar volume (f-ll)
60.0
40.0
20.0
0.0
0
5
10
15
20
25
30
Trial
Figure 2: Simulations of bee foraging behavior using predictive Hebbian learning. A)
Reinforcement neuron output as a function of nectar volume for a fixed concentration of
nectar[9, 10]. B) Proportion of visits to blue flowers. Each trial represents approximately
40 flower visits averaged over 5 real bees and exactly 40 flower visits for a single model
bee. Trials 1 - 15 for the real and model bees had blue flowers as the constant type, the
remaining trials had yellow flowers as constant. At the beginning of each trial, wYand w B
were set to 0.5 consistent with evidence that information from past foraging bouts is not
used[14]. The real bees were more variable than the model bees - sources of stochasticity
such as the two-dimensional feeding ground were not represented. The real bees also had a
slight preference for blue flowers [21]. Note the slower drop for A = 0.1 when the flowers
are switched.
and (ii) there is no provision for allowing a sensory event to predict the future delivery of
reinforcement. The latter problem makes equation 1 inconsistent with a substantial volume
of data on classical and instrumental conditioning [16]. Adding a postsynaptic factor to
equation 1 does not alter these conclusions [17].
This inadequacy suggests that another form of learning rule and a model in which P has a
direct input from rt. Assume that the firing rate of P is sensitive only to changes in its input
over time and habituates to constant or slowly varying input, like magnocellular ganglion
cells in the retina [18]. Under this assumption, the output of P, bt. reflects a temporal
derivative of its net input, approximated by:
(2)
where y is a factor that controls the weighting of near against distant rewards. We take
y = 1 for the current discussion.
In the presence of the reinforcement, the weights w B and w Yare adjusted according to the
simple correlational rule:
(3)
This permits the weights onto P to act as predictions of the expected reward consequent
on landing on a flower and can also be derived in a more general way for the prediction of
future values of any scalar quantity [19].
601
602
Montague, Dayan, and Sejnowski
A
B
. - 100.0
~
30.0
'-'
Q.)
0...
80.0
~
-
Q.)
~
.~
>
8
C'-l
.....
.-.->
8
60.0
~
20.0
?0
~
40.0
8--?lv=2
<r-----(> v = 8
b------i!. V = 30
20.0
C'-l
0 .0
0 .0
2 .0
4.0
6.0
>
10.0
A= 0 . 1
o
+ A= 0.9
0 .0
0.0
2.0
4.0
6 .0
Mean
Mean
Figure 3: Tradeoff between the mean and variance of nectar delivery. A) Method of
selecting indifference points. The indifference point is taken as the first mean for a given
variance (bold v in legend) for which a stochastic trial demonstrates the indifference. This
method of calculation tends to bias the indifference points to the left. B) Indifference plot
for model and real bees. Each point represents the (mean, variance) pair for which the
bee sampled each flower type equally. The circles are for A 0.1 and the pluses are for
A 0.9.
=
=
When the bee actually lands on a flower and samples the nectar, R influences the output of
P through its fixed connection (Fig. 1). Suppose that just prior to sampling the nectar the
bee switched to viewing a blue flower, for example. Then, since T t -l
0, lit would be
Tt - x~_1 w~_I. In this way, the term x~_1 w~_1 is a prediction of the value of T t and the
difference Tt - x~_1 wt 1 is the error in that prediction. Adjusting the weight w~ according
to the correlational rule in equation 3 allows the weight w~, through P's outputs, to report
to the rest of the brain the amount of reinforcement Tt expected from blue flowers when
they are sensed.
=
As the model bee flies between flowers, reinforcement from nectar is not present (Tt = 0)
and lit is proportional to V t - V t- 1. w B and w Y can again be used as predictions but through
modulation of action choice. For example, suppose the learning process in equation 3 sets
w Y less than w B? In flight, switching from viewing yellow flowers to viewing blue flowers
causes lit to be positive and biases the activity in any action selection units driven by
outgoing connections from B. This makes the bee more likely than chance to land on or
steer towards blue flowers. This discussion is not offered as an accurate model of action
choice, rather, it simply indicates how output from a diffuse system could also be used to
influence action choice.
The biological assumptions of this neural architecture are explicit: (i) the diffusely projecting neuron changes its firing according to the temporal difference in its inputs; (ii) the
output of P is used to adjust its weights upon landing; and (iii) the output otherwise biases
the selection of actions by modulating the activity of its target neurons.
For the particular case of the bee, both the learning rule described in equation 3 and the
biasing of action selection described above can be further simplified for the purposes of a
Foraging in an Uncertain Environment Using Predictive Hebbian Learning
simple demonstration. As mentioned above, significant learning about a particular flower
color may occur only in the 1 - 2 seconds just prior to an encounter [21, 14]. This
is tantamount to restricting weight changes to each encounter with the reinforcer which
allows only the sensory input just preceding the delivery or non-delivery of r t to drive
synaptic plasticity. We therefore make the learning rule punctate, updating the weights on
a flower by flower basis. During each encounter with the reinforcer in the environment, P
produces a prediction error cSt = rt - V t -l where rt is the actual reward at time t, and the
lX~_l
last flower color seen by the bee at time t, say blue, causes a prediction V t -l =
of future reward rt to be made through the weight w~_l and the input activity
l' The
weights are then updated using a form of the delta rule[20]:
wt
xt
(4)
where A is a time constant and controls the rate of forgetting. In this rule, the weights from
the sensory input onto P still mediate a prediction of r; however, the temporal component
for choosing how to steer and when to land has been removed.
We model the temporal biasing of actions such as steering and landing with a probabilistic
algorithm that uses the same weights onto P to choose which flower is actually visited
on each trial. At each flower visit, the predictions are used directly to choose an action,
according to:
e~(WYxY)
q(Y) = e~(wBxB) + ell(wYxY)
(5)
where q(Y) is the probability of choosing a yellow flower. Values of J.L > 0 amplify the
difference between the two predictions so that larger values of J.L make it more likely that
the larger prediction will result in choice toward the associated flower color. In the limit as
J.L ---+ 00 this approaches a winner-take-all rule. In the simulations, J.L was varied from 2.8 to
6.0 and comparable results obtained. Changing J.L alters the magnitude of the weights that
develop onto neuron P since different values of J.L enforce different degrees of competition
between the predictions.
To apply the model to the foraging experiment, it is necessary to specify how the amount of
nectar in a particular flower gets reported to P. We assume that the reinforcement neuron
R delivers its signal rt as a saturating function of nectar volume (Fig. 2A). Harder and
Real [10] suggest just this sort of decelerating function of nectar volume and justify it on
biomechanical grounds. Fig. 2B shows the behavior of model bees compared with that of
real bees [9] in the experiment testing the extent to which they prefer a constant reward to
a variable reward of the same long-term mean. Further details are presented in the figure
legend.
The behavior of the model matched the observed data for A = 0.9 suggesting that the real
bee utilizes information over a small time window for controlling its foraging [9]. At this
value of A, the average proportion of visits to blue was 85% for the real bees and 83%
for the model bees. The constant and variable flower types were switched at trial 15 and
both bees switched flower preference in 1 - 3 subsequent visits. The average proportion
of visits to blue changed to 23% and 20%, respectively, for the real and model bee. Part of
the reason for the real bees' apparent preference for blue may come from inherent biases.
Honey bees, for instance, are known to learn about shorter wavelengths more quickly than
others [21]. In our model, A is a measure of the length of time over which an observation
exerts an influence on flower selection rather than being a measure of the bee's time horizon
in terms of the mean rate of energy intake [9, 10].
603
604
Montague, Dayan, and Sejnowski
Real bees can be induced to forage equally on the constant and variable flower types if the
mean reward from the variable type is made sufficiently large, as in Fig. 3B. For a given
variance, the mean reward was increased until the bees appeared indifferent between the
flowers. In this experiment, the constant flower type contained 0.5J.11 of nectar. The data
for the real bee is shown as points connected by a solid line in order to make clear the
envelope of the real data. The indifference points for A = 0.1 (circles) and A = 0.9 (pluses)
also demonstrate that a higher value of A is again better at reproducing the bee's behavior.
The model captured both the functional relationship and the spread of the real data.
The diffuse neurotransmitter system reports prediction errors to control learning and bias
the selection of actions. Distributing such a signal diffusely throughout a large set of
target structures permits this prediction error to influence learning generally as a factor in a
correlational or Hebbian rule. The same signal, in its second role, biases activity in an action
selection system to favor rewarding behavior. In the model, construction of the prediction
error only requires convergent input from sensory representations onto a neuron or neurons
whose output is a temporal derivative of its input. The output of this neuron can also be
used as a secondary reinforcer to associate other sensory stimuli with the predicted reward.
We have shown how this relatively simple predictive learning system closely simulates the
behavior of bumble bees in a foraging task.
Acknowledgements
This work was supported by the Howard Hughes Medical Institute, the National Institute
of Mental Health, the UK Science and Engineering Research Council, and computational
resources from the San Diego Supercomputer Center. We would like to thank Patricia
Churchland, Anthony Dayan, Alexandre Pouget, David Raizen, Steven Quartz and Richard
Zemel for their helpful comments and criticisms.
References
[1] Konorksi, 1. Conditioned reflexes and neuron organization, (Cambridge, England,
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[2] Quartz, SR, Dayan, P, Montague, PR, Sejnowski, Tl. (1992) Society for Neurosciences
Abstracts. 18, 210.
[3] Montague, PR, Dayan, P, Nowlan, Sl, Pouget, A, Sejnowski, Tl. (1993) In Advances
in Neural Information Processing Systems 5, Sl Hanson, ID Cowan, CL Giles, editors,
(San Mateo CA: Morgan Kaufmann), pp. 969-976.
[4] Morrison, IH and Magistretti, Pl. Trends in Neurosciences, 6, 146 (1983).
[5] Wise, RA. Behavioral and Brain Sciences, 5,39 (1982).
[6] Cole, Bl and Robbins, TW. Neuropsychopharmacology, 7, 129 (1992).
[7] Schultz, W. Seminars in the Neurosciences, 4, 129 (1992).
[8] Hammer, M, thesis, FU Berlin (1991).
[9] Real, LA. Science, 253, pp 980 (1991).
Foraging in an Uncertain Environment Using Predictive Hebbian Learning
[10] Real, LA. Ecology, 62,20 (1981); Harder, LD and Real, LA. Ecology, 68(4), 1104
(1987); Real, LA, Ellner, S, Harder, LD. Ecology, 71(4), 1625 (1990).
[11] Berry, DA and Fristedt, B. Bandit Problems: Sequential Allocation of Experiments.
(London, England: Chapman and Hall, 1985).
[12] Gould, JL. In Foraging Behavior, AC Kamil, JR Krebs and HR Pulliam, editors, (New
York, NY: Plenum, 1987), p 479.
[13] Krebs, JR, Kacelnik, A, Taylor, P. Nature" 275, 27 (1978), Houston, A, Kacelnik,
A, McNamara, J. In Functional Ontogeny, D McFarland, editor, (London: Pitman,
1982).
[14] Menzel, R and Erber, 1. Scientific American, 239(1), 102.
[15] Carew, TJ, Hawkins RD, Abrams 1W and Kandel ER. Journal of Neuroscience, 4(5),
1217 (1984).
[16] Mackintosh, NJ. Conditioning and Associative Learning. (Oxford, England: Oxford
University Press, 1983). Sutton, RS and Barto, AG. Psychological Review, 882, 135
(1981). Sutton, RS and Barto, AG. Proceedings of the Ninth Annual Conference of
the Cognitive Science Society. Seattle, WA (1987).
[17] Reeke, GN, Jr and Sporns, O. Annual Review of Neuroscience. 16,597 (1993).
[18] Dowling, JE. The Retina. (Cambridge, MA: Harvard University Press, 1987).
[19] The overall algorithm is a temporal difference (TO) learning rule and is related to
an algorithm Samuel devised for teaching a checker playing program, Samuel, AL.
IBM Journal of Research and Development, 3,211 (1959). It was first suggested in
its present form in Sutton, RS, thesis, University of Massachusetts (1984); Sutton and
Barto [1] showed how it could be used for classical conditioning; Barto, AG, Sutton,
RS and Anderson, CWo IEEE Transactions on Systems, Man, and Cybernetics, 13,
834 (1983) used a variant of it in a form of instrumental conditioning task; Barto,
AG, Sutton, RS, Watkins, CJCH, Technical Report 89-95, (Computer and Information
Science, University of Massachusetts, Amherst, MA, 1989); Barto, AG, Bradtke, SJ,
Singh, SP, Technical Report 91-57, (Computer and Information Science, University of
Massachusetts, Amherst, MA, 1991) showed its relationship to dynamic programming,
an engineering method of optimal control.
[20] Rescorla, RA and Wagner, AR. In Classical Conditioning II: Current Research and
Theory, AH Black and WF Prokasy, editors, (New York, NY: Appleton-CenturyCrofts, 1972), p 64; Widrow, B and Stearns, SD. Adaptive Signal Processing, (Englewood Cliffs, NJ: Prentice-Hall, 1985).
[21] Menzel, R, Erber, J and Masuhr, J. In Experimental Analysis of Insect Behavior, LB
Browne, editor, (Berlin, Germany: Springer-Verlag, 1974), p 195.
605
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7,027 | 801 | Use of Bad Training Data For Better
Predictions
Tal Grossman
Complex Systems Group (T13) and CNLS
LANL, MS B213 Los Alamos N.M. 87545
Alan Lapedes
Complex Systems Group (T13)
LANL, MS B213 Los Alamos N.M. 87545
and The Santa Fe Institute, Santa Fe, New Mexico
Abstract
We show how randomly scrambling the output classes of various
fractions of the training data may be used to improve predictive
accuracy of a classification algorithm. We present a method for
calculating the "noise sensitivity signature" of a learning algorithm
which is based on scrambling the output classes. This signature can
be used to indicate a good match between the complexity of the
classifier and the complexity of the data. Use of noise sensitivity
signatures is distinctly different from other schemes to avoid overtraining, such as cross-validation, which uses only part of the training data, or various penalty functions, which are not data-adaptive.
Noise sensitivity signature methods use all of the training data and
are manifestly data-adaptive and non-parametric. They are well
suited for situations with limited training data.
1
INTRODUCTION
A major problem of pattern recognition and classification algorithms that learn from
a training set of examples is to select the complexity of the model to be trained.
How is it possible to avoid an overparameterized algorithm from "memorizing"
the training data? The dangers inherent in over-parameterization are typically
343
344
Grossman and Lapedes
illustrated by analogy to the simple numerical problem of fitting a curve to data
points drawn from a simple function. If the fit is with a high degree polynomial then
prediction on new points, i.e. generalization, can be quite bad, although the training
set accuracy is quite good. The wild oscillations in the fitted function, needed to
acheive high training set accuracy, cause poor predictions for new data. When
using neural networks, this problem has two basic aspects. One is how to choose
the optimal architecture (e.g. the number oflayers and units in a feed forward net),
the other is to know when to stop training. Of course, these two aspects are related:
Training a large net to the highest training set accuracy usually causes overfitting.
However, when training is stopped at the "correct" point (where train-set accuracy
is lower), large nets are generalizing as good as, or even better than, small networks
(as observed e.g. in Weigend 1994). This prompts serious consideration of methods
to avoid overparameterization. Various methods to select network architecture or
to decide when to stop training have been suggested. These include: (1) use of
a penalty function (c.!. Weigend et al. 1991). (2) use of cross validation (Stone
1974). (3) minimum description length methods (Rissanen 1989), or (4) "pruning"
methods (e.g. Le Cun et al. 1990).
Although all these methods are effective to various degrees, they all also suffer some
form of non-optimality:
(1) various forms of penalty function have been proposed and results differ between
them. Typically, using a penalty function is generally preferable to not using one.
However, it is not at all clear that there exists one "correct" penalty function and
hence any given penalty function is usually not optimal. (2) Cross validation holds
back part of the training data as a separate valdiation set. It therefore works best in
the situation where use of smaller training sets, and use of relatively small validation
sets, still allows close approximation to the optimal classifier. This is not likely to
be the case in a significantly data-limited regime. (3) MDL methods may be viewed
as a form of penalty function and are subject to the issues in point (1) above. (4)
pruning methods require training a large net, which can be time consuming, and
then "de-tuning" the large network using penalty functions. The issues expressed
in point(l) above apply.
We present a new method to avoid overfitting that uses "noisy" training data where
some of the output classes for a fraction of the data are scrambled. We describe
how to obtain the "noise sensitivity signature" of a classifier (with its learning
algorithm), which is based on the scrambled data. This new methodology is not
computationally cheap, but neither is it prohibitively expensive. It can provide an
alternative to methods (1 )-( 4) above that (i) can test any complexity parameter of
any classifying algorithm (i.e. the architecture, the stopping criterion etc.) (ii) uses
all the training data, and (iii) is data adaptive, in contrast to fixed penalty/pruning
functions.
2
A DETAILED DESCRIPTION OF THE METHOD
Define a "Learning Algorithm" L(S, P), as any procedure which produces a classifier
f(~), which is a (discrete) function over a given input space X (~ E X). The input
of the learning algorithm L is a Training Set S and a set of parameters P. The
training set S is a set of M examples, each example is a pair of an input instance ~i
Use of Bad Training Data for Better Predictions
=
and the desired output Yi associated with it (i
l..M). We assume that the desired
output represents an unknown "target function" f* which we try to approximate,
i.e. Yi
f*(:ni). The set of parameters P includes all the relevant parameters of the
specific learning algorithm and architecture used. When using a feed-forward neural
network classifier this set usually includes the size of the network, its connectivity
pattern, the distribution of the initial weights and the learning parameters (e.g.
the learning rate and momentum term size in usual back-propagation). Some of
these parameters determine the "complexity" of the classifiers produced by the
learning algorithm, or the set of functions f that are realizable by L. The number
of hidden units in a two layer perceptron, for example, determines the number
of free parameters of the model (the weights) that the learning algorithm will fit
to tbe data (the training set). In general, the output of L can be any classifier:
a neural network, a decision tree, boolean formula etc. The classifier f can also
depend on some random choices, like the initial choice of weights in many network
lenrning algortihm. It can also depend, like in pruning algorithms on any "stopping
crite~'ion" which may also influence its complexity.
=
2.1
PRODUCING
ff
The classification task is given as the training set S. The first step of our method
is to prepare a set of noisy, or partially scrambled realizations of S. We define S:
as one partiCUlar such realization, in which for fraction P of the M examples tne
desired ou.tpu.t values (classes) are changed. In this work we consider only binary
classification tasks, which means that we choose pM examples at random for which
= 1 - Yi? For each noise level p and set of n such realizations S; (f.L l..n) is
prepared, each with a different random choice of scrambled examples. Practically,
8-10 noise levels in the range p = 0.0 - 0.4, with n "" 4 - 10 realizations of
for
each level are enough. The second step is to apply the learning algorithm to each
of the different
to produce the corresponding classifiers, which are the boolean
functions ff
L(S;, P).
=
yf
S:
=
2.2
S:
NOISE SENSITIVITY MEASURES
Using the set of ff, three quantities are measured for each noise level p:
? The average performance on the original (noise free) training set S. We
define the average noise-free error as
1
Ej(p) = Mn
n
M
I: L If;(:ni) I/o
Yil
(1)
i
And the noise-free pereformance, or score as Qj(p) = 1 - Ej(p).
? In a similar way, we define the average error on the noisy training-sets S::
1
En(P) = Mn
n
M
I/o
\
L ~ If;(:ni) - yfl
f;
(2)
Note that the error of each classifier
is measured on the training set
by which it was created. The noisy-set performance is then defined as
Qn(P) = 1 - En(P)?
345
346
Grossman and Lapedes
? The average functional distance between classifiers. The functional distance
between two classifiers, or boolean functions, d(J, g) is the probability of
I(z) #- g(z). For a uniform input distribution, it is simply the fraction of
the input space X for which I(z) #- g(z). In order to approximate this
quantity, we can use another set of examples. In contrast with validation
set methods, these examples need not be classified, i.e. we only need a set of
inputs z, without the target outputs y, so we can usually use an "artificial"
set of m random inputs. Although, in principle at least, these z instances
should be taken from the same distribution as the original task examples.
The approximated distance between two classifiers is therefore
1 m
d(J, g) = m ~ I/(Zi) - g(zi)1
,
We then calculate the average distance, D(p), between the n classifiers
obtained for each noise level p:
(3)
It
n
D(p) = n(n 2_ 1)
L d(J:, I;)
(4)
IJ.>V
3
NOISE SENSITIVITY BEHAVIOR
Observing the three quantities Q,(p), Qn(P) and D(p), can we distinguish between
an overparametrized classifier and a "well tuned" one? Can we use this data in order
to choose the best generalizer out of several candidates? Or to find the right point
to stop the learning algorithm L in order to achieve better generalization?
Lets
estimate how the plots of Q" Qn and D vs. p, which we call the "Noise Sensitivity
Signature" (NSS) of the algorithm L, look like in several different scenarios.
3.1
D(p)
The average functional distance between realizations, D(p), measures the sensitivity of the classifier (or the model) to noise. An over-parametrized architecture is
expected to be very sensitive to noise since it is capable of changing its classification boundary to learn the scrambled examples. Different realizations of the noisy
training set will therefore result in different classifiers.
On the other hand, an under-parametrized classifier should be stable against at
least a small amount of noise. Its classification boundary will not change when
a few examples change their class. Note, however, that if the training set is not
very "dense", an under-parametrized architecture can still yield different classifiers,
even when trained on a noise free training set (e.g. when using BP with different initial weights). Therefore, it may be possible to observe some "background
variance", i.e. non-zero average distance for small (down to zero) noise levels for
under-parametrized classifiers.
Similar considerations apply for the two quantities Q,(p) and Qn(P). When the
training set is large enough, an under-parametrized classifier cannot "follow" all
Use of Bad Training Data for Better Predictions
the changed examples. Therefore most of them just add to the training error.
Nevertheless, its performance on the noise free training set, Qf(P), will not change
much. As a result, when increasing the noise level P from zero (where Qf(P)
Qn(P)), we should find Qf (p) > Qn(P) up to a high noise level - where the decision
boundary has changed enough so the error on the original training set becomes
larg '~r than the error on the actual noisy set. The more parameters our model has,
the sooner (i.e. smaller p) it will switch to the Qf(P) < Qn(P) state. If a network
starts with Qf(P)
Qn(P) and then exhibits a behavior with Qf(P) < Qn(P), this
is a signature of overparameterization.
=
=
3.3
THE TRAINING SET
In addition to the set of parameters P and the learning algorithm itself, there
is another important factor in the learning process. This is the training set S.
The dependence on M, the number of examples is evident. When M is not large
enough, the training set does not provide enough data in order to capture the full
complexity of the original task. In other words, there are not enough constraints
- to approximate well the target function f*. Therefore overfitting will occur for
smaller classifier complexity and the optimal network will be smaller.
4
EXPERIMENTAL RESULTS
To demonstrate the possible outcomes of the method described above in several
cases, we have performed the following experiment . A random neural network
"teacher" was created as the target function f*. This is a two layer percept ron
with 20 inputs, 5 hidden units and one output. A set of M random binary input
examples was created and the teacher network was used to classify the training
examples. Namely, a desired output Yi was obtained by recording the output of
the teacher net when input :l:i was presented to the network, and the output was
calculated by applying the usual feed forward dynamincs:
(5)
This binary threshold update rule is applied to each of the network's units j, i.e
the hidden and the output units. The weights of the teacher were chosen from a
uniform distribution [-1,1]. No threshold (bias weights) were used.
St
The set of scrambled training sets
was produced as explained above and different
network architectures were trained on it to produce the set of classifiers jl1o. The
learning networks are standard two layer networks of sigmoid units, trained by conjugate gradient back-propagation, using a quadratic error function with tolerance,
i.e. if the difference between an output of the net and the desired 0 or 1 target is
smaller than the tolerance (taken as 0.2 in our experiment) it does not contribute
to the error. The tolerance is, of course, another parameter which may influences
the complexity of the resulting network, however, in this experiment it is fixed.
The quantities Qf(P), Qn(P) and D(p) were calculated for networks with 1,2,3, .. 7
hidden units (1 hidden unit means just a perceptron, trained with the same error
function). In our terminology, the architecture specification is part of the set of
347
348
Grossman and Lapedes
hidden units
1
2
3
4
5
6
7
400
0.81
0.81
0.78
0.77
0.74
0.74
0.71
0.04
0.04
0.02
0.03
( 0.03
( 0.01
( 0.01
Training Set Size
1024
700
0.81 0.001) 0.82
0.84 0.05
0.86
0.82 0.06
0.90
0.81 0.05
0.90
0.87
0.79 0.03
0.89
0.80 0.05
0.76 0.02
0.85
0. 0011
0.04
0.03
0.03
0.04
0.03
0.05
Table 1: The prediction rate for 1..7 hidden units, averaged on 4 nets that were
trained on the noisefree training set of size M = 400,700,1024 (the standard deviation is given in parenthesis).
parameters P that is input to the learning algorithm L. The goal is to identify the
"correct" architecture according to the behavior of QJ, Qn and D with p.
=
The experiment was done with three training set sizes M
400, 700 and 1024.
Another set of m = 1000 random examples was used to calculate D. As an "external control" this set was also classified by the teacher network and was used to
measure the generalization (or prediciton rate) of the different learning networks.
The prediction rate, for the networks trained on the noise free training set (averaged over 4 networks, trained with different random initial weights) is given for
the 1 to 7 hidden unit architectures, for the 3 sizes of M, in Table 1. The noise
sensitivity signatures of three architectures trained with M = 400 (1,2,3 hidden
units) and with M = 1024 examples (2,4,6 units) are shown in Figure 1. Compare
these (representative) results with the expected behaviour of the NSS as described
qualitatively in the previous section.
5
CONCLUSIONS and DISCUSSION
We have introduced a method of testing a learning model (with its learning algorithm) against a learning task given as a finite set of examples, by producing and
characterizing its "noise sensitivity signature". Relying on the experimental results
presented here, and similar results obtained with other (less artificial) learning tasks
and algorithms, we suggest some guidelines for using the NSS for model tuning:
1. If D(p) approaches zero with p -+ 0, or if QJ(p) is significantly better than
Qn(P) for noise levels up to 0.3 or more - the network/model complexity can be
safely inreased.
2. If QJ(p) < Qn(P) already for small levels of noise (say 0.2 or less) - reduce the
network complexity.
3. In more delicate situations: a "good" model will have at least a trace of concavity
in D(p). A clearly convex D(p) probably indicates an over-parametrized model. In
a "good" model choice, Qn (p) will follow Q J (p) closely, from below, up to a high
noise level.
Use of Bad Training Data for Better Predictions
04
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02
oL-__L -_ _
o
oos
~
01
__
~
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015
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02
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0 25
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03
~
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035
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I
01
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005
045
02
025
400 IlX~. 2 hrd:len UIlIIs
03
035
04
1024 exa~9S 4 hidden units
08
-
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o
005
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025
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04
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?
i
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?0L---O~OS---0~1---0~15--~
02--~02~5---0~
3 ---0~35---0~4--~045
Figure 1: The signatures (Q and D vs. p) of networks with 1,2,3 hidden units (top
to bottom) trained on M=400 examples (left), and networks with 2,4,6 hidden units
trained on M=1024 examples. The (noisy) training set score Qn(P) is plotted with
full line, the noise free score Qf(P) with dotted line, and the average functional
distance D(p) with error bars (representing the standard deviation of the distance).
349
350
Grossman and Lapedes
5.1
Advanatages of the Method
1. The method uses all the data for training. Therefore we can extract all the
available information. Unlike validation set methods - there is no need to spare
part of the examples for testing (note that classified examples are not needed for
the functional distance estimation). This may be an important advantage when
the data is limited. As the experiment presented here shows: taking 300 examples
out of the 1024 given, may result in choosing a smaller network that will give
inferior prediction (see table 1). Using "delete-1 cross-validation" will minimize
this problem but will need at least as much computation as the NSS calculation in
order to achieve reliable prediction estimation.
2. It is an "external" method, i.e. independent of the classifier and the training
algorithm. It can be used with neural nets, decision trees, boolean circuits etc. It
can evaluate different classifiers, algorithms or stopping/prunning criteria.
5.2
Disadvantages
1. Computationally expensive (but not prohibitively so). In principle one can use
just a few noise levels to reduce computational cost.
2. Presently requires a subjective decision in order to identify the signature, unlike
cross-validation methods which produce one number. In some situations, the noise
sensitivity signature gives no clear distinction between similar architectures. In
these cases, however, there is almost no difference in their generalization rate.
Acknowledgements
We thank David Wolpert, Michael Perrone and Jerom Friedman for many iluminating discussions and usefull comments. We also thank Rob Farber for his invaluable
help with software and for his assistance with the Connection Machine.
Referencess
Le Cun Y., Denker J.S. and Solla S. (1990), in Adv. in NIPS 2, Touretzky D.S. ed.
(Morgan Kaufmann 1990) 598.
Rissanen J. (1989), Stochastic Complezity in Statistical Inquiry (World Scientific
1989).
Stone M. (1974), J.Roy.Statist.Soc.Ser.B 36 (1974) 11I.
Wiegend A.S. (1994), in the Proc. of the 1993 Connectionist Models Summer School,
edited by M.C. Mozer, P. Smolensky, D.S. Touretzky, J.L. Elman and A.S. Weigend,
pp. 335-342 (Erlbaum Associates, Hillsdale NJ, 1994).
Wiegend A.S., Rummelhart D. and Huberman B.A. (1991), in Adv. in NIPS 3,
Lippmann et al. eds. (Morgen Kaufmann 1991) 875.
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7,028 | 802 | Constructive Learning Using Internal
Representation Conflicts
Laurens R. Leerink and Marwan A. J abri
Systems Engineering & Design Automation Laboratory
Department of Electrical Engineering
The University of Sydney
Sydney, NSW 2006, Australia
Abstract
We present an algorithm for the training of feedforward and recurrent neural networks. It detects internal representation conflicts
and uses these conflicts in a constructive manner to add new neurons to the network . The advantages are twofold: (1) starting with
a small network neurons are only allocated when required; (2) by
detecting and resolving internal conflicts at an early stage learning
time is reduced. Empirical results on two real-world problems substantiate the faster learning speed; when applied to the training
of a recurrent network on a well researched sequence recognition
task (the Reber grammar), training times are significantly less than
previously reported .
1
Introduction
Selecting the optimal network architecture for a specific application is a nontrivial
task, and several algorithms have been proposed to automate this process. The
first class of network adaptation algorithms start out with a redundant architecture
and proceed by pruning away seemingly unimportant weights (Sietsma and Dow,
1988; Le Cun et aI, 1990). A second class of algorithms starts off with a sparse
architecture and grows the network to the complexity required by the problem.
Several algorithms have been proposed for growing feedforward networks. The
upstart algorithm of Frean (1990) and the cascade-correlation algorithm of Fahlman
(1990) are examples of this approach.
279
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Leerink and Jabri
The cascade correlation algorithm has also been extended to recurrent networks
(Fahlman, 1991), and has been shown to produce good results. The recurrent
cascade-correlation (RCC) algorithm adds a fully connected layer to the network
after every step, in the process attempting to correlate the output of the additional
layer with the error. In contrast, our proposed algorithm uses the statistical properties of the weight adjustments produced during batch learning to add additional
units.
The RCC algorithm will be used as a baseline against which the performance of
our method will be compared. In a recent paper, Chen et al (1993) presented an
algorithm which adds one recurrent neuron with small weights every N epochs.
However, no significant improvement in training speed was reported over training
the corresponding fixed size network, and the algorithm will not be further analyzed.
To the authors knowledge little work besides the two mentioned papers have applied
constructive algorithms to recurrent networks.
In the majority of our empirical studies we have used partially recurrent neural
networks, and in this paper we will focus our attention on such networks. The motivation for the development of this algorithm partly stemmed from the long training
times experienced with the problems of phoneme and word recognition from continuous speech. However, the algorithm is directly applicable to feedforward networks.
The same criteria and method used to add recurrent neurons to a recurrent network
can be used for adding neurons to any hidden layer of a feed-forward network.
2
Architecture
In a standard feedforward network, the outputs only depend on the current inputs,
the network architecture and the weights in the network. However, because of the
temporal nature of several applications, in particular speech recognition, it might
be necessary for the network to have a short term memory.
Partially recurrent networks, often referred to as Jordan (1989) or Elman (1990)
networks, are well suited to these problems. The architecture examined in this
paper is based on the work done by Robinson and Fallside (1991) who have applied
their recurrent error propagation network to continuous speech recognition.
A common feature of all partially recurrent networks is that there is a special set
of neurons called context units which receive feedback signals from a previous time
step. Let the values of the context units at time t be represented by C(t). During
normal operation the input vector at time t are applied to the input nodes I(t), and
during the feedforward calculation values are produced at both the output nodes
O(t + 1) and the context units C(t + 1). The values of the context units are then
copied back to the input layer for use as input in the following time step.
Several training algorithms exist for training partially recurrent neural networks,
but for tasks with large training sets the back-propagation through time (Werbos,
1990) is often used. This method is computationally efficient and does not use
any approximations in following the gradient. For an application where the time
information is spread over T. input patterns, the algorithm simply duplicates the
network T times - which results in a feedforward network that can be trained by a
variation of the standard backpropagation algorithm.
Constructive Learning Using Internal Representation Conflicts
3
The Algorithm
For partially recurrent networks consisting of input, output and context neurons,
the following assertions can be made:
? The role of the context units in the network is to extract and store all
relevant prior information from the sequence pertaining to the classification
problem.
? For weights entering context units the weight update values accumulated
during batch learning will eventually determine what context information
is stored in the unit (the sum of the weight update values is larger than the
initial random weights).
? We assume that initially the number of context units in the network is
insufficient to implement this extraction and storage of information (we
start training with a small network). Then, at different moments in time
during the recognition of long temporal sequences, a context unit could be
required to preserve several different contexts.
? These conflicts are manifested as distinct peaks in the distribution of the
weight update values during the epoch.
All but the last fact follows directly from the network architecture and requires no
further elaboration. The peaks in the distribution of the weight update values are a
result of the training algorithm attempting to adjust the value of the context units in
order to provide a context value that will resolve short-term memory requirements.
After the algorithm had been developed, it was discovered that this aspect of the
weight update values had been used in the past by Wynne-Jones (1992) and in
the Meiosis Networks of Hanson (1990). The method of Wynne-Jones (1992) in
particular is very closely related; in this case principal component analysis of the
weight updates and the Hessian matrix is used to detect oscillating nodes in fully
trained feed-forward networks. This aspect of backpropagation training is fully
discussed in Wynne-Jones (1992), to which reader is referred for further details.
The above assertions lead to the proposed training algorithm, which states that if
there are distinct maxima in the distribution of weight update values of the weights
entering a context unit, then this is an indication that the batch learning algorithm
requires this context unit for the storage of more than one context.
If this conflict can be resolved, the network can effectively store all the contexts
required, leading to a reduction in training time and potentially an increase III
performance .
The training algorithm is given below (the mode of the distribution is defined as
the number of distinct maxima):
For all context units {
Set N = modality ot the distribution ot weight update values;
It N > 1 then {
Add N-1 new context units to the network which are identical
(in terms ot weighted inputs) to the current context unit.
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Leerink and Jabri
Adjust each of these N context units (including the
original) by the weight update value determined by each
maxima (the average value of the mode).
Adjust all weights leaving these N context units so that the
addition of the new units do not affect any subsequent layers
(division by N). This ensures that the network retains all
previously acquired knowledge.
}
}
The main problem in the implementation of the above algorithm is the automatic
detection of significant maxima in the distribution of weight updates. A standard
statistical approach for the determination of the modality (the number of maxima)
of a distribution of noisy data is to fit a curve of a certain predetermined order to
the data. The maxima (and minima) are then found by setting the derivative to
zero. This method was found to be unsuitable mainly because after curve fitting it
was difficult to determine the significance of the detected peaks.
It was decided that only instances of bi-modality and tri-modality were to be iden-
tified, each corresponding to the addition of one or two context units. The following
heuristic was constructed:
? Calculate the mean and standard deviation of the weight update values.
? Obtain the maximum value in the distribution.
? If there are any peaks larger than 60% of the maxima outside one standard
deviation of the mean, regard this as significant.
This heuristic provided adequate identification of the modalities. The distribution
was divided into three areas using the mean ? the standard deviation as boundaries.
Depending on the number of maxima detected, the average within each area is used
to adjust the weights.
4
Discussion
According to our algorithm it follows that if at least one weight entering a context
unit has a multi-modal distribution, then that context unit is duplicated. In the
case where multi-modality is detected in more than one weight, context units were
added according to the highest modality.
Although this algorithm increases the computational load during training, the standard deviation of the weight updates rapidly decreases as the network converges.
The narrowing of the distribution makes it more difficult to determine the modality. In practice it was only found useful to apply the algorithm during the initial
training epochs, typically during the first 20.
During simulations in which strong multi-modalities were detected in certain nodes,
frequently the multi-modalities would persist in the newly created nodes. In this
Constructive Learning Using Internal Representation Conflicts
manner a strong bi-modality would cause one node to split into two, the two nodes
to grow to four, etc. This behaviour was prevented by disabling the splitting of
a node for a variable number of epochs after a multi-modality had been detected.
Disabling this behaviour for two epochs provided good results.
5
Simulation Results
The algorithm was evaluated empirically on two different tasks:
? Phoneme recognition from continuous multi-speaker speech usmg the
TIMIT (Garofolo, 1988) acoustic-phonetic database .
? Sequence Recognition: Learning a finite-state grammar from examples of
valid sequences.
For the phoneme recognition task the algorithm decreased training times by a factor
of 2 to 10, depending on the size of the network and the size of the training set.
The sequence recognition task has been studied by other researchers in the past, notably Fahlman (1991). Fahlman compared the performance of the recurrent cascade
correlation (RCC) network with that of previous results by Cleeremans et al (1989)
who used an Elman (1990) network. It was concluded that the RCC algorithm
provides the same or better performance than the Elman network with less training
cycles on a smaller training set. Our simulations have shown that the recurrent
error propagation network of Robinson and Fallside (1991), when trained with our
constructive algorithm and a learning rate adaptation heuristic, can provide the
same performance as the RCC architecture in 40% fewer training epochs using a
training set of the same size. The resulting network has the same number of weights
as the minimum size RCC network which correctly solves this problem.
Constructive algorithms are often criticized in terms of efficiency, i.e. "Is the increase in learning speed due to the algorithm or just the additional degrees of
freedom resulting from the added neuron and associated weights?". To address this
question several simulations were conducted on the speech recognition task, comparing the performance and learning time of a network with N fixed context units
to that of a network with small number of context units and growing a network
with a maximum of N context units. Results indicate that the constructive algorithm consistently trains faster, even though both networks often have the same
final performance.
6
Summary
In this paper the statistical properties of the weight update values obtained during
the training of a simple recurrent network using back-propagation through time
have been examined. An algorithm has been presented for using these properties to
detect internal representation conflicts during training and to use this information
to add recurrent units to the network. Simulation results show that the algorithm
decreases training time compared to networks which have a fixed number of context
units. The algorithm has not been applied to feedforward networks, but can III
principle be added to all training algorithms that operate in batch mode.
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References
Chen, D., Giles, C.L., Sun, G.Z., Chen, H.H., Lee, Y.C., Goudreau, M.W. (1993).
Constructive Learning of Recurrent Neural Networks. In 1993 IEEE International
Conference on Neural Networks, 111:1196-1201. Piscataway, NJ: IEEE Press.
Cleeremans, A., Servan-Schreiber, D., and McClelland, J.L. (1989). Finite State
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Elman, J .L. (1990). Finding Structure in Time. Cognitive Science 14:179-21l.
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2, 524-532. San Mateo, CA: Morgan Kaufmann.
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Feedforward Neural Networks. Neural Computation 2:198-209.
Garofolo, J.S. (1988). Getting Started with the DARPA TIMIT CD-ROM: an
Acoustic Phonetic Continuous Speech Database. National Institute of Standards
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Hanson, S.J. (1990). Meiosis Networks. In D. S. Touretzky (ed.), Advances in Neural Information Processing Systems 2, 533-541, San Mateo, CA: Morgan Kaufmann.
Jordan, M.1. (1989). Serial Order: A Parallel, Distributed Processing Approach. In
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Hillsdale: Erlbaum.
Le Cun, Y., J .S. Denker, and S.A Solla (1990). Optimal Brain Damage. In D. S.
Touretzky (ed.), Advances in Neural Information Processing Systems 2, 598-605.
San Mateo, CA: Morgan Kaufmann.
Reber, A.S. (1967). Implicit learning of artificial grammars. Journal of Verbal
Learning and Verbal Behavior 5:855-863.
Robinson, A.J. and Fallside F. (1991). An error propagation network speech recognition system. Computer Speech and Language 5:259-274.
Sietsma, J. and RJ.F Dow (1988). Neural Net Pruning-\Vhy and How. In IEEE
International Conference on Neural Networks. (San Diego 1988), 1:325-333.
Wynne-Jones, M. (1992) Node Splitting: A Constructive Algorithm for FeedForward Neural Networks. In D. S. Touretzky (ed.), Advances in Neural Information Processing Systems 4, 1072-1079. San Mateo, CA: Morgan Kaufmann.
Werbos, P.J. (1990). Backpropagation Through Time, How It Works and How to
Do It. Proceedings of the IEEE, 78:1550-1560.
| 802 |@word simulation:5 nsw:1 moment:1 reduction:1 initial:2 selecting:1 past:2 current:2 comparing:1 stemmed:1 subsequent:1 predetermined:1 wynne:4 update:13 fewer:1 short:2 provides:1 detecting:1 node:9 constructed:1 fitting:1 manner:2 acquired:1 notably:1 behavior:1 elman:5 frequently:1 growing:2 multi:6 brain:1 detects:1 researched:1 little:1 resolve:1 provided:2 what:1 developed:1 finding:1 nj:1 temporal:2 every:2 unit:28 rcc:6 engineering:2 might:1 garofolo:2 studied:1 examined:2 mateo:4 sietsma:2 bi:2 decided:1 practice:1 implement:1 backpropagation:3 area:2 empirical:2 significantly:1 cascade:6 word:1 storage:2 context:30 attention:1 starting:1 automaton:1 splitting:2 variation:1 diego:1 us:2 rumelhart:1 recognition:11 werbos:2 persist:1 database:2 narrowing:1 role:1 electrical:1 calculate:1 cleeremans:2 ensures:1 connected:1 cycle:1 sun:1 solla:1 decrease:2 highest:1 mentioned:1 complexity:1 trained:3 depend:1 division:1 efficiency:1 resolved:1 darpa:1 represented:1 train:1 distinct:3 pertaining:1 detected:5 artificial:1 outside:1 heuristic:3 larger:2 grammar:3 noisy:1 final:1 seemingly:1 advantage:1 sequence:6 indication:1 net:1 adaptation:2 relevant:1 rapidly:1 getting:1 requirement:1 produce:1 oscillating:1 converges:1 depending:2 recurrent:21 frean:2 school:1 disabling:2 solves:1 sydney:2 strong:2 c:1 indicate:1 laurens:1 closely:1 australia:1 hillsdale:1 behaviour:2 normal:1 automate:1 early:1 applicable:1 schreiber:1 weighted:1 focus:1 improvement:1 consistently:1 mainly:1 contrast:1 baseline:1 detect:2 accumulated:1 typically:1 initially:1 hidden:1 classification:1 development:1 special:1 extraction:1 abri:1 identical:1 jones:4 report:1 connectionist:1 duplicate:1 preserve:1 national:1 iden:1 consisting:1 freedom:1 detection:1 adjust:4 analyzed:1 necessary:1 criticized:1 instance:1 giles:1 assertion:2 servan:1 retains:1 deviation:4 conducted:1 erlbaum:1 reported:2 stored:1 peak:4 international:2 lee:1 off:1 cognitive:1 derivative:1 leading:1 automation:1 start:3 parallel:1 timit:2 phoneme:3 who:2 kaufmann:4 identification:1 produced:2 researcher:1 touretzky:4 ed:5 against:1 lebiere:1 associated:1 newly:1 duplicated:1 knowledge:2 back:3 feed:2 modal:1 done:1 evaluated:1 though:1 just:1 stage:1 implicit:1 correlation:6 dow:2 propagation:5 mode:3 grows:1 entering:3 laboratory:1 during:12 speaker:1 substantiate:1 criterion:1 common:1 empirically:1 discussed:1 significant:3 mellon:1 ai:1 automatic:1 language:1 had:3 etc:1 add:7 recent:1 store:2 certain:2 phonetic:2 manifested:1 morgan:4 minimum:2 additional:3 determine:3 redundant:1 signal:1 resolving:1 rj:1 technical:1 faster:2 determination:1 calculation:1 long:2 elaboration:1 divided:1 serial:1 prevented:1 reber:2 cmu:1 usmg:1 receive:1 addition:2 decreased:1 grow:1 leaving:1 concluded:1 allocated:1 modality:12 ot:3 operate:1 tri:1 jordan:2 feedforward:9 iii:2 split:1 affect:1 fit:1 architecture:10 upstart:2 speech:9 proceed:1 hessian:1 cause:1 adequate:1 useful:1 unimportant:1 mcclelland:1 reduced:1 exist:1 correctly:1 carnegie:1 four:1 sum:1 reader:1 layer:5 copied:1 nontrivial:1 meiosis:2 aspect:2 speed:3 attempting:2 department:1 according:2 piscataway:1 smaller:1 cun:2 computationally:1 previously:2 eventually:1 operation:1 apply:1 denker:1 away:1 batch:4 original:1 unsuitable:1 added:3 question:1 damage:1 fallside:3 gradient:1 maryland:1 majority:1 rom:1 besides:1 insufficient:1 difficult:2 potentially:1 design:1 implementation:1 neuron:8 finite:2 nist:1 extended:1 discovered:1 required:4 hanson:2 conflict:9 acoustic:2 robinson:3 address:1 below:1 pattern:1 including:1 memory:2 tified:1 technology:1 leerink:4 created:1 started:1 extract:1 epoch:6 prior:1 fully:3 degree:1 principle:1 cd:1 summary:1 fahlman:6 last:1 verbal:2 institute:1 sparse:1 distributed:1 regard:1 feedback:1 curve:2 boundary:1 world:1 valid:1 author:1 forward:2 made:1 san:5 correlate:1 pruning:2 marwan:1 continuous:4 nature:1 ca:4 constructing:1 jabri:3 significance:1 spread:1 main:1 motivation:1 referred:2 experienced:1 load:1 specific:1 goudreau:1 adding:1 effectively:1 chen:3 suited:1 simply:1 adjustment:1 partially:5 twofold:1 determined:1 principal:1 called:1 partly:1 internal:6 constructive:10 |
7,029 | 803 | Learning Curves: Asymptotic Values and
Rate of Convergence
Corinna Cortes, L. D. Jackel, Sara A. Solla, Vladimir Vapnik,
and John S. Denker
AT&T Bell Laboratories
Holmdel, NJ 07733
Abstract
Training classifiers on large databases is computationally demanding. It is desirable to develop efficient procedures for a reliable
prediction of a classifier's suitability for implementing a given task,
so that resources can be assigned to the most promising candidates
or freed for exploring new classifier candidates. We propose such
a practical and principled predictive method. Practical because it
avoids the costly procedure of training poor classifiers on the whole
training set, and principled because of its theoretical foundation.
The effectiveness of the proposed procedure is demonstrated for
both single- and multi-layer networks.
1
Introd uction
Training classifiers on large data.bases is computationally demanding. It is desirable
to develop efficient procedures for a reliable prediction of a classifier's suitability
for implementing a given task. Here we describe such a practical and principled
predictive method.
The procedure applies to real-life situations with huge databases and limited resources. Classifier selection poses a problem because training requires resources especially CPU-cycles, and because there is a combinatorical explosion of classifier
candidates. Training just a few of the many possible classifiers on the full database
might take up all the available resources, and finding a classifier particular suitable
for the task requires a search strategy.
327
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Cortes, Jackel, SoBa, Vapnik, and Denker
test
error
-,~.......-----------------
...........
~~~~,~-----------
10,000
30,000
50,000
training
set size
Figure 1: Test errors as a function of the size of the training set for three different
classifiers. A classifier choice based on best test error at training set size 10 = 10,000
will result in an inferior classifier choice if the full database contains more than
15,000 patterns.
The naive solution to the resource dilemma is to reduce the size of the database to
1 10 , so that it is feasible to train all classifier candidates. The performance of the
classifiers is estimated from an independently chosen test set after training. This
makes up one point for each classifier in a plot of the test error as a function of the
size 1 of the training set. The naive search strategy is to keep the best classifier at
10 , under the assumption that the relative ordering of the classifiers is unchanged
when the test error is extrapolated from the reduced size 10 to the full database size.
Such an assumption is questionable and could easily result in an inferior classifier
choice as illustrated in Fig. 1.
=
Our predictive method also utilizes extrapolation from medium sizes to large sizes
of the training set, but it is based on several data points obtained at various sizes
of the training set in the intermediate size regime where the computational cost of
training is low. A change in the representation of the measured data points is used
to gain confidence in t.he extrapolation.
2
A Predictive Method
Our predictive method is based on a simple modeling of the learning curves of a
classifier. By learning curves we mean the expectation value of the test and training
errors as a function of the training set size I. The expectat.ion value is taken over
all the possible ways of choosing a training set of a given size.
A typical example of learning curves is shown in Fig. 2. The test error is always
larger than the training error, but asymptotically t.hey reach a common value, a.
We model the errors for large siz?'s of the training s?'t as power-law decays to the
Learning Curves: Asymptotic Values and Rate of Convergence
error
a?? ..?. ???? ?.?
~~-
?~.~?~.~.~.~.~.~?~Trr.P.~
_------------training error
training set size, I
Figure 2: Learning curves for a typical classifier. For all finite values of the
training set size I the test error is larger t han the training error. Asymptotically
they converge to the same value a.
asymptotic error value, a:
b
['test = a + ler
c
and
['train = a - 1i3
where I is the size of the training set, and a and f3 are positive exponents. From
these two expressions the sum and difference is formed:
['test
+ ['train
['test - ['train
If we make the assumption
0'=
b
c
2a + ler - 1i3
(1)
b
ler
(2)
c
+ 1i3
f3 and b = c the equation (1) and (2) reduce to
['test
+ [train
[test - [train
2a
2b
ler
(3)
These expressions suggest a log-log representation of the sum and difference of the
test and training errors as a function of the the training set size I, resulting in
two straight lines for large sizes of the training set: a constant "-' log(2a) for the
sum, and a straight line with slope -a and intersection log(b + c) "-' log(2b) for the
difference, as shown in Fig. 3.
=
The assumption of equal amplitudes b c of the two convergent terms is a convenient but not crucial simplification of the model. \Ve find experimentally that for
classifiers where this approximation does not hold, the difference ['test - ['train
still forms a straight line in a log-log-plot. From this line the sum s
b+c
can be extracted as the intersection, as indicated on Fig. 3. The weighted sum
=
329
330
Cortes, Jackel, Solla, Vapnik, and Denker
log(error)
log(b+c)
log(2b)
.. ..
....
..
log(?tesl +?train) -log(2a)
log(trainlng set size, l)
Figure 3: Within the validity of the power-law modeling of the test and training
errors, the sum and difference between the two errors as a function of training set
size give two straight lines in a log-log-plot: a constant"" log(2a) for the sum, and a
straight line with slope -0' and intersection log(b + c) ,..., log(2b) for the difference.
c . Etest + b . Etrain will give a constant for an appropriate choice of band c, with
b + c s.
=
The validity of the above model was tested on numerous boolean classifiers with
linear decision surfaces. In all experiments we found good agreement with the model
and we were able to extract reliable estimates of the three parameters needed to
model the learning curves: the asymptotic value a, and the power 0', and amplitude
b of the power-law decay. An example is shown in Fig. 4, (left). The considered
task is separation of handwritten digits 0-4 from the digits 5-9. This problem is
unrealizable with the given database and classifier.
The simple modeling of the test and training errors of equation (3) is only assumed
to hold for large sizes of the training set, but it appears to be valid already at
intermediate sizes, as seen in Fig. 4, (left). The predictive model suggested here is
based on this observation, and it can be illustrated from Fig. 4, (left): with test
and training errors measured for I ~ 2560 it is possible to estimate the two straight
lines, extract approximate values for the three parameters which characterize the
learning curves, and use the resulting power-laws to extrapolate the learning curves
to the full size of the database.
The algorithm for the predictive method is therefore as follows:
1.
Measure Etest and Etrain for intermediate sizes of the training set.
2.
Plot 10g(Etest
3.
Estimate the two straight lines and extract the asymptotic value a
the amplitude b, and the exponent 0'.
4.
Extrapolate the learning curves to the full size of the database.
+ Etrain)
and 10g(Etest - Etrain) versus log I.
Learning Curves: Asymptotic Values and Rate of Convergence
log (error)
error
0.25
-1
+
-
points used for prediction
???? predicted learning curves
0.2
0.15
-2
0.1
-3r---------~~----o
1
2 log (1/ 256)
I
256
2560
25600
?
I
0.05
I
...... ~. -4-??...-.am
.?. _??
A.~-?-A? -~
training error
0+------------2560
7680
15360
training set size, I
Figure 4:
Left: Test of the model for a 256 dimensional boolean classifier trained by minimizing a mean squared error. The sum and difference of the test and training errors
are shown as a function of the normalized training set size in a log-log-plot (base
10). Each point is the mean with standard deviation for ten different choices of a
training set of the given size. The straight line with a
1, corresponding to a 1/1
decay, is shown as a reference.
Right: Prediction of learning curves for a 256 dimensional boolean classifier trained
by minimizing a mean squared error. Measured errors for training set size of
I ~ 2560 are used to fit the two proposed straight lines in a log-log plot. The
three parameters which characterize the learning curves are extracted and used for
extrapolation.
=
A prediction for a boolean classifier with linear decision surface is illustrated in
Fig. 4, (right). The prediction is excellent for this type of classifiers because the
sum and difference of the test and training errors converge quickly to two straight
lines in a log-log-plot. Unfortunately, linear decision surfaces are in general not
adequate for many real-life applications.
The usefulness of the predictive method proposed here can be judged from its performance on real-life sophisticated multi-layer networks. Fig. 5 demonstrates the
validity of the model even for a fully-connected multi-layer network operating in its
non-linear regime to implement an unrealizable digit recognition task. Already for
intermediate sizes of the training set the sum and difference between the test and
training errors are again observed to follow straight lines.
The predictive method was finally tested on sparsely connected multi-layer networks. Fig. 6, (left), shows the test and training errors for two networks trained
for the recognition of handwritten digits. The network termed "old" is commonly
referred to as LeNet [LCBD+90]. The network termed "new" is a modification of
LeN et with additional feature maps. The full size of the database is 60,000 patterns,
331
332
Cortes, Jackel, SoHa, Vapnik, and Denker
log (error)
E lest
+ E train
-1
-2
-3
log ( 11100)
,
1000
10000
.
100000 I
Figure 5: Test of the model for a fully-connected 100-10-10 network. The sum
and the difference of the test and training error are shown as a function of the
normalized training set size in a log-log-plot. Each point is the mean with standard
deviation for 20 different choices of a training set of the given size.
a 50-50 % mixture of the NIST 1 training and test sets.
After training on 12,000 patterns it becomes obvious that the new network will outperform the old network when trained on the full database, but we wish to quantify
the expected improvement. If our predictive method gives a good quantitative
estimate of the new network's test error at 60,000 patterns, we can decide whether
three weeks of training should be devoted to the new architecture.
A log-log-plot based on the three datapoints from the new network result in values
for the three parameters that determine the power-laws used to extrapolate the
learning curves of the new network to the full size of the database, as illustrated in
Fig. 6, (right). The predicted test error at the full size of the database I = 60,000
is less than half of the test error for the old architecture, which strongly suggest
performing the training on the full database. The result of the full training is also
indicated in Fig. 6, (right). The good agreement between predicted and measured
values illustrates the power and applicability of the predictive method proposed
here to real-life applications.
3
Theoretical Foundation
The proposed predictive method based on power-law modeling of the learning curves
is not just heuristic. A fair amount of theoretical work has been done within the
framework of statistical mechanics [SST92] to compute learning curves for simple
classifiers implementing unrealizable rules with non-zero asymptotic error value. A
key assumption of this theoretical approach is that the number of weights in the
network is large.
1
National Institute for Standards and Technology, Special Database 3.
Learning Curves: Asymptotic Values and Rate of Convergence
error
error
0.03 ?
. . ? : old network
- : new network
0.02
: new network
- - - : new network predicted
0.02
.................. a
.
......
C;
0.01
0.01
,
o
20
30 40 50 60
training set size, 111000
--.- ..... ---------- - - 't:)
... -- ..------------n
---
o ~~---------------------.
20 30 40 50 60
training set size, 111000
Figure 6:
Left: Test (circles) and training ( triangles) errors for two networks. The "old" network is what commonly is referred to as LeNet. The network termed "new" is a
modification of LeNet with additional feature maps. The full size of the database
is 60,000 patterns, and it is a 50-50 % mixture of the NIST training and test set.
Right: Test (circles) and training (triangles) errors for the new network. The figure
shows the predicted values of the learning curves in the range 20,000 - 60,000 training patterns for the "new" network, and the actually measured values at 60,000
patterns.
The statistical mechanical calculations support a symmetric power-law decay of the
expected test and training errors to their common asymptotic value. The powerlaws describe the behavior in the large I regime, with an exponent a which falls in
the interval 1/2 ~ a ~ 1. Our numerical observations and modeling of the test and
training errors are in agreement with these theoretical predictions.
We have, moreover, observed a correlation between the exponent a and the asymptotic error value a not accounted for by any of the theoretical models considered so
far. Fig. 7 shows a plot of the exponent a versus the asymptotic error a evaluated
for three different tasks. It appears from this data that the more difficult the target
rule, the smaller the exponent, or the slower the learning. A larger generalization
error for intermediate training set sizes is in such cases due to the combined effect
of a larger asymptotic error and a slower convergence. Numerical results for classifiers of both smaller and larger input dimension support the explanation that this
correlation might be due to the finite size of the input dimension of the classifier
(here 256).
4
Summary
In this paper we propose a practical and principled method for predicting the suitability of classifiers trained on large databases. Such a procedure may eliminate
333
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Cortes, Jackel, Solla, Vapnik, and Denker
exponent,
(X.
?
0.9
~I
0.8
0.7
0.6
0.
0
0.1
0.2
asymptotic
error, a
Figure 1: Exponent of extracted power-law decay as a function of asymptotic
error for three different tasks. The un-realizability of the tasks, as characterized by
the asymptotic error a, can be changed by tuning the strength of a weight-decay
constraint on the norm of the weights of the classifier.
poor classifiers at an early stage of the training procedure and allow for a more
intelligent use of computational resources.
The method is based on a simple modeling of the expected training and test errors,
expected to be valid for large sizes of the training set. In this model both error measures are assumed to follow power-law decays to their common asymptotic
error value, with the same exponent and amplitude characterizing the power-law
convergence.
The validity of the model has been tested on classifiers with linear as well as nonlinear decision surfaces. The free parameters of the model are extracted from data
points obtained at medium sizes of the training set, and an extrapolation gives good
estimates of the test error at large size of the training set.
Our numerical studies of learning curves have revealed a correlation between the
exponent of the power-law decay and the asymptotic error rate. This correlation is
not accounted for by any existing theoretical models, and is the subject of continuing
research.
References
[LCBD+90] Y. Le Cun, B. Boser, J. S. Denker, D. Henderson, R. E. Howard,
W. Hubbard, and L. D. Jackel. Handwritten digit recognition with a
back-propagation network. In Advances in Neural Information Processing Systems, volume 2, pages 396-404. Morgan Kaufman, 1990.
[SST92]
H. S. Seung, H. Sompolinsky, and N. Tishby. Statistical mechanics of
learning from examples. Physical Review A, 45:6056-6091, 1992.
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7,030 | 804 | Asynchronous Dynamics of Continuous
Time Neural Networks
Xin Wang
Computer Science Department
University of California at Los Angeles
Los Angeles, CA 90024
Qingnan Li
Department of Mathematics
University of Southern California
Los Angeles, CA 90089-1113
Edward K. Blum
Department of Mathematics
University of Southern California
Los Angeles, CA 90089-1113
ABSTRACT
Motivated by mathematical modeling, analog implementation and
distributed simulation of neural networks, we present a definition of
asynchronous dynamics of general CT dynamical systems defined
by ordinary differential equations, based on notions of local times
and communication times. We provide some preliminary results
on globally asymptotical convergence of asynchronous dynamics
for contractive and monotone CT dynamical systems. When applying the results to neural networks, we obtain some conditions
that ensure additive-type neural networks to be asynchronizable.
1
INTRODUCTION
Neural networks are massively distributed computing systems. A major issue in parallel and distributed computation is synchronization versus asynchronization (Bertsekas and Tsitsiklis, 1989). To fix our idea, we consider a much studied additive-type
model (Cohen and Grossberg, 1983; Hopfield, 1984; Hirsch, 1989) of a continuoustime (CT) neural network of n neurons, whose dynamics is governed by
n
Xi(t)
= -ajXi(t) + L WijO'j (Jlj Xj (t)) + Ii,
i
= 1,2, ... , n,
(1)
j=1
493
494
Wang. Li. and Blum
with neuron states Xi (t) at time t, constant decay rates ai, external inputs h, gains
neuron activation functions Uj and synaptic connection weights Wij. Simulation and implementation of idealized models of neural networks such as (1) on
centralized computers not only limit the size of networks, but more importantly
preclude exploiting the inherent massive parallelism in network computations. A
truly faithful analog implementation or simulation of neural networks defined by
(1) over a distributed network requires that neurons follow a global clock t, communicate timed states Xj(t) to all others instantaneously and synchronize global
dynamics precisely all the time (e.g., the same Xj(t) should be used in evolution of
all Xi(t) at time t). Clearly, hardware and software realities make it very hard and
sometimes impossible to fulfill these requirements; any mechanism used to enforce
such synchronization may have an important effect on performance of the network. Moreover, absolutely insisting on synchronization contradicts the biological
manifestation of inherent asynchrony caused by delays in nerve signal propagation,
variability of neuron parameters such as refractory periods and adaptive neuron
gains. On the other hand, introduction of asynchrony may change network dynamics, for example, from convergent to oscillatory. Therefore, validity of asynchronous
dynamics of neural networks must be assessed in order to ensure desirable dynamics
in a distributed environment.
JJj,
Motivated by the above issues, we study asynchronous dynamics of general CT dynamical systems with neural networks in particular. Asynchronous dynamics has
been thoroughly studied in the context of iterative maps or discrete-time (DT) dynamical systems; see, e.g., (Bertsekas and Tsitsiklis, 1989) and references therein.
Among other results are that P-contractive maps on Rn (Baudet, 1978) and continuous maps on partially ordered sets (Wang and Parker, 1992) are asynchronizable,
i.e., any asynchronous iterations of these maps will converge to the fixed points
under synchronous (or parallel) iterations. The synchronization issue has also been
addressed in the context of neural networks. In fact, the celebrated DT Hopfield
model (Hopfield, 1982) adopts a special kind of asynchronous dynamics: only one
randomly chosen neuron is allowed to update its state at each iterative step. The
issue is also studied in (Barhen and Gulati, 1989) for CT neural networks. The
approach there is, however, to convert the additive model (1) into a DT version
through the Euler discretization and then to apply the existing result for contractive mappings in (Baudet, 1978) to ensure the discretized system to be asynchronizable. Overall, studies for asynchronous dynamics of CT dynamical systems are
still lacking; there are even no reasonable definitions for what it means, at least to
our knowledge.
In this paper, we continue our studies on relationships between CT and DT dynamical systems and neural networks (Wang and Blum, 1992; Wang, Blum and Li,
1993) and concentrate on their asynchronous dynamics. We first extend a concept
of asynchronous dynamics of DT systems to CT systems, by identifying the distinction between synchronous and asynchronous dynamics as (i) presence or absence of
a common global clock used to synchronize the dynamics of the different neurons
and (ii) exclusion or inclusion of delay times in communication between neurons,
and present some preliminary results for asynchronous dynamics of contractive and
monotone CT systems.
Asynchronous Dynamics of Continuous Time Neural Networks
2
MATHEMATICAL FORMULATION
To be general, we consider a CT dynamical system defined by an n-dimensional
system of ordinary differential equations,
(2)
where Ii : Rn --+ R are continuously differentiable and x(t) E Rn for all t in R+ (the
set of all nonnegative real numbers). In contrast to the asynchronous dynamics
given below, dynamics of this system will be called synchronous. An asynchronous
scheme consists of two families of functions Ci : R+ --+ R+ and rj : R+ --+ R+,
i, j
1, ... , n, satisfying the following constraints: for any t > 0,
=
(i) Initiation: Ci(t) ~ 0 and rJ(t) ~ 0;
(ii) Non-starvation:
Ci'S
are differentiable and l\(t)
(iii) Liveness: limt_oo Ci(t) =
00
> 0;
and limt_oo rJ(t) =
00;
(iv) Accessibility: rj(t) ~ Cj(t).
Given an asynchronous scheme ({cd, {rJ}), the associated asynchronous dynamics
of the system (2) is the solution of the following parametrized system:
(3)
We shall call this system an asynchronized system of the original one (2).
The functions Ci(t) should be viewed as respective "local" times (or clocks) of components i, as compared to the "global" time (or clock) t. As each component i
evolves its state according to its local time Ci(t), no shared global time t is needed
explicitly; t only occurs implicitly. The functions rj(t) should be considered as time
instants at which corresponding values Xi of components j are used by component
i; hence the differences (ci(t) - rj(t? ~ 0 can be interprated as delay times in
communication between the components j and i. Constraint (i) reflects the fact
that we are interested in the system dynamics after some global time instance, say
0; constraint (ii) states that the functions Ci are monotone increasing and hence the
local times evolve only forward; constraint (iii) characterizes the live ness property
of the components and communication channels between components; and, finally,
constraint (iv) precludes the possibility that component i accesses states x j ahead
of the local times Cj(t) of components j which have not yet been generated.
Notice that, under the assumption on monotonicity of Ci(t), the inverses C;l(t) exist
and the asynchronized system (3) can be transformed into
(4)
by letting Yi(t) = Xi( Ci(t? and y} (t) = Xj (rJ(t? = Yj (c;l (rJ(t? for i, j = 1,2, ... , n.
The vector form of (4) can be given by
iJ = C F[Y]
f
(5)
495
496
Wang, Li, and Blum
where yet) = [Yl (t), "" Yn(t)]T, C' = diag(dcl (t)/dt, "" dcn(t)/dt) , F
y [Y;] and
=
_
F[Y]
=
/1 cYi(t) , yHt), "" y~(t))
hcYr(t),
y~(t), "" y~(t))
[
,
fn (i/'l (t), y~(t), ""
= [/1, ""
fn]T,
1
'
y~(t))
Notice that the complication in the way F applies to Y ~imply means ,t hat every
component i will use possibly different "global" states [Yi(t) , y2(t) , "" y~(t)] , This
peculiarity makes the equation (5) fit into none ofthe categories of general functional
1, "., n are equal,
differential equations (Hale, 1977), However, if rJ(t) for i
all the components will use a same global state y
[yHt) , y~(t), .. " y~(t)] and
the asynchronized system (5) assumes a form of retarded functional differential
equations,
=
=
(6)
iJ = c' FcY),
We shall call this case uniformly-delayed, which will be a main consideration in the
next section where we discuss asynchronizable systems,
The system (5) includes some special cases. In a no communication delay situation,
rj(t) Cj(t) for all i and the system (5) reduces to iJ C' F(y), This includes the
simplest case where the local times Ci(t) are taken as constant-time scalings cit of
the global time t; specially, when all Ci(t) = t the system goes back to the original
one (2), If, on the other hand, all the local time~ are identi~al to the global time t
and the communication times take the form of rJ(t) t - OJ(t) one obtains a most
general delayed system
=
=
=
(7)
where the state Yj(t) of component j may have different delay times O)(t) for different other components i.
Finally, we should point out that the above definitions of asynchronous schemes and
dynamics are analogues of their counterparts for DT dynamical systems (Bertsekas
and Tsitsiklis, 1989; Blum, 1990), Usually, an asynchronous scheme for a DT
system defined by a map f : X -+ X, where X Xl X X 2 X '" X X n , consists of a
1, ,.. , n} of subset~ of discrete times (N) at which components
family {Ti ~ N I i
i update their states and a family {rJ : N -+ N Ii
1,2"", n} of communication
times, Asynchronous dynamics (or chaotic iteration, relaxation) is then given by
=
X.(t
I
+ 1) = {
=
=
fi(xl(rt(t)), "', xn(r~(t)))
Xi(t)
if t E ~
otherwise.
Notice that the sets Ti can be interpreted as local times of components i . In fact,
one can define local time functions Ci : N -+ N as Ci(O) = 0 and Ci(t + 1) = Ci(t) + 1
if t E 11 and Ci(t) otherwise. The asynchronous dynamics can then be defined by
Xi(t
+ 1) - Xi(t) = (Ci(t + 1) - ci(t))(fi(xl(rf(t)), ... ,Xn(r~(t))) - Xi(t)),
which is analogous to the definition given in (4).
Asynchronous Dynamics of Continuous Time Neural Networks
3
ASYNCHRONIZABLE SYSTEMS
In general, we consider a CT dynamical system as asynchronizable ifits synchronous
dynamics (limit sets and their asymptotic stability) persists for some set of asynchronous schemes. In many cases, asynchronous dynamics of an arbitrary CT system will be different from its synchronous dynamics, especially when delay times
in communication are present. An example can be given for the network (1) with
symmetric matrix W. It is well-known that (synchronous) dynamics of such networks is quasi-convergent, namely, all trajectories approach a set of fixed points
(Hirsch, 1989). But when delay times are taken into consideration, the networks
may have sustained oscillation when the delays exceed some threshold (Marcus and
Westervelt, 1989). A more careful analysis on oscillation induced by delays is given
in (Wu, 1993) for the networks with symmetric circulant weight matrices.
Here, we focus on asynchronizable systems. We consider CT dynamical systems on
Rn of the following general form
Ax(t) = -x(t) + F(x(t?
(8)
where x(t) ERn, A = diag(a1,a2, ... ,an ) with aj > 0 and F = [Ji] E G 1(Rn). It
is easy to see that a point x E Rn is a fixed point of (8) if and only if x is a fixed
point of the map F. Without loss of generality, we assume that 0 is a fixed point
of the map F. According to (5), the asynchronized version of (8) for an arbitrary
asynchronous scheme ({ cd, {rj}) is
Ay
where jj
3.1
= G'( -y + F[Y]),
(9)
= (jjtct), jj~(t), ... , y~(t)].
Contractive Systems
Our first effort attempts to obtain a result similar to the one for P-contractive
maps in (Baudet, 1978). We call the system (8) strongly P-contractive if there is a
symmetric and invertible matrix S such that IS- 1 F(Sx)1 < Ixl for all x E Rn and
IS- 1 F(Sx)1 Ixl only for x 0; here Ixl denotes the vector with components Ixil
and < is component-wise.
=
=
Theorem 1 If the system (8) is strongly P-contractive, then it is asynchronizable
Ci(t) for all
for any asynchronous schemes without self time delays (i. e., rf (t)
i=1,2, ... ,n).
=
Proof. It is not hard to see that synchronous dynamics of a strongly P-contractive
system is globally convergent to the fixed point O. Now, consider the transformation
z = A- 1 y and the system for z
Ai
=G'( -z + S-1 F[SZ]) =G'( -z + G[Z]),
=
where G[Z]
S-1 FS[Z]. This system has the same type of dynamics as (9).
Define a function E : R+ x Rn --+ R+ by E(t) = z T (t)Az(t)j2, whose derivative
with respect to t is
E = z TG' (-z + G(Z? < IIG'II (-z Tz + IzlT IG(Z)!) < IIG'II( -z Tz + IzlT Izl) ::; O.
497
498
Wang, Li, and Blum
Hence E is an energy function and the asynchronous dynamics converges to the
fixed point O.
0
Our second result is for asynchronous dynamics of contractive systems with no
communication delay. The system (8) is called contractive if there is a real constant
o ~ a < 1 such that
IIF(x) - F(y)1I ~
for all x, y E Rn; here
II . II
allz - yll
denotes the usual Euclidean norm on Rn.
Theorem 2 If the system (8) is contractive, then it is asynchronizable for asynchronous schemes with no communication delay.
Proof. The synchronous dynamics of contractive systems is known to be globally
convergent to a unique fixed point (Kelly, 1990). For an asynchronous scheme with
no communication delay, the system (8) is simplified to Ali G' ( -y + F(y?. We
consider again the function E
y T Ay/2, which is an energy function as shown
below.
=
=
E = YT G' (-y + F(y?
~
IIG/II( -lIyll2 + lIyIlIlF(y)ID <
O.
o
Therefore, the asynchronous dynamics converges to the fixed point O.
For the additive-type neural networks (1), we have
Corollary 1 Let the network (1) have neuron activation functions
type with 0 < uHz) ~ SUPzER ui(z) = 1. If it satisfies the condition
Ui
of sigmoidal
(10)
=
where M
diag(J-ll, ... , J-ln), then it is asynchronizable for any asynchronous
schemes with no communication delay.
Proof. The condition (10) ensures the map F(x)
contractive.
= A-I Wu(M x) + A- 1 I
to be
0
Notice that the condition (10) is equivalent to many existing ones on globally asymptotical stability based on various norms of matrix W, especially the contraction condition given in (Kelly, 1990) and some very recent ones in (Matsuoka, 1992). The
condition (10) is also related very closely to the condition in (Barhen and Gulati,
1989) for asynchronous dynamics of a discretized version of (1) and the condition
in (Marcus and Westervelt, 1989) for the networks with delay.
We should emphasize that the results in Theorem 2 and Corollary 1 do not directly
follow from the result in (Kelly, 1990); this is because local times Ci(t) are allowed
to be much more general functions than linear ones Ci t.
3.2
Monotone Systems
A binary relation ~ on Rn is called a partial order if it satisfies that, for all x, y, z E
x ~ x; (ii) x ~ y and y ~ x imply x = y; and (iii) x -< y and y -< z
imply x -< z. For a partial order ~ on Rn , define ~ on Rn by x ~ y iff x < y
and Xi # Yi for all i
1, .. " n. A map F : Rn -I- Rn is monotone if x ~ y implies
Rn, (i)
=
Asynchronous Dynamics of Continuous Time Neural Networks
F(x) -< F(y). A CT dynamical system of the form (2) is monotone if Xl ~ X2 implies
the trajectories Xl(t), X2(t) with Xl(O) = Xl and X2(0) = X2 satisfy Xl(t) ::5 X2(t)
for all t ~ 0 (Hirsch, 1988).
Theorem 3 If the map F in (8) is monotone, then the system (8) is asynchronizable for uniformly-delayed asynchronous schemes, provided that all orbits x(t) have
compact orbit closure and there is a to > 0 with x(to) ~ x(O) or x(to) ~ x(O).
Proof. This is an application of a Henry's theorem (see Hirsch, 1988) that implies that the asynchronized system (9) in the no communication delay situation
is monotone and Hirsch's theorem (Hirsch, 1988) that guarantees the asymptotic
convergence of monotone systems to fixed points.
0
Corollary 2 If the additive-type neural network (1) with sigmoidal activation functions is cooperative (i.e., Wij > 0 for i # j (Hirsch, 1988 and 1989)), then it is
asynchronizable for uniformly-delayed asynchronous schemes, provided that there is
a to > 0 with x(to) ~ x(O) or x(to) ~ x(O).
Proof. According to (Hirsch, 1988), cooperative systems are monotone. As the
network has only bounded dynamics, the result follows from the above theorem. 0
4
CONCLUSION
By incorporating the concepts of local times and communication times, we have
provided a mathematical formulation of asynchronous dynamics of continuous-time
dynamical systems. Asynchronized systems in the most general form haven't been
studied in theories of dynamical systems and functional differential equations. For
contractive and monotone systems, we have shown that for some asynchronous
schemes, the systems are asynchronizable, namely, their asynchronizations preserve
convergent dynamics of the original (synchronous) systems. When applying these
results to the additive-type neural networks, we have obtained some special conditions for the networks to be asynchronizable.
We are currently investigating more general results for asynchronizable dynamical
systems, with a main interest in oscillatory dynamics.
References
G. M. Baudet (1978). Asynchronous iterative methods for multiprocessors. Journal
of the Association for Computing Machinery, 25:226-244.
J. Barhen and S. Gulati (1989). "Chaotic relaxation" in concurrently asynchronous
neurodynamics. In Proceedings of International Conference on Neural Networks,
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Bertsekas and Tsitsiklis (1989). Parallel and Distributed Computation: Numerical
Methods. Englewood Cliffs, NJ: Prentice Hall.
E. K. Blum (1990). Mathematical aspects of outer-product asynchronous contentaddressable memories. Biological Cybernetics, 62:337-348, 1990.
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E. K. Blum and X. Wang (1992). Stability of fixed-points and periodic orbits, and
bifurcations in analog neural networks. Neural Networks, 5:577-587.
J. Hale (1977). Theory of Functional Differential Equations. New York: SpringerVerlag.
M. W. Hirsch (1988). Stability and convergence in strongly monotone dynamical
systems. J. reine angew. Math., 383:1-53.
M. W. Hirsch (1989). Convergent activation dynamics in continuous time networks.
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J. Hopfield (1984) . Neurons with graded response have collective computational
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D. G. Kelly (1990). Stability in contractive nonlinear neural networks. IEEE Trans.
Biomedi. Eng., 37:231-242.
Q. Li (1993). Mathematical and Numerical Analysis of Biological Neural Networks.
Unpublished Ph.D. Thesis, Mathematics Department, University of Southern California.
C. M. Marcus and R. M. Westervelt (1989). Stability of analog neural networks
with delay. Physical Review A, 39(1):347-359.
K. Matsuoka (1992) . Stability conditions for nonlinear continuous neural networks
with asymmetric connection weights. Neural Networks, 5:495-500 .
J. M. Ortega and W. C. Rheinboldt (1970). Iterative solution of nonlinear equations
in several variables. New York: Academic Press.
X. Wang, E. K. Blum, and Q. Li (1993). Consistency on Local Dynamics and
Bifurcation of Continuous-Time Dynamical Systems and Their Discretizations. To
appear in the AMS proceedings of Symposia in Applied Mathematics, Mathematics
of Computation 1943 - 1993, Vancouver, BC, August, 1993, edited by W. Gautschi.
X. Wang and E. K. Blum (1992). Discrete-time versus continuous-time neural
networks. Computer and System Sciences, 49:1-17 .
X. Wang and D. S. Parker (1992). Computing least fixed points by asynchronous
iterations and random iterations. Technical Report CSD-920025, Computer Science
Department, UCLA.
J .-H. Wu (1993). Delay-Induced Discrete Waves of Large Amplitudes in Neural Networks with Circulant Connection Matrices. Preprint, Department of Mathematics
and Statistics, York University.
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7,031 | 805 | ?
Probabilistic Anomaly Detection In
Dynamic Systems
Padhraic Smyth
Jet Propulsion Laboratory 238-420
California Institute of Technology
4800 Oak Grove Drive
Pasadena, CA 91109
Abstract
This paper describes probabilistic methods for novelty detection
when using pattern recognition methods for fault monitoring of
dynamic systems. The problem of novelty detection is particularly acute when prior knowledge and training data only allow one
to construct an incomplete classification model. Allowance must
be made in model design so that the classifier will be robust to
data generated by classes not included in the training phase. For
diagnosis applications one practical approach is to construct both
an input density model and a discriminative class model. Using
Bayes' rule and prior estimates of the relative likelihood of data
of known and unknown origin the resulting classification equations
are straightforward. The paper describes the application of this
method in the context of hidden Markov models for online fault
monitoring of large ground antennas for spacecraft tracking, with
particular application to the detection of transient behaviour of
unknown origin.
1
PROBLEM BACKGROUND
Conventional control-theoretic models for fault detection typically rely on an accurate model ofthe plant being monitored (Patton, Frank, and Clark, 1989). However,
in practice it common that no such model exists for complex non-linear systems.
The large ground antennas used by JPL's Deep Space Network (DSN) to track
825
826
Smyth
Jet Prcpllslon Laboratory
Mission
Control
Figure 1: Block diagram of typical Deep Space Network downlink
planetary spacecraft fall into this category. Quite detailed analytical models exist
for the electromechanical pointing systems. However, these models are primarily
used for determining gross system characteristics such as resonant frequencies; they
are known to be a poor fit for fault detection purposes.
We have previously described the application of adaptive pattern recognition methods to the problem of online health monitoring of DSN antennas (Smyth and Mellstrom, 1992; Smyth, in press). Rapid detection and identification of failures in the
electromechanical antenna pointing systems is highly desirable in order to minimize
antenna downtime and thus minimise telemetry data loss when communicating with
remote spacecraft (see Figure 1). Fault detection based on manual monitoring of
the various antenna sensors is neither reliable or cost-effective.
The pattern-recognition monitoring system operates as follows. Sensor data such as
motor current, position encoder, tachometer voltages, and so forth are synchronously sampled at 50Hz by a data acquisition system. The data are blocked off into
disjoint windows (200 samples are used in practice) and various features (such as
estimated autoregressive coefficients) are extracted; let the feature vector be fl.
The features are fed into a classification model (every 4 seconds) which in turn provides posterior probability estimates of the m possible states of the system given the
estimated features from that window, p(wdfl). WI corresponds to normal conditions,
the other Wi'S, 1 ~ i ~ m, correspond to known fault conditions.
Finally, since the system has "memory" in the sense that it is more likely to remain
in the current state than to change states, the posterior probabilities need to be
correlated over time. This is achieved by a standard first-order hidden Markov
Probabilistic Anomaly Detection in Dynamic Systems
model (HMM) which models the temporal state dependence. The hidden aspect
of the model reflects the fact that while the features are directly observable, the
underlying system states are not, i.e., they are in effect "hidden." Hence, the purpose
of the HMM is to provide a model from which the most likely sequence of system
states can be inferred given the observed sequence of feature data.
The classifier portion of the model is trained using simulated hard ware faults. The
feed-forward neural network has been the model of choice for this application because of its discrimination ability, its posterior probability estimation properties
(Richard and Lippmann, 1992; Miller, Goodman and Smyth, 1993) and its relatively simple implementation in software. It should be noted that unlike typical
speech recognition HMM applications, the transition probabilities are not estimated
from data but are designed into the system based on prior knowledge of the system mean time between failure (MTBF) and other specific knowledge of the system
configuration (Smyth, in press).
2
LIMITATIONS OF THE DISCRIMINATIVE MODEL
The model described above assumes that there are m known mutually exclusive and
exhaustive states (or "classes") of the system, WI, ... ,Wm . The mutually exclusive
assumption is reasonable in many applications where multiple simultaneous failures
are highly unlikely. However, the exhaustive assumption is somewhat impractical.
In particular, for fault detection in a complex system such as a large antenna, there
are thousands of possible fault conditions which might occur. The probability of
occurrence of any single condition is very small, but nonetheless there is a significant
probability that at least one of these conditions will occur over some finite time.
While the common faults can be directly modelled it is not practical to assign model
states to all the other minor faults which might occur.
As discussed in (Smyth and Mellstrom, 1992; Smyth 1994) a discriminative model
directly models P(Wi I~), the posterior probabilities of the classes given the feature
data, and assumes that the classes WI, . . . ,W m are exhaustive. On the other hand, a
generative model directly models the probability density function of the input data
conditioned on each class, p(~IWi)' and then indirectly determines posterior class
probabilities by application of Bayes' rule. Examples of generative classifiers include
parametric models such as Gaussian classifiers and memory-based methods such as
kernel density estimators. Generative models are by nature well suited to novelty
detection whereas discriminative models have no built-in mechanism for detecting
data which are different to that on which the model was trained. However, there
is a trade-off; because generative models typically are doing more modelling than
just searching for a decision boundary, they can be less efficient (than discriminant
methods) in their use of the data. For example, generative models typically scale
poorly with input dimensionality for fixed training sample size.
3
HYBRID MODELS
A relatively simple and practical approach to the novelty detection problem is to
use both a generative and discriminative classifier (an idea originally suggested to
the author by R. P. Lippmann). An extra "m+ lth" state is added to the model to
827
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Smyth
cover "all other possible states" not accounted for by the known m states. In this
framework, the posterior estimates of the discriminative classifier are conditioned
on the event that the data come from one of the m known classes .
Let the symbol w{1 ,.. .,m} denote the event that the true system state is one of the
known states, let Wm+l be the unknown state, and let p(wm+1I~) be the posterior
probability that the system is in an unknown state given the data. Hence, one can
estimate the posterior probability of individual known states as
(1)
where Pd(wd~,w{1,,, . ,m}) is the posterior probability estimate of state i as provided
by a discriminative model, i.e., given that the system is in one of the known states.
The calculation of p(wm+ll~) can be obtained via the usual application of Bayes'
rule if P(~lwm+d, p(wm+d, and P(~IW{l, ,, . ,m}) are known:
(
I(}) -
P Wm+l -
-
P(~lwm+dp(wm+d
P(I
~ wm+dp(wm+d + P(~Iw{1, ...,m}) ""m'
L...Ji p(Wi)
(2)
Specifying the prior density P(~lwm+d, the distribution of the features conditioned
on the occurrence of the unknown state, can be problematic. In practice we have
used non-informative Bayesian priors for P(~lwm+d over a bounded space of feature
values (details are available in a technical report (Smyth and Mellstrom, 1993)) ,
although the choosing of a prior density for data of unknown origin is basically
ill-posed. The stronger the constraints which can be placed on the features the
narrower the resulting prior density and the better the ability of the overall model
to detect novelty. If we only have very weak prior information, this will translate
into a weaker criterion for accepting points which belong to the unknown category.
The term P(W m+l) (in Equation (2)) must be chosen based on the designer's prior
belief of how often the system will be in an unknown state - a practical choice is
that the system is at least as likely to be in an unknown failure state as any of the
known failure states.
The P(~IW{l, ,, .,m}) term in Equation (2) is provided directly by the generative model. Typically this can be a mixture of Gaussian component densities or a kernel
density estimate over all of the training data (ignoring class labels) . In practice,
for simplicity of implementation we use a simple Gaussian mixture model. Furthermore, because of the afore-mentioned scaling problem with input dimensions, only
a subset of relatively significant input features are used in the mixture model. A
less heuristic approach to this aspect of the problem (with which we have not yet
experimented) would be to use a method such as projection pursuit to project the
data into a lower dimensional subspace and perform the input density estimation in
this space. The main point is that the generative model need not necessarily work
in the full dimensional space of the input features.
Integration of Equations (1) and (2) into the hidden Markov model scheme is straightforward and is not derived here - the HMM now has an extra state, "unknown." The choice oftransition probabilities between the unknown and other states
is once again a matter of design choice. For the antenna application at least, many
of the unknown states are believed to be relatively brief transient phenomena which
Probabilistic Anomaly Detection in Dynamic Systems
last perhaps no longer than a few seconds: hence, the Markov matrix is designed
to reflect these beliefs since the expected duration of any state d[wd (in units of
sampling intervals) must obey
1
I - PH
is the self-transition probability of state
d[wd = - -
where
4
Pii
(3)
Wi.
EXPERIMENTAL RESULTS
For illustrative purposes the experimental results from 2 particular models are compared. Each was applied to monitoring the servo pointing system of a DSN 34m
antenna at Goldstone, California. The models were implemented within Lab View
data acquisition software running in real-time on a Macintosh II computer at the antenna site. The models had previously been trained off-line on data collected some
months earlier. 12 input features were used consisting of estimated autoregressive
coefficients and variance terms from each window of 200 samples of multichannel
data. For both models a discriminative feedforward neural network model (with 8
hidden units, sigmoidal hidden and output activation functions) was trained (using conjugate-gradient optimization) to discriminate between a normal state and 3
known and commonly occurring fault states (failed tachometer, noisy tachometer,
and amplifier short circuit - also known as "compensation loss"). The network
output activations were normalised to sum to 1 in order to provide posterior class
probability estimates.
Model (a) used no HMM and assumed that the 4 known states are exhaustive, i.e.,
it just used the feedforward network. Model (b) used a HMM with 5 states, where a
generative model (a Gaussian mixture model) and a flat prior (with bounds on the
feature values) were used to determine the probability of the 5th state (as described
by Equations (1) and (2)). The same neural network as in model (a) was used as a
discriminator for the other 4 known states. The generative mixture model had 10
components and used only 2 of the 12 input features, the 2 which were judged to be
the most sensitive to system change. The parameters of the HMM were designed
according to the guidelines described earlier. Known fault states were assumed to
be equally likely with 1 hour MTBF's and with 1 hour mean duration. Unknown
faults were assumed to have a 20 minute MTBF and a 10 second mean duration.
Both HMMs used 5-step backwards smoothing, i.e., the probability estimates at
any time n are based on all past data up to time n and future data up to time
n + 5 (using a larger number of backward steps was found empirically to produce
no effect on the estimates).
Figures 2 (a) and (b) show each model's estimates (as a function of time) that
the system is in the normal state. The experiment consisted of introducing known
hardware faults into the system in a controlled manner after 15 minutes and 45
minutes, each of 15 minutes duration.
Model (a) 's estimates are quite noisy and contain a significant number of potential
false alarms (highly undesirable in an operational environment). Model (b) is much
more stable due to the smoothing effect of the HMM. Nonetheless, we note that
between the 8th and 10th minutes, there appear to be some possible false alarms:
829
830
Smyth
- - Discriminative model, no HMM
.. ' ' I' ~ . .
l'
Probability
of nonnal 0.6
cmditionl
0.4
I
0.2
0
0
l?trom1~mof
In
20
taclKmJc1l:r fault
~~~~f
nonnal candiuom
40
~
SO
60
Imrod ctiooof
Time minutes)
alIIUlCIl&&tim lou fault
- - Hybrid model. with HMM
,
rr0.8
Probability
of nonnal 0.6
cmditionl
0.4
0.2
o
0
l~~ctimof
In
tac:homcliCl' fault
20
30
tim of
Rcsum1
nonna1 CCJnditiom
~
60
SO
ctioo of
Time minutell)
c:om'DCllHlim la-. fault
Figure 2: Estimated posterior probability of normal state (a) using no HMM and
the exhaustive assumption (normal + 3 fault states), (b) using a HMM with a
hybrid model (normal + 3 faults + other state).
these data were classified into the unknown state (not shown). On later inspection
it was found that large transients (of unknown origin) were in fact present in the
original sensor data and that this was what the model had detected, confirming
the classification provided by the model. It is worth pointing out that the model
without a generative component (whether with or without the HMM) also detected
a non-normal state at the same time, but incorrectly classified this state as one of
the known fault states (these results are not shown).
Also not shown are the results from using a generative model alone, with no discriminative component. While its ability to detect unknown states was similar to
the hybrid model, its ability to discriminate between known states was significantly
worse than the hybrid model.
The hybrid model has been empirically tested on a variety of other conditions where
various "known" faults are omitted from the discriminative training step and then
Probabilistic Anomaly Detection in Dynamic Systems
presented to the model during testing: in all cases, the anomalous unknown state
was detected by the model, i.e., classified as a state which the model had not seen
before.
5
APPLICATION ISSUES
The model described here is currently being integrated into an interactive antenna
health monitoring software tool for use by operations personnel at all new DSN
antennas. The first such antenna is currently being built at the Goldstone (California) DSN site and is scheduled for delivery to DSN operations in late 1994. Similar
antennas, also equipped with fault detectors of the general nature described here,
will be constructed at the DSN ground station complexes in Spain and Australia in
the 1995-96 time-frame.
The ability to detect previously unseen transient behaviour has important practical
consequences: as well as being used to warn operators of servo problems in realtime, the model will also be used as a filter to a data logger to record interesting
and anomalous servo data on a continuous basis. Hence, potentially novel system
characteristics can be recorded for correlation with other antenna-related events
(such as maser problems, receiver lock drop during RF feedback tracking, etc.) for
later analysis to uncover the true cause of the anomaly. A long-term goal is to
develop an algorithm which can automatically analyse the data which have been
classified into the unknown state and extract distinct sub-classes which can be
added as new explicit states to the HMM monitoring system in a dynamic fashion.
Stolcke and Omohundro (1993) have described an algorithm which dynamically
creates a state model for HMMs for the case of discrete-valued features. The case
of continuous-valued features is considerably more subtle and may not be solvable
unless one makes significant prior assumptions regarding the nature of the datagenerating mechanism.
6
CONCLUSION
A simple hybrid classifier was proposed for novelty detection within a probabilistic framework . Although presented in the context of hidden Markov models for
fault detection, the proposed scheme is perfectly general for generic classification
applications. For example, it would seem highly desirable that fielded automated
medical diagnosis systems (such as various neural network models which have been
proposed in the literature) should always contain a "novelty-detection" component
in order that novel data are identified and appropriately classified by the system.
The primary weakness of the methodology proposed in this paper is the necessity
for prior knowledge in the form of densities for the feature values given the unknown
state. The alternative approach is not to explicitly model the the data from the
unknown state but to use some form of thresholding on the input densities from the
known states (Aitchison, Habbema, and Kay, 1977; Dubuisson and Masson, 1993).
However, direct specification of threshold levels is itself problematic. In this sense,
the specification of prior densities can be viewed as a method for automatically
determining the appropriate thresholds (via Equation (2)).
831
832
Smyth
As a final general comment, it is worth noting that online learning systems must
use some form of novelty detection. Hence, hybrid generative-discriminative models
(a simple form of which has been proposed here) may be a useful framework for
modelling online learning.
Acknowledgements
The author would like to thank Jeff Mellstrom, Paul Scholtz, and Nancy Xiao for
assistance in data acquisition and analysis. The research described in this paper
was performed at the Jet Propulsion Laboratory, California Institute of Technology,
under a contract with the National Aeronautics and Space Administration and was
supported in part by ARPA under grant number NOOOl4-92-J-1860
References
R. Patton, P. Frank, and R. Clark (eds.), Fault Diagnosis in Dynamic Systems:
Theory and Application, New York, NY: Prentice Hall, 1989.
P. Smyth and J. Mellstrom, 'Fault diagnosis of antenna pointing systems using
hybrid neural networks and signal processing techniques,' in Advances in Neural
Information Processing Systems 4, J. E. Moody, S. J. Hanson, R. P. Lippmann
(eds.), San Mateo, CA: Morgan Kaufmann, pp.667-674, 1992.
P. Smyth, 'Hidden Markov models for fault detection in dynamic systems,' Pattern
Recognition, vo1.27, no.l, in press.
M. D. Richard and R. P. Lippmann, 'Neural network classifiers estimate Bayesian
a posteriori probabilities,' Neural Computation, 3(4), pp.461-483, 1992.
J. Miller, R. Goodman, and P. Smyth, 'On loss functions which minimize to conditional expected values and posterior probabilities,' IEEE Transactions on Information Theory, vo1.39, no.4, pp.1404-1408, July 1993.
P. Smyth, 'Probability density estimation and local basis function neural networks,'
in Computational Learning Theory and Natural Learning Systems, T. Petsche, M.
Kearns, S. Hanson, R. Rivest (eds.), Cambridge, MA: MIT Press, 1994.
P. Smyth and J. Mellstrom, 'Failure detection in dynamic systems: model construction without fault training data,' Telecommuncations and Data Acquisition
Progress Report, vol. 112, pp.37-49, Jet Propulsion Laboratory, Pasadena, CA,
February 15th 1993.
A. Stokke and S. Omohundro, 'Hidden Markov model induction by Bayesian merging,' in Advances in Neural Information Processing Systems 5, C. L. Giles, S. J.
Hanson and J. D. Cowan (eds.), San Mateo, CA: Morgan Kaufmann, pp.11-18,
1993.
J. Aitchison, J. D. F. Habbema, and J. W. Kay, 'A critical comparison of two
methods of statistical discrimination,' Applied Statistics, vo1.26, pp.15-25, 1977.
B. Dubuisson and M. Masson, 'A statistical decision rule with incomplete knowledge
about the classes,' Pattern Recognition, vo1.26 , no.l, pp.155-165, 1993.
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7,032 | 806 | Observability of Neural Network
Behavior
Max Garzon
Fernanda Botelho
sarzonmOherme ?. msci.mem.t.edu
botelhoflherme ?. msci.mem.t.edu
Institute for Intelligent Systems
Department of Mathematical Sciences
Memphis State University
Memphis, TN 38152 U.S.A.
Abstract
We prove that except possibly for small exceptional sets, discretetime analog neural nets are globally observable, i.e. all their corrupted pseudo-orbits on computer simulations actually reflect the
true dynamical behavior of the network. Locally finite discrete
(boolean) neural networks are observable without exception.
1
INTRODUCTION
We address some aspects of the general problem of implementation and robustness
of (mainly recurrent) autonomous discrete-time neural networks with continuous
activation (herein referred to as analog networks) and discrete activation (herein,
boolean networks). There are three main sources of perturbations from ideal operation in a neural network. First, the network's parameters may have been contaminated with noise from external sources. Second, the network is being implemented
in optics or electronics (digital or analog) and inherent measurement limitations
preclude the use of perfect information on the network's parameters. Third, as has
been the most common practice so far, neural networks are simulated or implemented on a digital device, with the consequent limitations on precision to which
net parameters can be represented. Finally, for these or other reasons, the activation
functions (e.g. sigmoids) of the network are not known precisely or cannot be evaluated properly. Although perhaps negligible in a single iteration, these perturbations
are likely to accumulate under iteration, even in feedforward nets. Eventually, they
may, in fact, distort the results of the implementation to the point of making the
simulation useless, if not misleading.
455
456
Garzon and Botelho
There is, therefore, an important difference between the intended operation of an
idealized neural network and its observable behavior. This is a classical problem in
systems theory and it has been addressed in several ways. First, the classical notions of distinguishability and observability in control theory (Sontag, 1990) which
roughly state that every pair of system's states are distinguishable by different outputs over evolution in finite time. This is thus a notion of local state observability.
More recently, several results have established more global notions of identifiability of discrete-time feedfoward (Sussmann, 1992; Chen, Lu, Hecht-Nelson, 1993)
and continuous-time recurrent neural nets (Albertini and Sontag, 1993a,b), which
roughly state that for given odd activation functions (such as tanh), the weights
of a neural network are essentially uniquely determined (up to permutation and
cell redundancies) by the input/output behavior of the network. These notions do
assume error-free inputs, weights, and activation functions.
In general, a computer simulation of an orbit of a given dynamical system in the
continuuum (known as a pseudo-orbit) is, in fact, far from the orbit in the ideal
system. Motivated by this problem, Birkhoff introduced the so-called shadowing
property. A system satisfies the shadowing propertyif all pseudo-orbits are uniformly
approximated by actual orbits so that the former capture the long-term behavior
of the system. Bowen showed that sufficiently hyperbolic systems in real euclidean
spaces do have the shadowing property (Bowen, 1978). However, it appears difficult
even to give a characterization of exactly which maps on the interval possess the
property -see e.g. (Coven, Kan, Yorke, 1988). Precise definitions of all terms can
be found in section 2.
By comparison to state observability and identifiability, the shadowing property is
a type of global observability of a system through its dynamical behavior. Since
autonomous recurrent networks can be seen as dynamical systems, it is natural
to investigate this property. Thus, a neural net is observable in the sense that its
behavior (i.e. the sequence of its ideal actions on given initial conditions) can be
observed on computer simulations or discrete implementations, despite inevitable
concomitant approximations and errors.
The purpose of this paper is to explore this property as a deterministic model for
perturbations of neural network behavior in the presence of arbitrary small errors
from various sources. The model includes both discrete and analog networks. In
section 4 we sketch a proof that locally finite boolean neural networks (even with
an infinite number of neurons)' are all observable in this sense. This is not true
in general for analog networks, and section 3 is devoted to sketching necessary and
sufficient conditions for the relatively few analog exceptions for the most common
transfer functions: hard-thresholds, a variety of sigmoids (hyperbolic tangent, logistic, etc.) and saturated linear maps. Finally, section 5 discusses the results and
poses some other problems worthy of further research.
2
DEFINITIONS AND MAIN RESULTS
This section contains notation and precise definitions in a general setting, so as to
include discrete-time networks both with discrete and continuous activations.
Let
f :X
~
X be a continuous map of a compact metric space with metric 1*,
* I.
Observability of Neural Network Behavior
The orbit of x E X is the sequence {x, f(x), ... , fk(x) ... }, i.e. a sequence of points
{xkh~o for which X k + 1 = f(x k ), for all k ~ o. Given a number 6 > 0, a 6-pseudoorbit is a sequence {xk} so that the distances If(xk), xk+11 < 6 for all k ~ o.
Pseudo-orbits arise as trajectories of ideal dynamical processes contaminated by
errors and noise. In such cases, especially when errors propagate exponentially,
it is important to know when the numerical process is actually representing some
meaningful trajectory of the real process.
Definition 2.1 The map f on a metric space X is (globally) observable (equivalently] has the shadowing property] or is traceable) if and only if for every f > 0
there exists a 6 > 0 so that any 6 -pseudo-orbit {xk} is f-approximated by the orbit]
under f] of some point z E X] i.e. Ixk, fk(z) I < f for all k > o.
One might observe that computer simulations only run for finite time. On compact
spaces (as is the case below)' observability can be shown to be equivalent to a similar
property of shadowing finite pseudo-orbits.
'Analog neural network' here means a finite number n of units (or cells), each of
which is characterized by an activation (sometimes called output) function Ui :
R -+ R, and weight matrix W of synaptic strengths between the various units.
Units can assume real-valued activations Xi, which are updated synchronously and
simultaneously at discrete instants of time, according to the equation
+ 1) -
udL wi,ixi(t)].
(1)
i
The total activation of the network at any time is hence given by a vector x in
euclidean space Rn, and the entire network is characterized by a global dynamics
Xi(t
T(x)
u[W x],
(2)
where W x denotes ordinary product and u is the map acting as Ui along the ith
component. This component in a vector x is denoted Xi (as opposed to xk, the kth
term of a sequence). The unit hypercube in Rn is denoted In. An analog network
is then defined as a dynamical system in a finite-dimensional euclidean space and
one may then call a neural network (globally) observable if its global dynamics is an
observable dynamical system. Likewise for boolean networks, which will be defined
precisely in section 4.
We end this section with some background facts about observability on the continuum. It is perhaps surprising but a trivial remark that the identity map of the real
interval is not observable in this sense, since orbits remain fixed, but pseudo-orbits
may drift away from the original state and can, in fact, be dense in the interval.
Likewise, common activation functions of neural networks (such as hard thresholds
and logistic maps) are not observable. For linear maps, observability has long been
known to be equivalent to hyperbolicity (all eigenvalues>. have 1>'1 =f:. 1). Composition of observable maps is usually not observable (take, for instance, a hyperbolic
homeomorphism and its inverse). In contrast, composition of linear and nonobservable activation functions in neural networks are, nevertheless, observable. The
main take-home message can be loosely summarized as follows .
Theorem 2.1 Except for a negligible fraction of exceptions, discrete-time analog
neural nets are observable. All discrete (boolean) neural networks are observable.
457
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Garzon and Botelho
3
ANALOG NEURAL NETWORKS
This section contains (a sketch) of necessary and sufficient conditions for analog
networks to be observable for common types of activations functions.
3.1
HARD-THRESHOLD ACTIVATION FUNCTIONS
It is not hard to give necessary and sufficient conditions for observability of nets
with discrete activation functions of the type
.-
{~
if 1.? ~ Oi
else.
where Oi is a theshold characterizing cell i.
1 : Rn -+ Rn with finite range is observable il and only il it
is continuous at each point of its range.
Lemma 3.1 A map
PROOF. The condition is clearly sufficient. If 1 is continuous at every point of its
range, small enough perturbations X k + 1 of an image I(x k ) have the same image
I(x k+ l ) = l(f(x k )) and hence, for 8 small enough, every 8-pseudo-orbit is traced
by the first element of the pseudo-orbit. Conversely, assume 1 is not continuous at a
poin t of its range 1(XO). Let xl, x 2, ... be a sequence constant under 1 whose image
does not converge to 1(I(xO)) (such a sequence can always be so chosen because
the range is discrete). Let
c=
~
min
2.z,yER ...
I/(x), f(y)l.
For a given 8 > 0 the pseudo-orbit xo, xk, f(xk), 12(xk ), ... is not traceable for k
large enough. Indeed, for any z within ?-distance of xO, either f(z) =I f(xO), in which
case this distance is at least ?, or else they coincide, in which case 1/2(z), l(xk)1 > ?
anyway by the choice of xk. 0
Now we can apply Lemma 3.1 to obtain the following characterization.
Theorem 3.1 A discrete-time neural net T with weight matrix W := (Wij) and
threshold vector 0 is observable if and only ~f for every y in the range
OJ for every i (1 ::; i ::; n).
3.2
01 T, (W Y)i =I
SIGMOIDAL ACTIVATION FUNCTIONS
In this section, we establish the observability of arbitrary neural nets with a fairly
general type of sigmoidal activation functions, as defined next.
Definition 3.1 A map
(j :
R
-+
R is sigmoidal if it is strictly increasing, bounded
(above and below), and continuously differentiable.
Important examples are the logistic map
a?(1.?) ,
1
---:--~
- 1 + exp( -J.L1.?) ,
Observability of Neural Network Behavior
the arctan and the hyperbolic tangent maps
adu) = arctan(J.tu)
, adu) =tanh(u) =
exp(u) - exp(-u)
()
().
exp u + exp -u
Note that, in particular, the range of a sigmoidal map is an open and bounded
interval, which without loss of generality, can be assumed to be the unit interval
I. Indeed, if a neural net has weight matrix Wand activation function a which is
conjugate to an activation function a' by a conjugacy ~, then
a
0
W
--.J
a' ~W ~-1
where denotes conjugacy. One can, moreover, assume that the gain factors in the
sigmoid functions are all J.t = 1 (multiply the rows of W).
--.J
Theorem 3.2 Every neural networks with a sigmoidal activation function has a
strong attractor, and in particular, it is observable.
PROOF. Let a neural net with n cells have weight matrix Wand sigmoidal a.
Consider a parametrized family {T,L}", of nets with sigmoidals given by a", := J.ta.
It is easy to see that each T", (J.t > 0) is conjugate to T. However, W needs to be
replaced by a suitable conjugation with a homeomorphism ~w By Brouwer's fixed
point theorem, T,L has a fixed point p* in In. The key idea in the proof is the fact
that the dynamics of the network admits a Lyapunov function given by the distance
from p*. Indeed,
II
T",(x) - T,L(P*) II~ sup IJT", I II x - p*
y
II,
where J denotes jacobian. Using the chain rule and the fact that the derivatives of
~'" and aiL are bounded (say, below by b and above by B), the Jacobian satisfies
IJT,L(Y) I ~ J.tn(bB)nIWI,
where IW I denotes the determinant of W. Therefore we can choose J.t small enough
that the right-hand side of this expression is less than 1 for arbitrary y, so that T",
is a contraction. Thus, the orbit of the first element in any ?-pseudo-orbit ?-traces
the orbit. 0
3.3
SATURATED-LINEAR ACTIVATION FUNCTIONS
The case of the nondifferentiable saturated linear sigmoid given by the piecewise
linear map
0,
{ u,
I,
for u < 0
for 0 ~ u ~ 1
for u > 1
(3)
presents some difficulties. First, we establish a general necessary condition for
observability, which follows easily for linear maps since shadowing is then equivalent
to hyperbolicity.
Theorem 3.3 If T leaves a segment of positive length pointwise fixed, then T
not observable.
lS
459
460
Garzon and Botelho
Although easy to see in the case of one-dimensional systems due to the fact that
the identity map is not observable, a proof in higher dimensions requires showing
that a dense pseudo-orbit in the fixed segment is not traceable by points outside
the segment. The proof makes use of an auxiliary result.
Lemma 3.2 A linear map L : Rn - Rn, acts along the orbit of a point x in the
unit hypercube either as an attractor to 0, a repellor to infinity, or else as a rigid
rotation or reflection.
en - en
PROOF. By passing to the complexification L' :
of L and then to a
conjugate, assume without loss of generality that L has a matrix in Jordan canonical
form with blocks either diagonal or diagonal with the first upper minor diagonal of
Is. It suffices to show the claim for each block, since the map is a cartesian product
of the restrictions to the subspaces corresponding to the blocks. First, consider the
diagonal case. If the eigenvalues P,I < 1 (P,I > I, respectively), clearly the orbit
Lk(x) _ 0 (II Lk(x) 11- 00). If P,I = I, L acts as a rotation. In the nondiagonal
case, it is easy to see that the iterates of x = (XlJ .. " x m ) are given by
t-l
t
Lt(x)
.-
L (k) ).t-k Xk + + L (k) ).t-k Xk +
1
k=O
2
+ ... + ).txm'
(4)
k=O
The previous argument for the diagonal block still applies for 1).1 =I 1. If 1).1 = 1
and if at least two components of x E In are nonzero, then they are positive and
again I L(x) 11- 00. In the remaining case, L acts as a rotation since it reduces to
multiplication of a single coordinate of x by).. 0
PROOF OF THEOREM 3.3. Assume that T = u 0 Land T leaves invariant a
segment xy positive length. Suppose first that L leaves invariant the same segment
as well. By Lemma 3.2, a pseudo-orbit in the interior of the hypercube In cannot
be traced by the orbit of a point in the hypercube. If L moves the segment xy
invariant under T, we can aSsume without loss of generality it lies entirely on a
hyperplane face F of In and the action of u on L(xy) is just a projection over F.
But in that case, the action of T on the segment is a (composition of two) linear
map(s) and the same argument applies. 0
We point out that, in particular, T may not be observable even if W is hyperbolic.
The condition in Theorem 3.3 is, in fact, sufficient. The proof is more involved and
is given in detail in (Garzon & Botelho, 1994). WIth Theorem 3.3 one can then
determine relatively simple necessary and sufficient conditions for observability (in
terms of the eigenvalues and determinants of a finite number of linear maps). They
establish Theorem 2.1 for saturated-linear activation functions.
4
BOOLEAN NETWORKS
This section contains precise definitions of discrete (boolean) neural networks and
a sketch of the proof that they are observable in general.
Discrete neural networks have a finite number of activations and their state sets are
endowed with an addition and multiplication. The activation function OJ (typically
Observability of Neural Network Behavior
a threshold function) can be given by an arbitrary boolean table, which may vary
from cell to cell. They can, moreover, have an infinite number of cells (the only case
of interest here, since finite booolean networks are trivially observable). However,
since the activation set if is finite, it only makes sense to consider locally finite
networks, for which every cell i only receives input from finitely many others.
A total state is now usually called a configuration. A configuration is best thought
of as a bi-infinite sequence x := XIX2X3 .?? consisting of the activations of all cells
listed in some fixed order. The space of all configurations is a compact metric space
if endowed with any of a number of equivalent metrics, such as lx, YI := 2;'" where
m = inf{i : Xi =1= Yd. In this metric, a small perturbation of a configuration is
obtained by changing the values of x at pixels far away from Xl.
The simplest question about observability in a general space concerns the shadowing
of the identity function. Observability of the identity happens to be a property
characteristic of configuration spaces. Recall that a totally disconnected topological
space is one in which the connected component of every element is itself.
Theorem 4.1 The idenh'ty map id of a compact metric space X is observable iff
X is totally disconnected.
The first step in the proof of Theorem 4.3 below is to characterize observability of
linear boolean networks (i.e. those obeying the superposition principle).
Theorem 4.2 Every linear continuous map has the shadowing property.
For the other step we use a global decomposition T = F 0 L of the global dynamics
of a discrete network as a continuous transformation of configuration space due to
(Garzon & Franklin, 1990). The reader is referred to (Garzon & Botelho, 1992) for
a detailed proof of all the results in this section.
Theorem 4.3 Every discrete (boolean) neural network is observable.
5
CONCLUSION AND OPEN PROBLEMS
It has been shown that the particular combination of a linear map with an activation function is usually globally observable, despite the fact that neither of them
is observable and the fact that, ordinarily, composition destroys observability. Intuitively, this means that observing the input/output behavior of a neural network
will eventually give away the true nature of the network's behavior, even if the
network perturbs its behavior slighlty at each step of its evolution. In simple terms,
such a network cannot fool all the people all of the time.
The results are valid for virtually every type of autonomous first-order network
that evolves in discrete-time, whether the activations are boolean or continuous.
Several results follow from this characterization. For example, in all likelihood there
exist observable universal neural nets, despite the consequent undecidability of their
computational behavior. Also, neural nets are thus a very natural set of primitives
for approximation and implementation of more general dynamical systems. These
and other consequences will be explored elsewhere (Botelho & Garzon, 1994).
461
462
Garzon and Botelho
Natural questions arise from these results. First, whether observability is a general
property of most analog networks evolving in continuous time as well. Second, what
other type of combinations of non observable systems of more general types creates
observability, i.e. to what extent neural networks are peculiar in this regard. For
example, are higher-order neural networks observable? Those with sigma-pi units?
Finally, there is the broader question of robustness of neural network implementations, which bring about inevitable errors in input and/or weights. The results in
this paper give a deeper explanation for the touted robustness and fault-tolerance
of neural network solutions. But, further, they also seem to indicate that it may be
possible to require that neural net solutions have observable behavior as well, without a tradeoff in the quality of the solution. An exact formulation of this question
is worthy of further research.
Acknow ledgements
The work of the first author was partially done while on support from NSF grant
CCR-9010985 and CNRS-France.
References
F. Albertini and E.D. Sontag. (1993) Identifiability of discrete-time neural networks.
In Proc. European Control Conference, 460-465. Groningen, The Netherlands:
Springer-Verlag.
F. Albertini and E.D. Sontag. (1993) For neural networks, function determines
form. Neural Networks 6(7): 975-990.
F. Botelho and M. Garzon. (1992) Boolean Neural Nets are Observable, Memphis
State University: Technical Report 92-18.
F. Botelho and M. Garzon. (1994) Generalized Shadowing Properties. J. Random
and Computat?onal Dynamics, in print.
R. Bowen. (1978) On Axiom A diffeomorphisms. In CBMS Regional Conference
Ser?es ?n Math. 35. Providence, Rhode Island: American Math. Society.
A.M. Chen, H. Lu, and R. Hecht-Nielsen, (1993) On the Geometry of Feedforward
Neural Network Error Surfaces. Neural Computat?on 5(6): 910-927.
E. Coven, 1. Kan, and J. Yorke. (1988) Pseudo-orbit shadowing in the family of
tent maps. Trans. AMS 308: 227-241.
M. Garzon and S. P. Franklin.
Complex Systems 4(5): 509-518.
(1990) Global dynamics in neural networks II.
M. Garzon and F. Botelho. (1994) Observability of Discrete-time Analog Networks,
preprint.
E.D. Sontag.
(1990) Mathemat~?cal Control Theory: Deterministic Fin?teDimens?onal Dynam?cal Systems. New York: Springer-Verlag.
H. Sussmann. (1992) Uniqueness of the Weights for Minimal Feedforward Nets with
a Given Input-Output Map. Neural Networks 5(4): 589-593.
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elsewhere:1 free:1 side:1 deeper:1 institute:1 characterizing:1 face:1 tolerance:1 regard:1 dimension:1 valid:1 author:1 coincide:1 far:3 bb:1 observable:33 compact:4 global:7 mem:2 assumed:1 xi:4 continuous:11 table:1 nature:1 transfer:1 european:1 complex:1 main:3 dense:2 noise:2 arise:2 referred:2 en:2 precision:1 obeying:1 xl:2 lie:1 third:1 jacobian:2 theorem:13 showing:1 explored:1 admits:1 consequent:2 concern:1 exists:1 yer:1 sigmoids:2 cartesian:1 chen:2 lt:1 distinguishable:1 likely:1 explore:1 partially:1 applies:2 springer:2 kan:2 satisfies:2 determines:1 identity:4 hard:4 determined:1 except:2 uniformly:1 infinite:3 acting:1 hyperplane:1 lemma:4 called:3 total:2 e:1 meaningful:1 exception:3 people:1 support:1 |
7,033 | 807 | Memory-Based Methods for Regression
and Classification
Thomas G. Dietterich and Dietrich Wettschereck
Department of Computer Science
Oregon State University
Corvallis, OR 97331-3202
Chris G. Atkeson
MIT AI Lab
545 Technology Square
Cambridge, MA 02139
Andrew W. Moore
School of Computer Science
Carnegie Mellon University
Pittsburgh, PA 15213
Memory-based learning methods operate by storing all (or most) of the training data
and deferring analysis of that data until "run time" (i.e., when a query is presented
and a decision or prediction must be made). When a query is received, these
methods generally answer the query by retrieving and analyzing a small subset of
the training data-namely, data in the immediate neighborhood of the query point.
In short, memory-based methods are "lazy" (they wait until the query) and "local"
(they use only a local neighborhood). The purpose of this workshop was to review
the state-of-the-art in memory-based methods and to understand their relationship
to "eager" and "global" learning algorithms such as batch backpropagation.
There are two essential components to any memory-based algorithm: the method
for defining the "local neighborhood" and the learning method that is applied to
the training examples in the local neighborhood.
We heard several talks on issues related to defining the "local neighborhood". Federico Girosi and Trevor Hastie reviewed "kernel" methods in classification and regression. A kernel function K(d) maps the distance d from the query point to a
training example into a real value. In the well-known Parzen window approach, the
kernel is a fixed-width gaussian, and a new example is classified by taking a weighted
vote of the classes of all training examples, where the weights are determined by
the gaussian kernel. Because of the "local" shape of the gaussian, distant training
examples have essentially no influence on the classification decision. In regression
problems, a common approach is to construct a linear regression fit to the data,
where the squared error from each data point is weighted by the kernel.
=
Hastie described the kernel used in the LOESS method: K(d) (1_d 3)3 (0::; d::; 1
and K(d) = 0 otherwise). To adapt to the local density of training examples, this
kernel is scaled to cover the kth nearest neighbor. Many other kernels have been
explored, with particular attention to bias and variance at the extremes of the
1165
1166
Dietterich, Wettschereck, Atkeson, and Moore
training data. Methods have been developed for computing the effective number of
parameters used by these kernel methods.
Girosi pointed out that some "global" learning algorithms (e.g., splines) are equivalent to kernel methods. The kernels often have informative shapes. If a kernel
places most weight near the query point, then we can say that the learning algorithm is local, even if the algorithm performs a global analysis of the training data
at learning time. An open problem is to determine whether multi-layer sigmoidal
networks have equivalent kernels and, if so, what their shapes are.
David Lowe described a classification algorithm based on gaussian kernels. The
kernel is scaled by the mean distance to the k nearest neighbors. His Variablekernel Similarity Metric (VSM) algorithm learns the weights of a weighted Euclidean
distance in order to maximize the leave-one-out accuracy of the algorithm. Excellent
results have been obtained on benchmark tasks (e.g., NETtalk) .
Patrice Simard described the tangent distance method. In optical character recognition, the features describing a character change as that character is rotated, translated, or scaled. Hence, each character actually corresponds to a manifold of points
in feature space. The tangent distance is a planar approximation to the distance between two manifolds (for two characters). Using tangent distance with the nearest
neighbor rule gives excellent results in a zip code recognition task.
Leon Bottou also employed a sophisticated distance metric by using the Euclidean
distance between the hidden unit activations of the final hidden layer in the Bell
Labs "LeNet" character recognizer. A simple linear classifier (with weight decay)
was constructed to classify each query. Bottou also showed that there is a tradeoff
between the quality of the distance metric and the locality of the learning algorithm.
The tangent distance is a near-perfect metric, and it can use the highly local firstnearest-neighbor rule. The hidden layer of the LeNet gives a somewhat better
metric, but it requires approximately 200 "local" examples. With the raw features,
LeNet itself requires all of the training examples.
We heard several talks on methods that are local but not lazy. John Platt described
his RAN (Resource Allocating Network) that learns a linear combination of radial
basis functions by iterative training on the data. Bernd Fritzke described his improvements to RAN. Stephen Omohundro explained model merging, which initially
learns local patches and, when the data justifies, combines primitive patches into
larger high-order patches. Dietrich Wettschereck presented BNGE, which learns a
set of local axis-parallel rectangular patches.
Finally, Andrew Moore, Chris Atkeson, and Stefan Schaal described integrated
memory-based learning systems for control applications. Moore's system applies
huge amounts of cross-validation to select distance metrics, kernels, kernel widths,
and so on. Atkeson advocated radical localism-all algorithm parameters should be
determined by lazy, local methods. He described algorithms for obtaining confidence
intervals on the outputs of local regression as well as techniques for outlier removal.
One method seeks to minimize the width of the confidence intervals.
Some of the questions left unanswered by the workshop include these: Are there inherent computational penalties that lazy methods must pay (but eager methods can
avoid)? How about the reverse? For what problems are local methods appropriate?
| 807 |@word dietterich:2 hence:1 lenet:3 question:1 open:1 moore:4 seek:1 nettalk:1 width:3 kth:1 distance:12 chris:2 manifold:2 omohundro:1 performs:1 code:1 relationship:1 activation:1 must:2 john:1 common:1 distant:1 informative:1 shape:3 girosi:2 purpose:1 recognizer:1 he:1 mellon:1 corvallis:1 cambridge:1 ai:1 benchmark:1 short:1 weighted:3 stefan:1 pointed:1 mit:1 immediate:1 defining:2 gaussian:4 sigmoidal:1 avoid:1 similarity:1 constructed:1 retrieving:1 david:1 showed:1 namely:1 bernd:1 combine:1 schaal:1 improvement:1 reverse:1 multi:1 integrated:1 somewhat:1 zip:1 initially:1 hidden:3 employed:1 window:1 determine:1 maximize:1 stephen:1 memory:6 issue:1 classification:4 what:2 loess:1 art:1 adapt:1 developed:1 cross:1 construct:1 technology:1 prediction:1 axis:1 regression:5 essentially:1 metric:6 scaled:3 classifier:1 platt:1 inherent:1 control:1 unit:1 spline:1 kernel:17 review:1 tangent:4 removal:1 local:16 interval:2 analyzing:1 operate:1 approximately:1 huge:1 validation:1 highly:1 extreme:1 near:2 fritzke:1 storing:1 allocating:1 fit:1 hastie:2 bias:1 backpropagation:1 understand:1 tradeoff:1 neighbor:4 taking:1 euclidean:2 advocated:1 whether:1 bell:1 confidence:2 radial:1 classify:1 penalty:1 wait:1 made:1 cover:1 atkeson:4 influence:1 generally:1 heard:2 subset:1 equivalent:2 map:1 amount:1 primitive:1 attention:1 global:3 eager:2 rectangular:1 pittsburgh:1 answer:1 rule:2 iterative:1 density:1 reviewed:1 his:3 unanswered:1 carnegie:1 parzen:1 obtaining:1 squared:1 excellent:2 bottou:2 pa:1 recognition:2 simard:1 wettschereck:3 run:1 place:1 oregon:1 vsm:1 patch:4 decision:2 lowe:1 ran:2 lab:2 layer:3 pay:1 parallel:1 learns:4 minimize:1 square:1 accuracy:1 variance:1 explored:1 decay:1 basis:1 translated:1 workshop:2 essential:1 raw:1 merging:1 talk:2 leon:1 optical:1 justifies:1 effective:1 department:1 classified:1 query:8 combination:1 locality:1 neighborhood:5 trevor:1 character:6 larger:1 say:1 deferring:1 otherwise:1 federico:1 lazy:4 explained:1 outlier:1 applies:1 itself:1 corresponds:1 patrice:1 final:1 resource:1 ma:1 dietrich:2 describing:1 sophisticated:1 actually:1 change:1 determined:2 planar:1 appropriate:1 batch:1 until:2 vote:1 select:1 thomas:1 perfect:1 leave:1 rotated:1 include:1 andrew:2 radical:1 quality:1 nearest:3 school:1 received:1 |
7,034 | 808 | Training Neural Networks with
Deficient Data
Volker Tresp
Siemens AG
Central Research
81730 Munchen
Germany
[email protected]
Subutai Ahmad
Interval Research Corporation
1801-C Page Mill Rd.
Palo Alto, CA 94304
[email protected]
Ralph N euneier
Siemens AG
Central Research
81730 Munchen
Germany
[email protected]
Abstract
We analyze how data with uncertain or missing input features can
be incorporated into the training of a neural network. The general solution requires a weighted integration over the unknown or
uncertain input although computationally cheaper closed-form solutions can be found for certain Gaussian Basis Function (GBF)
networks. We also discuss cases in which heuristical solutions such
as substituting the mean of an unknown input can be harmful.
1
INTRODUCTION
The ability to learn from data with uncertain and missing information is a fundamental requirement for learning systems. In the "real world" , features are missing
due to unrecorded information or due to occlusion in vision, and measurements are
affected by noise. In some cases the experimenter might want to assign varying
degrees of reliability to the data.
In regression, uncertainty is typically attributed to the dependent variable which is
assumed to be disturbed by additive noise. But there is no reason to assume that
input features might not be uncertain as well or even missing competely.
In some cases, we can ignore the problem: instead of trying to model the relationship between the true input and the output we are satisfied with modeling the
relationship between the uncertain input and the output. But there are at least two
128
Training Neural Networks with Deficient Data
reasons why we might want to explicitly deal with uncertain inputs. First, we might
be interested in the underlying relationship between the true input and the output
(e.g. the relationship has some physical meaning). Second, the problem might be
non-stationary in the sense that for different samples different inputs are uncertain
or missing or the levels of uncertainty vary. The naive strategy of training networks
for all possible input combinations explodes in complexity and would require sufficient data for all relevant cases. It makes more sense to define one underlying true
model and relate all data to this one model. Ahmad and Tresp (1993) have shown
how to include uncertainty during recall under the assumption that the network
approximates the "true" underlying function. In this paper, we first show how input uncertainty can be taken into account in the training of a feedforward neural
network. Then we show that for networks of Gaussian basis functions it is possible
to obtain closed-form solutions. We validate the solutions on two applications.
2
THE CONSEQUENCES OF INPUT UNCERTAINTY
Consider the task of predicting the dependent variable l y E ~ from the input
vector x E ~M consisting of M random variables. We assume that the input
data {(xklk = 1,2, ... , K} are selected independently and that P(x) is the joint
probability distribution of x. Outputs {(yklk = 1,2, ... , K} are generated following
the standard signal-plus-noise model
yk
= /(xk) + (k
=
where {(klk 1,2, ... , K} denote zero-mean ran'dom variables with probability density Pc(t:). The best predictor (in the mean-squared sense) of y given the input x
is the regressor defined by E(ylx)
J y P(ylx) dx f(x), where E denotes the
expectation. Unbiased neural networks asymptotically (K -+ 00) converge to the
regressor.
=
=
To account for uncertainty in the independent variable we assume that we do not
have access to x but can only obtain samples from another random vector z E ~M
with
zk = xk + Ok
=
where {Ok Ik 1,2, ... , K} denote independent random vectors containing M random
variables with joint density P6(6).2
= 1,2, ... , K} approximates
E(ylz) = P~z) y P(ylx) P(zlx) P(x) dydx = P~z) /(x) P6(Z - x) P(x) dx.
A neural network trained with data {(zk, yk)lk
J
J
(1)
Thus, in general E(ylz) # /(z) and we obtain a biased solution. Consider the
case that the noise processes can be described by Gaussians Pc(() = G((j 0, O'Y) and
P6(6) G(Oj 0, 0') where, in our notation, G(Xj m, s) stands for
=
G(x' m s) ,
,
1 1 M (x? - m?)2
- (211')M/2
n:;l
Sj
exp[-- ""
2~
J
s]
J]
lOur notation does not distinguish between a random variable and its realization.
this point, we assume that P6 is independent of x.
2 At
129
130
Tresp, Ahmad, and Neuneier
?
f(x)
E(y!x)
y
t
E(ylz)
I
F \j
./ \
~
Figure 1: The top half of the figure shows the probabilistic model. In an example,
the bottom half shows E(Ylx)
f( x) (continuous), the input noise distribution
(dotted) and E(ylz ) (dashed).
=
where m, s are vectors with the same dimensionality as x (here M). Let us take a
closer look at four special cases.
Certain input. If t7
= 0 (no input noise), the integral collapses and E(ylz) = fez).
Uncertain input. If P(x) varies much more slowly than P(zlx), Equation 1 described the convolution of f(x) with the noise process P6(Z - x). Typical noise
processes will therefore blur or smooth the original mapping (Figures 1). It is
somewhat surprising that the error on the input results in a (linear) convolution
integral. In some special cases we might be able to recover f( x) from an network
trained on deficient data by deconvolution, although one should use caution since
deconvolution is very error sensitive.
Unknown input. If t7j - 00 then the knowledge of Zj does not give us any information about Xj and we can consider the jth input to be unknown. Our formalism
therefore includes the case of missing inputs as special case. Equation 1 becomes
an integral over the unknown dimensions weighted by P(x) (Figure 2).
Linear approximation. If the approximation
(2)
=
is valid, the input noise can be transformed into output noise and E(ylz)
fez).
This results can also be derived using Equation 1 if we consider that a convolution of
a linear function with a symmetrical kernel does not change the function. This result
tells us that if f(x) is approximately linear over the range where P6(6) has significant
Training Neural Networks with Deficient Data
'."r---~-~--'
'.2
...
=
Figure 2: Left: samples yk
f(xt, x~) are shown (no output noise). Right: with
one input missing, P(yIX1) appears noisy.
amplitude we can substitute the noisy input and the network will still approximate
f(x). Similarly, the mean mean(xi) of an unknown variable can be substituted
for an unknown input, if f(x) is linear and xi is independent of the remaining
input variables. But in all those cases, one should be aware of the potentially large
additional variance (Equation 2).
3
MAXIMUM LIKELIHOOD LEARNING
In this section, we demonstrate how deficient data can be incorporated into the
training of feedforward networks. In a typical setting, we might have a number of
complete data, a number of incomplete data and a number of data with uncertain
features. Assuming independent samples and Gaussian noise, the log-likelihood I
for a neural network NNw with weight vector W becomes
K
K
1= 2:logP(zk,yk)
k=1
= 2: log
k=1
J
G(yk jNNw(x),(1Y) G(zk jX ,(1k) P(x) dx.
Note that now, the input noise variance is allowed to depend on the sample k. The
gradient of the log-likelihood with respect to an arbitrary weight Wi becomes3
~ 8IogP(zk, yk)
8w. = L...J
8w'
l
k=1
l
01
J(yk - NNw (x))
8N:~(x)
1
~
1
= ((1y)2 k=1'
L...J P(zk yk) X
G(yk;NNw(x),(1Y) G(zk;X,(1k) P(x) dx.
(3)
First, realize that for a certain sample k ((1k --+ 0): 8IogP(zk,yk)/8wi =
(yk _ N N w(zk))/((1Y)2 8N Nw(zk)/8wi which is the gradient used in normal backpropagation. For uncertain data, this gradient is replaced by an averaged gradient. The integral averages the gradient over possible true inputs x weighted
by the probability of P(xlzk,yk) = P(zkl x ) P(ykl x ) P(x)/p(zk,yk). The term
3This equation can also be obtained via the EM formalism. A similar equation was
obtained by Buntine and Weigend (1991) for binary inputs.
131
132
Tresp, Ahmad, and Neuneier
P(ykl x ) == G(ykjNNw(x),D''') is of special importance since it weights the gradient higher when the network prediction NNw (x) agrees with the target yk. This
term is also the main reason why heuristics such as substituting the mean value
for a missing variable can be harmful: if, at the substituted input, the difference
between network prediction and target is large, the error is also large and the data
point contributes significantly to the gradient although it is very unlikely that the
substitutes value was the true input.
In an implementation, the integral needs to be approximated by a finite sum (i. e.
Monte-Carlo integration, finite-difference approximation etc.). In the experiment
described in Figure 3, we had a 2-D input vector and the data set consisted of both
complete data and data with one missing input. We used the following procedure
1. Train the network using the complete data. Estimate (U Il )2. We used (U II )2 ~
(Ec /(K - H?, where Ec is the training error after the network was trained with
only the complete data, and H is the number of hidden units in the network.
2. Estimate the input density P(x) using Gaussian mixtures (see next section).
3. Include the incomplete training patterns in the training.
4. For every incomplete training pattern
? Let z~ be the certain input and let zt be the missing input, and z1c = (z~, zt) .
? Approximate (assuming -1/2 < Xj < 1/2, the hat stands for estimate)
8 log P(z~, y1c)
8Wi
:::::
1 1
J (ulI)2
1
p(z~
,
J/2
~
y1c) . L..J ?y1c - N Nw(z~, j / J? x
J=-J/2
where
4
GAUSSIAN BASIS FUNCTIONS
The required integration in Equation 1 is computationally expensive and one would
prefer closed form solutions. Closed form solutions can be found for networks which
are based on Gaussian mixture densities. 4 Let's assume that the joint density is
given by
N
P(x) ==
L G(x; Ci, Si) P(Wi),
i=l
where Ci is the location of the center of the ith Gaussian and and Sij corresponds to
the width of the ith Gaussian in the jth dimension and P(Wi) is the prior probability
of Wi. Based on this model we can calculate the expected value of any unknown
4Gaussian mixture learning with missing inputs is also addressed by Ghahramani and
Jordan (1993). See also their contribution in this volume.
Training Neural Networks with Deficient Data
0.1
0.1
28c
28c,
225m
125c
125c,
128m
28c
225m
28c,
225m
mean
subst
Figure 3: Regression. Left: We trained a feedforward neural network to predict the
housing price from two inputs (average number of rooms, percent of lower status
population (Tresp, Hollatz and Ahmad (1993?. The training data set contained
varying numbers of complete data points (c) and data points with one input missing
(m). For training, we used the method outlined in Section 3. The test set consisted
of 253 complete data. The graph (vertical axis: generalization error) shows that by
including the incomplete patterns in the training, the performance is significantly
improved. Right: We approximated the joint density by a mixture of Gaussians.
The incomplete patterns were included by using the procedure outlined in Section 4. The regression was calculated using Equation 4. As before, including the
incomplete patterns in training improved the performance. Substituting the mean
for the missing input (column on the right) on the other hand, resulted in worse
performance than training of the network with only complete data.
i
_.
-.-
0.86
1??84
-
1)
io.n
ic:
? 0.82
I
0.74 r-----.-----~--__,
0.7
0.8
60.68
tf!.
1000
2000
3000
# of data with miss. feat.
234
5
# of missing features
Figure 4: Left: Classification performance as a function of the number of missing
features on the task of 3D hand gesture recognition using a Gaussian mixtures
classifier (Equation 5). The network had 10 input units, 20 basis functions and 7
output units. The test set contained 3500 patterns. (For a complete description
of the task see (Ahmad and Tresp, 1993).) Class-specific training with only 175
complete patterns is compared to the performance when the network is trained with
an additional 350, 1400, and 3325 incomplete patterns. Either 1 input (continuous)
or an equal number of 1-3 (dashed) or 1-5 (dotted) inputs where missing. The
figure shows clearly that adding incomplete patterns to a data set consisting of
only complete patterns improves performance. Right: the plot shows performance
when the network is trained only with 175 incomplete patterns. The performance
is relatively stable as the number of missing features increases.
133
134
Tresp, Ahmad, and Neuneier
variable
1993)
XU
from any set of known variables xn using (Tresp, Hollatz and Ahmad,
E(xUlxn) =
E;
E.=1
ciG(xn; ci,si) P(w.)
G(xnj cf, sf) P(Wi)
(4)
Note, that the Gaussians are projected onto the known dimensions. The last equation describes the normalized basis function network introduced by Moody and
Darken (1989).
Classifiers can be built by approximating the class-specific data distributions
P(xlclassi) by mixtures of Gaussians. Using Bayes formula, the posterior class
probability then becomes
P(classi)P(xlclassi)
P( 1 I)
c ass, x = "L..J; P( class; )P(xI
'
class;)
(5)
We now assume that we do not have access to x but to z where, again, P(zlx) =
G(z; x, 0'). The log-likelihood of the data now becomes
K
N
K
N
1 = L:log jL:G(X;Ci,Si)P(Wi) G(zk;x,O'k) dx = LlogLG(zk;ci,Sf)P(wi)
k=1
i=1
k=1
.=1
=
where (Sf;)2
s~; + (O'j)2. We can use the EM approach (Dempster, Laird and
Rubin, 1977) to obtain the following update equations. Let Ci;, s.; and P(w.) denote
current parameter estimates and let Of; = (Ci;(O'j)2 + z;s~;)/(Sf;? and Df; =
?O'j)2 s~;)/(Sf;)2. The new estimates (indicated by a hat) can be obtained using
P(wdz k )
G(zk; Ci, Sf) P(Wi)
K
P(w.)
(6)
Ef=1 G(Zk;C;, Sf) pew;)
1 L:
K
P(w.lz k)
(7)
A
k=1
K
k
k
Ek=10i; P(wdz )
EKk=1 P(Wil z k)
A
Ci;
(8)
A
A2
si;
Ef=1[Df;
+ (Of; K
Ci;)2] P(wdz k )
Ek=1 P(Wi Izk)
A
(9)
These equations can be solved by alternately using Equation 6 to estimate P(wdz k )
and Equations 7 to 9 to update the parameter estimates. If uk
0 for all k (only
certain data) we obtain the well known EM equations for Gaussian mixtures (Duda
and Hart (1973), page 200). Setting 0': = 00 represents the fact that the jth input
is missing in the kth data point and Of; = Cij, Dfj = s~;. Figure 3 and Figure 4
show experimental results for a regression and a classification problem.
=
Training Neural Networks with Deficient Data
5
EXTENSIONS AND CONCLUSIONS
We can only briefly address two more aspects. In Section 3 we only discussed
regression. We can obtain similar results for classification problems if the costfunction is a log-likelihood function (e.g. the cross-entropy, the signal-plus-noise
model is not appropriate). Also, so far we considered the true input to be unobserved
data. Alternatively the true inputs can be considered unknown parameters. In this
case, the goal is to substitute the maximum likely input for the unknown or noisy
input. We obtain as log-likelihood function
~[_~ (y1:
I
(X
L...J
2
- N N w (x1c)? _ ~ ~ (x;- zt? + I P( 1:)]
(qy)2
2~
(q~)2
og X .
1:=1
1=1
1
The l.earning frocedure consists of finding optimal values for network weights wand
true mputs x .
6
CONCLUSIONS
Our paper has shown how deficient data can be included in network training. Equation 3 describes the solution for feedforward networks which includes a computationallyexpensive integral. Depending on the application, relatively cheap approximations might be feasible. Our paper hinted at possible pitfalls of simple heuristics. Particularly attractive are our results for Gaussian basis functions which allow
closed-form solutions.
References
Ahmad, S. and Tresp, V. (1993). Some solutions to the missing feature problem in vision.
In S. J. Hanson, J. D. Cowan and C. 1. Giles, (Eds.), Neural Information Processing
Systems 5. San Mateo, CA: Morgan Kaufmann.
Buntine, W. L. and Weigend, A. S. (1991). Bayesian Back-Propagation. Complex systems,
Vol. 5, pp. 605-643.
Dempster, A. P., La.ird, N. M. and Rubin, D. B. (1977). Maximum likelihood from incomplete data via the EM algorithm. J. Royal Statistical Society Series B, 39, pp. 1-38.
Duda, R. O. and Hart, P. E. (1973). Pattern Classification and Scene Analysis.
Wiley and Sons, New York.
John
Ghahramani, Z. and Jordan, M. I. (1993). Function approximation via density estimation
using an EM approach. MIT Computational Cognitive Sciences, TR 9304.
Moody, J. E. and Darken, C. (1989). Fast learning in networks oflocally-tuned processing
units. Neural Computation, Vol. 1, pp. 281-294.
Tresp, V., Hollatz J. and Ahmad, S. (1993). Network structuring and tra.ining using rulebased knowledge. In S. J. Hanson, J. D. Cowan and C. L. Giles, (Eds.), Neural Information
Processing Systems 5. San Mateo, CA: Morgan Kaufmann.
Tresp, V., Ahmad, S. and Neuneier, R. (1993). Uncerta.inty in the Inputs of Neural
Networks. Presented at Neural Networks for Computing 1993.
135
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7,035 | 809 | Bayesian Self-Organization
Alan L. Yuille
Division of Applied Sciences
Harvard University
Cambridge, MA 02138
Stelios M. Smirnakis
Lyman Laboratory of Physics
Harvard University
Cambridge, MA 02138
Lei Xu *
Dept. of Computer Science
HSH ENG BLDG, Room 1006
The Chinese University of Hong Kong
Shatin, NT
Hong Kong
Abstract
Recent work by Becker and Hinton (Becker and Hinton, 1992)
shows a promising mechanism, based on maximizing mutual information assuming spatial coherence, by which a system can selforganize itself to learn visual abilities such as binocular stereo. We
introduce a more general criterion, based on Bayesian probability
theory, and thereby demonstrate a connection to Bayesian theories of visual perception and to other organization principles for
early vision (Atick and Redlich, 1990). Methods for implementation using variants of stochastic learning are described and, for the
special case of linear filtering, we derive an analytic expression for
the output.
1
Introduction
The input intensity patterns received by the human visual system are typically
complicated functions of the object surfaces and light sources in the world. It
*Lei Xu was a research scholar in the Division of Applied Sciences at Harvard University
while this work was performed.
1001
1002
Yuille, Smimakis, and Xu
seems probable, however, that humans perceive the world in terms of surfaces and
objects (Nakayama and Shimojo, 1992). Thus the visual system must be able to
extract information from the input intensities that is relatively independent of the
actual intensity values. Such abilities may not be present at birth and hence must
be learned. It seems, for example, that binocular stereo develops at about the age
of two to three months (Held, 1981).
Becker and Hinton (Becker and Hinton, 1992) describe an interesting mechanism
for self-organizing a system to achieve this. The basic idea is to assume spatial
coherence of the structure to be extracted and to train a neural network by maximizing the mutual information between neurons with disjoint receptive fields. For
binocular stereo, for example, the surface being viewed is assumed flat (see (Becker
and Hinton, 1992) for generalizations of this assumption) and hence has spatially
constant disparity. The intensity patterns, however, do not have any simple spatial
behaviour. Adjusting the synaptic strengths of the network to maximize the mutual
information between neurons with non-overlapping receptive fields, for an ensemble of images, causes the neurons to extract features that are spatially coherent thereby obtaining the disparity [fig. 1].
maximize I (a;b)
(: I : I ~ I~ )
Figure 1: In Hinton and Becker's initial scheme (Becker and Hinton, 1992), maximization of mutual information between neurons with spatially disjoint receptive
fields leads to disparity tuning, provided they train on spatially coherent patterns
(i.e. those for which disparity changes slowly with spatial position)
Workers in computer vision face a similar problem of estimating the properties of
objects in the world from intensity images. It is commonly stated that vision is illposed (Poggio et al, 1985) and that prior assumptions about the world are needed
to obtain a unique perception. It is convenient to formulate such assumptions by
the use of Bayes' theorem P(SID)
P(DIS)P(S)/ P(D). This relates the proba-
=
Bayesian Self-Organization
bility P(SID) of the scene S given the data D to the prior probability of the scene
P(S) and the imaging model P(DIS) (P(D) can be interpreted as a normalization
constant) . Thus a vision theorist (see (Clark and Yuille, 1990), for example) determines an imaging model P(DIS), picks a set of plausible prior assumptions about
the world P(S) (such as natural constraints (Marr, 1982)), applies Bayes' theorem,
and then picks an interpretation S* from some statistical estimator of P(SID) (for
example, the maximum a posteriori (MAP) estimator S* = ARG{M AXsP(SID)}.)
An advantage of the Bayesian approach is that, by nature of its probabilistic formulation, it can be readily related to learning with a teacher (Kersten et aI, 1987). It is
unclear, however, whether such a teacher will always be available. Moreover, from
Becker and Hinton's work on self-organization, it seems that a teacher is not always
necessary. This paper proposes a way for generalizing the self-organization approach, by starting from a Bayesian perspective, and thereby relating it to Bayesian
theories of vision . The key idea is to force the activity distribution of the outputs to
be close to a pre-specified prior distribution Pp(S). We argue that this approach is
in the same spirit as (Becker and Hinton, 1992), because we can choose the prior distribution to enforce spatial coherence, but it is also more general since many other
choices of the prior are possible. It also has some relation to the work performed by
Atick and Redlich (Atick and Redlich, 1990) for modelling the early visual system.
We will take the viewpoint that the prior Pp(S) is assumed known in advance by
the visual system (perhaps by being specified genetically) and will act as a selforganizing principle. Later we will discuss ways that this might be relaxed.
2
Theory
We assume that the input D is a function of a signal L that the system wants
to determine and a distractor N [fig.2]. For example L might correspond to the
disparities of a pair of binocular stereo images and N to the intensity patterns. The
distribution of the inputs is PD(D) and the system assumes that the signal L has
distribution Pp(L).
Let the output of the system be S = G(D, ,) where G is a function of a set
of parameters, to be determined. For example, the function G(D, ,) could be
represented by a multi-layer perceptron with the , 's being the synaptic weights.
By approximation theory, it can be shown that a large varidy of neural networks
can approximate any input-output function arbitrarily well given enough hidden
nodes (Hornik et aI, 1989) .
The aim of self-organizing the network is to ensure that the parameters, are chosen
so that the outputs S are as close to the L as possible. We claim that this can be
achieved by adjusting the parameters, so as to make the derived distribution of the
outputs PDD(S : ,) = f 8(S - G(D, ,))PD (D)[dD] as close as possible to Pp(S).
This can be seen to be a consistency condition for a Bayesian theory as from Bayes
formula we obtain the equation:
J
P(SID)PD(D)[dD]
=
J
P(DIS)Pp(S)[dD] = Pp(S).
(1)
1003
1004
Yuille, Smimakis, and Xu
which is equivalent to our condition, provided we choose to identify P(SID) with
6(S - C(D, -y?.
To make this more precise we must define a measure of similarity between the two
distributions Pp(S) and PDD(S : -y). An attractive measure is the Kullback-Leibler
distance (the entropy of PDD relative to Pp):
I( L(-y) =
D=
F(~,N)
J
PDD(S : -y) log
PDD(S:-y)
Pp(S) [dS].
(2)
S=G(D,r)
~(~)
Figure 2: The parameters -yare adjusted to minihu~e the Kullback-Leibler distance between the prior (Pp) distribution of the true signal (E) and the derived
distribution (PDD) of the network output (8).
This measure can be divided into two parts: (i) - I PDD(S : -y) log Pp(S)[dS] and
(ii) I PDD(S : -y) log PDD(S : -y)[dS). The second term encourages variability of the
output while the first term forces similarity to the prior distribution.
Suppose that Pp(S) can be expressed as a Markov random field (i.e. the spatial
distribution of Pp(S) has a local neighbourhood structure, as is commonly assumed
in Bayesian models of vision). Then, by the Hammersely-Clifford theorem, we can
write Pp(S) = e-fJEp(S) /Z where Ep(S) is an energy function with local connections
(for example, Ep(S)
Li(S, - Si+1)2), {3 is an inverse temperature and Z is a
normalization constant.
=
Then the first term can be written (Yuille et ai, 1992) as
-J
PDD(S : -y) log Pp(S)[d8)
={3{Ep(G(D, -Y?)D + log Z.
(3)
Bayesian Self-Organization
We can ignore the log Z term since it is a constant (independent of ,). Minimizing the first term with respect to , will therefore try to minimize the energy
of the outputs averaged over the inputs - (Ep(G(D,')))D - which is highly desirable (since it has a close connection to the minimal energy principles in (Poggio
et aI, 1985, Clark and Yuille, 1990)). It is also important, however, to avoid the
trivial solution G(D,,) = 0 as well as solutions for which G(D,,) is very small
for most inputs. Fortunately these solutions are discouraged by the second term:
J PDD(D,,) log PDD(D, ,)[dD], which corresponds to the negative entropy of the
derived distribution of the network output. Thus, its minimization with respect to
, is a maximum entropy principle which will encourage variability in the outputs
G( D,,) and hence prevent the trivial solutions.
3
Reformulating for Implementation.
Our theory requires us to minimize the Kullback-Leibler distance, equation 2, with
respect to ,. We now describe two ways in which this could be implemented using
variants of stochastic learning. First observe that by substituting the form of the
derived distribution into equation 2 and integrating out the 5 variable we obtain:
" J
J\L({) =
PD(D) log
PDD(G(D,,) : ,)
Pp(G(D,,)) [dD].
(4)
Assuming a representative sample {DJ.t : JJ fA} of inputs we can approximate K L(,)
by LJ.ttA log[PDD(G(DJ.t,,) : ,)/ Pp(G(DJ.t, ,))]. We can now, in principle, perform
stochastic learning using backpropagation: pick inputs DJ.t at random and update
the weights, using log[PDD(G(DJ.t,,): ,)/Pp(G(DJ.t,,))] as the error function.
To do this, however, we need expressions for PDD(G(DJ.t,,) : ,) and its derivative with repect to,. If the function G(D,,) can be restricted to being 1-1 (increasing the dimensionality of the output space if necessary) then we can obtain
(Yuille et aI, 1992) analytic expressions PDD(G(D,,) :,) = PD(D)/I det(oG/oD)1
and (ologPDD(G(D,,) : ,)/0,)
-(oG/OD)-1(02G/oDo,), where [-1] denotes
the matrix inverse. Alternatively we can perform additional sampling to estimate
PDD(G(D,,):,) and (ologPDD(G(D,,): ,)/0,) directly from their integral representations. (This second approach is similar to (Becker and Hinton, 1992) though
they are only concerned with estimating the first and second moments of these
distributions. )
=
4
Connection to Becker and Hinton.
The Becker and Hinton method (Becker and Hinton, 1992) involves maximizing the
mutual information between the output of two neuronal units 5 1 ,52 [fig.l]. This is
given by :
where the first two terms correspond to maximizing the entropies of 51 and 52
while the last term forces 51 :::::: 52.
1005
1006
Yuille, Smirnakis, and Xu
By contrast, our version tries to minimize the quantity:
- S2) our second term will force S1 ~ S2
and our first term will maximize the entropy of the joint distribution of Sl, S2. We
argue that this is effectively the same as (Becker and Hinton, 1992) since maximizing the joint entropy of Sl, S2 with Sl constrained to equal S2 is equivalent to
maximizing the individual entropies of SI and S2 with the same constraint.
If we then ensure that Pp (S 1, S2) = 6(S 1
To be more concrete, we consider Becker and Hinton's implementation of the mutual
information maximization principle in the case of units with continuous outputs.
They assume that the outputs of units 1, 2 are Gaussian 1 and perform steepest
descent to maximize the symmetrized form of the mutual information between SI
and S2:
where VO stands for variance over the set of inputs. They assume that the difference
between the two outputs can be expressed as un correlated additive noise, SI =
S2 + N. We reformalize their criterion as maximizing EBH(V(S2), V(N)) where
E BH (V(S2), V(N)) = log{V(S2)
+ V(N)} + log V(S2) -
210g V(N).
(6)
For our scheme we make similar assumptions about the distributions of SI and
S2. We see that < logPDD(SI,S2) >= -log{< si >< S~ > - < S1S2 >2} =
-log{V(S2)V(N)} (since < S1S2 >=< (S2 + N)S2 >= V(S2) and < Sf >=
V(S2) + V(N)). Using the prior distribution PP(Sl' S2) ~ e- r (Sl-S2)2 our criterion
corresponds to minimizing EYSX(V(S2), V(N)) where:
Ey SX(V(S2), V(N)) = -log V(S2) - log V(N)
+ rV(N).
(7)
It is easy to see that maximizing E BH (V(S2), V(N)) will try to make V(S2) as
large as possible and force V(N) to zero (recall that, by definition, V(N) ~ 0).
Minimizing our energy will try to make V(S2) as large as possible and will force
V(N) to 1/r (recall that r appears as the inverse of the variance of a Gaussian
prior distribution for SI - S2 so making r large will force the prior distribution to
approach 6(Sl - S2).) Thus, provided r is very large, our method will have the
same effect as Becker and Hinton's.
5
Application to Linear Filtering.
We now describe an analysis of these ideas for the case of linear filtering. Our
approach will be contrasted with the traditional Wiener filter approach.
1 We
assume for simplicity that these Gaussians have zero mean.
Bayesian Self-Organization
Consider a process ofthe form D(i) = ~(i)+N(i) where D(i) denotes the input to
the system, ~(i) is the true signal which we would like to predict, and N(i) is the
n?ise corrupting the signal. The resulting Wiener filter Aw (i) has fourier transform
Aw = ~~ , ~/?h:: , ~ + ~N,N) where ~~,~ and ~N,N are the power spectrum of the
signal and the noise respectively.
By contrast, let us extract a linear filter Ab by applying our criterion. In the case
that the noise and signal are independent zero mean Gaussian distributions this
filter can be calculated explicitly (Yuille et aI, 1992). It has fourier transform with
squared magnitude given by IAbl2 = ~!:,~/(~~,~ + ~N,N) . Thus our filter can be
thought of as the square root of the Wiener filter.
It is important to realize that although our derivation assumed additive Gaussian
noise our system would not need to make any assumptions about the noise distribution. Instead our system would merely need to assume that the filter was linear and
then would automatically obtain the "correct" result for the additive Gaussian noise
case. We conjecture that the system might detect non-Gauusian noise by finding it
impossible to get zero Kullback-Liebler distance with the linear ansatz.
6
Conclusion
The goal of this paper was to introduce a Bayesian approach to self-organization
using prior assumptions about the signal as an organizing principle. We argued that
it was a natural generalization of the criterion of maximizing mutual information
assuming spatial coherence (Becker and Hinton, 1992) . Using our principle it should
be possible to self-organize Bayesian theories of vision, assuming that the priors
are known, the network is capable of representing the appropriate functions and
the learning algorithm converges. There will also be problems if the probability
distributions of the true signal and the distractor are too similar .
If the prior is not correct then it may be possible to detect this by evaluating
the goodness of the Kullback-Leibler fit after learning 2. This suggests a strategy
whereby the system increases the complexity of the priors until the Kullback-Leibler
fit is sufficiently good (this is somewhat similar to an idea proposed by Mumford
(Mumford, 1992)). This is related to the idea of competitive priors in vision (Clark
and Yuille, 1990). One way to implement this would be for the prior probability
itself to have a set of adjustable parameters that would enable it to adapt to different
classes of scenes. We are currently (Yuille et aI, 1992) investigating this idea and
exploring its relationships to Hidden Markov Models.
Ways to implement the theory, using variants of stochastic learning, were described.
We sketched the relation to Becker and Hinton .
As an illustration of our approach we derived the filter that our criterion would give
for filtering out additive Gaussian noise (possibly the only analytically tractable
case). This had a very interesting relation to the standard Wiener filter.
2This is reminiscent of Barlow's suspicious coincidence detectors (Barlow, 1993), where
we might hope to determine if two variables x & yare independent or not by calculating
the Kullback-Leibler distance between the joint distribution P(x, y) and the product of
the individual distributions P( x) P(y).
1007
1008
Yuille, Smirnakis, and Xu
Acknowledgements
We would like to thank DARPA for an Air Force contract F49620-92-J-0466. Conversations with Dan Kersten and David Mumford were highly appreciated.
References
J.J. Atick and A.N. Redlich. "Towards a Theory of Early Visual Processing".
Neural Computation . Vol. 2, No.3, pp 308-320. Fall. 1990.
H.B. Barlow. "What is the Computational Goal of the Neocortex?" To appear in:
Large scale neuronal theories of the brain. Ed. C. Koch. MIT Press. 1993.
S. Becker and G.E. Hinton. "Self-organizing neural network that discovers surfaces
in random-dot stereograms". Nature, Vol 355. pp 161-163. Jan. 1992.
J .J. Clark and A.L. Yuille. Data Fusion for Sensory Information Processing
Systems. Kluwer Academic Press . Boston/Dordrecht/London. 1990.
R. Held. "Visual development in infants". In The encyclopedia of neuroscience,
vol. 2. Boston: Birkhauser. 1987.
K. Hornik, S. Stinchocombe and H. White. "Multilayer feed-forward networks are
universal approximators". Neural Networks 4, pp 251-257. 1991.
D. Kersten, A.J. O'Toole, M.E . Sereno, D.C. Knill and J .A. Anderson. "Associative
learning of scene parameters from images". Optical Society of America, Vol. 26,
No. 23, pp 4999-5006. 1 December, 1987.
D. Marr . Vision. W.H . Freeman and Company. San Francisco . 1982.
D. Mumford. "Pattern Theory: a unifying perspective".
Preprint. Harvard University. 1992.
Dept.
Mathematics
K. Nakayama and S. Shimojo. "Experiencing and Perceiving Visual Surfaces".
Science. Vol. 257, pp 1357-1363. 4 September. 1992.
T. Poggio, V. Torre and C. Koch. "Computational vision and regularization theory" . Nature, 317, pp 314-319. 1985.
A.L. Yuille, S.M. Smirnakis and L. Xu. "Bayesian Self-Organization". Harvard
Robotics Laboratory Technical Report . 1992.
PART IX
SPEECH AND SIGNAL
PROCESSING
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7,036 | 81 | 830
Invariant Object Recognition Using a Distributed Associative Memory
Harry Wechsler and George Lee Zimmerman
Department or Electrical Engineering
University or Minnesota
Minneapolis, MN 55455
Abstract
This paper describes an approach to 2-dimensional object recognition. Complex-log conformal mapping is combined with a distributed associative memory to create a system
which recognizes objects regardless of changes in rotation or scale. Recalled information
from the memorized database is used to classify an object, reconstruct the memorized version of the object, and estimate the magnitude of changes in scale or rotation. The system
response is resistant to moderate amounts of noise and occlusion. Several experiments, using real, gray scale images, are presented to show the feasibility of our approach.
Introduction
The challenge of the visual recognition problem stems from the fact that the projection of an object onto an image can be confounded by several dimensions of variability
such as uncertain perspective, changing orientation and scale, sensor noise, occlusion, and
non-uniform illumination. A vision system must not only be able to sense the identity of an
object despite this variability, but must also be able to characterize such variability -- because the variability inherently carries much of the valuable information about the world.
Our goal is to derive the functional characteristics of image representations suitable for invariant recognition using a distributed associative memory. The main question is that of
finding appropriate transformations such that interactions between the internal structure
of the resulting representations and the distributed associative memory yield invariant
recognition. As Simon [1] points out, all mathematical derivation can be viewed simply as
a change of representation, making evident what was previously true but obscure. This
view can be extended to all problem solving. Solving a problem then means transforming it
so as to make the solution transparent .
We approach the problem of object recognition with three requirements:
classification, reconstruction, and characterization. Classification implies the ability to distinguish objects that were previously encountered. Reconstruction is the process by which
memorized images can be drawn from memory given a distorted version exists at the input. Characterization involves extracting information about how the object has changed
from the way in which it was memorized. Our goal in this paper is to discuss a system
which is able to recognize memorized 2-dimensional objects regardless of geometric distortions like changes in scale and orientation, and can characterize those transformations.
The system also allows for noise and occlusion and is tolerant of memory faults.
The following sections, Invariant Representation and Distributed Associative
Memory, respectively, describe the various components of the system in detail. The Experiments section presents the results from several experiments we have performed on real
data. The paper concludes with a discussion of our results and their implications for future
research.
? American Institute of Physics 1988
831
1. Invariant Representation
The goal of this section is to examine the various components used to produce the
vectors which are associated in the distributed associative memory. The block diagram
which describes the various functional units involved in obtaining an invariant image
representation is shown in Figure 1. The image is complex-log conformally mapped so that
rotation and scale changes become translation in the transform domain . Along with the
conformal mapping, the image is also filtered by a space variant filter to reduce the effects
of aliasing. The conformally mapped image is then processed through a Laplacian in order
to solve some problems associated with the conformal mapping. The Fourier transform of
both the conformally mapped image and the Laplacian processed image produce the four
output vectors. The magnitude output vector I-II is invariant to linear transformations of
the object in the input image. The phase output vector <1>2 contains information concerning the spatial properties of the object in the input image.
1.1 Complex-Log Mapping and Space Variant Filtering
The first box of the block diagram given in Figure 1 consists of two components:
Complex-log mapping and space variant filtering. Complex-log mapping transforms an
image from rectangular coordinates to polar exponential coordinates. This transformation
changes rotation and scale into translation. If the image is mapped onto a complex plane
then each pixel (x,y) on the Cartesian plane can be described mathematically by z = x +
jy. The complex-log mapped points ware described by
w =In{z) =In(lzl} +jiJ z
(1)
Our system sampled 256x256 pixel images to construct 64x64 complex-log mapped
images. Samples were taken along radial lines spaced 5.6 degrees apart. Along each radial
line the step size between samples increased by powers of 1.08. These numbers are derived
from the number of pixels in the original image and the number of samples in the
complex-log mapped image. An excellent examination of the different conditions involved
in selecting the appropriate number of samples for a complex-log mapped image is given in
[2J. The non-linear sampling can be split into two distinct parts along each radial line. Toward the center of the image the samples are dense enough that no anti-aliasing filter is
needed. Samples taken at the edge of the image are large and an anti-aliasing filter is
necessary. The image filtered in this manner has a circular region around the center which
corresponds to an area of highest resolution. The size of this region is a function of the
number of angular samples and radial samples. The filtering is done, at the same time as
the sampling, by convolving truncated Bessel functions with the image in the space
domain. The width of the Bessel functions main lobe is inversely proportional to the eccentricity of the sample point.
A problem associated with the complex-log mapping is sensitivity to center
misalignment of the sampled image. Small shifts from the center causes dramatic distortions in the complex-log mapped image. Our system assumes that the object is centered in
the image frame. Slight misalignments are considered noise. Large misalignments are considered as translations and could be accounted for by changing the gaze in such a way as
to bring the object into the center of the frame. The decision about what to bring into the
center of the frame is an active function and should be determined by the task. An example of a system which could be used to guide the translation process was developed by
Anderson and Burt [3J. Their pyramid system analyzes the input image at different tem-
00
c..:>
~
Inverse
Processing
and
Reconstruction
.
Image
~
I
Compl".lo,
Mapping
and
Space Variant
Filtering
I
I
I
~-?-FO",i"
II
'
1ransform
I
2
-1-1
I
I
-~
Laplacian
Fourier
Transform
2
_~I
Distributed
Associative
Memory
~
Rotation
and
Scale
Estimation
I-II
Classification
Figure 1. Block Diagram of the System.
833
poral and spatial resolution levels. Their smart sensor was then able to shift its fixation
such that interesting parts of the image (ie . something large and moving) was brought into
the central part of the frame for recognition .
1.2 Fourier Transform
The second box in the block diagram of Figure 1 is the Fourier transform. The
Fourier transform of a 2-dimensional image f(x,y) is given by
F(u,v) =
j j
f(x,y)e-i(ux+vy) dx dy
(2)
-00 -00
and can be described by two 2-dimensional functions corresponding to the magnitude
IF(u,v)1 and phase <l>F(u,v). The magnitude component of the Fourier trans~rm which is
invariant to translatIOn, carries much of the contrast information of the image . The phase
component of the Fourier transform carries information about how things ar} placed in an
image. Translation of f(x,y) corresponds to the addition of a linear phase cpmponent. The
complex-log mapping transforms rotation and scale into translation and tije magnitude of
the Fourier transform is invariant to those translations so that I-II ivill not change
significantly with rotation and scale of the object in the image .
1.3 Laplacian
The Laplacian that we use is a difference-of-Gaussians (DOG) approximation to the
function as given by Marr [4).
'V 2G
2
2
=h [1 - r2/2oo 2) e{ -r /200 }
(3)
'1rtT
The result of convolving the Laplacian with an image can be viewed as a two step process.
The image is blurred by a Gaussian kernel of a specified width oo. Then the isotropic
second derivative of the blurred image is computed. The width of the Gaussian kernel is
chosen such that the conformally mapped image is visible -- approximately 2 pixels in our
experiments. The Laplacian sharpens the edges of the object in the image and sets any region that did not change much to zero. Below we describe the benefits from using the Laplacian.
The Laplacian eliminates the stretching problem encountered by the complex-log
mapping due to changes in object size. When an object is expanded the complex-log
mapped image will translate . The pixels vacated by this translation will be filled with
more pixels sampled from the center of the scaled object. These new pixels will not be
significantly different than the displaced pixels so the result looks like a stretching in the
complex-log mapped image . The Laplacian of the complex-log mapped image will set the
new pixels to zero because they do not significantly change from their surrounding pixels.
The Laplacian eliminates high frequency spreading due to the finite structure of the
discrete Fourier transform and enhances the differences between memorized objects by accentuating edges and de-emphasizing areas of little change.
2. Distributed Associative Memory (DAM)
The particular form of distributed associative memory that we deal with in this paper is a memory matrix which modifies the flow of information. Stimulus vectors are associated with response vectors and the result of this association is spread over the entire
memory space . Distributing in this manner means that information about a small portion
of the association can be found in a large area of the memory. New associations are placed
834
over the older ones and are allowed to interact. This means that the size of the memory
matrix stays the same regardless of the number of associations that have been memorized.
Because the associations are allowed to interact with each other an implicit representation
of structural relationships and contextual information can develop, and as a consequence a
very rich level of interactions can be captured. There are few restrictions on what vectors
can be associated there can exist extensive indexing and cross-referencing in the memory.
Distributed associative memory captures a distributed representation which is context
dependent. This is quite different from the simplistic behavioral model [5].
The construction stage assumes that there are n pairs of m-dimensional vectors that
are to be associated by the distributed associative memory. This can be written as
"l.K:::+.
IV~
1
= -r.
1
~or 1?
I'
= 1 , ... ,n
(4)
-d
h ?th stlmu
. I us vector an d -d
h .th correspon d?mg response Vech
were
s. enotes tel
r. enotes tel
tor. W~ want to construct a memory matrix M such that when the kth stimulus vector S;
is projected onto the space defined by M the resulting projection will be the corresponding
More specifically we want to solve the following equation:
response vector
r;.
(5)
MS=R
- 11 s2
11 ? ??11 ?
?
~
S = [ s1
h
were
S ] an d R = [ -r 1 11 -r 2 11 ???11 r]. A
umque
soIutlOn
lor
t h?IS equation does not necessarily n exist for any arbitrary gr~up of associations that might be
chosen. Usually, the number of associations n is smaller than m, the length of the vector to
be associated, so the system of equations is underconstrained. The constraint used to solve
for a unique matrix M is that of minimizing the square error, IIMS - RJ1 2, which results in
the solution
(6)
where S+ is known as the Moore-Penrose generalized inverse of S [6J.
The recall operation projects an unknown stimulus vector
M. The resulting projection yields the response vector r
r =Ms
s onto
the memory space
(7)
If the memorized stimulus vectors are independent and the unknown stimulus vector s is
one of the memorized vectors
then the recalled vector will be the associated response
If the memorized stimulus vectors are dependent, then the vector recalled by
vector
one of the memorized stimulus vectors will contain the associated response vector and
some crosstalk from the other stored response vectors.
r;.
S;,
The recall can be viewed as the weighted sum of the response vectors. The recall
begins by assigning weights according to how well the unknown stimulus vector matches
with the memorized stimulus vector using a linear least squares classifier. The response
vectors are multiplied by the weights and summed together to build the recalled response
vector. The recalled response vector is usually dominated by the memorized response vector that is closest to the unknown stimulus vector.
Assume that there are n associations in the memory and each of the associated
stimulus and response vectors have m elements. This means that the memory matrix has
m 2 elements. Also assume that the noise that. is added to each element of a memorized
835
stimulus vector
memory is then
IS
independent, Zero mean, with a variance of O'~ The recall from the
1
(8)
where tt is the input noise vector and t1 is the output noise vector. The ratio of the average output noise variance to the averagg input noise variance is
0'2o/0'.12
1
[MMT]
= -Tr
m
(9)
For the autoassociative case this simplifies to
(10)
This says that when a noisy version of a memorized input vector is applied to the memory
the recall is improved by a factor corresponding to the ratio of the number of memorized
vectors to the number of elements in the vectors. For the heteroassociative memory matrix a similar formula holds as long as n is less than m [7].
(11)
Fault tolerance is a byproduct of the distributed nature and error correcting capabilities of the distributed associative memory. By distributing the information, no single
memory cell carries a significant portion of the information critical to the overall performance of the memory.
3. Experiments
In this section we discuss the result of computer simulations of our system. Images
of objects are first preprocessed through the sUbsystem outlined in section 2. The output of
such a subsystem is four vectors: I-I , <1>1' 1-1 2, and <1>2' We construct the memory by associating the stimulus vector I-II with ?he response vector <1>2 for each object in the database.
To perform a recall from the meJIlory the.. unknown image is preprocessed by the same_subsystem to produce the vectors I-II' <1>1' 1-12, and <1>2' The resulting stimulus vector I-I is
projected onto the m~mory matrix to produce a respOJlse vector which is an ~stimatel of
the memorized phase <1>2' The estimated phase vector cI> 2 and the magnitude I-II ate used
to reconstruct the memorized object. The difference between the estimated phase <1>2 and
the unknown phase <1>2 is used to estimate the amount of rotation and scale experienced by
the object.
The database of images consists of twelve objects: four keys, four mechanical parts,
and four leaves. The objects were chosen for their essentially two-dimensional structure.
Each object was photographed using a digitizing video camera against a black background. We emphasize that all of the images used in creating and testing the recognition
system were taken at different times using various camera rotations and distances. The images are digitized to 256x256, eight bit quantized pixels, and each object covers an area of
about 40x40 pixels. This small object size relative to the background is necessary due to
the non-linear sampling of the complex-log mapping. The objects were centered within the
frame by hand. This is the source of much of the noise and could have been done automatically using the object's center of mass or some other criteria determined by the task. The
orientation of each memorized object was arbitrarily chosen such that their major axis
836
was vertical. The 2-dimensional images that are the output from the invariant representation subsystem are scanned horizontally to form the vectors for memorization. The database used for these experiments is shown in Figure 2.
Figure 2. The Database of Objects Used in the Experiments
a) Original
b) Unknown
c) Recall: rotated 135?
d) Memory:6
SNR: -3.37 Db
Figure 3. :Recall Using a Rotated and scaled key
The first example of the operation of our system is shown in Figure 3. Figure 3a) is
the image of one of the keys as it was memorized. Figure 3b) is the unknown object
presented to our system. The unknown object in this caSe is the same key that has been
rotated by 180 degrees and scaled. Figure 3c) is the recalled, reconstructed image. The
837
rounded edges of the recalled image are artifacts of the complex-log mapping. Notice that
the reconstructed recall is the unrotated memorized key with some noise caused by errors
in the recalled phase. Figure 3d) is a histogram which graphically displays the
classification vector which corresponds to S+S. The histogram shows the interplay between
the memorized images and the unknown image. The" 6" on the bargraph indicates which
of the twelve classes the unknown object belongs. The histogram gives a value which is
the best linear estimate of the image relative to the memorized objects. Another measure,
the signal-to-noise ratio (SNR), is given at the bottom of the recalled image. SNR compares the variance of the ideal recall after processing with the variance of the difference
between the ideal and actual recall. This is a measure of the amount of noise in the recall.
The SNR does not carry rr.uch information about the q"Jality of the recall image because
the noise measured by the SNP.. is jue to many factors such as misalignment of the center,
changing reflections, and dependence between other memorized objects -- each affecting.
quality in a variety of ways. Rotation and scale estimate~ are made using a vector_ D
corresponding to the dlll'erence between the unknown vector <1>2 and the recalled vector <I> 2'
In an ideal situation D will be a plane whose E;radient indicates the exact amount of r:.otation and scale the recalled object has experienced. In our system the recalled vector <I> 2 is
corrupted with noise which means rotation...and scale have to be estim:ned. The estimate is
made by letting the first order difference D at each point in the plane vote for a specified
range of rotation or scale.
a) Original
b) Unknown
c) Recall
d) Memory:4
Figure 4 Recall Using Scaled and Rotated" S" with Occlusion
Figure 4 is an example of occlusion. The unknown object in this case is an "s"
curve which is larger and slightly tilted from the memorized "s" curve. A portion of the
bottom curve was occluded. The resulting reconstruction is very noisy but has filled in the
missing part of the bottom curve. The noisy recall is reflected in both the SNR and the interplay betw~en the memories shown by the hi~togram.
a) Ideal recall
b) 30% removed
c) 50% removed
d) 75% removed
Figure 5. Recall for Memory Matrix Randomly Set to Zero
Figure 5 is the result of randomly setting the elements of the memory matrix to
838
zero. Figure 5a) shows is the ideal recall. Figure 5b) is the recall after 30 percent of the
memory matrix has been set to zero. Figure 5c) is the recall for 50 percent and Figure 5d)
is the recall for 75 percent. Even when 90 percent of the memory matrix has been set to
zero a faint outline of the pin could still be seen in the recall. This result is important in
two ways. First, it shows that the distributed associative memory is robust in the presence
of noise. Second, it shows that a completely connected network is not necessary and as a
consequence a scheme for data compression of the memory matrix could be found.
4. Conclusion
In this paper we demonstrate a computer vIsIon system which recognIzes 2dimensional objects invariant to rotation or scale. The system combines an invariant
representation of the input images with a distributed associative memory such that objects
can be classified, reconstructed, and characterized. The distributed associative memory is
resistant to moderate amounts of noise and occlusion. Several experiments, demonstrating
the ability of our computer vision system to operate on real, grey scale images, were
presented.
Neural network models, of which the di~tributed associative memory is one example,
were originally developed to simulate biological memory. They are characterized by a
large number of highly interconnected simple processors which operate in p2..rallel. An excellent review of the many neural network models is given in [8J. The distrib-uted associative memory we use is linear, and as a result there are certain desirable properties which
will not be exhibited by our computer vision system. For example, feedback through our
system will not improve recall from the memory. Recall could be improved if a non-linear
element, such as a sigmoid function, is introduced into the feedback loop. Non-linear neural networks, such as those proposed by Hopfield [9] or Anderson et. al. [10J, can achieve
this type of improvement because each memorized pattern js associated with sta~le points
in an energy space. The price to be paid for the introduction of non-linearities into a
memory system is that the system will be difficult to analyze and can be unstable. Implementing our computer vision system using non-linear distributed associative memory is a
goal of our future research.
We are presently extending our work toward 3-dimensional object recognition. Much
of the present research in 3-dimensional object recognition is limited to polyhedral, nonoccluded objects' in a clean, highly controlled environment. Most systems are edge based
and use a generate-and-test paradigm to estimate the position and orientation of recognized objects. We propose to use an approach based on characteristic views [llJ or aspects
[12J which suggests that the infinite 2-dimensional projections of a 3-dimensional object
can be grouped into a finite number of topological equivalence classes. An efficie:.t 3dimensional recognition system would require a parallel indexing method to search for object models in the presence of geometric distortions, noise, and occlusion. Our object recognition system using distributed associative memory can fulfill those requirements with
respect to characteristic views.
Referenees
[lJ Simon, H. A., (1984), The Seienee of the Artifldal (2nd ed.), MIT Press.
[2J Massone, L., G. Sandini, and V. Tagliasco (1985), "Form-invariant" topological mapping strategy for 2D shape recognition, CVGIP, 30, 169-188.
[3J Anderson, C. H., P. J. Burt, and G. S. Van Der Wal (1985), Change detection and
tracking using pyramid transform techniques, Proe. of the SPIE Conferenee on
Intelligenee, Robots, and Computer Vision, Vol. 579, 72-78.
839
Marr, D. (1982), Vision, W. H. Freeman, 1982.
Hebb, O. D. (1949), The Organization of Behavior, New York: Wiley.
Kohonen, T. (1984), Self-Organization and Associative-Memories, Springer-Verlag.
Stiles, G. S. and D. L. Denq (1985), On the effect of noise on the Moore-Penrose generalized inverse associative memory, IEEE Trans. on PAMI, 7, 3,358-360.
[8J MCClelland, J. L., and D . E. Rumelhart, and the PDP Research Group (Eds.) (1986),
Parallel Distributed, Processing, Vol. 1, 2, MIT Press.
[9] Hopfield, J. J. (1982), Neural networks and physical systems with emergent collective
computational abilities, Proc. Natl. Acad. Sci. USA, 79, April 1982.
[10J Anderson, J. A., J. W. Silversteir., S. A. Ritz, and R. S. Jones (1977), Distinctive
features, categorical perception, and probability learning: some applications of a
neural model, Psychol. Rev., 84,413-451.
[11] Chakravarty, I., and H. Freeman (1982), Characteristic views as a basis for 3-D object
recognition, Proc. SPIE on Robot Vision, 336,37-45.
[12] Koenderink, J. J., and A . J . Van Doorn (1979), Internal representation of solid shape
with respect to vision, Bioi. Cybern., 32,4,211-216.
[4]
[5]
[6J
[7]
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7,037 | 810 | A Hybrid Radial Basis Function Neurocomputer
and Its Applications
Steven S. Watkins
ECE Department
Paul M. Chau
ECE Department
UCSD
La Jolla. CA. 92093
UCSD
La Jolla, CA. 92093
Raoul Tawel
JPL
Caltech
Pasadena. CA. 91109
Bjorn Lambrigtsen
JPL
Caltech
Pasadena. CA. 91109
Mark Plutowski
CSE Department
UCSD
La Jolla. CA. 92093
Abstract
A neurocomputer was implemented using radial basis functions and a
combination of analog and digital VLSI circuits. The hybrid system
uses custom analog circuits for the input layer and a digital signal
processing board for the hidden and output layers. The system combines
the advantages of both analog and digital circuits. featuring low power
consumption while minimizing overall system error. The analog circuits
have been fabricated and tested, the system has been built, and several
applications have been executed on the system. One application
provides significantly better results for a remote sensing problem than
have been previously obtained using conventional methods.
1.0 Introduction
This paper describes a neurocomputer development system that uses a radial basis
function as the transfer function of a neuron rather than the traditional sigmoid function.
This neurocOOlputer is a hybrid system which has been implemented with a combination
of analog and digital VLSI technologies. It offers the low-power advantage of analog
circuits operating in the subthreshold region and the high-precision advantage of digital
circuits. The system is targeted for applications that require low-power operation and use
input data in analog form, particularly remote sensing and portable computing
applications. It has already provided significantly better results for a remote sensing
850
A Hybrid Radial Basis Function Neurocomputer and Its Applications
,--- ----
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Figure I: Radial Basis Function Network
NEURoN
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Figure 2: Mapping of RBF Network to Hardware
Analog Board
=
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Figure 3: The RBF Neurocomputer Development System
851
852
Watkins, Chau, Tawel, Lambrigsten, and Plutowski
climate problem than have been previously obtained using conventional methods.
Figure 1 illustrates a radial basis functioo (RBF) network. Radial basis functions have
been used to solve mapping and function estimation problems with positive results
(Moody and Darken. 1989; Lippman, 1991). When coupled with a dynamic neuron
allocation algorithm such as Platt's RANN (platt. 1991). RBF networks can usually be
trained much more quickly than a traditional sigmoidal. back-propagation network.
RBF networlcs have been implemented with completely-analog (platt, Anderson and Kirk.
1993), c<mpletely-digital (Watkins. Chau and Tawel, Nov.? 1992). and with hybrid analogi
digital approaches (Watkins. Chau and Tawel, Oct., 1992). The hybrid approach is optimal
for applications which require low power consumption and use input data that is naturally
in the analog domain while also requiring the high precision of the digital domain.
2.0 System Architecture and Benefits
Figure 2 shows the mapping of the RBF network to hardware. Figure 3 shows the
neurocomputer development system. The system consists of a PC controller, a DSP board
with a Motorola 56000 DSP chip and a board with analog multipliers. The benefits of the
hybrid approach are lower-cost parallelism than is possible with a completely-digital
system, and more precise computation than is possible with a completely-analog system.
The parallelism is available for low cost in terms of area and power, when the inputs are in
the analog domain. When comparing a single analog multiplier to a 100bit fixed point
digital multiplier, the analog cell uses less than one-quarter the area and approximately
five orders of magnitude less power. When comparing an array of analog multipliers to a
Motorola 56000 DSP chip, 1000 Gilbert multipliers can fit in an area about half the size of
the DSP chip, while consuming .003% of the power.
The restriction of requiring analog inputs is placed on the system. because if the inputs
were digital, the high cost of D to A conversion would remove the low cost benefit of the
system. lbis restriction causes the neurocomputer to be taIgeted for applications using
inputs that are in the analog domain, such as remote sensing applications that use
microwave or infrared sensors and speech recognitioo applications that use analog filters.
The hybrid system reduces the overall system error when compared with a completelyanalog solution. The digital circuits compute the hidden and output layers with 24 bits of
precision while analog circuits are limited to about 8 bits of precision. Also the RANN
algorithm requires a large range of width variatioo for the Gaussian function and this is
more easily achieved with digital computation. Completely analog solutions to this
problem are severely limited by the voltage rails of the chip.
3.0 Circuits
Several different analog circuit approaches were explored as possible implementations of
the network. Mter the dust settled, we chose to implement only the input layer with analog
circuits because it offers the greatest opportunity for parallelism, providing parallel
performance benefits at a low cost in terms of area and power. The input layer requires
more than 0 UP) computations (where N is the number of neurons). while the hidden and
output layers require only 0 (N) computations (because there is one hidden layer
computatioo per neuron and the number of outputs is either one or very small).
A Hybrid Radial Basis Function Neurocomputer and Its Applications
The analog circuits used in the input layer are Gilbert multipliers (Mead. 1989). 'The
circuits were fabricated with 2.0 micron. double-poly, P-well. CMOS technology. The
Gilbert cell performs the operation of multiplying two voltage differences: (Vi-V2)x(V3V4). In this system. Vi =V3 and V2=V4. which causes the circuit to compute the square of
the difference between a stored weight and the input. The current outputs of the Gilbert
cells in a row are wired together to sum their currents. giving a sum of squared errors. This
current is converted to a voltage. fed to an A to D converter and then passed to the DSP
board where the hidden and output layers are computed. The radial basis function
(Gaussian) of the hidden layer is computed by using a lookup table. The system uses the
fast multiply/accumulate operation of the DSP chip to compute the output layer.
4.0 Applications
The low-power feature of the hybrid system makes it attractive for applications where
power consumption is a prime consideration, such as satellite-based applications and
portable computing (using battery power). The neurocomputer has been applied to three
problems: a remote sensing climate problem. the Mackey-Glass chaotic time series
estimation and speech phoneme recognitim. The remote sensing application falls into the
satellite category. The Mackey-Glass and speech recognition applications are potentially
portable. Systems fa these applications are likely to have inputs in the analog domain
(eliminating the need for D to A conversion. as already discussed) making it feasible to
execute them on the hybrid neurocomputer.
4.1 The Remote Sensing Application
The remote sensing problem is an inverse mapping problem that uses microwave energy
in different bands as input to predict the water vapor content of the atmosphere at different
altitudes. Water vapor content is a key parameter for predicting weather in the tropics and
mid-latitudes (Kakar and Lambrigtsen. 1984). The application uses 12 inputs and 1 output.
The system input is naturally in analog form. the result of amplified microwave signals, so
no D to A conversion of input data is required. Others have used neural networks with
success to perform a similar inverse mapping to predict the temperature gradient of the
atmosphere CMotteler et al .. 1993). Section 5 details the improved results of the RBF
network over conventional methods. Since water vapor content is a very important
compment of climate models. improved results in predicted water vapor content means
improved climate models.
Remote sensing problems require satellite hardware where power consumptim is always a
major constraint.The low-power nature of the hybrid network would allow the network to
be placed on board a satellite. With future EOS missions requiring several thousand
sensors. the on-board network would reduce the bandwidth requirements of the data being
sent back to earth. allowing the reduced water vapor content data to be transmitted rather
than the raw sensor data. This data bandwidth reduction could be used either to send back
more meaningful data to further improve climate models. or to reduce the amount of data
transmitted. saving energy.
4.2 The Mackey-Glass Application
The Mackey-Glass chaotic time series application uses several previous time sample
values to predict the current value of a time series which was generated by the MackeyGlass delay-difference equation. It was used because it has proved to be difficult for
853
854
Watkins, Chau, Tawel, Lambrigsten, and Plutowski
sigmoidal neural networks (platt. 1991). The applicatioo uses 4 inputs and 1 output. The
Mackey-Glass time series is representative of time series found in medical applications
such as detecting arrhythmias in heartbeats. It could be advantageous to implement this
application with portable hardware.
4.3 The Speech Phoneme Recognition Application
The speech phoneme recognition problem used the same data as Waibel (Waibel et ai.?
1989) to learn to recognize the acoustically similar phonemes of b. d and g. The
application uses 240 inputs and 3 outputs. The speech phoneme recognition problem
represents a sub problem of the more difficult continuous speech recognition problem.
Speech recognition applications also represent opportunities for portable computing.
5.0 Results
5.1 The Remote Sensing Application
Using the RBF neural network 00 the remote sensing climate problem produced
significantly better results than had been previously obtained using conventional statistical
methods (Kakar and Lambrigtsen. 1984). The input layer of the RBF network was
implemented in two different ways: 1) it was simulated with 32-bit floating point precision
to represent a digital input layer. and 2) it was implemented with the analog Gilbert
multipliers as the input layer. Both implementations produced similar results.
At an altitude corresponding to 570 mb pressure, the RBF neural network with a digital
input layer produced results with .33 absolute rms error vs. .42 rms error for the best
results using conventional methods. This is an improvement of 21 %. Figure 4 shows the
plot of retrieved vs. actual water vapor content for both the RBF network and the
conventional method. Using the hybrid neurocomputer with the analog input layer for the
data at 570 mb pressure produced results with .338 rms error. This is an improvement of
19.5% over the conventional method. Using the analog input layer produced nearly as
much improvement as a completely-digital system. demonstrating the feasibility of
placing the network on board a satellite. Similar results were obtained for other altitudes.
The RBF network also was compared to a sigmoidal network using back propagation
learning enhanced with line-search capability (to automatically set step-size). Both
networks used eight neurons in the hidden layer. As Figure 5 shows. the RBF network
learned much faster than the sigmoidal network.
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Figure 4: Comparison of Retrieved vs. Actual Water Vapor Content for 570 mb Pressure
for RBF Network and Conventional Statistical Method
A Hybrid Radial Basis Function Neurocomputer and Its Applications
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5.2 The Mackey-Glass Application
The RBF network was not compared to any non-neural network method for the MackeyGlass time series estimation. It was only compared to a traditional sigmoidal networlc
using back propagation learning enhanced with line search. Both networks used four neurons. As Figure 6 shows. applying the RBF neural network to the Mackey-Glass chaotic
time series estimation produced much faster learning than the sigmoidal network. The
RBF network with a digital input layer and the RBF hybrid network with an analog input
layer both produced similar results in dropping to an rms error of about .025 after only 5
minutes of training on a PC using a 486 CPU.
Using the digital input layer. the RBF network reached a minimum absolute rms error of
.017. while the sigmoidal network reached a minimum absolute rms error of .025. This is
an improvement of 32% over the sigmoidal network. Using the hybrid neurocomputer
with the analog input layer produced a minimum absolute rms error of .022. This is an
improvement of 12% over the sigmoidal network
855
856
Watkins, Chau, Tawel, Lambrigsten, and Plutowski
5.3 The Speech Phoneme Recognition Application
The RBF network was not compared to any non-neural network method for the speech
phooeme recognition problem. It was only compared to Waibel's Tme Delay Neural
Network (IDNN) (Waibel et al .. 1989). The IDNN uses a topology matched to the timevarying nature of speech with two hidden layers of eight and three neurons respectively.
The RBF network used a single hidden layer with the number of neurons varying between
eight and one hundred.
The IDNN achieved a 98% accuracy on the test set discriminating between the phooemes
b. d and g. The RBF network achieved over 99% accuracy in training. but was only able to
achieve an 86% accuracy on the test set. To obtain better results. it is clear that the
topology of the RBF network needs to be altered to more closely match Waibel's IDNN.
However. this topology will complicate the VLSI implementation.
5.4 The Feasibility of Using the Analog Input Layer
One potential problem with using an analog input layer is that every individual hybrid
RBF neurocomputer might need to be trained on a problem. rather than being able to use a
common set of weights obtained from another RBF neurocomputer (which had been
previously trained). This potential problem exists because every analog circuit is unique
due to variation in the fabrication process. A set of experiments was designed to test this
possibility.
The remote sensing application and the Mackey-Glass application were trained and tested
two different ways: 1) hardware-trainedlhardware-tested. that is. the analog input layer
was used for both training and testing; 2) software-trainedlhardware-tested. that is the
analog input layer was simulated with 32-bit floating point precision for training and then
the analog hardware was used for testing . .The hardwarelhardware results provided a
benchmark. The softwarelhardware results demonstrated the feasibility of having a
standard set of weights that are not particular to a given set of analog hardware. For both
the remote sensing and the Mackey-Glass applications. the rms error performance was
only slightly degraded by using weights learned during software simulation. The remote
sensing results degraded by only .Oll in terms of absolute rms error. and the MackeyGlass results degraded by only .002 in terms of absolute rms error. The results of the
experiment indicate that each individual hybrid RBF neurocomputer only needs to be
calibrated. not trained.
6.0 Conclusions
A low-power. hybrid analog/digital neurocomputer development system was constructed
using custom hardware. The system implements a radial basis function (RBF) network
and is targeted for applications that require low power consumption and use analog data as
their input. particularly remote sensing and portable applications. Several applications
were executed and results were obtained for a remote sensing application that are superior
to any previous results. Comparison of the results of a completely-digital simulation of the
RBF network and the hybrid analog/digital RBF network demonstrated the feasibility of
the hybrid approach.
A Hybrid Radial Basis Function Neurocomputer and Its Applications
Acknowledgments
The research described in this paper was performed at the Center for Space
Microelectronics Technology. Jet Propulsion Laboratory. California Institute of
Technology, and was sponsored by the National Aerooautics and Space Admjnjstration.
One of the authors. Steven S. Watkins. acknowledges the receipt of a Graduate Student
Researcher's Center Fellowship from the Natiooal Aeronautics and Space Administration.
Useful discussions with Silvio Eberhardt, Roo Fellman. Eric Fossum. Doug Kerns.
Fernando Pineda, John Platt, and Anil Thakoor are also gratefully acknowledged.
References
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of Atmospheric Moisture Profiles by Microwave Radiometry," Journal of Climate and
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R. P. Lippman. "A Critical Overview of Neural Network Pattern Oassifiers." Proceedings
of the IEEE Neural Networks for Signal Processing Workshop, 1991, Princeton. NJ.? pp.
266-275.
Carver Mead, Analog VLSI and Neural Systems. Addison-Wesley. 1989, pp. 90-94.
J. Moody and C. Darken. "Fast Learning in Networks of Locally-Tuned Processing
Units," Neural Computation, vol. 1. no. 2, Summer 1989. pp. 281-294.
Howard Motteler, lA. Gualtieri. LL. Strow and Larry McMillin. "Neural Networks for
Atmospheric Retrievals," NASA Goddard Conference on Space Applications of Artificial
Intelligence. 1993, pp. 155-167.
John Platt, "A Resource-Allocating Neural Network for Function Interpolation," Neural
Computation, vol. 3. no. 2, Summer 1991, pp. 213-225.
John Platt. Janeen Anderson and David B. Kirk. "An Analog VLSI Qrip for Radial Basis
Functions," NIPS 5. 1993, pp. 765-772.
Alexander Waibel. T. Hanazawa. G. Hinton. K. Shikano and K. Lang. "Phoneme
Recognition Using Tune-Delay Neural Networks." IEEE International Conference on
Acoustics, Speech and Signal Processing, May 1989, pp. 393-404.
Steve Watkins, Paul Chau and Raoul Tawel. "A Radial Basis Functioo Neurocomputer
with an Analog Input Layer." Proceedings of the IJCNN, Beijing. China. November 1992.
pp. ill 225-230.
Steve Watkins. Paul Chau and Raoul Tawel, "Different Approaches to Implementing A
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7,038 | 811 | Generalization Error and The Expected
Network Complexity
Chuanyi Ji
Dept. of Elec., Compt. and Syst Engl' .
Rensselaer Polytechnic Inst.itu( e
Troy, NY 12180-3590
[email protected]
Abstract
For two layer networks with n sigmoidal hidden units, the generalization error is
shown to be bounded by
O(E~)
l N)
K + O( (EK)d
N
og
,
where d and N are the input dimension and the number of training samples, respectively. E represents the expectation on random number K of hidden units
(1 :::; I\ :::; n). The probability Pr(I{ = k) (1 :::; k :::; n) is (kt.erl11ined by a prior
distribution of weights, which corresponds to a Gibbs distribtt! ion of a regularizeI'.
This relationship makes it possible to characterize explicitly how a regularization
term affects bias/variance of networks. The bound can be obtained analytically
for a large class of commonly used priors. It can also be applied to estimate the
expected net.work complexity Ef{ in practice. The result provides a quantitative
explanation on how large networks can generalize well .
1
Introduction
Regularization (or weight-decay) methods are widely used in supervised learning by
adding a regularization term t.o an energy function. Although it is well known that
such a regularization term effectively reduces network complexity by introducing
more bias and less variance[4] to the networks, it is not clear whether and how the
information given by a regularization term can be used alone to characterize the
effective network complexity and how the estimated effective network complexity
relates to the generaliza.tion error . This research attempts to provide answers to
t.hese questions for two layer feedforward networks with sigmoidal hidden units.
367
368
Ji
Specifically) the effective network complexity is ch(lJ'act.erized by the expected nUI11bel' of hidden units determined by a Gibbs dist.ribution corresponding to a regula L'ization tenl1. The generalization error can then be bounded by the expected network
complexity) and thus be tighter than the original bound given by Barron[2]. The
new bound shows explicitly) through a bigger approximation error and a smaller
estimation error I how a regularization term introduces more bias and less varia nce
to the networks. It therefore provides a quantitative explanation on how a network
larger than necessary can also generalize well under certain conditions) which can
not be explained by the existing learning theory[9].
For a class of commonly-used regularizers) the expecced network complexity can
be obtained in a closed form. It is then used to estimate the expected network
complexity for Gaussion mixture model[6].
Background and Previous Results
2
A relationship has been developed by Barron[2] between generalization error and
network complexity for two layer net.works used for function approximation. "Ve
will briefly describe this result in this section and give our extension subsequently.
Consider a class of two layer networks of fixed architecture with n sigmoidal hidden
units a.nd one (linear) output unit. Let fn(x; w) = twF)91(wP)T x) be a n eiW01'k
1=1
wP)
function) where wEen is the network weight vcctor comprising both Wf2) and
for 1 ::; l ::; n. w}l) and W}2) are the incoming weights to the l-th hidden unit and
the weight from the l-th hidden unit to the output) respectively. en ~ Rn(d+1) is
t.he weight space for n hidden unit.s (and input dimension d) . Each sigmoid unit
!JI(Z) is assumed to be of tanh type: !J/(z) --+ ?1 as z --+ ?oo for 1 ::; I :S n 1.
The input is xED ~ Rd. '''' ithout loss of generality) D is assumed to be a unit
hypercube in R d ) i.e.) all the components of x are in [?-1) 1].
Let f( x) be a target function defined in the sa.me domain D and satisfy some
smoot.hness conditions [2]. Consider N training samples independently drawn from
some distribution p(:/.:): (x1)f(:I:1)), ... ) (xN)f(;t.v)). Define an energy function e)
where e = f1 + A LTI.~~(1U) . Ln ,N(W) is a regularization term as a function of tv
for a. fixed II . A is a const.ant. . C1 is a quadratic error function on N training
lV
samples :
e1
=
J: i=l
'L,(fn(Xi;W) -
function such t.hat
'ttl
')
f(Xi)t? Let fll,l'.,r(x;-t'iJ) be t.he (optimal) network
.
minimizes t.he energy function e: tV = arg min e. The gen-
=
wEen
=
eralization error Eg is defined to be the squared L'2 norm E9
Ell f - fn,N 112
EJU(x) - fn,N(X; w))2dp(x)) where E is the expectation over all training sets of
D
size N drawn from the same distributioll. Thus) the generalization error measnres
the mean squared distance between the unknown function an' I the best network
function that can be obtained for training sets of size N . The generalization error
1 In the previous ,\'ork by Barron) t.he sigmoillal hidden units atC' '1,( ~)+1. It is easy t.o
show that his results are applica.ble to the class of .t!1(Z))S we consider h;re.
Generalization Error and the Expected Network Complexity
Eg is shown[2] to be bounded as
Eg ::; O(Rn,N),
(1)
where Rn ,N, called the index of resol vability [2], can be expressed as
Rn ,N = min
wEen
{II .f _ in
112
+ Ln,~( tv)},
(2)
where III is the clipped fn(x; tv) (see [2]). The index of resolvability can be further
bounded as Rn,N :::; O(~) + O(',~~logN). Therefore, the generalization error IS
bounded as
1
E!! :::; 0(;;)
+ O(
nd
N logN),
(3)
where O(~) and 0(';1 logN) are t.he bounds for approxima.tion error (bia.s) and
esti ;:l.lnt.ion error (varia.nce), respectively.
In addition, t.he hOllnd for E9 can be minimized if all additional regularization term
LN (71) is used in the energy function to minimize the number of hidden units, i.e.,
r=N
Eg :::; O( V dlogN ).
3
Open Questions and Motivations
Two open questions, which can not be answered by the previous result, are of the
primary interest of this work.
I) How do large networks generalize?
The largc networks refer to those wit.h a rat.io ~~ to he somewhat big, where TV
and N are the t.ot.al number of independent.ly modifiable weights (lV ~ nel, for
11 lcugc) and the number of training samples, respectively. Networks tra.ined with
reglll<Hization t.erms may fall int.o this category. Such large networks are found
(0 Jw abk to gen eralize well sometimes. JImH'H'J', when '~~{ is big, the bonnel in
Eqll ahon (~:l) is t.oo loose t.o bOllnd the actual generaliza t.ion error meaningfully.
Therefme. for the large networks, the tot.al number of hidden ullits n ma.y no longer
be a. good est.imate for network complexity. Efforts have been made to develop
measures on effective net.work complexity both analytically and cmpirically[1][5][10] .
These measures depend on training data as well as a regularization term in an
implicit way which make it difficult to see direct. effects of a regulariza.tion term on
generaliza.tion error. This naturally leads t.o our second question.
2) Is it possible to characterize network complexit.y for a cLI~~ of networks using
only the information given by a regularizat.ion term:!? How t.o relat.e the estimated
network complexity rigorously with generalization error?
In practice, when a regularization term (L I1 .N(W)) is used to penalize the m;l~llitude
of weights, it effectively minimizes the number of hidden units as ,,,,'ell even til' '1lgb a.n
additional regularization term LN(n) is not used. This is dne to the fact tbll. some
of the hidden units may only operate in the lineal' region of a sigmoid when their
2This was posed as an open problem hy Solia. ei..al. [8]
369
370
Ji
incoming weights are small and inputs are bounded. Therefore, a Ln,N(W) term can
effectively act like a LN(n) term that reduces the effective number of hidden units,
and thus result in a degenerate parameter space whose degrees of freedom is fewer
than nd. This fact was not taken into consideration in the previous work, and as
shown later in this work, will lead to a tighter bound on Rn,N.
In what follows, we will first define the expected network complexity, then use it to
bound the generalization error.
4
The Expected Network C0111plexity
For reasons that will hecome apparent, we choose to define the effective complexity
of a feedforward two layer network as the expected number of hidden unit.s EE
(1 :::; J{ :::; 11) ,vhich are effectively nonlinear, i.e. operating outside t.he central
linear regions of their sigmoid response function g(.::). '''''e define the linear region
as an interval 1 z 1< b with b a positive constant.
Consider the presynaptic input:: = wiT x to a hidden unit g(z), where Wi is the
incoming weight vector for the unit. Then the unit is considered to be effectively
linear if 1z 1< b for all xED. This will happen if 1 Zl 1< b, where z' = wiT x' with
x' being any vertex of the unit hypercube D. This is b~cause 1 z I:::; wiT X, where x
is the vertex of D whose elements are t.he 8gn functions of the elements of Wi.
Next, consider network weights as random variaJ)lcs wit.h a distribution p(w) =
Aex1J( - Ln,N (tv)), ,,,hich corresponds t.o a. Gibbs distribution of a regularization
term wit.h a normalizing constant. A. Consider the vector ;'1;' to be a random vector
also wit.h eqnally probable l~s ,Hld -l's. Then I::' 1< b will be a random event. The
probability for this hidden unit to be effectively nonlin0.ill' equals to 1- Pr(1 z 1< b),
which can be completely determined by the distributions of weights p( 'W) and x'
(equally probable). Let. f{ be the number of hidden units which are effectively
nonlinear. Then t.he probability, Pr(K = k) (1 :::; k :::; n), can be determined
through a joint probabilit.y of k hidden units that are operating beyond the central
linear region of sigmoid fUllctions. The expected network complexity, EI<, can then
be obtained through Pr(I< = k), which is determined by the Gibbs distribution of
LN,n (w). The motivation on utilizing such a Gibbs distribution comes from the fact
that Rk,N is independent of training samples but dependent. of a regularization term
which corresponds to a prior distribution of weights. Using sHch a formulation, as
will be shown later, the effect of a regularization term on bias and va riance ca.n be
characterized explicitly.
5
A New Bound for The Generalization Error
To develop a t.ightcr houucl for the generalizat.ion error, we consider subspa.ces of
t.he weights indexed by different number of effectively nonlinc(lr hidden units: 8 1 ~
8 2 . .. ~ 8 n . For ead, 8 j , there are j out of 11 hidden unit.s which are effectively
nonlinear fo], 1 :; j :::; n. '1'11e1'e1'ore, the index ofl'esolvability T?71,N ca.n be expressed
as
(4)
Generalization Error and the Expected Network Complexity
where each Rk,N = min {II
wEe"
f -
in
112 + Ln.~(w)}. Next let us consider the number
of effectively nonlinear units to be random. Since the minimum is no bigger than
the average, we have
(5)
where the expectation is taken over the random variable J{ utilizing the probability
Pr(I{ = k). For each K , however, the t,yO terms in Rf(,N can be bounded as
by the t.rian.gle ine4uality, where fn-l":,n is the actuallletwork function with n - J{
hidden units operating in the region bounded by the constant b, and ff( is the
correspondillg network funct.ion which t.rea ts the 11 - J{ units as linear units. In
addition, we have
.
Ln,N(W) ::; O(II.fn-K,n - jI{
')
I{d
W) + O( N logN),
(7)
\vhere the f-irst term also results from the triangle inequality, and the second term
is obtained by cliscretizing the degenerate parameter space e J{ using similar techl1lques as in [2]3. Applying Taylor expansion on the t.erm \\ fn-K,n - ff( \\2, \\'e
have
\\ fn-K,n - ff{ \\2 ::; O(b13(n - K)).
(8)
Putting Equations (5) (6) (7) and (8) into Equation (1), \\'(' have
1
(EK)d
Eg ::; O(E !{) + O( N logN)
+ O(b
6
(11 - EX))
()
+ o(b)),
(9)
where EX is the expected number of hidden units which are effectively nonlinear.
If b ::; O( -\-), we have
n3
1
Eg ::; O(E J()
6
+ O(
(EI{)d
N logN) .
(10)
A Closed Fornl Expression For a Class of Regularization
Ternls
For commonly used regularization terms, how can \"e actually find the probability
distribution of the number of (nonlinear) hidden units Pr(I{ = k)? And how shall
we evaluate EK and E J( ?
As a simple example, we consider a special class of prior distrihutions for iid weights,
i.e, p( w) = TIiP( Wi), where W.i are the "i<'ments of wEen. This corresponds to
a large class of regularization terms ,,'hicIt minimize the magnitudes of individual
weights indepcndently[7].
Consider each weight as a random variable with zero mean and a common variance
(J. Then for large input dimension el,
is approximately normal with zero-mean
v7zZ'
3 Deta.ils
\Yill be given ill
iL
longer version of the pa.per in prepa.ra.tion.
371
372
Ji
and varia.nce (J by the Central Limit Theorem[3]. Let q denote the probability that
a. unit is effectively nonlinear. We have
q = 2Q(-x
where Q( -;1.:)
b
r,)'
(11 )
(Jyd
:;l
= );- J e- T ely.
Next consider the probability that
J(
out of n
-co
hidden units are nonlinear. Based
x', I( has a binomial distribution
011
Pr(I{ = I.:) =
where 1
< k < n.
our (independence) assumptions on w' a.nd
( 71.)
k qk (1 -
q) n - /.; ,
Then
EX = nq.
1
1
E}, = - +~,
\
n
where ~ =
n-1
L
(12)
(1:3)
(14)
.
HI - qr-~ + (1 - qt?
Then the generalization error Eo satisfies
i=1
1
Eg. :::; 0(n
7
+~)
nqd
+ O(-N logN)
(15)
Application
As an example for applica.t.ions of t.he tJleoretical results, the expected network complexity EJ{ is estimat.ed for G<:tussian mixture model used for time-series prediction
(details can he found in [6]) 4.
In genera.l, llsillg only a prior dist.ribut.ion of ,,,eights to est.ima.te the network COlllplexit.y EJ{ may lead to a less accurate measure on the effective net.work complexiLy
than incorporat.ing informat.ion on training data also. However, if parameters of a
regularization term also get optimized during training, as shown in this example ,
the resulting Gibbs prior distribution of weights may lead to a good estimate of the
effective number of hidden units.
Specifically, the corresponding Gibbs distribution p( 'W) of the weights from the
Gaussion mixture is iicl, which consists of a linear combination of eight Gaussia.n
distributions. This function results in a skewed distribntion with a sharp peak
around the zero (see [6]). The mean and variance of the presynaptic inputs z t.o
the hidden units can thus be estimated as 0.02 and 0.04, respectively. The other
parameters used are n = 8, d = 12. b = 0.6 is chosen. Then q ~ 004 is obtained
through Equation (11). The effective network complexity is EJ{ ~ 3 (or 4). The
empirical result(10], which estima.tes the effective number of hidden units using the
dominated eigenvalues at the hidden layer, results in about ;3 effective hidden units.
4 Strictly speaking , the theoretical resnlts deal with l'egulariza tion terms with discrete
weight.s. It. can a.nd ha.s been extended to continuous weight.s by D.F. McCaffrey and A .R.
Galla.nt. Details are beyond the content of this paper.
Generalization Error and the Expected Network Complexity
5r---------.----------r---------.----------r-------~
4.5
4
variance
0.5
increase in bias
0.2
0.6
0.4
0.8
q
Figure 1: Illustration of an increase .6.. in bias and variance Bqn as a function of q.
A sca.ling fadar J3 = 0.25 is used for t.he convenience of the plot. 11 = 20 is chosen.
8
Discussions
Is this new bound for the generalization tighter than the old one which takes no
account of l1etwork-weight.-dependent information? If so . what does it tell us?
Compared wit.h the bOllnd in Equation (3), the new bound results in an increase .6..
in approximation error (bias), and qn instea.d of n as ~sLimatjon errol' (variallce).
These two terms are plotted as functions of q in Figure (1). Since q is a. function of
(J which characterizes how strongly the magnitude of the weights is penalized, the
larger the (J, the less the weights get penalized, the larger the q, the more hidden
uni ts are likely to be effectively nonlinear, thus the smaller the bias and larger the
variance. ,\Vhen q = 1, all the hidden units are effectively nonlinear and the new
bound reduces to the old one. This shows ho",- a regulariza.tion t.erm directly affects
bias / variance.
'\i\Then the estimation error dominates, the bound for the generalization error will be
proportional to nq inst.ead of n. The value of 1'/,I}, however, depends on the choice of
a. For small (J, the new bound can be much tighter than the old one, especially for
large netwOl'ks with n large but nq small. This will provide a practical method to
cstilllate gCltcrnlizn.tion errol' for large nctworks as well as an explanation of when
rllld why hn~e networks can generalize ,,-ell.
How tight the bound really is depends on how well Ln,l\ (lL!) is chosen. Let no denote
t.he optimallll1ll1ber of (nonlinear) hidden units needeJ to approximate I(x). If
Ln,N(W) is chosen so that. the corresponding 1J(W) is almost a delta. function a.t no,
t.hen ERK,i\' ~ Rno,N, which gives a. very tight bound. Otherwise, if, for insta.nce,
373
374
Ii
Ln,N(W) penalizes network complexity so little that ERJ(,N :=:::: Rn,N, the bound
will be as loose as the original one. It should also be noted that an exact value for
the bound cannot be obtained unless some information on the unknown function f
itself is available.
For commonly used regularization terms, the expected network complexity can be
estimated through a close form expression. Such expected network complexity is
shown to be a good estimate for the actual network complexity if a Gibbs prior
distribution of weights also gets optimized through training, and is also sharply
peaked. More research will be done to evaluate the applica.bility of the theoretical
results.
A cknow ledgeluent
The support of National Science Foundation is gratefully acknowledged.
References
[1] S. Amari and N. Murata, "Statistical Theory of Learning Curves under Entropic Loss Criterion," Neural Computation, 5, 140-153, 1993.
[2] A. Barron, "Approximation a.nd Estimation Bounds for Artificial Neural Networks," Proc. of The 4th Workshop on Computational Learning Theory, 243249, 1991.
[3] Vv. Feller, An Introduction to Probability Theory and Its Applications, New
York: John \Viley and Sons, 1968.
[4] S. Geman, E. Bienenstock, and R. Doursat, "Neural Networks and the
Bias/Variance Dilemma," Neural Comp1tiation, 4, 1-58, 1992.
[5] J. Moody, "Generalization, vVeight Decay, and Architecture Selection for Nonlinear Learning Systems," Proc. of Neural Information Processing Systems,
1991.
[6] S.J. Nowlan, and G.E. Hinton, "Simplifying Neural Networks by Soft \Veight
Sha.ring," Neural computation, 4,473-493(1992).
[7] R. Reed, "Pruning Algorithms-A Survey," IEEE Trans. Neural Networks Vol.
4, 740-'i'47, (1993).
[8] S. Solla, "The Emergence of Generalization Ability in Learning," Presented at
NIPS92.
[9] V. Vapnik, "Estimation of Dependences Based on Empirical Data," SpringerVerlag, New York, 1982.
[10] A.S . V\'eigend and D.E . Rumelhart, "The Effective Dimension of the Space of
Hidden Units," Proc. of International Joint Conference on Ne1tral Networks,
1992.
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7,039 | 812 | Analyzing Cross Connected Networks
Thomas R. Shultz
Department of Psychology &
McGill Cognitive Science Centre
McGill University
Montreal, Quebec, Canada H3A IB 1
[email protected]
and
Jeffrey L. Elman
Center for Research on Language
Department of Cognitive Science
University of California at San Diego
LaJolla, CA 92093-0126 U.S.A.
[email protected]
Abstract
The non-linear complexities of neural networks make network solutions
difficult to understand. Sanger's contribution analysis is here extended to
the analysis of networks automatically generated by the cascadecorrelation learning algorithm. Because such networks have cross
connections that supersede hidden layers, standard analyses of hidden
unit activation patterns are insufficient. A contribution is defined as the
product of an output weight and the associated activation on the sending
unit, whether that sending unit is an input or a hidden unit, multiplied
by the sign of the output target for the current input pattern.
Intercorrelations among contributions, as gleaned from the matrix of
contributions x input patterns, can be subjected to principal
components analysis (PCA) to extract the main features of variation in
the contributions. Such an analysis is applied to three problems,
continuous XOR, arithmetic comparison, and distinguishing between
two interlocking spirals. In all three cases, this technique yields useful
insights into network solutions that are consistent across several
networks.
1
INTRODUCTION
Although neural network researchers are typically impressed with the performance
achieved by their learning networks, it often remains a challenge to explain or even
characterize such performance. The latter difficulties stem principally from the complex
non-linear properties of neural nets and from the fact that information is encoded in a form
that is distributed across many weights and units. The problem is exacerbated by the fact
that multiple nets generate unique solutions depending on variation in both starting states
and training patterns.
Two techniques for network analysis have been applied with some degree of success,
focusing respectively on either a network's weights or its hidden unit activations. Hinton
(e.g., Hinton & Sejnowski, 1986) pioneered a diagrammatic analysis that involves
plotting a network's learned weights. Occasionally, such diagrams yield interesting
insights but often, because of the highly distributed nature of network representations, the
most notable features of such analyses are the complexity of the pattern of weights and its
variability across multiple networks learning the same problem.
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Statistical analysis of the activation patterns on the hidden units of three layered feedforward nets has also proven somewhat effective in understanding network performance.
The relations among hidden unit activations, computed from a matrix of hidden units x
input patterns, can be subjected to either cluster analysis (Elman, 1990) or PCA (Elman,
1989) to determine the way in which the hidden layer represents the various inputs.
However, it is not clear how this technique should be extended to multi-layer networks or
to networks with cross connections.
Cross connections are direct connections that bypass intervening hidden layers. Cross
connections typically speed up learning when used in static back-propagation networks
(Lang & Witbrock, 1988) and are an obligatory and ubiquitous feature of some generative
learning algorithms, such as cascade-correlation (Fahlman & Lebiere, 1990). Generative
algorithms construct their own network topologies as they learn. In cascade-correlation,
this is accomplished by recruiting new hidden units into the network, as needed, installing
each on a separate layer. In addition to layer-to-layer connections, each unit in a cascadecorrelation network is fully cross connected to all non-adjacent layers downstream.
Because such cross connections carry so much of the work load, any analysis restricted to
hidden unit acti vations provides a partial picture of the network solution at best.
Generative networks seem to provide a number of advantages over static networks,
including more principled network design, leaner networks, faster learning, and more
realistic simulations of hwnan cognitive development (Fahlman & Lebiere, 1990; Shultz,
Schmidt, Buckingham, & Mareschal, in press). Thus, it is important to understand how
these networks function, even if they seem impervious to standard analytical tools.
2
CONTRIBUTION ANALYSIS
One analytical technique that might be adapted for multi-layer, cross connected nets is
contribution analysis (Sanger, 1989). Sanger defined a contribution as the triple product
of an output weight, the activation of a sending unit, and the sign of the output target for
that input. He argued that contributions are potentially more informative than either
weights alone or hidden unit activations alone. A large weight may not contribute much
if it is connected to a sending unit with a small activation. Likewise, a large sending
activation may not contribute much if it is connected via a small weight. In contrast,
considering a full contribution, using both weight and sending activation, would more
likely yield valid comparisons.
Sanger (1989) applied contribution analysis to a small version of NETtalk, a net that
learns to convert written English into spoken English (Sejnowski & Rosenberg, 1987).
Sanger's analysis began with the construction of an output unit x hidden unit x input
pattern array of contributions. Various two-dimensional slices were taken from this threedimensional array, each representing a particular output unit or a particular hidden unit.
Each two-dimensional slice was then subjected to PCA, yielding information about either
distributed or local hidden unit responsibilities, depending on whether the focus was on an
individual output unit or individual hidden unit, respectively.
3 CONTRIBUTION ANALYSIS FOR MULTI? LAYER,
CROSS CONNECTED NETS
We adapted contribution analysis for use with multi-layered, cross connected cascadecorrelation nets. Assume a cascade-correlation network with j units (input units + hidden
units) and k output units, being trained with i input patterns. There are j x k output
weights in such a network, where an output weight is defined as any weight connected to
Analyzing Cross-Connected Networks
an output unit. A contribution c for a particular ijk combination is defined as
Cijk = Wjk aij 2tki
(1)
where Wjk is the weight connecting sending unit j with output unit k, aij is the activation
of sending unit j given input pattern i, and tki is the target for output unit k given input
pattern i. The term 2tki adjusts the sign of the contribution so that it provides a measure
of correctness. That is, positive contributions push the output activation towards the
target, whereas negative contributions push the output activation away from the target. In
cascade-correlation, sigmoid output units have targets of either -0.5 or +0.5. Hence,
mUltiplying a target by 2 yields a positive sign for positive targets and a negative sign for
negative targets. Our term 2tki is analogous to Sanger's (1989) term 2tik - 1, which is
appropriate for targets of 0 and I, commonly used in back-propagation learning.
In contrast to Sanger's (1989) three-dimensional array of contributions (output unit x
hidden unit x input pattern). we begin with a two-dimensional output weight (k * j) x
input pattern (i) array of contributions. This is because we want to include all of the
contributions coming into the output units, including the cross connections from more
than one layer away. Since we begin with a two-dimensional array. we do not need to
employ the somewhat cumbersome slicing technique used by Sanger to isolate particular
output or hidden units. Nonetheless. as will be seen, our technique does allow the
identification of the roles of specific contributions.
4
PRINCIPAL COMPONENTS ANALYSIS
Correlations among the various contributions across input patterns are subjected to PCA.
PCA is a statistical technique that identifies significant dimensions of variation in a
multi-dimensional space (Flury, 1988). A component is a line of closest fit to a set of
points in multi-dimensional space. The goal of PCA is to summarize a multivariate data
set using as few components as possible. It does this by taking advantage of possible
correlations among the variables (contributions, in our case).
We apply PCA to contributions, as defined in Equation I, taken from networks learning
three different problems: continuous XOR, arithmetic comparisons. and distinguishing
between interlocking spirals. The contribution matrix for each net, as described in section
3, is subjected to PCA using 1.0 as the minimum eigenvalue for retention. Varimax
rotation is applied to improve the interpretability of the solution. Then the scree test is
applied to eliminate components that fail to account for much of the variance (Cattell,
1966). In cases where components are eliminated. the analysis is repeated with the correct
number of components. again with a varimax rotation. Component scores for the retained
components are plotted to provide an indication of the function of the components.
Finally. component loadings for the various contributions are examined to determine the
roles of the contributions from hidden units that had been recruited into the networks.
5
APPLICATION TO THE CONTINUOUS XOR PROBLEM
The simplicity of binary XOR and the small number of training patterns (four) renders
application of contribution analysis superfluous. However, it is possible to construct a
continuous version of the XOR problem that is more suitable for contribution analysis.
We do this by dividing the input space into four quadrants. Input values are incremented
in steps of 0.1 starting from 0.0 up to 1.0, yielding 100 x, y input pairs. Values of x up
to 0.5 combined with values of y above 0.5 produce a positive output target (0.5), as do
values of x above 0.5 combined with values of y below 0.5. Input pairs in the other two
quadrants yield a negative output target (-0.5).
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Shultz and Elman
Three cascade-correlation nets are trained on this problem. Each of the three nets generates
a unique solution to the continuous XOR problem, with some variation in number of
hidden units recruited. PCA of contributions yields different component loadings across
the three nets and different descriptions of components. Yet with all of that variation in
detail, it is apparent that all three nets make the same three distinctions that are afforded
by the training patterns. The largest distinction is that which the nets are explicitly
trained to make, between positive and negative outputs. Two components are sufficient to
describe the representations. Plots of rotated component scores for the 100 training
patterns cluster into four groups of 25 points, each cluster corresponding to one of the
four quadrants described earlier. Component loadings for the various contributions on the
two components indicate that the hidden units play an interactive and distributed role in
separating the input patterns into their respective quadrants.
6
APPLICATION TO COMPARATIVE ARITHMETIC
A less well understood problem than XOR in neural net research is that of arithmetic
operations, such as addition and multiplication. What has a net learned when it learns to
add, or to multiply, or to do both operations? The non-linear nature of multiplication
makes it particularly interesting as a network analysis problem. The fact that several
psychological simulations using neural nets involve problems of linear and non-linear
arithmetic operations enhances interest in this sort of problem (McClelland, 1989; Shultz
et al., in press).
We designed arithmetic comparison tasks that provided interesting similarities to some of
the psychological simulations. In particular, instead of simply adding or multiplying, the
nets learn to compare sums or products to some value and then output whether the sum or
product is greater than, less than, or equal to that comparative value.
The addition and multiplication tasks each involve three linear input units. The first two
input units each code a randomly selected integer in the range from 0 to 9, inclusive. The
third input unit codes a randomly selected comparison integer. For addition problems, the
comparison values are in the range of 0 to 19, inclusive; for multiplication the range is 0
to 82, inclusive. Two output units code the results of the comparison. Target outputs of
0.5 and -0.5 represent that the results of the arithmetic operation are greater than the
comparison value, targets of -0.5 and 0.5 represent less than, and targets of 0.5 and 0.5
represent equal to. For problems involving both addition and multiplication, a fourth
input unit codes the type of arithmetic operation to be performed: 0 for addition, 1 for
multiplication.
Nets trained on either addition or multiplication have 100 randomly selected training
patterns, with the restriction that 45 of them have correct answers of greater than, 45 have
correct answers of less than, and 10 have correct answers of equal to. The latter constraints
are designed to reduce the natural skew of comparative values in the high direction on
multiplication problems. Nets trained on both addition and multiplication have 100
randomly selected addition problems and 100 randomly selected multiplication problems,
subject to the constraints just described. We trained three nets on addition, three on
multiplication, and three on both addition and multiplication.
6.1
RESULTS FOR ADDITION
PCA of contributions in all three addition nets yield two significant components. In each
of the three nets, the component scores form three clusters, representing the three correct
answers. In all three nets, the first component distinguishes greater than from less than
answers and places equal to answers in the middle; the second component distinguishes
Analyzing Cross-Connected Networks
equal to from unequal to answers. The primary role of the hidden unit in these nets is to
distinguish equality from inequality. The hidden unit is not required to perform addition
per se in these nets, which have additive activation functions.
6.2
RESUL TS FOR MULTIPLICATION
PCA applied to the contributions in the three multiplication nets yields from 3 to 4
significant components. Plots of rotated component scores show that the first component
separates greater than from less than outputs, placing equal to outputs in the middle.
Other components further differentiate the problems in these categories into several
smaller groups that are related to the particular values being multiplied. Rotated
component loadings indicate that component 1 is associated not only with contributions
coming from the bias unit and the input units, but also with contributions from some
hidden units. This underscores the need for hidden units to capture the non-linearities
inherent to multiplication.
6.3
RESULTS FOR BOTH ADDITION AND MULTIPLICATION
PCA of contributions yields three components in each of the three nets taught to do both
addition and multiplication. In addition to the familiar distinctions between greater than,
less than, and equal to outputs found in nets doing either addition or multiplication, it is
of interest to determine whether nets doing both operations distinguish between adding
and multiplying.
Figure 1 shows the rotated component scores for net 1. Components 1 and 2 (accounting
for 30.2% and 21.9% of the variance, respectively) together distinguish greater than
answers from the rest. Component 3, accounting for 20.2% of the variance, separates
equal to answers from less than answers and multiplication from addition for greater than
answers. Together, components 2 and 3 separate multiplication from addition for less than
answers. Results for the other two nets learning both multiplication and addition
comparisons are essentially similar to those for net 1.
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Figure 1. Rotated component scores for a net doing both addition and multiplication.
6.4
DISCUSSION OF COMPARATIVE ARITHMETIC
As with continuous XOR, there is considerable variation among networks learning
comparative arithmetic problems. Varying numbers of hidden units are recruited by the
networks and different types of components emerge from PCA of network contributions.
In some cases, clear roles can be assigned to particular components, but in other cases,
separation of input patterns relies on interactions among the various components.
1121
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Shultz and Elman
Yet with all of this variation, it is apparent that the nets learn to separate arithmetic
problems according to features afforded by the training set. Nets learning either addition or
multiplication differentiate the problems according to answer types: greater than, less
than, and equal to. Nets learning both arithmetic operations supplement these answer
distinctions with the operational distinction between adding and multiplying.
7
APPLICATION TO THE TWO-SPIRALS PROBLEM
We next apply contribution analysis to a particularly difficult discrimination problem
requiring a relatively large number of hidden units. The two-spirals problem requires the
net to distinguish between two interlocking spirals that wrap around their origin three
times. The standard version of this problem has two sets of 97 continuous-valued x, y
pairs, each set representing one of the spirals. The difficulty of the two-spirals problem is
underscored by the finding that standard back-propagation nets are unable to learn it
(Wieland, unpublished, cited in Fahlman & Lebiere, 1990). The best success to date on
the two-spirals problem was reported with cascade-correlation nets, which learned in an
average of 1700 epochs while recruiting from 12 to 19 hidden units (Fahlman & Lebiere,
1990). The relative difficulty of the two-spirals problem is undoubtedly due to its high
degree of non-linearity. It suited our need for a relatively difficult, but fairly well
understood problem on which to apply contribution analysis. We ran three nets using the
194 continuous x, y pairs as inputs and a single sigmoid output unit, signaling -0.5 for
spiral 1 and 0.5 for spiral 2.
Because of the relative difficulty of interpreting plots of component scores for this
problem, we focus primarily on the extreme component scores, defined as less than -lor
greater than 1. Those x, y input pairs with extreme component scores on the first two
components for net 1 are plotted in Figure 2 as filled points on the two spirals. There are
separate plots for the positive and negative ends of each of the two components. The fllled
points in each quadrant of Figure 2 define a shape resembling a tilted hourglass covering
approximately one-half of the spirals. The positive end of component 1 can be seen to
focus on the northeast sector of spiral 1 and the southwest sector of spiral 2. The negative
end of component 1 has an opposite focus on the northeast sector of spiral 2 and the
southwest sector of spiral 1. Component 2 does precisely the opposite of component 1:
its positive end deals with the southeast sector of spiral 1 and the northwest sector of
spiral 2 and its negative end deals with the southeast sector of spiral 2 and the northwest
sector of spiral 1. Comparable plots for the other two nets show this same hourglass
shape, but in a different orientation.
The networks appear to be exploiting the symmetries of the two spirals in reaching a
solution. Examination of Figure 2 reveals the essential symmetries of the problem. For
each x, y pair, there exists a corresponding -x, -y pair 180 degrees opposite and lying on
the other spiral. Networks learn to treat these mirror image points similarly, as revealed
by the fact that the plots of extreme component scores in Figures 2 are perfectly
symmetrical across the two spirals. If a point on one spiral is plotted, then so is the
corresponding point on the other spiral, 180 degrees opposite and at the same distance out
from the center of the spirals. If a trained network learns that a given x, y pair is on spiral
1, then it also seems to know that the -x, -y pair is on spiral 2. Thus, it make good sense
for the network to represent these opposing pairs similarly.
Recall that contributions are scaled by the sign of their targets, so that all of the products
of sending activations and output weights for spiral 1 are multiplied by -1. This is to
ensure that contributions bring output unit activations close to their targets in proportion
Analyzing Cross-Connected Networks
to the size of the contribution. Ignoring this scaling by target, the networks possess
sufficient information to separate the two spirals even though they represent points of the
two spirals in similar fashion. The plot of the extreme component scores in Figure 2
suggests that the critical information for separating the two spirals derives mainly from
the signs of the input activations.
Because scaling contributions by the sign of the output target appears to obscure a full
picture of network solutions to the two-spirals problem, there may be some value in
using unsealed contributions in network analysis. Use of unscaled contributions also
could be justified on the grounds that the net has no knowledge of targets as it represents
a particular problem; target information is only used in the error correction process. A
disadvantage of using un scaled contributions is that one cannot distinguish contributions
that facilitate vs. contributions that inhibit reaching a relatively error free solution.
The symmetry of these network representations suggests a level of systematicity that is,
on some accounts, not supposed to be possible in neural nets (Fodor & Pylyshyn, 1988).
Whether this representational symmetry reflects systematicity in performance is another
matter. One empirical prediction would be that as a net learns that x, y is on one spiral, it
also learns at about the same time that -x, -y is on the other spiral. If confirmed, this
would demonstrate a clear case of systematic cognition in neural nets.
8
GENERAL DISCUSSION
Performing PCA on network contributions is here shown to be a useful technique for
understanding the performance of networks constructed by the cascade-correlation learning
algorithm. Because cascade-correlation nets typically possess multiple hidden layers and
are fully cross connected, they are difficult to analyze with more standard methods
emphasizing activation patterns on the hidden units alone. Examination of their weight
patterns is also problematic, particularly in larger networks, because of the highly
distributed nature of the net's representations.
Analyzing contributions, in contrast to either hidden unit activations or weights, is a
naturally appealing solution. Contributions capture the influence coming into output
units both from adjacent hidden units and from distant, cross connected hidden and input
units. Moreover, because contributions include both sending activations and connecting
weights, they are not unduly sensitive to one at the expense of the other.
In the three domains examined in the present paper, PCA of the network contributions
both confirm some expected results and provide new insights into network performance.
In all cases examined, the nets succeed in drawing all of the important distinctions in their
representations that are afforded by the training patterns, whether these distinctions
concern the type of output or the operation being performed on the input. In combination
with further experimentation and analysis of network weights and activation patterns, this
technique could help to provide an account of how networks accomplish whatever it is
they learn to accomplish.
It might be of interest to apply the present technique at various points in the learning
process to obtain a developmental trace of network performance. Would all networks
learning under the same constraints progress through the same stages of development, in
terms of the problem distinctions they are able to make? This would be of particular
interest to network simulations of human cognitive development, which has been claimed
to be stage-like in its progressions.
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The present technique could also be useful in predicting the results of lesioning
experiments on neural nets. If the role of a hidden unit can be identified by its association
with a particular principal component, then it could be predicted that lesioning this unit
would impair the function served by the component.
Acknowledgments
This research was supported by the Natural Sciences and Engineering Research Council of
Canada and the MacArthur Foundation. Helpful comments were provided by Scott
Fahlman, Denis Mareschal, Yuriko Oshima-Takane, and Sheldon Tetewsky.
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Touretzky, G. Hinton, & T. Sejnowski (Eds)., Proceedings of the Connectioni.rt
Models Summer School, (pp. 52-59). Mountain View, CA: Morgan Kaufmann.
McClelland, 1. L. (1989). Parallel distributed processing: Implications for cognition and
development. In Morris, R. G. M. (Ed.), Para/lei distributed processing: Implications
for psychology and neurobiology, pp. 8-45. Oxford University Press.
Rumelhart, D. E., Hinton, G. E., & Williams, R. J. (1986). Learning internal
representations by error propagation. In D. E. Rumelhart & J. L. McClelland (Eds.),
Parallel distributed processing: Explorations in the microstructure of cognition.
Volume 1: Foundations, pp. 318-362. Cambridge, MA: MIT Press.
Sanger, D. (1989). Contribution analysis: A technique for assigning responsibilities to
hidden units in connectionist networks. Connection Science, I, 115-138.
Sejnowski, T. J., & Rosenberg, C. R. (1987). Parallel networks that learn to pronounce
English text. Complex Systems, I, 145-168.
Shultz, T. R., Schmidt, W. C., Buckingham, D., & Mareschal, D. (In press). Modeling
cognitive development with a generative connectionist algorithm. In G. Halford & T.
Simon (Eds.), Developing cognitive competence: New approaches to process
mndeling. Hillsdale, NJ: Erlbaum.
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| 812 |@word middle:2 version:3 proportion:1 loading:4 seems:1 simulation:4 accounting:2 carry:1 score:12 current:1 activation:21 lang:2 buckingham:2 written:1 yet:2 assigning:1 tilted:1 additive:1 realistic:1 informative:1 distant:1 shape:2 plot:7 designed:2 hourglass:2 discrimination:1 alone:3 generative:4 selected:5 half:1 v:1 pylyshyn:2 provides:2 contribute:2 denis:1 lor:1 constructed:1 direct:1 acti:1 behavioral:1 expected:1 elman:10 multi:6 automatically:1 considering:1 begin:2 provided:2 linearity:2 moreover:1 what:1 mountain:2 psych:1 cijk:1 spoken:1 finding:2 nj:1 interactive:1 scaled:2 whatever:1 unit:63 appear:1 positive:8 retention:1 understood:2 local:1 treat:1 engineering:1 tki:4 analyzing:5 oxford:1 approximately:1 might:2 examined:3 suggests:2 range:3 pronounce:1 unique:2 acknowledgment:1 impressed:1 signaling:1 empirical:1 cascade:9 quadrant:5 lesioning:2 cannot:1 close:1 layered:2 influence:1 restriction:1 interlocking:3 center:3 resembling:1 williams:1 starting:2 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7,040 | 813 | Functional Models of Selective Attention
and Context Dependency
Thomas H. Hildebrandt
Department of Electrical Engineering and Computer Science
Room 304 Packard Laboratory
19 Memorial Drive West
Lehigh University
Bethlehem PA 18015-3084
[email protected]
Scope
This workshop reviewed and classified the various models which have emerged from
the general concept of selective attention and context dependency, and sought to
identify their commonalities. It was concluded that the motivation and mechanism of these functional models are "efficiency" and ''factoring'', respectively. The
workshop focused on computational models of selective attention and context dependency within the realm of neural networks. We treated only ''functional'' models; computational models of biological neural systems, and symbolic or rule-based
systems were omitted from the discussion.
Presentations
Thomas H. Hildebrandt presented the results of his recent survey of the literature on functional models of selective attention and context dependency. He
set forth the notions that selective attention and context dependency are equivalent, that the goal of these methods is to reduce computational requirements, and
that this goal is achieved by what amounts to factoring or a divide-and-conquer
technique which takes advantage of nonlinearities in the problem.
Daniel S. Levine (University of Texas at Arlington) showed how the gated dipole
structure often used in the ART models can be used to account for time-dependent
phenomena such as habituation and overcompensation. His adjusted model appropriately modelled the public's adverse reaction to "New Coke".
Lev Goldfarb (University of New Brunswick) presented a formal model for inductive learning based on symbolic transformation systems and parametric distance
functions as an alternative to the commonly used algebraic transformation system
and Euclidean distance function. The drawbacks of the latter system were briefly
discussed, and it was shown how this new formal system can give rise to learning
models which overcome these problems.
1180
Functional Models of Selective Attention and Context Dependency
Chalapathy Neti (IBM, Boca Raton) presented a model which he has used to
increase signal-to-noise ratio (SNR) in noisy speech signals. The model is based on
adaptive filtering of frequency bands with a constant frequency to bandwidth ratio.
This thresholding in the wavelet domain gives results which are superior to similar
methods operating in the Adaptive Fourier domain. Several types of signal could
be detected with SNRs close to Odb.
Paul N. Refenes (University of London Business School) demonstrated the need
to take advantage of contextual information in attempting to model the capital
markets. There exist some fundamental economic formulae, but they hold only in
the long term. The desire to model events on a finer time scale requires reference
to significant factors within a smaller window. To do this effectively requires the
identification of appropriate short-term indicators, as mere overparameterization
has been shown to lead to negative results.
Jonathan A. Marshall (University of North Carolina) reviewed the EXIN model,
which correctly encodes partially overlapping patterns as distinct activations in
the output layer, while allowing the simultaneous appearance of nonoverlapping
patterns to give rise to multiple activations in the output layer. The model thus
produces a factored representation of complex scenes.
Albert Nigrin (American University) presented a model, similar in concept to the
EXIN model. It correctly handles synonymous inputs by means of cross-inhibition
of the links connecting the synonyms to the target node.
Thomas H. Hildebrandt also presented a model for adaptive classification based
on decision feedback equalization. The model shifts the decision boundaries of
the underlying classifier to compensate shifts in the statistics of the input. On
handwritten character classification, it outperformed an identical classifier which
used only static decision boundaries.
Summary
According to Hildebrandt's first talk, the concepts underlying selective attention are
quite broad and generally applicable. Large nonlinearities in the problem permit the
use of problem subdivision or factoring (by analogy with the factoring of a Boolean
equation). Factoring is a good method for reducing the complexity of nonlinear
systems.
The talks by Levine and Refenes showed that context enters naturally into the description, formulation, and solution ofreal-world modelling problems. Those by Neti
and Hildebrandt showed that specific reference to temporal context can result in
immediate performance gains. The presentations by Marshall and Nigrin provided
models for appropriately encoding contexts involving overlapping and synonymous
patterns, respectively. The talk by Goldfarb indicates that abandoning assumptions
regarding linearity ab initio may lead to more powerful learning systems. Refer to
[1] for further information.
References
[1] Hildebrandt, Thomas H. Neural Network Models for Selective Attention and
Context Dependency. Submitted to Neural Networks, December 1993.
1181
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7,041 | 814 | Surface Learning with Applications to
Lipreading
Christoph Bregler *.**
*Computer Science Division
University of California
Berkeley, CA 94720
Stephen M. Omohundro **
**Int. Computer Science Institute
1947 Center Street Suite 600
Berkeley, CA 94704
Abstract
Most connectionist research has focused on learning mappings from
one space to another (eg. classification and regression). This paper
introduces the more general task of learning constraint surfaces.
It describes a simple but powerful architecture for learning and
manipulating nonlinear surfaces from data. We demonstrate the
technique on low dimensional synthetic surfaces and compare it to
nearest neighbor approaches. We then show its utility in learning
the space of lip images in a system for improving speech recognition
by lip reading. This learned surface is used to improve the visual
tracking performance during recognition.
1
Surface Learning
Mappings are an appropriate representation for systems whose variables naturally
decompose into "inputs" and "outputs)). To use a learned mapping, the input variables must be known and error-free and a single output value must be estimated for
each input. Many tasks in vision, robotics, and control must maintain relationships
between variables which don't naturally decompose in this way. Instead, there is
a nonlinear constraint surface on which the values of the variables are jointly restricted to lie. We propose a representation for such surfaces which supports a wide
range of queries and which can be naturally learned from data.
The simplest queries are "completion queries)). In these queries, the values of certain
variables are specified and the values (or constraints on the values) of remaining
43
44
Bregler and Omohundro
Figure 1: Using a constraint surface to reduce uncertainty in two variables
~.
Figure 2: Finding the closest point in a surface to a given point.
variables are to be determined. This reduces to a conventional mapping query if the
"input" variables are specified and the system reports the values of corresponding
"output" variables. Such queries can also be used to invert mappings, however, by
specifying the "output" variables in the query. Figure 1 shows a generalization in
which the variables are known to lie with certain ranges and the constraint surface
is used to further restrict these ranges.
For recognition tasks, "nearest point" queries in which the system must return the
surface point which is closest to a specified sample point are important (Figure
2). For example, symmetry-invariant classification can be performed by taking the
surface to be generated by applying all symmetry operations to class prototypes (eg.
translations, rotations, and scalings of exemplar characters in an OCR system). In
our representation we are able to efficiently find the globally nearest surface point
in this kind of query.
Other important classes of queries are "interpolation queries" and "prediction
queries". For these, two or more points on a curve are specified and the goal is to interpolate between them or extrapolate beyond them. Knowledge of the constraint
surface can dramatically improve performance over "knowledge-free" approaches
like linear or spline interpolation.
In addition to supporting these and other queries, one would like a representation
which can be efficiently learned. The training data is a set of points randomly
drawn from the surface. The system should generalize from these training points
to form a representation of the surface (Figure 3). This task is more difficult than
mapping learning for several reasons: 1) The system must discover the dimension of
the surface, 2) The surface may be topologically complex (eg. a torus or a sphere)
Surface Learning with Applications to Lipreading
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Figure 3: Surface Learning
and may not support a single set of coordinates, 3) The broader range of queries
discussed above must be supported.
Our approach starts from the observation that if the data points were drawn from
a linear surface, then a principle components analysis could be used to discover the
dimension of the linear space and to find the best-fit linear space of that dimension.
The largest principle vectors would span the space and there would be a precipitous
drop in the principle values at the dimension of the surface. A principle components
analysis will no longer work, however, when the surface is nonlinear because even
a I-dimensional curve could be embedded so as to span all the dimensions of the
space.
If a nonlinear surface is smooth, however, then each local piece looks more and
more linear under magnification. If we consider only those data points which lie
within a local region, then to a good approximation they come from a linear surface
patch. The principle values can be used to determine the most likely dimension of
the surface and that number of the largest principle components span its tangent
space (Omohundro, 1988). The key idea behind our representations is to "glue"
these local patches together using a partition of unity.
We are exploring several implementations, but all the results reported here come
from a represenation based on the "nearest point" query. The surface is represented as a mapping from the embedding space to itself which takes each point
to the nearest surface point. K-means clustering is used to determine a initial set
of "prototype centers" from the data points. A principle components analysis is
performed on a specified number of the nearest neighbors of each prototype. These
"local peA" results are used to estimate the dimension of the surface and to find
the best linear projection in the neighborhood of prototype i. The influence of these
local models is determined by Gaussians centered on the prototype location with a
variance determined by the local sample density. The projection onto the surface
is determined by forming a partition of unity from these Gaussians and using it to
form a convex linear combination of the local linear projections:
(1)
This initial model is then refined to minimize the mean squared error between the
45
46
Bregler and Omohundro
a)
b)
Figure 4: Learning a I-dimensional surface. a) The surface to learn b) The local
patches and the range of their influence functions, c) The learned surface
training samples and the nearest surface point using EM optimization and gradient
descent.
2
Synthetic Examples
To see how this approach works, consider 200 samples drawn from a I-dimensional
curve in a two-dimensional space (Figure 4a). 16 prototype centers are chosen by kmeans clustering. At each center, a local principle components analysis is performed
on the closest 20 training samples. Figure 4b shows the prototype centers and the
two local principle components as straight lines. In this case, the larger principle
value is several times larger than the smaller one. The system therefore attempts
to construct a one-dimensional learned surface. The circles in Figure 4b show the
extent of the Gaussian influence functions for each prototype. Figure 4c shows the
resulting learned suface. It was generated by randomly selecting 2000 points in the
neighborhood of the surface and projecting them according to the learned model.
Figure 5 shows the same process applied to learning a two-dimensional surface
embedded in three dimensions.
To quantify the performance of this learning algorithm, we studied the effect of the
different parameters on learning a two-dimensional sphere in three dimensions. It
is easy to compare the learned results with the correct ones in this case. Figure 6a
shows how the empirical error in the nearest point query decreases as a function
of the number of training samples. We compare it against the error made by a
nearest-neighbor algorithm. With 50 training samples our approach produces an
error which is one-fourth as large. Figure 6b shows how the average size of the local
principle values depends on the number of nearest neighbors included. Because
this is a two-dimensional surface, the two largest values are well-separated from the
third largest. The rate of growth of the principle values is useful for determining
the dimension of the surface in the presence of noise.
Surface Learning with Applications to Lipreading
Figure 5: Learning a two-dimensional surface in the three dimensions a) 1000 random samples on the surface b) The two largest local principle components at each
of 100 prototype centers based on 25 nearest neighbors.
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dimensions. a) Mean squared error of closest point querries as function of the number of samples for the learned surface vs. nearest training point b) The mean square
root of the three principle values as a function of number of neighbors included in
each local PCA .
47
48
Bregler and Omohundro
a
b
Figure 7: Snakes for finding the lip contours a) A correctly placed snake b) A snake
which has gotten stuck in a local minimum of the simple energy function.
3
Modelling the space of lips
We are using this technique as a part of system to do "lipreading". To provide
features for "vise me classification" (visemes are the visual analog of phonemes), we
would like the system to reliably track the shape of a speaker's lips in video images.
It should be able to identify the corners of the lips and to estimate the bounding
curves robustly under a variety of imaging and lighting conditions. Two approaches
to this kind of tracking task are "snakes" (Kass, et. aI, 1987) and "deformable
templates" (Yuille, 1991). Both of these approaches minimize an "energy function"
which is a sum of an internal model energy and an energy measuring the match to
external image features.
For example, to use the "snake" approach for lip tracking, we form the internal
energy from the first and second derivatives of the coordinates along the snake,
prefering smoother snakes to less smooth ones. The external energy is formed
from an estimate of the negative image gradient along the snake. Figure 7a shows
a snake which has correctly relaxed onto a lip contour. This energy function is
not very specific to lips, however. For example, the internal energy just causes
the snake to be a controlled continuity spline. The "lip- snakes" sometimes relax
onto undesirable local minima like that shown in Figure 7b. Models based on
deformable templates allow a researcher to more strongly constrain the shape space
(typically with hand-coded quadratic linking polynomials), but are difficult to use
for representing fine grain lip features.
Our approach is to use surface learning as described here to build a model of the
space of lips. We can then replace the internal energy described above by a quantity
computed from the distance to the learned surface in lip feature space.
Our training set consists of 4500 images of a speaker uttering random words l .
The training images are initially "labeled" with the conventional snake algorithm.
Incorrectly aligned snakes are removed from the database by hand. The contour
shape is parameterized by the x and y coordinates of 40 evenly spaced points along
the snake. All values are normalized to give a lip width of 1. Each lip contour is
IThe data was collected for an earlier lipreading system described in (Bregler, Hild,
Manke, Waibel 1993)
Surface Learning with Applications to Lipreading
(Ja
C7b
~d
e
Figure 8: Two principle axes in a local patch in lip space. a, b, and c are configurations along the first principle axis, while d, e, and f are along the third axis.
a
b
c
Figure 9: a) Initial crude estimate of the contour b) An intermediate step in the
relaxation c) The final contour.
therefore a point in an 80-dimensional "lip- space". The lip configurations which
actually occur lie on a lower dimensional surface embedded in this space. Our
experiments show that a 5-dimensional surface in the 80-dimensional lip space is
sufficient to describe the contours with single pixel accuracy in the image. Figure 8
shows some lip models along two of the principle axes in the local neighborhood of
one of the patches. The lip recognition system uses this learned surface to improve
the performance of tracking on new image sequences.
The tracking algorithm starts with a crude initial estimate of the lip position and
size. It chooses the closest model in the lip surface and maps the corresponding
resized contour back onto the estimated image position (Figure 9a). The external
image energy is taken to be the cumulative magnitude of graylevel gradient estimates along the current contour. This term has maximum value when the curve
is aligned exactly on the lip boundary. We perform gradient ascent in the contour
space, but constrain the contour to lie in the learned lip surface. This is achieved by
reprojecting the contour onto the lip surface after each gradient step. The surface
thereby acts as the analog of the internal energy in the snake and deformable template approaches. Figure 9b shows the result after a few steps and figure 9c shows
the final contour. The image gradient is estimated using an image filter whose width
is gradually reduced as the search proceeds.
The lip contours in successive images in the video sequence are found by starting
with the relaxed contour from the previous image and performing gradient ascent
49
50
Bregler and Omohundro
with the altered external image energies. Empirically, surface-based tracking is far
more robust than the "knowledge-free" approaches. While we have described the
approach in the context of contour finding, it is much more general and we are
currently extending the system to model more complex aspects of the image.
The full lipreading system which combines the described tracking algorithm and a
hybrid connectionist speech recognizer (MLP /HMM) is described in (Bregler and
Konig 1994). Additionally we will use the lip surface to interpolate visual features
to match them with the higher rate auditory features.
4
Conclusions
We have presented the task of learning surfaces from data and described several important queries that the learned surfaces should support: completion, nearest point,
interpolation, and prediction. We have described an algorithm which is capable of
efficiently performing these tasks and demonstrated it on both synthetic data and
on a real-world lip-tracking problem. The approach can be made computationally
efficient using the "bumptree" data structure described in (Omohundro, 1991). We
are currently studying the use of "model merging" to improve the representation
and are also applying it to robot control.
Acknowledgements
This research was funded in part by Advanced Research Project Agency contract
#NOOOO 1493 C0249 and by the International Computer Science Institute. The
database was collected with a grant from Land Baden Wuerttenberg (Landesschwerpunkt Neuroinformatik) at Alex Waibel's institute.
References
C. Bregler, H. Hild, S. Manke & A. Waibel. (1993) Improving Connected Letter
Recognition by Lipreading. In Proc. of Int. Conf. on Acoustics, Speech, and Signal
Processing, Minneapolis.
C. Bregler, Y. Konig (1994) "Eigenlips" for Robust Speech Recognition. In Proc.
of Int. Conf. on Acoustics, Speech, and Signal Processing, Adelaide.
M. Kass, A. Witkin, and D. Terzopoulos. (1987) SNAKES: Active Contour Models,
in Proc. of the First Int. Conf. on Computer Vision, London.
S. Omohundro. (1988) Fundamentals of Geometric Learning. University of Illinois
at Urbana-Champaign Technical Report UIUCDCS-R-88-1408.
S. Omohundro. (1991) Bumptrees for Efficient Function, Constraint, and Classification Learning. In Lippmann, Moody, and Touretzky (ed.), Advances in Neural
Information Processing Systems 3. San Mateo, CA: Morgan Kaufmann.
A. Yuille. (1991) Deformable Templates for Face Recognition, Journal of Cognitive
Neuroscience, Volume 3, Number 1.
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7,042 | 815 | Hoo Optimality Criteria for LMS and
Backpropagation
Babak Hassibi
Information Systems Laboratory
Stanford University
Stanford, CA 94305
Ali H. Sayed
Dept. of Elec. and Compo Engr.
University of California Santa Barbara
Santa Barbara, CA 93106
Thomas Kailath
Information Systems Laboratory
Stanford University
Stanford, CA 94305
Abstract
We have recently shown that the widely known LMS algorithm is
an H OO optimal estimator. The H OO criterion has been introduced,
initially in the control theory literature, as a means to ensure robust performance in the face of model uncertainties and lack of
statistical information on the exogenous signals. We extend here
our analysis to the nonlinear setting often encountered in neural
networks, and show that the backpropagation algorithm is locally
H OO optimal. This fact provides a theoretical justification of the
widely observed excellent robustness properties of the LMS and
backpropagation algorithms. We further discuss some implications
of these results.
1
Introduction
The LMS algorithm was originally conceived as an approximate recursive procedure
that solves the following problem (Widrow and Hoff, 1960): given a sequence of n x 1
input column vectors {hd, and a corresponding sequence of desired scalar responses
{di }, find an estimate of an n x 1 column vector of weights w such that the sum
of squared errors, L:~o Idi w1 2 , is minimized. The LMS solution recursively
hi
351
352
Hassibi. Sayed. and Kailath
updates estimates of the weight vector along the direction of the instantaneous gradient of the squared error. It has long been known that LMS is an approximate
minimizing solution to the above least-squares (or H2) minimization problem. Likewise, the celebrated backpropagation algorithm (Rumelhart and McClelland, 1986)
is an extension of the gradient-type approach to nonlinear cost functions of the form
2:~o Id i - hi (W ) 12 , where hi ( .) are known nonlinear functions (e. g., sigmoids). It
also updates the weight vector estimates along the direction of the instantaneous
gradients.
We have recently shown (Hassibi, Sayed and Kailath, 1993a) that the LMS algorithm is an H<Xl-optimal filter, where the H<Xl norm has recently been introduced
as a robust criterion for problems in estimation and control (Zames, 1981). In general terms, this means that the LMS algorithm, which has long been regarded as
an approximate least-mean squares solution, is in fact a minimizer of the H<Xl error
norm and not of the JI2 norm. This statement will be made more precise in the
next few sections. In this paper, we extend our results to a nonlinear setting that
often arises in the study of neural networks, and show that the backpropagation
algorithm is a locally H<Xl-optimal filter. These facts readily provide a theoretical
justification for the widely observed excellent robustness and tracking properties of
the LMS and backpropagation algorithms, as compared to, for example, exact least
squares methods such as RLS (Haykin, 1991).
In this paper we attempt to introduce the main concepts, motivate the results, and
discuss the various implications. \Ve shall, however, omit the proofs for reasons of
space. The reader is refered to (Hassibi et al. 1993a), and the expanded version of
this paper for the necessary details.
2
Linear HOO Adaptive Filtering
\Ve shall begin with the definition of the H<Xl norm of a transfer operator. As
will presently become apparent, the motivation for introducing the H<Xl norm is to
capture the worst case behaviour of a system.
Let h2 denote the vector space of square-summable complex-valued causal sequences
{fk, 0 :::; k < oo}, viz.,
<Xl
h2 = {set of sequences {fk} such that
f; fk < oo}
k=O
L
=
with inner product < {Ik}, {gd >
2:~=o f; gk ,where * denotes complex
conjugation. Let T be a transfer operator that maps an input sequence {ud to an
output sequence {yd. Then the H<Xl norm of T is equal to
IITII<Xl =
sup
utO,uEh 2
IIyl12
II u l1 2
where the notation 111/.112 denotes the h 2 -norm of the causal sequence
{ttd,
viz.,
2 ~<Xl *
Ilull:? = L...Jk=o ttkUk
The H<Xl norm may thus be regarded as the maximum energy gain from the input
u to the output y.
Hoc Optimality Criteria for LMS and Backpropagation
Suppose we observe an output sequence {dd that obeys the following model:
di
= hT W + Vi
(1)
where hT = [hi1 hi2
hin ] is a known input vector, W is an unknown weight
vector, and {Vi} is an unknown disturbance, which may also include modeling errors.
We shall not make any assumptions on the noise sequence {vd (such as whiteness,
normally distributed, etc.).
Let Wi = F(d o, di, ... , di) denote the estimate of the weight vector W given the
observations {dj} from time 0 up to and including time i. The objective is to
determine the functional F, and consequently the estimate Wi, so as to minimize a
certain norm defined in terms of the prediction error
ei = hT W - hT Wi-1
which is the difference between the true (uncorrupted) output hT wand the predicted output hT Wi -1. Let T denote the transfer operator that maps the unknowns
{w - W_1, {vd} (where W-1 denotes an initial guess of w) to the prediction errors
{ed. The HOO estimation problem can now be formulated as follows.
Problem 1 (Optimal HOC Adaptive Problem) Find an Hoc -optimal estimation strategy Wi
F(d o, d 1, ... , d i ) that minimizes IITlloc' and obtain the resulting
=
!~
= inf IITII!:, = inf
:F
:F
(2)
sup
w,vEh 2
=
where Iw - w_11 2
(w - w-1f (w - W-1), and J1- is a positive constant that reflects
apriori knowledge as to how close w is to the initial guess W-1 .
Note that the infimum in (2) is taken over all causal estimators F. The above
problem formulation shows that HOC optimal estimators guarantee the smallest
prediction error energy over all possible disturbances offixed energy. Hoc estimators
are thus over conservative, which reflects in a more robust behaviour to disturbance
variation.
Before stating our first result we shall define the input vectors {hd exciting if, and
only if,
N
lim
N-+oc
L hT hi =
00
i=O
Theoreln 1 (LMS Algorithm) Consider the model (1), and suppose we wish to
minimize the Hoc norm of the transfer operator from the unknowns w - W-1 and
Vi to the prediction errors ei. If the input vectors hi are exciting and
o < J1- < i~f h:h.
(3)
tit
then the minimum H oo norm is !Opt = 1. In this case an optimal Hoo estimator is
given by the LA-IS alg01'ithm with learning rate J1-, viz.
(4)
353
354
Hassibi, Sayed, and Kailath
In other words, the result states that the LMS algorithm is an H oo -optimal filter.
Moreover, Theorem 1 also gives an upper bound on the learning rate J-t that ensures
the H oo optimality of LMS. This is in accordance with the well-known fact that
LMS behaves poorly if the learning rate is too large.
Intuitively it is not hard to convince oneself that "'{opt cannot be less than one. To
this end suppose that the estimator has chosen some initial guess W-l. Then one
may conceive of a disturbance that yields an observation that coincides with the
output expected from W-l, i.e.
hT W-l = hT W + Vi = di
In this case one expects that the estimator will not change its estimate of w, so that
Wi
W-l for all i. Thus the prediction error is
=
ei
= hTw -
hTwi-l
= hTw -
hTw-l
= -Vi
and the ratio in (2) can be made arbitrarily close to one.
The surprising fact though is that "'{opt is one and that the LMS algorithm achieves
it. What this means is that LMS guarantees that the energy of the prediction
error will never exceed the energy of the disturbances. This is not true for other
estimators. For example, in the case of the recursive least-squares (RLS) algorithm,
one can come up with a disturbance of arbitrarily small energy that will yield a
prediction error of large energy.
To demonstrate this, we consider a special case of model (1) where hi is now a
scalar that randomly takes on the values + 1 or -1. For this model J-t must be less
than 1 and we chose the value J-t
.9. We compute the Hoo norm of the transfer
operator from the disturbances to the prediction errors for both RLS and LMS. We
also compute the worst case RLS disturbance, and show the resulting prediction
errors. The results are illustrated in Fig. 1. As can be seen, the H OO norm in
the RLS case increases with the number of observations, whereas in the LMS case
it remains constant at one. Using the worst case RLS disturbance, the prediction
error due to the LMS algorithm goes to zero, whereas the prediction error due to
the RLS algorithm does not. The form of the worst case RLS disturbance is also
interesting; it competes with the true output early on, and then goes to zero.
=
We should mention that the LMS algorithm is only one of a family of HOO optimal
estimators. However, LMS corresponds to what is called the central solution, and
has the additional properties of being the maximum entropy solution and the risksensitive optimal solution (Whittle 1990, Glover and Mustafa 1989, Hassibi et al.
1993b).
If there is no disturbance in (1) we have the following
Corollary 1 If in addition to the assumptions of Theorem 1 there is no disturbance
in {1J, then LMS guarantees II e II~:::; J-t-1Iw - w_11 2 , meaning that the prediction
error converges to zero.
Note that the above Corollary suggests that the larger J-t is (provided (3) is satisfied)
the faster the convergence will be.
Before closing this section we should mention that if instead of the prediction error
one were to consider the filtered error ej,i = hjw - hjwj, then the HOO optimal
estimator is the so-called normalized LMS algorithm (Hassibi et al. 1993a).
Hoo Optimality Criteria for LMS and Backpropagation
2.5 . - - - - - - - - - - ' a' - = - - - - - - - - ,
1
0.98
0.96
0.94
0.92
0.5L-------------J
o
50
0.9
0
50
0.5 r - - - - - - > -(d)
=--------,
(e)
0.5
\,
o
1"'-"
"
-0.5
-l~---------~
o
50
-1L-------------------~
o
50
Figure 1: Hoo norm of transfer operator as a function of the number of observations
for (a) RLS, and (b) LMS. The true output and the worst case disturbance signal
(dotted curve) for RLS are given in (c). The predicted errors for the RLS (dashed)
and LMS (dotted) algorithms corresponding to this disturbance are given in (d).
The LMS predicted error goes to zero while the RLS predicted error does not.
3
Nonlinear HOO Adaptive Filtering
In this section we suppose that the observed sequence {dd obeys the following
nonlinear model
(5)
where hi (.) is a known nonlinear function (with bounded first and second order
derivatives), W is an unknown weight vector, and {vd is an unknown disturbance
sequence that includes noise and/or modelling errors. In a neural network context
the index i in hi (.) will correspond to the nonlinear function that maps the weight
vector to the output when the ith input pattern is presented, i.e., hi(W) h(x(i), w)
where x(i) is the ith input pattern. As before we shall denote by Wi = :F(do, ... , di)
the estimate of the weight vector using measurements up to and including time i,
and the prediction error by
=
I
ei
= hi(w) -
hi(Wi-1)
Let T
{ W -
denote the transfer operator that maps the unknowns/disurbances
W -1 , { vd} to the prediction errors {e;}.
Problem 2 (Optimal Nonlinear HOO Adaptive Problem) Find
an Hoo-optimal estimation strategy Wi = :F(d o, d 1 , . .. , d i ) that minimizes
IITllooI
355
356
Hassibi, Sayed, and Kailath
and obtain the resulting
i'~
= inf
:F
IITII~
= inf
:F
(6)
sup
w,vEh2
Currently there is no general solution to the above problem, and the class of nonlinear functions hi(.) for which the above problem has a solution is not known (Ball
and Helton, 1992).
To make some headway, though, note that by using the mean value theorem (5)
may be rewritten as
di
= hi(wi-d + ~~ T (wi_d.(w -
Wi-I)
+ Vi
(7)
where wi-l is a point on the line connecting wand Wi-I. Theorem 1 applied to (7)
shows that the recursion
(8)
=
will yield i'
1. The problem with the above algorithm is that the wi's are
not known. But it suggests that the i'opt in Problem 2 (if it exists) cannot be
less than one. Moreover, it can be seen that the backpropagation algorithm is an
approximation to (8) where wi is replaced by Wi. To pursue this point further we
use again the mean value theorem to write (5) in the alternative form
ohi T
) 1
T 02hi(_
di = hi(wi-d+ ow (wi-d?(w-Wi-l +2(W-Wi-d . ow 2 wi-d?(w-Wi-d+Vi
(9)
where once more Wi-l lies on the line connecting Wi-l and w. Using (9) and
Theorem 1 we have the following result.
Theorem 2 (Backpropagation Algorithm) Consider the model (5) and the
backpropagation algorithm
Wi
= Wi-l + J.L ohi
Ow (wi-d(di -
(10)
hi(wi-d)
then if the ~~i (Wi- d are exciting, and
. f - - : : T =1- - - - - - o < J.L < In
i
(11)
ill!..
) ill!..(
ow (Wi-I?
ow wi-l )
then for all nonzero w, v E h 2:
II ~~i
w_112+ II Vi + !(w -
II~
T (wi-d(w - wi-d
-----------~~=-~--~~--~~---------------
J.L-11w where
wi_d T ~:::J (wi-d?(w - Wi-I) II~
<
-
1
Hoo Optimality Criteria for LMS and Backpropagation
v;
The above result means that if one considers a new disturbance
= Vi + ~ (w Wi_I)T ~::J (Wi-I).(W - Wi-I), whose second term indicates how far hi(w) is from a
first order approximation at point Wi-I, then backpropagation guarantees that the
energy of the linearized prediction error ~~ T (wi-d(w - Wi-I) does not exceed the
energy of the new disturbances W - W-l and
v:.
It seems plausible that if W-I is close enough to w then the second term in v~ should
be small and the true and linearized prediction errors should be close, so that we
should be able to bound the ratio in (6). Thus the following result is expected,
where we have defined the vectors {hd persistently exciting if, and only if, for all
a E
nn
Theorem 3 (Local Hoc Optimality) Consider the model (5) and the backpropagation algorithm (10). Suppose that the ~':: (Wi-I) are persistently exciting, and
that (11) is satisfied. Then for each ( > 0, there exist cSt, ch > 0 such that for all
Iw - w-ti < cSt and all v E h2 with IVil < 82, we have
, 12
II I 2
Il-Ilw - w_112+ II v
ej
II~
< 1+(
-
The above Theorem indicates that the backpropagation algorithm is locally HOC
optimal. In other words for W-l sufficiently close to w, and for sufficiently small
disturbance, the ratio in (6) can be made arbitrarily close to one. Note that the
conditions on wand Vi are reasonable, since if for example W is too far from W-l,
or if some Vi is too large, then it is well known that backpropagation may get stuck
in a local minimum, in which case the ratio in (6) may get arbitrarily large.
As before (11) gives an upper bound on the learning rate Il, and indicates why
backpropagation behaves poorly if the learning rate is too large.
If there is no disturbance in (5) we have the following
Corollary 2 If in addition to the assumptions in Theorem 3 there is no disturbance
in (5), then for every ( > 0 there exists a 8 > 0 such that for all Iw - w-il < 8,
the backpropagation algorithm will yield II e' II~:::; 1l- 18(1 + (), meaning that the
prediction error converges to zero. Moreover Wi will converge to w.
Again provided (11) is satisfied, the larger Il is the faster the convergence will be.
4
Discussion and Conclusion
The results presented in this paper give some new insights into the behaviour of
instantaneous gradient-based adaptive algorithms. We showed that ifthe underlying
observation model is linear then LMS is an HOC optimal estimator, whereas if the
underlying observation model is nonlinear then the backpropagation algorithm is
locally HOC optimal. The HOC optimality of these algorithms explains their inherent
robustness to unknown disturbances and modelling errors, as opposed to other
estimation algorithms for which such bounds are not guaranteed.
357
358
Hassibi, Sayed, and Kailath
Note that if one considers the transfer operator from the disturbances to the prediction errors, then LMS (backpropagation) is H OO optimal (locally), over all causal
estimators. This indicates that our result is most applicable in situations where
one is confronted with real-time data and there is no possiblity of storing the training patterns. Such cases arise when one uses adaptive filters or adaptive neural
networks for adaptive noise cancellation, channel equalization, real-time control,
and undoubtedly many other situations. This is as opposed to pattern recognition,
where one has a set of training patterns and repeatedly retrains the network until
a desired performance is reached.
Moreover, we also showed that the H oo optimality result leads to convergence proofs
for the LMS and backpropagation algorithms in the absence of disturbances. We
can pursue this line of thought further and argue why choosing large learning rates
increases the resistance of backpropagation to local minima, but we shall not do so
due to lack of space.
In conclusion these results give a new interpretation of the LMS and backpropagation algorithms, which we believe should be worthy of further scrutiny.
Acknowledgements
This work was supported in part by the Air Force Office of Scientific Research, Air
Force Systems Command under Contract AFOSR91-0060 and in part by a grant
from Rockwell International Inc.
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7,043 | 816 | Optimal Stopping and Effective Machine
Complexity in Learning
Changfeng Wang
Department of SystE'IIlS Sci. (Iud Ell/!,.
UJliversity of PPIIIlsylv1I.Ili(l
Philadelphin, PA, U.S.A. I!HlJ4
Salltosh S. Venkatesh
Dp?artn}(,llt (If Elf'drical EugiJlPprinJ!,
UIIi v('rsi ty (If Ppnllsyl va nia
Philadelphia, PA, U.S.A. 19104
J. Stephen Judd
Siemens Corporate Research
755 College Rd. East,
Princeton, NJ, U.S.A. 08540
Abstract
We study tltt' problem of when to stop If'arning a class of feedforward networks
- networks with linear outputs I1PUrOIl and fixed input weights - when they are
trained with a gradient descent algorithm on a finite number of examples. Under
general regularity conditions, it is shown that there a.re in general three distinct
phases in the generalization performance in the learning process, and in particular,
the network has hetter gt'neralization pPTformance when learning is stopped at a
certain time before til(' global miniIl111lu of the empirical error is reachert. A notion
of effective size of a machine is rtefil1e<i and used to explain the trade-off betwf'en
the complexity of the marhine and the training error ill the learning process.
The study leads nat.urally to a network size selection critt'rion, which turns Ol1t to
be a generalization of Akaike's Information Criterioll for the It'arning process. It if;
shown that stopping Iparning before tiJt' global minimum of the empirical error has
the effect of network size splectioll.
1
INTRODUCTION
The primary goal of learning in neural nets is to find a network that gives valid generalization. In
achieving this goal, a central issue is the trade-off between the training error and network complexity.
This usually reduces to a problem of network size selection, which has drawn much research effort in
recent years. Various principles, theories, and intuitions, including Occam's razor, statistical model
selection criteria such as Akaike's Information Criterion (AIC) [11 and many others [5, 1, 10,3,111 all
quantitatively support the following PAC prescription: between two machines which have the same
empirical error, the machine with smaller VC-dimf'nsion generalizes better. However, it is noted
that these methods or criteria do not npcpssarily If'ad to optimal (or llearly optimal) generalization
performance. Furthermore, all of these m<.'thods are valid only at th~ global minimum of thf' empirical
error function (e.g, the likelihood function for AIC), and it is not clear by these methods how the
generalization error is f'ffected by network complexity or, more generally, how a network generalizes
during the learning process. This papPI acldrf'f;sPs these issues .
303
304
Wang, Venkatesh, and Judd
Recently, it has often been observed that when a network is 'trained by a gradient descent
algorithm, there exists a critical region in the training epochs where the trained network generalizes
best, and after that region the generalization error will increase (frequently called over-training). Our
numerical experiments with gradient-type algorithms in training feedforward networks also indicate
that in this critical region, as long as the network is large enougb to learn the examples, the size
of the network plays little role in the (hest) generalization performance of the network. Does this
mean we must revise Occam's principle? How should one define the complexity of a network and go
about tuning it to optimize geIlNalization performance? When should one stop learning? Although
relevant learning processes wen' treatccJ by nUlll<'TOIIS authors [2, 6, 7, 4], the formal theoretical
studies of these problems are abeyant.
Under rather general regularity conditions (Section 1), we give in Section 2 a theorem which
relates the generalization error at each epoch of learning to that at the global minimum of the
training error. Its consequence is that for any linear machine whose VC-dimension is finite but large
enough to learn the target concept, the number of iterations needed for the best generalization to
occur is at the order of the logarithm of the sample size, rather than at the global minimum of
the training error; it also provides bounds on the improvement expected. Section 3 deals with the
relation between the size of the machine and generalization error by appealing to the concept of
effective size. Section 4 concerns the application of these results to the problem of network size
selection, where the AIC is generalized to cover the time evolution of the learning process. Finally,
we conclude the paper with comments on practical implementation and further research in this
direction.
2
THE LEARNING MACHINE
The machine we considf'f acc.epts input v('ctors X from an arbitrary input space and produc('s scalar
outputs
d
Y
= 2: 1./;,(X)n', + ? = 1/J(X)'o:* + ?.
(1)
.=1
Here, 0:* = (0:*1, . . . ,0:' d)' is a fixed vect.or of real weights, for eac.h i, 1./;,(X) is a fixed real fUBction
of the inpnts, with 1/J(X) = (1/JI (X), . . . ,t/Jd(X)), the corresponding vedor of functions, and ~ is a
random noise term. The machine (1) can be thought of as a feedforward nenral network with a fixed
front end and variable weights at the output. In particular, the functions 1/J; can represent fixed
polynomials (higher-order or sigma-pi neural networks), radial basis functions with fixed centers, a
fixed hidden-layer of sigmoidal neurons, or simply a linear map. In this context, N. J. Nilsson [8)
has called similar structures cI>-machines.
We consider the problem of learning from examples a relationship between a random variable Y
and an n-dimensional random vector X. We assume that this function is given by (1) for some fixed
integer d, the random vector X and random variable ~ are defined on the same probability space,
that E [~IX) = 0, and (12(X) = Var{?lX) = constant < 00 almost surely. The smallest eigenvalue of
the matrix 1fJ(x)1fJ(x ) is assumed to be bounded from below by the inverse of some square integrable
function.
Note that it can be shown that the VC-dirnension of the class of cI>-machines with d neurons
is d under the last assumption. The learning-theoretic properties of the system will be determined
largely by the eigen structure of cI>. Accordingly, let >'1 ~ >'2 ~ ... ~ >'d denote the eigenvalues of cI>.
The goal of the learning is to finei the true concept 0: given independently drawn examples (X, y)
from (1). Given any hypothesis (vector) W = (WI, ... ,Wd)' for consideration as an approximation
to the true concept 0:, the performance measure we use is the mean-square prediction (or ensemble)
error
(2)
?(W) = E (Y -1/J(X)'w(
Note that the true concept
0:*
is the mean-square solution
0:*
= argmin?(w)
tv
= cI>-IE
(1/J(X)y),
(3)
Optimal Stopping and Effective Machine Complexity in Learning
and the minimum predict.ion error is given by ?(0) = lllinw E.(w)
= u'l.
Let 11. be the nUmbE'f of samples of (X,} -). WE' assume that an independent, ic\entkally
distributed sample (X(1),y(J), ... , (x(n),y(n), generated according to the joint distribution
of (X, Y) induced by (1), is provided to thE:' IE:'arner. To simplify notation, define thE' matrix
'It == [",,(X(l) . . . ""(X(",) ) and the corresponding vector of outputR y = (y(l), . . . , y(n))'. In
analogy with (2) define the empirical error 011 the sample by
Let a denote the hypothesis vector for which t.he empirical error
'Vw?(o) = O. Analogously with (3) we can thell show that
011
the sample is minimized:
(4)
where cj, = t- 'It 'It' is the empirical covariallre matrix, whirh is almost surely nonsingular for large n.
The terms in (4) are the empirical counterparts of the ensemble averages in (3).
The gradient descent algorithm is givf'n by:
(5)
where 0 = (01,02,
we can get
. . . ,03 )"
t is the number of iterations, and
a,
where ~(t) = (I -
?ci?t,
clIld
00
= (I -
~(t?o
?
is the rate of learning. From this
+ ~(t.)oo,
(6)
is the initial weight vector.
The limit of Ot is n when t. goes to infinity, provided ci> is positive definite and the learning rate
small enough (Le., smaller than the smallest eigenvalue of ci?. This implies that the gradient
descent algorithm converges to the least squarE'S solution, starting from any point in Rn.
? is
3
3.1
GENERALIZATION DYNAMICS AND STOPPING TIME
MAIN THEOREM OF GENERALIZATION DYNAMICS
Even if the true concept (i.e., the precise relation between Y and X in the current problem) is in the
class of models we consider, it is usually hopeless to find it using only a finite number of examples,
except in some trivial cases. Our goal is hf'nce less ambitious; we seek to find the best approximation
of the true concept, the approach entailing a minimization of the training or empirical error, and
then taking the global minimum of the empirical error a as the approximation. As we have seen the
procedure is unbiased and consistent. Does this then imply that training should always be carried
out to the limit? Surprisingly, the answer is 110. This assertion follows from the next theorem.
Theorem 3.1 Let Mn > 0 be an arbitrary f'eal constant (possibly depending on 11.), and suppose
a.~s1tm.ptions Al to A.'I af'C satisfied; then the rwnrralizatioll dynamics in the training process are
gOllerned "y the follolllinq rquatioll.:
uniformly for all initial weight ver.iors, no in the d-dim.ensional ball {n*
and for all t > O.
+ 8 : 11811
~ M n , 8 E R d },
0
305
306
Wang, Venkatesh, and Judd
3.2
THREE PHASES IN GENERALIZATION
By Theorem 3.1, the mean gelleralil':ation <'nor at each epoch of the t.raining proc?'ss is characterized
by the following function:
?J(t) ==
t ['\,8~(1
~(1
- f'\,)2' - 20- 2 (1 - ?,\;)t[1n
2
- f,\;)'I] .
.
,=1
The analysis of the evolution of generalization with training is facilitated by treating ?J(.) as a
function of a continuotls tinlf' parameter f. \Ve will show that. there are three distinct phases in
generalization dynamics. These results are givpn in the following in form by several corollaries of
Theorem 3.1.
Without loss of geIH'rality, w<' assnllJ(' th<' init.ial w<'ight ve-ctm is pickerl 11)1 in a region with
1", = 11I(l/1:.~?.x?r1)f, the-II for all 0 S; t < f, _
11811 ~ Mn = 0(11?), and in particular, 181 = O(I/ n ). L<'t
2111(1;'t'1~ ?.x.r1)' we have 0::; 7', < ~,and thus
d
L '\i"; (1 :.I
1
"T, ?> 0(-1n'"
f,\r/) 2' = O( - -12
.=-1
:.I
3 -. )
= -20n
L (1 -
'
tI
f'\;) .
;= 1
The quantity 8; (1 - ?A;) 2t in the fi rst term of t he above inequalities is related to the elimination of
initial error, and can be defined as the approximation error (or fitting error); the last term is related
to the effective complexity of the network at t (in fact, an order O( ~) shift of the complexity (,rror).
The definition and observations here will be discussed in more detail ill the next section.
We call the learning process during the time interval 0 ~ t S; tl the first
Siuce ill this interval ?J(t) = 0(,,-2,?,) is n lIlollutollkally df'Cfeasillg function of i,
error decreases monotonically ill the first phase of It'arning. At the end of first
?J(tl) = O( ~), therefore the generalization error is ?(nt,) = [(nco) + O( ~). As a
statements we have the following mroJlary.
phase of learning.
the g('neralil':atiou
phase of learning
snmmary of these
Corollary 3.2 In the first phase of learning, the complexity error is dominated by the approximation
error, and within an order of O( ~) I the generalization eTTor decrea.5es monotonically in the lrarnin.q
process to ? (noo) + O( ~) at the end of first pha.~e.
0
For f > t 1 , we can show by Thp.orem 3.1 t ha t t It(' g<'llcralizatioll dynamics is given by thp. following
equation, where 8" == n(tl) - n~,
20?a(at,+t) = ?a(ao) - - 2
n
when~ p~
L(l - f'\i) ,
cI
[
1
1 - - (1
2
,=1
2
+ Pi) (1
- f'\i)
,]
_1
+ O(n 2),
== ,\j8;(tl )n./0- 2 , which is, with probahility a.pproaching one, of ordPf O(nO).
Without causing confusion, WP. stillnse ?J(-) for the new time-varying part of the gf'neralization
error. The function ?J(.) has much more complex behavior after tl than in the first phase of learninr;.
As we will see, it decreases for some time, and finally begins to increase again. In particular, we
found the best generalization at tha.t t where ?J( t) is minimized. (It is noted that 8tl is a random
variable now, and the following statements of the generalization dynamics are ill the sense of with
probability approaching one as n -+ 00 .)
Define the optimal stopping tim<': f",ill == argmin{?(a,) : t E [D,oo!}, i.e., the epoch corresponding to the smallest gPllPralization Pfror. Then we can prove the following corollaries:
Corollary 3.3 The optimal stoppin.q time t",ill = O(ln 71.), p1'Ovided 0- 2
>
D.
In particular, the
following inequalities hold:
2
2
.
In(l+p,)
d
In(I+Pj)
b h
tf = t\ + nlIn, In(I/[1 -,.x,I) an ttL = tl + max, 111(1/[1-,,\.)) are ot
finite real numbers. Th at is, the smallest generalization occurs before the global minimum
of the empirical err07' is 1?eachcd.
.,
111 tere
1. tt S; tmin ~ tl/,
Optimal Stopping and Effective Machine Complexity in Learning
2. ?C) (tmcking the gencmlizntio7! eTTor) decreases monotonically for t
monotonically tn zero for t > tu; fuf'thermore, tmin is unique if tt +
<
tf and increases
:x > tu.
In 2
In(I/[I- < I? -
3.
_",2
"d 1 --L,.,
,. 0t=
< ?(tmin) <
-
H.pi -
_",2
n
-2!L[...h.~d ]1' where'" = 11l(1-<~I) and
1+1' 1'+1 +p'
' I n ( I - ( d)'
(12 _
"d
- 0t=1
p2
i'
In accordance with our earlier definitions, We' call the learning proeess during the time intl'rval
between tl and t" til(' s('cond pitas(' of l('aruinl1;; and the rest of timl' til(' third phasf' of learning.
According to Corollary 3.3, for t > tlL sufficiently large, the gell('ralization error is uniformly
better than at the globalminimuIn, a, of the empirieal error, although minimum generalization error
is achieved betwel'n t f and tu. The generalization error is redllced hy. at least. - ",2
AJ1'
,. -2!L
1+1' [...h.+
l' I n+p
over that for a if we stop training at. a prop<'f time. For a fixed nUlnlwr of it'aming examples,
the larger is the ratio d/lI, the larger is til(' improvement in generalization error if the algorithm is
stopped before the glohal minimum n? is reariwd .
4
THE EFFECTIVE SIZE OF THE MACHINE
Our concentration on dynamics and our seeming disregard for complexity do not conflict with the
learning-theoretic focus on VC-dimension; in fact, the two attitudes fit nicely together . This section
explains the generalization dynamics by introducing the the concept of effective complexity of the
machine. It is argued that early stopping in drect sets the l'ffective size of the network to a value
smaller than its VC-dimension.
The effective size of the machine at time t is defined to be d(t) == L~=1 [1 - (1 - d.,)fJ2, which
increases monotonically to d, the VC-dimensioll of the network, as t -+ 00. This definition is justified
after the following theorem:
Theorem 4.1 Under the a.5sumptions of Them'em 3.1, the following equation holds uniformly for
nil no such that 1151 ~ 111n,
(7)
o
In the limit of learning, we have by letting t
-+ 00
in the above equation,
2
?(a) =?(a*)+ ~d+O(n-~)
n
(8)
Hence, to an order of O(n-1.5), the generalization error at the limit of training breaks into two parts:
the approximation error ?(0 0 ) , and the complexity error ~0'2 . Clearly, the latter is proportional to
d, the VC-dimension of the network. For all d's larger than necessary, ?(a*) remains a constant,
and the generalization error is determined solely by ~. The term ?(a.,t) differs from ?(0*) only in
terms of initial error, and is identified to be the approximation error at t. Comparison of the above
2
two equations thus shows that it is reasonable to define ':. d(t) as the complexity error at t, and
justifies the definition of d(t) as the effective size of the machine at the same time. The quantity
d(t) captures the notion of the degree to which the capacity of the machine is used at t. It depends
on the machine parameters, the a.lgorithm being IIsed , and the marginal distribution of X. Thus, we
see from (7) that the generalization error at epoch t falls into the same two parts as it does at the
limit: the approximation error (fitting error) and the complexity error (determined by the effective
size of the machine).
As we have show in the last section, during the first phase of learning, the complexity error is of
higher order in n compared to the fitting error during the first phase of learning, if the initial error
is of order O(nO) or largN . Thus derrpase of til(' fitting error (which is proportional to the training
error, as we will see in the next section) illl plies the decrpase of the generalization error. However,
307
308
Wang, Venkatesh, and Judd
when the fitting error is brought down to the order O( ~), thE' decreas~ of fitting error will no longer
imply th~ decreasE' of the' genc>rali?:ation error. In fact, by the ahoVf' t.heorem , the generali?:ation
error at t + tl can be written as
The fitting error and the complexity error compete at order O( ~) during the second phase oflearning.
After the second the phase of icarning, th(' complexity error dominates the fitting error, still at tilE'
order of O( ~) . Furthermore, if we define K == 1 ~.
d~lp2 J', then by the above equation and (3.3),
we have
[#I
Corollary 4.2 At the optimal 8topping time,
holds,
flip
following u1J11er bound (m the generalization error
Since K is a quantity of order 0(71,?), (1 - K)d is strictly smaller than d. Thus stopping training at
tmin has the same effed as using a smaller machine of size less than (1 - K)d and carrying training
out to the limit! A more detailed analysis reveals how the effective size of the machine is affected
by each neuron in thE' learning process (omitted dne to the space limit).
REMARK:
The concept of effE'ctive size of the machine can be defined similarly for an arbitrary
starting point. However, to compare the degree to which the capacity of the machine has been used
at t, one must specify at what distance between the hypothesis a and the truth o' is such comparison started. While each point in the d-diuwnsional Euclidean space can be rega.rded as a hypothesis
(machine) about 0*, it is intuitively dear that earh of these machines has a different capacity to
approximate. it. But it is r('asonable to think that all of the machines that a.re 011 the same sphere
{a : 10 - 0*1 = r}, for each ,. > 0, haW' the same capacity in approximating 0*. Thus, to compare
the capacity being llsed at t, we mllst specify a sl)('cifk sphere as the starting point; defining the
effective size of the marhillc at t withont spedfying the starting sphere is clearly meallingless. As
we have seen, r ~
is found to be a good choice for our purposes.
7;
5
NETWORK SIZE SELECTION
The next theorem relates the generalization error and training error at E'ach epoch of learning, and
forms the basis for choosing the optimal stopping time as well as the best size of the machine during
the learning process. In the limit of the learning process, the criterion reduces to the well-known
Akaike Information Criterion (AIC) for statistical model selection. Comparison of the two criteria
reveals that our criterion will result in better generalization than AIC, since it incorporates the
information of each individual neuron rather than just the total number of neurons as in the Ale.
Theorem 5.1 A.9suming the learning algorithm converges, and the conditions of Theorem 3.1 are
satisfied; then the following equation holds:
? ((t,)
IIIhr.rr r(d, t) = 2~~_
2:7-1[J
= (1 + () (] ?E ?
1I (
0, )
+ r( d, t) + 0 ( ~ )
(9)
o
-- (1 -- rAj)'1
A(~('ording to this th('orl'lIl, We' find an M;.YIIlJllotically unbiased estimate of ?(u,) to ht' ?,,(0,) +
C(d, t) when (J"2 is known . This results in the following criterion for finding the optimal stopping
time and network size:
min{?n(at)
+ C(d, t) : d, t = 1,2, .. .}
(10)
Optimal Stopping and Effective Machine Complexity in Learning
When t goes to infinity, the above criterion becomes:
: d = 1,2, . . . }
(11)
n
which is the AIC for choosing the b!:'st siz!:' of networks. Therefore, (10) can be viewed as an
extension of the AIC to the learning process.
To understand the differences, consider the case
when ~ has standard normal distribution N(O, (12) . Under this assumption, the Maximum Likelihood
(ML) estimation of the weiglJt vectors is the saine as the Mean Square estimation. The AIC was
obtained by minimizing E !::~:: i~l, the K ullback-Leibler distance of the density function f 0 M L (X)
with aML being the ML estimation of n and that of the true density 10' This is equivalent. to
minimizing Iimt--+ooE(Y - lo,(X))2 = E(Y - fOML(X))2 (assuming the limit and the expectation
are interchangeable) . Now it is dear that while AIC chooses networks only at the limit of learning,
(10) does this in the whole learning procef1s. Observe that the matrix 4' is now exactly the Fisher
Information Matrix of the density function f.,(X), and Ai is a measure of the capacity of 'ljJi in
fitting the relation b!:'tween X and Y. Therefore Hllr criterion incorporates the information about
each specific neuron provided by the Fisher Information Matrix, which is a measure of how well
the data fit the model. This implies that there are two aspects in finding the trade-off between the
model complexity and the empirical error in order to minimize the generalization error: one is to
have the smallest number of neurons and the other is to minimize the utilization of each neuron .
The AIC (and in fact most statistical model selection criteria) are aimed at the former, while our
criterion incorporates the two aspects at the same time. We have seen in the earlier discussions that
for a given number of neurons, this is done by using the capacit.y of each neuron in fitting the data
only to the degree 1 - (1 - fA,)t",;" rather than to its limit.
min{?,,(&)
6
+ 2(12d
CONCLUDING REMARKS
To the best of our knowledge, the results described in this paper provide for the first time a precise
language to describe overtraining phenomena in [('arning machin!:'s such as neural networks. We
have studied formally the generalization process of a linear machine when it is trained with a
gradient descent algorithm . The concept of effective size of a machine was introduced to break the
generalization error into two parts: t.he approximation error and the error caused by a complexity
term which is proportional to effective size; the former decreases monotonically and the later increases
monotonically in the learning proress. When the machine is trained on a finite number of examples,
there are in general three distinct phases of l!:'arning according to the relative magnitude of the
fitting and complexity errors. In particular, there exists an optimal stopping time tmin = O(lnn)
for minimizing generalization error which occurs before the global minimum of the empirical error
is reached . These results lead to a generalization of the AIC in which the effect of certain network
parameters and time of learning are together taken into account in the network size selection process.
For practical application of neural networks , these results demonstrate that training a network
to its limits is not desirable. From the learning-theoretic- point of view, the concept of effective
dimension of a network t!:'Us us that we need more than thp VC-dimension of a machine to describe
the generalization properties of a machine, excppt in the limit of learning.
The generalization of the AIC reveals some unknown factf1 ill statistical model selection theory:
namely, the generalization error of a network is affeded not only by the number of parameters
but also by the degree to which each parametf'r is act.ually used in the learning process. Occam's
principle therefore stands in a subtler form: Make minimal ILse of the ca.pacity of a network for
encoding the information provided by learning samples.
Our results hold for weaker assumptions than were made herein about the distributions of X
The case of machines that have vector (rather than scalar) outputs is a simple generalization.
Also, our theorems have recently been generalized to the case of general nonlinear machines and are
not restricted to the squared error loss function.
and~.
While the problem of inferring a rule from the observational data has been studied for a long
time in learning theory as well as in other context sHch (IS in Linear and Nonlinear Regression, the
309
310
Wang, Venkatesh, and Judd
study of the problem as a dynamical process seems to open a new ave~ue for looking at the problem.
Many problems are open. For example, it is interesting to know what could be learned from a finite
number of examples in a finite number of itf'rations in the case where the size of the machine is not
small compared to the sample size.
Acknowledgments
C. Wang thanks Siemens Corporate Research for slIpport during the summer of 1992 whE'n t.his
research was initiated . Thp work of C. Wang aud S. Venkatesh has bf'en supported in part by thp
Air Force Office of Srif'lIt.ific Rpsparrh unrler grant. F49620-93-1-0120.
References
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7,044 | 817 | Grammatical Inference by
Attentional Control of Synchronization
in an Oscillating Elman Network
Bill Baird
Dept Mathematics,
U.C.Berkeley,
Berkeley, Ca. 94720,
[email protected]
Todd Troyer
Dept of Phys.,
U.C.San Francisco,
513 Parnassus Ave.
San Francisco, Ca. 94143,
[email protected]
Frank Eeckman
Lawrence Livermore
National Laboratory,
P.O. Box 808 (L-270),
Livermore, Ca. 94550,
[email protected]
Abstract
We show how an "Elman" network architecture, constructed from
recurrently connected oscillatory associative memory network modules, can employ selective "attentional" control of synchronization
to direct the flow of communication and computation within the
architecture to solve a grammatical inference problem.
Previously we have shown how the discrete time "Elman" network
algorithm can be implemented in a network completely described
by continuous ordinary differential equations. The time steps (machine cycles) of the system are implemented by rhythmic variation
(clocking) of a bifurcation parameter. In this architecture, oscillation amplitude codes the information content or activity of a module (unit), whereas phase and frequency are used to "softwire" the
network. Only synchronized modules communicate by exchanging amplitude information; the activity of non-resonating modules
contributes incoherent crosstalk noise.
Attentional control is modeled as a special subset of the hidden
modules with ouputs which affect the resonant frequencies of other
hidden modules. They control synchrony among the other modules and direct the flow of computation (attention) to effect transitions between two subgraphs of a thirteen state automaton which
the system emulates to generate a Reber grammar. The internal
crosstalk noise is used to drive the required random transitions of
the automaton.
67
68
Baird, Troyer, and Eeckman
1
Introduction
Recordings of local field potentials have revealed 40 to 80 Hz oscillation in vertebrate
cortex [Freeman and Baird, 1987, Gray and Singer, 1987]. The amplitude patterns
of such oscillations have been shown to predict the olfactory and visual pattern
recognition responses of a trained animal. There is further evidence that although
the oscillatory activity appears to be roughly periodic, it is actually chaotic when
examined in detail. This preliminary evidence suggests that oscillatory or chaotic
network modules may form the cortical substrate for many of the sensory, motor,
and cognitive functions now studied in static networks.
It remains be shown how networks with more complex dynamics can performs these
operations and what possible advantages are to be gained by such complexity. We
have therefore constructed a parallel distributed processing architecture that is inspired by the structure and dynamics of cerebral cortex, and applied it to the problem of grammatical inference. The construction views cortex as a set of coupled
oscillatory associative memories, and is guided by the principle that attractors must
be used by macroscopic systems for reliable computation in the presence of noise.
This system must function reliably in the midst of noise generated by crosstalk from
it's own activity. Present day digital computers are built of flip-flops which, at the
level of their transistors, are continuous dissipative dynamical systems with different attractors underlying the symbols we call "0" and "1". In a similar manner, the
network we have constructed is a symbol processing system, but with analog input
and oscillatory subsymbolic representations.
The architecture operates as a thirteen state finite automaton that generates the
symbol strings of a Reber grammar. It is designed to demonstrate and study the
following issues and principles of neural computation: (1) Sequential computation
with coupled associative memories. (2) Computation with attractors for reliable
operation in the presence of noise. (3) Discrete time and state symbol processing
arising from continuum dynamics by bifurcations of attractors. (4) Attention as
selective synchronization controling communication and temporal program flow. (5)
chaotic dynamics in some network modules driving randomn choice of attractors in
other network modules. The first three issues have been fully addressed in a previous
paper [Baird et. al., 1993], and are only briefly reviewed. ".le focus here on the last
two.
1.1
Attentional Processing
An important element of intra-cortical communication in the brain, and between
modules in this architecture, is the ability of a module to detect and respond to
the proper input signal from a particular module, when inputs from other modules
irrelevant to the present computation are contributing crosstalk noise. This is smilar
to the problem of coding messages in a computer architecture like the Connection
Machine so that they can be picked up from the common communication buss line
by the proper receiving module.
Periodic or nearly periodic (chaotic) variation of a signal introduces additional degrees of freedom that can be exploited in a computational architecture. We investigate the principle that selective control of synchronization, which we hypopthesize
to be a model of "attention", can be used to solve this coding problem and control
communication and program flow in an architecture with dynamic attractors.
The architecture illust.rates the notion that synchronization not only "binds" sen-
Grammatical Inference by Attentional Control of Synchronization
sory inputs into "objects" [Gray and Singer, 1987], but binds the activity of selected
cortical areas into a functional whole that directs behavior. It is a model of "attended activity" as that subset which has been included in the processing of the
moment by synchronization. This is both a spatial and temporal binding. Only the
inputs which are synchronized to the internal oscillatory activity of a module can
effect previously learned transitions of at tractors within it. For example, consider
two objects in the visual field separately bound in primary visual cortex by synchronization of their components at different phases or frequencies. One object may be
selectively attended to by its entrainment to oscillatory processing at higher levels
such as V4 or IT. These in turn are in synchrony with oscillatory activity in motor
areas to select the attractors there which are directing motor output.
In the architecture presented here, we have constrained the network dynamics so
that there exist well defined notions of amplitude, phase, and frequency. The network has been designed so that amplitude codes the information content or activity
of a module, whereas phase and frequency are used to "softwire" the network. An
oscillatory network module has a passband outside of which it will not synchronize with an oscillatory input. Modules can therefore easily be de synchronized
by perturbing their resonant frequencies. Furthermore, only synchronized modules
communicate by exchanging amplitude information; the activity of non-resonating
modules contributes incoherant crosstalk or noise. The flow of communication between modules can thus be controled by controlling synchrony. By changing the
intrinsic frequency of modules in a patterned way, the effective connectivity of the
network is changed. The same hardware and connection matrix can thus subserve
many different computations and patterns of interaction between modules without
crosstalk problems.
The crosstalk noise is actually essential to the function of the system. It serves as
the noise source for making random choices of output symbols and automaton state
transitions in this architecture, as we discuss later. In cortex there is an issue as to
what may constitute a source of randomness of sufficient magnitude to perturb the
large ensemble behavior of neural activity at the cortical network level. It does not
seem likely that the well known molecular fluctuations which are easily averaged
within one or a few neurons can do the job. The architecture here models the
hypothesis that deterministic chaos in the macroscopic dynamics of a network of
neurons, which is the same order of magnitude as the coherant activity, can serve
this purpose.
In a set of modules which is desynchronized by perturbing the resonant frequencies
of the group, coherance is lost and "random" phase relations result. The character
of the model time traces is irregular as seen in real neural ensemble activity. The behavior of the time traces in different modules of the architecture is similar to the temporary appearance and switching of synchronization between cortical areas seen in
observations of cortical processing during sensory/motor tasks in monkeys and humans [Bressler and Nakamura, 1993]. The structure of this apparently chaotic signal and its use in network learning and operation are currently under investigation.
2
Normal Form Associative Memory Modules
The mathematical foundation for the construction of network modules is contained
in the normal form projection algorithm [Baird and Eeckman, 1993]. This is a
learning algorithm for recurrent analog neural networks which allows associative
memory storage of analog patterns, continuous periodic sequences, and chaotic
69
70
Baird, Troyer, and Eeckman
attractors in the same network. An N node module can be shown to function
as an associative memory for up to N /2 oscillatory, or N /3 chaotic memory attractors [Baird and Eeckman, 1993]. A key feature of a net constructed by this
algorithm is that the underlying dynamics is explicitly isomorphic to any of a
class of standard, well understood nonlinear dynamical systems - a normal form
[Guckenheimer and Holmes, 1983].
The network modules of this architecture were developed previously as models of
olfactory cortex with distributed patterns of activity like those observed experimentally [Baird, 1990, Freeman and Baird, 1987]. Such a biological network is dynamically equivalent to a network in normal form and may easily be designed, simulated,
and theoretically evaluated in these coordinates. When the intramodule competition is high, they are "memory" or winner-take-all cordinates where attractors have
one oscillator at maximum amplitude, with the other amplitudes near zero. In figure two, the input and output modules are demonstrating a distributed amplitude
pattern ( the symbol "T"), and the hidden and context modules are two-attractor
modules in normal form coordinates showing either a right or left side active.
In this paper all networks are discussed in normal form coordinates. By analyzing the network in these coordinates, the amplitude and phase dynamics have a
particularly simple interaction. When the input to a module is synchronized with
its intrinsic oscillation, the amplitude of the periodic activity may be considered
separately from the phase rotation. The module may then be viewed as a static
network with these amplitudes as its activity.
To illustrate the behavior of individ ualnetwork modules, we examine a binary (twoattractor) module; the behavior of modules with more than two attractors is similar.
Such a unit is defined in polar normal form coordinates by the following equations
of the Hopf normal form:
rli
1l.i r li - Cdi
+ (d -
bsin(wclockt))rli r
5i + L wtlj cos(Oj -
Oli)
j
rOi
1l.j
r Oi - crg i + (d - bsin(wclockt))roirii
+L
wijlj cos(Oj - OOi)
j
Oli
Wi
+L
wt(Ij /1?li) sin(Oj - Oli)
j
OOi
Wi
+L
wij(Ij/rOi) sin(Oj - OOi)
j
The clocked parameter bsin(wclockt) is used to implement the discrete time machine
cycle of the Elman architecture as discussed later. It has lower frequency (1/10)
than the intrinsic frequency of the unit Wi.
Examination of the phase equations shows that a unit has a strong tendency
to synchronize with an input of similar frequency. Define the phase difference
cp = 00 - OJ = 00 - wJt between a unit 00 and it's input OJ. For either side of a
unit driven by an input of the same frequency, WJ = Wo, There is an attractor
at zero phase difference cp = 00 - OJ
and a repellor at cp
180 degrees. In
simulations, the interconnected network of these units described below synchronizes robustly within a few cycles following a perturbation. If the frequencies of
some modules of the architecture are randomly dispersed by a significant amount,
WJ - Wo #- 0, phase-lags appear first, then synchronization is lost in those units. An
oscillating module therefore acts as a band pass filter for oscillatory inputs.
=
?
=
Grammatical Inference by Attentional Control of Synchronization
When the oscillators are sychronized with the input, OJ - Oli = 0, the phase terms
cos(Oj - Oli)
cos(O)
1 dissappear. This leaves the amplitude equations rli
and rOi with static inputs E j wt;Ij and E j wijlj. Thus we have network modules
which emulate static network units in their amplitude activity when fully phaselocked to their input. Amplitude information is transmitted between modules, with
an oscillatory carrier.
=
=
For fixed values of the competition, in a completely synchronized system, the internal amplitude dynamics define a gradient dynamical system for a fourth order
energy fUllction. External inputs that are phase-locked to the module's intrinsic
oscillation simply add a linear tilt to the landscape.
For low levels of competition, there is a broad circular valley. When tilted by
external input, there is a unique equilibrium that is determined by the bias in tilt
along one axis over the other. Thinking of Tli as the "acitivity" of the unit, this
acitivity becomes a monotonically increasing function of input. The module behaves
as an analog connectionist unit whose transfer function can be approximated by a
sigmoid. We refer to this as the "analog" mode of operation of the module.
With high levels of competition, the unit will behave as a binary (bistable) digital
flip-flop element. There are two deep potential wells, one on each axis. Hence the
module performs a winner-take-all choice on the coordinates of its initial state and
maintains that choice "clamped" and independent of external input. This is the
"digital" or "quantized" mode of operation of a module. We think of one attractor
within the unit as representing "1" (the right side in figure two) and the other as
representing "0" .
3
Elman Network of Oscillating Associative Memories
As a benchmark for the capabilities of the system, and to create a point of contact to standard network architectures, we have constructed a discrete-time recurrent "Elman" network [Elman, 1991] from oscillatory modules defined by ordinary
differential equations. Previously we cons
structed a system which functions as the six Figure 1.
state finite automaton that perfectly recognizes or generates the set of strings defined by
the Reber grammar described in Cleeremans
et. al. [Cleeremans et al., 1989]. We found
the connections for this network by using the
backpropagation algorithm in a static network
that approximates the behavior of the amplitudes of oscillation in a fully synchronized dynamic network [Baird et al., 1993].
Here we construct a system that emulates
the larger 13 state automata similar (less one
state) to the one studied by Cleermans, et al
in the second part of their paper. The graph
of this automaton consists of two subgraph
branches each of which has the graph structure of the automaton learned as above, but
with different assignments of transition output symbols (see fig. 1).
T
71
72
Baird, Troyer, and Beckman
We use two types of modules in implementing the Elman network architecture shown
in figure two below. The input and output layer each consist of a single associative
memory module with six oscillatory attractors (six competing oscillatory modes),
one for each of the six symbols in the grammar. The hidden and context layers
consist of the binary "units" above composed of a two oscillatory attractors. The
architecture consists of 14 binary modules ill the hidden and context layers - three
of which are special frequency control modules. The hidden and context layers are
divided into four groups: the first three correspond to each of the two subgraphs plus
the start state, and the fourth group consists of three special control modules, each
of which has only a special control output that perturbs the resonant frequencies of
the modules (by changing their values in the program) of a particular state coding
group when it is at the zero attractor, as illustrated by the dotted control lines in
figure two. This figure shows control unit two is at the one attractor (right side
of the square active) and the hidden units coding for states of subgraph two are
in synchrony with the input and output modules. Activity levels oscillate up and
down through the plane of the paper. Here in midcycle, competition is high in all
modules.
Figure 2.
OSCILLATING ELMAN NETWORK
OUTPUT
INPUT
The discrete machine cycle of the Elman algorithm is implemented by the sinusoidal
variation (clocking) of the bifurcation parameter in the normal form equations that
determines the level of intramodule competition [Baird et al., 1993]. At the beginning of a machine cycle, when a network is generating strings, the input and context
layers are at high competition and their activity is clamped at the bottom of deep
basins of attraction. The hidden and output modules are at low competition and
therefore behave as a traditional feedforward network free to take on analog values.
In this analog mode, a real valued error can be defined for the hidden and output
units and standard learning algorithms like backpropagation can be used to train
the connections.
Then the situation reverses. For a Reber grammar there are always two equally possible next symbols being activated in the output layer, and we let the crosstalk noise
Grammatical Inference by Attentional Control of Synchronization
break this symmetry so that the winner-take-all dynamics of the output module can
chose one. High competition has now also "quantized" and clamped the activity in
the hidden layer to a fixed binary vector. Meanwhile, competition is lowered in the
input and context layers, freeing these modules from their attractors. An identity
mapping from hidden to context loads the binarized activity of the hidden layer
into the context layer for the next cycle, and an additional identity mapping from
the output to input module places the chosen output symbol into the input layer
to begin the next cycle.
4
Attentional control of Synchrony
We introduce a model of attention as control of program flow by selective synchronization. The attentional controler itself is modeled in this architecture as a special
set of three hidden modules with ouputs that affect the resonant frequencies of the
other corresponding three subsets of hidden modules. Varying levels of intramodule
competition control the large scale direction of information flow between layers of the
architecture. To direct information flow on a finer scale, the attention mechanism
selects a subset of modules within each layer whose output is effective in driving the
state transition behavior of the system.
By controling the patterns of synchronization within the network we are able to
generate the grammar obtained from an automaton consisting of two subgraphs
connected by a single transition state (figure 1). During training we enforce a segregation of the hidden layer code for the states of the separate subgraph branches of
the automaton so that different sets of synchronized modules learn to code for each
subgraph of the automaton. Then the entire automaton is hand constructed with
an additional hidden module for the start state between the branches. Transitions
in the system from states in one subgraph of the automaton to the other are made
by "attending" to the corresponding set of nodes in the hidden and context layers.
This switching of the focus of attention is accomplished by changing the patterns
of synchronization within the network which changes the flow of communication
between modules.
Each control module modulates the intrinsic frequency of the units coding for the
states a single su bgraph or the unit representing the start state. The control modules
respond to a particular input symbol and context to set the intrinsic frequency of
the proper subset of hidden units to be equal to the input layer frequency. As
described earlier, modules can easily be desynchronized by perturbing their resonant
frequencies. By perturbing the frequencies of the remaining modules away from the
input frequency, these modules are no longer communicating with the rest of the
network. Thus coherent information flows from input to output only through one
of three channels. Viewing the automata as a behavioral program, the control
of synchrony constitutes a control of the program flow into its subprograms (the
subgraphs of the automaton).
When either exit state of a subgraph is reached, the "B" (begin) symbol is then
emitted and fed back to the input where it is connected through the first to second
layer weight matrix to the attention control modules. It turns off the synchrony
of the hidden states of the subgraph and allows entrainment of the start state to
begin a new string of symbols. This state in turn activates both a "T" and a "P' in
the output module. The symbol selected by the crosstalk noise and fed back to the
input module is now connected to the control modules through the weight matrix.
It desynchronizes the start state module, synchronizes in the subset of hidden units
73
74
Baird. Troyer. and Eeckman
coding for the states of the appropriate subgraph, and establishes there the start
state pattern for that subgraph.
Future work will investigate the possibilities for self-organization of the patterns of
synchrony and spatially segregated coding in the hidden layer during learning. The
weights for entire automata, including the special attention control hidden units,
should be learned at once.
4.1
Acknowledgments
Supported by AFOSR-91-0325, and a grant from LLNL. It is a pleasure to acknowledge the invaluable assistance of Morris Hirsch, and Walter Freeman.
References
[Baird, 1990] Baird, B. (1990). Bifurcation and learning in network models of oscillating cortex. In Forest, S., editor, Emergent Computation, pages 365-384. North
Holland. also in Physica D, 42.
[Baird and Eeckman, 1993] Baird, B. and Eeckman, F. H. (1993). A normal form
projection algorithm for associative memory. In Hassoun, M. H., editor, Associative Neural Memories: Theory and Implementation, New York, NY. Oxford
University Press.
[Baird et al., 1993] Baird, B., Troyer, T., and Eeckman, F. H. (1993). Synchronization and gramatical inference in an oscillating elman network. In Hanson,
S., Cowan, J., and Giles, C., editors, Advances in Neural Information Processing
Systems S, pages 236-244. Morgan Kaufman.
[Bressler and Nakamura, 1993] Bressler, S. and Nakamura. (1993). Interarea synchronization in Macaque neocortex during a visual discrimination task. In Eeckman,F. H., and Bower, J., editors, Computation and Neural Systems, page 515.
Kluwer.
[Cleeremans et al., 1989] Cleeremans, A., Servan-Schreiber, D., and McClelland, J.
(1989). Finite state automata and simple recurrent networks. Neural Computation, 1(3):372-381.
[Elman, 1991] Elman, J. (1991). Distributed representations, simple recurrent networks and grammatical structure. Machine Learning, 7(2/3):91.
[Freeman and Baird, 1987] Freeman, W. and Baird, B. (1987). Relation of olfactory
EEG to behavior: Spatial analysis. Behavioral Neuroscience, 101:393-408.
[Gray and Singer, 1987] Gray, C. M. and Singer, W. (1987). Stimulus dependent
neuronal oscillations in the cat visual cortex area 17. Neuroscience [Supplj,
22:1301P.
[Guckenheimer and Holmes, 1983] Guckenheimer, J. and Holmes, D. (1983). Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields.
Springer, New York.
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7,045 | 818 | Grammatical Inference by
Attentional Control of Synchronization
in an Oscillating Elman Network
Bill Baird
Dept Mathematics,
U.C.Berkeley,
Berkeley, Ca. 94720,
[email protected]
Todd Troyer
Dept of Phys.,
U.C.San Francisco,
513 Parnassus Ave.
San Francisco, Ca. 94143,
[email protected]
Frank Eeckman
Lawrence Livermore
National Laboratory,
P.O. Box 808 (L-270),
Livermore, Ca. 94550,
[email protected]
Abstract
We show how an "Elman" network architecture, constructed from
recurrently connected oscillatory associative memory network modules, can employ selective "attentional" control of synchronization
to direct the flow of communication and computation within the
architecture to solve a grammatical inference problem.
Previously we have shown how the discrete time "Elman" network
algorithm can be implemented in a network completely described
by continuous ordinary differential equations. The time steps (machine cycles) of the system are implemented by rhythmic variation
(clocking) of a bifurcation parameter. In this architecture, oscillation amplitude codes the information content or activity of a module (unit), whereas phase and frequency are used to "softwire" the
network. Only synchronized modules communicate by exchanging amplitude information; the activity of non-resonating modules
contributes incoherent crosstalk noise.
Attentional control is modeled as a special subset of the hidden
modules with ouputs which affect the resonant frequencies of other
hidden modules. They control synchrony among the other modules and direct the flow of computation (attention) to effect transitions between two subgraphs of a thirteen state automaton which
the system emulates to generate a Reber grammar. The internal
crosstalk noise is used to drive the required random transitions of
the automaton.
67
68
Baird, Troyer, and Eeckman
1
Introduction
Recordings of local field potentials have revealed 40 to 80 Hz oscillation in vertebrate
cortex [Freeman and Baird, 1987, Gray and Singer, 1987]. The amplitude patterns
of such oscillations have been shown to predict the olfactory and visual pattern
recognition responses of a trained animal. There is further evidence that although
the oscillatory activity appears to be roughly periodic, it is actually chaotic when
examined in detail. This preliminary evidence suggests that oscillatory or chaotic
network modules may form the cortical substrate for many of the sensory, motor,
and cognitive functions now studied in static networks.
It remains be shown how networks with more complex dynamics can performs these
operations and what possible advantages are to be gained by such complexity. We
have therefore constructed a parallel distributed processing architecture that is inspired by the structure and dynamics of cerebral cortex, and applied it to the problem of grammatical inference. The construction views cortex as a set of coupled
oscillatory associative memories, and is guided by the principle that attractors must
be used by macroscopic systems for reliable computation in the presence of noise.
This system must function reliably in the midst of noise generated by crosstalk from
it's own activity. Present day digital computers are built of flip-flops which, at the
level of their transistors, are continuous dissipative dynamical systems with different attractors underlying the symbols we call "0" and "1". In a similar manner, the
network we have constructed is a symbol processing system, but with analog input
and oscillatory subsymbolic representations.
The architecture operates as a thirteen state finite automaton that generates the
symbol strings of a Reber grammar. It is designed to demonstrate and study the
following issues and principles of neural computation: (1) Sequential computation
with coupled associative memories. (2) Computation with attractors for reliable
operation in the presence of noise. (3) Discrete time and state symbol processing
arising from continuum dynamics by bifurcations of attractors. (4) Attention as
selective synchronization controling communication and temporal program flow. (5)
chaotic dynamics in some network modules driving randomn choice of attractors in
other network modules. The first three issues have been fully addressed in a previous
paper [Baird et. al., 1993], and are only briefly reviewed. ".le focus here on the last
two.
1.1
Attentional Processing
An important element of intra-cortical communication in the brain, and between
modules in this architecture, is the ability of a module to detect and respond to
the proper input signal from a particular module, when inputs from other modules
irrelevant to the present computation are contributing crosstalk noise. This is smilar
to the problem of coding messages in a computer architecture like the Connection
Machine so that they can be picked up from the common communication buss line
by the proper receiving module.
Periodic or nearly periodic (chaotic) variation of a signal introduces additional degrees of freedom that can be exploited in a computational architecture. We investigate the principle that selective control of synchronization, which we hypopthesize
to be a model of "attention", can be used to solve this coding problem and control
communication and program flow in an architecture with dynamic attractors.
The architecture illust.rates the notion that synchronization not only "binds" sen-
Grammatical Inference by Attentional Control of Synchronization
sory inputs into "objects" [Gray and Singer, 1987], but binds the activity of selected
cortical areas into a functional whole that directs behavior. It is a model of "attended activity" as that subset which has been included in the processing of the
moment by synchronization. This is both a spatial and temporal binding. Only the
inputs which are synchronized to the internal oscillatory activity of a module can
effect previously learned transitions of at tractors within it. For example, consider
two objects in the visual field separately bound in primary visual cortex by synchronization of their components at different phases or frequencies. One object may be
selectively attended to by its entrainment to oscillatory processing at higher levels
such as V4 or IT. These in turn are in synchrony with oscillatory activity in motor
areas to select the attractors there which are directing motor output.
In the architecture presented here, we have constrained the network dynamics so
that there exist well defined notions of amplitude, phase, and frequency. The network has been designed so that amplitude codes the information content or activity
of a module, whereas phase and frequency are used to "softwire" the network. An
oscillatory network module has a passband outside of which it will not synchronize with an oscillatory input. Modules can therefore easily be de synchronized
by perturbing their resonant frequencies. Furthermore, only synchronized modules
communicate by exchanging amplitude information; the activity of non-resonating
modules contributes incoherant crosstalk or noise. The flow of communication between modules can thus be controled by controlling synchrony. By changing the
intrinsic frequency of modules in a patterned way, the effective connectivity of the
network is changed. The same hardware and connection matrix can thus subserve
many different computations and patterns of interaction between modules without
crosstalk problems.
The crosstalk noise is actually essential to the function of the system. It serves as
the noise source for making random choices of output symbols and automaton state
transitions in this architecture, as we discuss later. In cortex there is an issue as to
what may constitute a source of randomness of sufficient magnitude to perturb the
large ensemble behavior of neural activity at the cortical network level. It does not
seem likely that the well known molecular fluctuations which are easily averaged
within one or a few neurons can do the job. The architecture here models the
hypothesis that deterministic chaos in the macroscopic dynamics of a network of
neurons, which is the same order of magnitude as the coherant activity, can serve
this purpose.
In a set of modules which is desynchronized by perturbing the resonant frequencies
of the group, coherance is lost and "random" phase relations result. The character
of the model time traces is irregular as seen in real neural ensemble activity. The behavior of the time traces in different modules of the architecture is similar to the temporary appearance and switching of synchronization between cortical areas seen in
observations of cortical processing during sensory/motor tasks in monkeys and humans [Bressler and Nakamura, 1993]. The structure of this apparently chaotic signal and its use in network learning and operation are currently under investigation.
2
Normal Form Associative Memory Modules
The mathematical foundation for the construction of network modules is contained
in the normal form projection algorithm [Baird and Eeckman, 1993]. This is a
learning algorithm for recurrent analog neural networks which allows associative
memory storage of analog patterns, continuous periodic sequences, and chaotic
69
70
Baird, Troyer, and Eeckman
attractors in the same network. An N node module can be shown to function
as an associative memory for up to N /2 oscillatory, or N /3 chaotic memory attractors [Baird and Eeckman, 1993]. A key feature of a net constructed by this
algorithm is that the underlying dynamics is explicitly isomorphic to any of a
class of standard, well understood nonlinear dynamical systems - a normal form
[Guckenheimer and Holmes, 1983].
The network modules of this architecture were developed previously as models of
olfactory cortex with distributed patterns of activity like those observed experimentally [Baird, 1990, Freeman and Baird, 1987]. Such a biological network is dynamically equivalent to a network in normal form and may easily be designed, simulated,
and theoretically evaluated in these coordinates. When the intramodule competition is high, they are "memory" or winner-take-all cordinates where attractors have
one oscillator at maximum amplitude, with the other amplitudes near zero. In figure two, the input and output modules are demonstrating a distributed amplitude
pattern ( the symbol "T"), and the hidden and context modules are two-attractor
modules in normal form coordinates showing either a right or left side active.
In this paper all networks are discussed in normal form coordinates. By analyzing the network in these coordinates, the amplitude and phase dynamics have a
particularly simple interaction. When the input to a module is synchronized with
its intrinsic oscillation, the amplitude of the periodic activity may be considered
separately from the phase rotation. The module may then be viewed as a static
network with these amplitudes as its activity.
To illustrate the behavior of individ ualnetwork modules, we examine a binary (twoattractor) module; the behavior of modules with more than two attractors is similar.
Such a unit is defined in polar normal form coordinates by the following equations
of the Hopf normal form:
rli
1l.i r li - Cdi
+ (d -
bsin(wclockt))rli r
5i + L wtlj cos(Oj -
Oli)
j
rOi
1l.j
r Oi - crg i + (d - bsin(wclockt))roirii
+L
wijlj cos(Oj - OOi)
j
Oli
Wi
+L
wt(Ij /1?li) sin(Oj - Oli)
j
OOi
Wi
+L
wij(Ij/rOi) sin(Oj - OOi)
j
The clocked parameter bsin(wclockt) is used to implement the discrete time machine
cycle of the Elman architecture as discussed later. It has lower frequency (1/10)
than the intrinsic frequency of the unit Wi.
Examination of the phase equations shows that a unit has a strong tendency
to synchronize with an input of similar frequency. Define the phase difference
cp = 00 - OJ = 00 - wJt between a unit 00 and it's input OJ. For either side of a
unit driven by an input of the same frequency, WJ = Wo, There is an attractor
at zero phase difference cp = 00 - OJ
and a repellor at cp
180 degrees. In
simulations, the interconnected network of these units described below synchronizes robustly within a few cycles following a perturbation. If the frequencies of
some modules of the architecture are randomly dispersed by a significant amount,
WJ - Wo #- 0, phase-lags appear first, then synchronization is lost in those units. An
oscillating module therefore acts as a band pass filter for oscillatory inputs.
=
?
=
Grammatical Inference by Attentional Control of Synchronization
When the oscillators are sychronized with the input, OJ - Oli = 0, the phase terms
cos(Oj - Oli)
cos(O)
1 dissappear. This leaves the amplitude equations rli
and rOi with static inputs E j wt;Ij and E j wijlj. Thus we have network modules
which emulate static network units in their amplitude activity when fully phaselocked to their input. Amplitude information is transmitted between modules, with
an oscillatory carrier.
=
=
For fixed values of the competition, in a completely synchronized system, the internal amplitude dynamics define a gradient dynamical system for a fourth order
energy fUllction. External inputs that are phase-locked to the module's intrinsic
oscillation simply add a linear tilt to the landscape.
For low levels of competition, there is a broad circular valley. When tilted by
external input, there is a unique equilibrium that is determined by the bias in tilt
along one axis over the other. Thinking of Tli as the "acitivity" of the unit, this
acitivity becomes a monotonically increasing function of input. The module behaves
as an analog connectionist unit whose transfer function can be approximated by a
sigmoid. We refer to this as the "analog" mode of operation of the module.
With high levels of competition, the unit will behave as a binary (bistable) digital
flip-flop element. There are two deep potential wells, one on each axis. Hence the
module performs a winner-take-all choice on the coordinates of its initial state and
maintains that choice "clamped" and independent of external input. This is the
"digital" or "quantized" mode of operation of a module. We think of one attractor
within the unit as representing "1" (the right side in figure two) and the other as
representing "0" .
3
Elman Network of Oscillating Associative Memories
As a benchmark for the capabilities of the system, and to create a point of contact to standard network architectures, we have constructed a discrete-time recurrent "Elman" network [Elman, 1991] from oscillatory modules defined by ordinary
differential equations. Previously we cons
structed a system which functions as the six Figure 1.
state finite automaton that perfectly recognizes or generates the set of strings defined by
the Reber grammar described in Cleeremans
et. al. [Cleeremans et al., 1989]. We found
the connections for this network by using the
backpropagation algorithm in a static network
that approximates the behavior of the amplitudes of oscillation in a fully synchronized dynamic network [Baird et al., 1993].
Here we construct a system that emulates
the larger 13 state automata similar (less one
state) to the one studied by Cleermans, et al
in the second part of their paper. The graph
of this automaton consists of two subgraph
branches each of which has the graph structure of the automaton learned as above, but
with different assignments of transition output symbols (see fig. 1).
T
71
72
Baird, Troyer, and Beckman
We use two types of modules in implementing the Elman network architecture shown
in figure two below. The input and output layer each consist of a single associative
memory module with six oscillatory attractors (six competing oscillatory modes),
one for each of the six symbols in the grammar. The hidden and context layers
consist of the binary "units" above composed of a two oscillatory attractors. The
architecture consists of 14 binary modules ill the hidden and context layers - three
of which are special frequency control modules. The hidden and context layers are
divided into four groups: the first three correspond to each of the two subgraphs plus
the start state, and the fourth group consists of three special control modules, each
of which has only a special control output that perturbs the resonant frequencies of
the modules (by changing their values in the program) of a particular state coding
group when it is at the zero attractor, as illustrated by the dotted control lines in
figure two. This figure shows control unit two is at the one attractor (right side
of the square active) and the hidden units coding for states of subgraph two are
in synchrony with the input and output modules. Activity levels oscillate up and
down through the plane of the paper. Here in midcycle, competition is high in all
modules.
Figure 2.
OSCILLATING ELMAN NETWORK
OUTPUT
INPUT
The discrete machine cycle of the Elman algorithm is implemented by the sinusoidal
variation (clocking) of the bifurcation parameter in the normal form equations that
determines the level of intramodule competition [Baird et al., 1993]. At the beginning of a machine cycle, when a network is generating strings, the input and context
layers are at high competition and their activity is clamped at the bottom of deep
basins of attraction. The hidden and output modules are at low competition and
therefore behave as a traditional feedforward network free to take on analog values.
In this analog mode, a real valued error can be defined for the hidden and output
units and standard learning algorithms like backpropagation can be used to train
the connections.
Then the situation reverses. For a Reber grammar there are always two equally possible next symbols being activated in the output layer, and we let the crosstalk noise
Grammatical Inference by Attentional Control of Synchronization
break this symmetry so that the winner-take-all dynamics of the output module can
chose one. High competition has now also "quantized" and clamped the activity in
the hidden layer to a fixed binary vector. Meanwhile, competition is lowered in the
input and context layers, freeing these modules from their attractors. An identity
mapping from hidden to context loads the binarized activity of the hidden layer
into the context layer for the next cycle, and an additional identity mapping from
the output to input module places the chosen output symbol into the input layer
to begin the next cycle.
4
Attentional control of Synchrony
We introduce a model of attention as control of program flow by selective synchronization. The attentional controler itself is modeled in this architecture as a special
set of three hidden modules with ouputs that affect the resonant frequencies of the
other corresponding three subsets of hidden modules. Varying levels of intramodule
competition control the large scale direction of information flow between layers of the
architecture. To direct information flow on a finer scale, the attention mechanism
selects a subset of modules within each layer whose output is effective in driving the
state transition behavior of the system.
By controling the patterns of synchronization within the network we are able to
generate the grammar obtained from an automaton consisting of two subgraphs
connected by a single transition state (figure 1). During training we enforce a segregation of the hidden layer code for the states of the separate subgraph branches of
the automaton so that different sets of synchronized modules learn to code for each
subgraph of the automaton. Then the entire automaton is hand constructed with
an additional hidden module for the start state between the branches. Transitions
in the system from states in one subgraph of the automaton to the other are made
by "attending" to the corresponding set of nodes in the hidden and context layers.
This switching of the focus of attention is accomplished by changing the patterns
of synchronization within the network which changes the flow of communication
between modules.
Each control module modulates the intrinsic frequency of the units coding for the
states a single su bgraph or the unit representing the start state. The control modules
respond to a particular input symbol and context to set the intrinsic frequency of
the proper subset of hidden units to be equal to the input layer frequency. As
described earlier, modules can easily be desynchronized by perturbing their resonant
frequencies. By perturbing the frequencies of the remaining modules away from the
input frequency, these modules are no longer communicating with the rest of the
network. Thus coherent information flows from input to output only through one
of three channels. Viewing the automata as a behavioral program, the control
of synchrony constitutes a control of the program flow into its subprograms (the
subgraphs of the automaton).
When either exit state of a subgraph is reached, the "B" (begin) symbol is then
emitted and fed back to the input where it is connected through the first to second
layer weight matrix to the attention control modules. It turns off the synchrony
of the hidden states of the subgraph and allows entrainment of the start state to
begin a new string of symbols. This state in turn activates both a "T" and a "P' in
the output module. The symbol selected by the crosstalk noise and fed back to the
input module is now connected to the control modules through the weight matrix.
It desynchronizes the start state module, synchronizes in the subset of hidden units
73
74
Baird. Troyer. and Eeckman
coding for the states of the appropriate subgraph, and establishes there the start
state pattern for that subgraph.
Future work will investigate the possibilities for self-organization of the patterns of
synchrony and spatially segregated coding in the hidden layer during learning. The
weights for entire automata, including the special attention control hidden units,
should be learned at once.
4.1
Acknowledgments
Supported by AFOSR-91-0325, and a grant from LLNL. It is a pleasure to acknowledge the invaluable assistance of Morris Hirsch, and Walter Freeman.
References
[Baird, 1990] Baird, B. (1990). Bifurcation and learning in network models of oscillating cortex. In Forest, S., editor, Emergent Computation, pages 365-384. North
Holland. also in Physica D, 42.
[Baird and Eeckman, 1993] Baird, B. and Eeckman, F. H. (1993). A normal form
projection algorithm for associative memory. In Hassoun, M. H., editor, Associative Neural Memories: Theory and Implementation, New York, NY. Oxford
University Press.
[Baird et al., 1993] Baird, B., Troyer, T., and Eeckman, F. H. (1993). Synchronization and gramatical inference in an oscillating elman network. In Hanson,
S., Cowan, J., and Giles, C., editors, Advances in Neural Information Processing
Systems S, pages 236-244. Morgan Kaufman.
[Bressler and Nakamura, 1993] Bressler, S. and Nakamura. (1993). Interarea synchronization in Macaque neocortex during a visual discrimination task. In Eeckman,F. H., and Bower, J., editors, Computation and Neural Systems, page 515.
Kluwer.
[Cleeremans et al., 1989] Cleeremans, A., Servan-Schreiber, D., and McClelland, J.
(1989). Finite state automata and simple recurrent networks. Neural Computation, 1(3):372-381.
[Elman, 1991] Elman, J. (1991). Distributed representations, simple recurrent networks and grammatical structure. Machine Learning, 7(2/3):91.
[Freeman and Baird, 1987] Freeman, W. and Baird, B. (1987). Relation of olfactory
EEG to behavior: Spatial analysis. Behavioral Neuroscience, 101:393-408.
[Gray and Singer, 1987] Gray, C. M. and Singer, W. (1987). Stimulus dependent
neuronal oscillations in the cat visual cortex area 17. Neuroscience [Supplj,
22:1301P.
[Guckenheimer and Holmes, 1983] Guckenheimer, J. and Holmes, D. (1983). Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields.
Springer, New York.
ADAPTIVE KNOT PLACEMENT FOR
NONPARAMETRIC REGRESSION
Hossein L. Najafi*
Department of Computer Science
University of Wisconsin
River Falls, WI 54022
Vladinlil' Cherkas sky
Department of Electrical Engineering
University of Minnesota
Minneapolis, Minnesota 55455
Abstract
Performance of many nonparametric methods critically depends
on the strategy for positioning knots along the regression surface.
Constrained Topological Mapping algorithm is a novel method that
achieves adaptive knot placement by using a neural network based
on Kohonen's self-organizing maps. We present a modification to
the original algorithm that provides knot placement according to
the estimated second derivative of the regression surface.
1
INTRODUCTION
Here we consider regression problems. Using mathematical notation, we seek to find
a function f of N - 1 predictor variables (denoted by vector X) from a given set of
n data points, or measurements, Zi
(Xi , Yi ) (i
1, ... , n) in N-dimensional
sample space:
=
Y = f(X)
=
+ error
(l)
where error is unknown (but zero mean) and its distribution may depend on X. The
distribution of points in the training set can be arbitrary, but uniform distribution
in the domain of X is often used.
? Responsible
for
correspondence,
[email protected].
Telephone
(715)
425-3769,
e-mail
247
248
Najafi and Cherkassky
The goal of this paper is to show how statistical considerations can be used to
improve the performance of a novel neural network algorithm for regression [eN91],
in order to achieve adaptive positioning of knots along the regression surface. By
estimating and employing the second derivative of the underlying function, the
modified algorithm is made more flexible around the regions with large second
derivative. Through empirical investigation, we show that this modified algorithm
allocates more units around the regions where the second derivative is large. This
increase in the local knot density introduces more flexibility into the model (around
the regions with large second derivative) and makes the model less biased around
these regions. However, no over-fitting is observed around these regions.
2
THE PROBLEM OF KNOT LOCATION
One of the most challenging problems in practical implementations of adaptive
methods for regression is adaptive positioning of knots along the regression surface.
Typical1y, knot positions in the domain of X are chosen as a subset of the training
data set, or knots are uniformly distributed in X. Once X-locations are fixed,
commonly used data-driven methods can be applied to determine the number of
knots. However, de Boor [dB78] showed that a polynomial spline with unequally
spaced knots can approximate an arbitrary function much better than a spline
with equally spaced knots. Unfortunately, the minimization problem involved in
determination of the optimal placement of knots is highly nonlinear and the solution
space is not convex [FS89). Hence, t.he performance of many recent algorit.hms that
include adaptive knot placement (e .g. MARS) is difficult to evaluate analytically. In
addition, it is well-known that when data points are uniform, more knots should be
located where the second derivative of the function is large. However, it is difficult to
extend these results for non-uniform data in conjunction with data-dependent noise.
Also, estimating the second derivative of a true function is necessary for optimal
knot placement. Yet, the function itself is unknown and its estimation depends on
the good placement of knots. This suggests the need for some iterative procedure
that alternates between function estimation(smoothing) and knot posit.ioning steps.
Many ANN methods effectively try to solve the problem of adaptive knot location using ad hoc strategies that are not statistically optimal. For example, local
adaptive methods [Che92) are generalizat.ion of kernel smoothers where the kernel functions and kernel centers are determined from the data by some adaptive
algorithm. Examples of local adaptive methods include several recently proposed
ANN models known as radial basis function (RBF) networks, regularization networks, networks with locally tuned units etc [BL88, MD89, PG90). When applied
to regression problems, all these methods seek to find regression estimate in the
(most general) form 2::=1 biHi(X, C i ) where X is the vector of predictor variable,
Ci is the coordinates of the i-th 'center' or 'bump', Hi is the response function of
the kernel type (the kernel width may be different for each center i), bi are linear
coefficients to be determined, and k is the total number of knots or 'centers'.
Whereas the general formulat.ion above assumes global opt.imizat.ion of an error measure for the training set with respect. to all parameters, i.e. center locations, kernel
width and linear coefficients, this is not practically feasible because the error surface
is generally non-convex and may have local minima [PG90, MD89). Hence most
Adaptive Knot Placement for Nonparametric Regression
practical approaches first solve the problem of center(knot) location and assume
identical kernel functions. Then the remaining problem of finding linear coefficients
bi is solved by using familiar methods of Linear Algebra [PG90] or gradient-descent
techniques [MD89]. It appears that the problem of center locations is the most
critical one for the local neural network techniques. Unfortunately, heuristics used
for center location are not based on any statistical considerations, and empirical
results are too sketchy [PG90, MD89]. In statistical methods knot locations are
typically viewed as free parameters of the model, and hence the number of knots
directly controls the model complexity. Alternatively, one can impose local regularization constraints on adjacent knot locations, so that neighboring knots cannot
move independently. Such an approach is effectively implemented in the model of
self-organization known as Kohonen's Self-Organizing Maps (SOM) [Koh84]. This
model uses a set of units ("knots") with neighborhood relations between units defined according to a fixed topological structure (typically 1D or 2D grid). During
training or self-organization, data points are presented to the map iteratively, one
at a time, and the unit closest to the data moves towards it, also pulling along its
topological neighbors.
3
MODIFIED CTM ALGORITHM FOR ADAPTIVE
KNOT PLACEMENT
The SOM model has been applied to nonparametric regression by Cherkassky and
Najafi [CN9I] in order to achieve adaptive positioning of knots along the regression surface. Their technique, called Constrained Topological Mapping (CTM), is a
modification of Kohonen's self-organization suitable for regression problems. CTM
interprets the units of the Kohonen map as movable knots of a regression surface.
Correspondingly, the problem of finding regression estimate can be stated as the
problem of forming an M - dimensional topological map using a set of samples
from N-dimensional sample space (where AI ~ N - 1) . Unfortunately, straightforward application of the Kohonen Algorithm to regression problem does not work
well [CN9I]. Because, the presence of noise in the training data can fool the algorithm to produce a map that is a multiple-valued function of independent variables
in the regression problem (1). This problem is overcome in the CTM algorithm,
where the nearest neighbor is found in the subspace of predictor variables, rather
than in the input(sample) space [CN9I].
We present next a concise description of the CTM algorithm. Using standard formulation (1) for regression, the training data are N-dimensional vectors Zi = (Xi
, Yi), where Y i is a noisy observation of an unknown function of N - 1 predictor
variables given by vector Xi. The CTM algorithm constructs an M - dimensional
topological map in N-dimensional sample space (M ~ N - 1) as follows:
o.
Initialize the M - dimensional t.opological map in N-dimensional sample
space.
1. Given an input vector Z in N-dimensional sample space, find the closest
(best matching) unit i in the subspace of independent val?iables:
II
Z*(k) - Wi
II =
Minj{IIZ* -
W;
II}
Vj E [I, ... ,L]
249
250
Najafi and Cherkassky
where Z? is the projection of the input vector onto the subspace of independent variables,
is the projection of the weight vector of unit j, and
k is the discrete time step.
2. Adjust the units' weights according to the following and return to 1:
Wi
'Vi
where /3( k) is the learning rate and Cj (k) is the neighborhood for unit
iteration k and are given by:
/3(k) = /30 x
(~~)
(k:.. )
(2)
i
at
1
,Cj(k)
= -----~~
(3)
o 5 ( IIi - ill )
exp'
/3(k) x So
where kmax is the final value of the time step (k max is equal to the product of
the training set size by the number of times it was recycled), /30 is the initial
1.0 and /3/
0.05 were
learning rate, and /3/ is the final learning rate (/30
used in all of our experiments), Iii - ill is the topological distance between
the unit i and the best matched unit i and So is the initial size of the map
(i.e., the number of units per dimension) .
=
=
Note that CTM method achieves placement of units (knots) in X-space according
to density of training data. This is due to the fact that X-coordinates of CTM units
during training follow the standard Kohonen self-organization algorithm [Koh84],
which is known to achieve faithful approximation of an unknown distribution. However, existing CTM method does not place more knots where the underlying function
changes rapidly. The improved strategy for CTM knot placement in X-space takes
into account estimated second derivative of a function as is described next.
The problem with estimating second derivative is that the function itself is unknown.
This suggests using an iterative strategy for building a model, i.e., start with a crude
model, estimate the second derivative based on this crude model, use the estimated
second derivative to refine the model, etc. This strategy can be easily incorporated
into the CTM algorithm due to its iterative nature. Specifically, in CTM method
the map of knots(i.e., the model) becomes closer and closer to the final regression
model as the training proceeds. Therefore, at each iteration, the modified algorithm
estimates the second derivative at the best matching unit (closest to the presented
data point in X-space), and allows additional movement of knots proportional to
this estimate. Estimating the second derivative from the map (instead of using the
training data) makes sense due to smoothing properties of CTM.
The modified CTM algorithm can be summarized as follows:
=
1. Present training sample Zi (Xi, Yi) to the map and find the closest (best
matching) unit i in the su bspace of independent variables to this data point.
(same as in the original CTM)
2. Move the the map (i.e., the best matching unit and all its neighbors) toward
the presented data point (same as in the original CTM)
Adaptive Knot Placement for Nonparametric Regression
3. Estimate average second derivative of the function at the best matching
unit based on the current positions of the map units.
4. Normalize this average second derivative to an interval of [0,1].
5. Move the map toward the presented data point at a rate proportional to
the estimated normalizes average second derivative and iterate.
For multivariate functions only gradients along directions given by the topological
structure ofthe map can be estimated in step 4. For example, given a 2-dimensional
mesh that approximates function I(XI, X2), every unit of the map (except the border
units for which there will be only one neighbor) has two neighboring units along
each topological dimension. These neighboring units can be used to approximate
the function's gradients along the corresponding topological dimension of the map.
These values along each dimension can then be averaged to provide a local gradient
estimate at a given knot.
In step 5, estimated average second derivative I" is normalized to [0,1] range using
1/Ji 1 - exp(lf"ll tan(T)) This is done because the value of second derivative is used
as the learning rate.
=
In step 6, the map is modified according to the following equation:
'Vj
(4)
It is this second movement of the map that allows for more flexibility around the
region of the map where the second derivative is large. The process described by
equation (4) is equivalent to pulling all units towards the data, with the learning
rate proportional to estimated second derivative at the best matched unit. Note
that the influence of the second derivative is gradually increased during the process
of self-organization by the factor (1-,B( k)). This factor account for the fact that the
map becomes closer and closer to the underlying function during self-orga.nization;
hence, providing a more reliable estimate of second deriva.tive.
4
EMPIRICAL COMPARISON
Performance of the two algorithms (original and modified CTM) was compared for
several low-dimensional problems. In all experiments the two algorithms used the
same training set of 100 data points for the univariate problems and 400 data points
for the 2-variable problems.
The training samples (Xi, Yi) were generated according to (1), with Xi randomly
drawn from a uniform distribution in the closed interval [-1,1]' and the error drawn
from the normal distribution N(O, (0.1)2). Regression estimates produced by the
self-organized maps were tested on a different set of n = 200 samples (test set)
generated in the same manner as the training set.
=j
We used the Average Residual, AR
~ L~=l [Yi - I(Xd]2, as the performance
measure on the test set. Here, I(X) is the piecewise linear estimate of the function
with knot locations provided by coordinates of the units of trained CTM. The Aver-
251
252
Najafi and Cherkassky
age Residual gives an indication of standard deviation of the overall generalization
error.
1.2
1
True function
Original CTM
~-.
0.8
Modified CTM -+--. 0.6
>(
t::;'
0.4
0.2
0 ~~~~~................:...............~.~~~~~
-0.2
-0.6
-0.8
-0.4
-0.2
o
0.2
0.4
0.6
0.8
x
1
Figure 1: A 50 unit map formed by the original and modified algorithm for the
Gaussian function.
1.2
1
True function
0.8
Original CTM
Modified CTM
~- .
-+--
Q 0.6
t::;'
0.4
0.2
o~~~_00!6I~~~;..J ............................................................ .
-0.2
I . . . . - _........_ _........_ _--'-_ _........._ _- ' -_ _..L.-_ _.L..-_........IL.-_---'
0.1
0.2
0.3
0.4
0.5
x
0.6
0.7
0.8
0.9
Figure 2: A 50 unit map formed by the original and modified algorithm for the step
function.
We used a gaussian function (f(x) = exp-64X 2 ) and a step function for our first set
of experiments. Figure 1 and 2 show the actual maps formed by the original and
modified algorithm for these functions. It is clear from these figures that the modified algorithm allocates more units around the regions where the second derivative
is large. This increase in the local knot density has introduced more flexibility into
the model around the regions with large second derivatives. As a result of this the
1
Adaptive Knot Placement for Nonparametric Regression
253
model is less biased around these regions. However, there is no over-fitting in the
regions where the second derivative is large.
0.29 r-----:"-___r---..,..---~---~---r__--___r--___,
0.28
Original CTM ~
0.27
Modified CTM +_.
0.26
0.25
c.::: 0.24
~
0.23
0.22
',~-~
0.21
~---- '"'+--__
..L
0.2
--aoj- __ +-- ---r____ + __ + __ _
- L ...... ... +---+
---y0.19
0.18 ""'-_ _--L._ _ _.........._ _----L_ _ _.........._ _ _' - - -_ _......L.._ _- - - - '
o
10
20
30
40
50
60
70
# of units per dimension
Figure 3: Average Residual error as a function of the size of the map for the 3dimensional Step function
0.55
0.5
Original CTM
Modified CTM
0.45
c.:::
~
~
+_.
0.4
0.35
0.3
----~---+ --- +- --+--- +- -- +- --+--+--+--+
0.25
0.2
0
10
20
30
40
50
60
# of units per dimension
Figure 4: Average Residual error as a function of the size of the map for the 3dimensional Sine function
To compare the behavior of the two algorithms in their predictability of structureless
data, we trained them on a constant function I(x) = a with eTT01' = N(O, (0.1)2).
This problem is known as smoothing pure noise in regression analysis. It has been
shown [CN9l] that the original algorithm handles this problem well and quality of
CTM smoothing is independent of the number of units in the map. Our experiments
70
254
Najafi and Cherkas sky
show that the modified algorithm performs as good as the original one in this
respect.
Finally, we used the following two-variable functions (step, and sine) to see how
well the modified algorithm performs in higher dimensional settings.
Ste : f(x
x) = {I for ((x~ < 0.5) 1\ (X2
0 otherwise
PI, 2
Sine: f(XI, X2)
< 0.5)) V ((Xl ~ 0.5) 1\ (X2 ~ 0.5))
= sin (27rJ(xt)2 + (X2)2)
The results of these experiments are summarized in Figure 3 and 4. Again we see
that the modified algorithm outperforms the original algorithm. Note that the above
example of a two-variable step function can be easily handled by recursive partitioning techniques such as CART [BFOS84]. However, recursive methods are sensitive to
coordinate rotation. On the other hand, CTM is a coordinate-independent method,
i.e. its performance is independent of any affine transformation in X-space.
References
[BFOS84] 1. Breiman, J.H. Friedman, R.A. Olshen, and C.J. Stone. Classification
and Regression Trees. Wadswordth, Belmont, CA, 1984.
[BL88]
D.S. Broomhead and D. Lowe. Multivariable functional interpolation
and adaptive networks. Complex Systems, 2:321-355, 1988.
[Che92]
V. Cherkassky. Neural networks and nonparametric regression. In S.Y.
Kung, F. Fallside, J .Aa. Sorenson, and C.A. Kamm, editors, Neural Networks for Signal Processing, volume II. IEEEE, Piscataway, NJ, 1992.
[CN91]
V. Cherkassky and H.L. Najafi. Constrained topological mapping for
nonparametric regression analysis. Neural Networks, 4:27-40, 1991.
[dB78]
C. de Boor. A Practical Guide to Splines. Springer-Verlag, 1978.
[FS89]
J .H. Friedman and B.W. Silverman. Flexible parsimonious smoothing
and additive modeling. Technometrics, 31(1):3-21, 1989.
T. Kohonen. Self-Organization and Associative Memory. SpringerVerlag, third edition, 1984.
[Koh84]
[MD89]
J. Moody and C.J. Darken. Fast learning in networks of locally tuned
processing units. Neural Computation, 1:281, 1989.
[PG90]
T. Poggio and F. Girosi. Networks for approximation and learning. Proceedings of the IEEE, 78(9):1481-1497, 1990.
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evaluate:1 tested:1 |
7,046 | 819 | Globally Trained Handwritten Word
Recognizer using Spatial Representation,
Convolutional Neural Networks and
Hidden Markov Models
Yoshua Bengio . .
Dept. Informatique et Recherche Operationnelle
Universite de Montreal
Montreal, Qc H3C-3J7
Yann Le Cun
AT&T Bell Labs
Holmdel NJ 07733
Donnie Henderson
AT&T Bell Labs
Holmdel NJ 07733
Abstract
We introduce a new approach for on-line recognition of handwritten words written in unconstrained mixed style. The preprocessor
performs a word-level normalization by fitting a model of the word
structure using the EM algorithm. Words are then coded into low
resolution "annotated images" where each pixel contains information about trajectory direction and curvature. The recognizer is a
convolution network which can be spatially replicated. From the
network output, a hidden Markov model produces word scores. The
entire system is globally trained to minimize word-level errors.
1
Introduction
Natural handwriting is often a mixture of different "styles", lower case printed,
upper case, and cursive. A reliable recognizer for such handwriting would greatly
improve interaction with pen-based devices, but its implementation presents new
*also, AT&T Bell Labs, Holmdel NJ 07733
937
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Bengio, Le Cun, and Henderson
technical challenges. Characters taken in isolation can be very ambiguous, but considerable information is available from the context of the whole word. We propose
a word recognition system for pen-based devices based on four main modules: a
preprocessor that normalizes a word, or word group, by fitting a geometrical model
to the word structure using the EM algorithm; a module that produces an "annotated image" from the normalized pen trajectory; a replicated convolutional neural
network that spots and recognizes characters; and a Hidden Markov Model (HMM)
that interprets the networks output by taking word-level constraints into account.
The network and the HMM are jointly trained to minimize an error measure defined
at the word level.
Many on-line handwriting recognizers exploit the sequential nature of pen trajectories by representing the input in the time domain. While these representations
are compact and computationally advantageous, they tend to be sensitive to stroke
order, writing speed, and other irrelevant parameters. In addition, global geometric
features, such as whether a stroke crosses another stroke drawn at a different time,
are not readily available in temporal representations. To avoid this problem we
designed a representation, called AMAP, that preserves the pictorial nature of the
handwriting.
In addition to recognizing characters, the system must also correctly segment the
characters within the words. One approach, that we call INSEG, is to recognize
a large number of heuristically segmented candidate characters and combine them
optimally with a postprocessor (Burges et al 92, Schenkel et al 93). Another approach, that we call OUTSEG, is to delay all segmentation decisions until after the
recognition, as is often done in speech recognition. An OUTSEG recognizer must
accept entire words as input and produce a sequence of scores for each character at
each location on the input. Since the word normalization cannot be done perfectly,
the recognizer must be robust with respect to relatively large distortions, size variations, and translations. An elastic word model -e.g., an HMM- can extract word
candidates from the network output. The HMM models the long-range sequential
structure while the neural network spots and classifies characters, using local spatial
structure.
2
Word Normalization
Input normalization reduces intra-character variability, simplifying character recognition. This is particularly important when recognizing entire words. We propose a
new word normalization scheme, based on fitting a geometrical model of the word
structure. Our model has four "flexible" lines representing respectively the ascenders line, the core line, the base line and the descenders line (see Figure 1). Points
on the lines are parameterized as follows:
y
= fk(X) = k(x -
XO)2
+ s(x -
xo)
+ YOk
(1)
where k controls curvature, s is the skew, and (xo,Yo) is a translation vector. The
parameters k, s, and Xo are shared among all four curves, whereas each curve has
its own vertical translation parameter YOk. First the set of local maxima U and
minima L of the vertical displacement are found. Xo is determined by taking the
average abscissa of extrema points. The lines of the model are then fitted to the
extrema: the upper two lines to the maxima, and the lower two to the minima.
The fit is performed using a probabilistic model for the extrema points given the
lines. The idea is to find the line parameters 8* that maximize the probability of
Globally Trained Handwritten Word Recognizer
---------'
Figure 1: Word Normalization Model: Ascenders and core curves fit y-maxima
whereas descenders and baseline curves fit y-minima. There are 6 parameters: a
(ascenders curve height relative to baseline), b (baseline absolute vertical position),
c (core line position), d (descenders curve position), k (curvature), s (angle).
generating the observed points.
0* = argmax log P(X
(J
I 0) + log P(O)
(2)
The above conditional distribution is chosen to be a mixture of Gaussians (one
per curve) whose means are the y-positions obtained from the actual x-positions
through equation 1:
3
P(Xi, Yi
1
0)
= log L
WkN(Yi; fk(xd, (J'y)
(3)
k=O
where N(x; J1, (J') is a univariate Normal distribution of mean J1 and standard deviation (J'. The Wk are the mixture parameters, some of which are set to 0 in order to
constrain the upper (lower) points to be fitted to the upper (lower) curves. They are
computed a-priori using measured frequencies of associations of extrema to curves
on a large set of words. The priors P(O) on the parameters are required to prevent
the collapse of the curves. They can be used to incorporate a-priori information
about the word geometry, such as the expected position of the baseline, or the
height of the word. These priors for each parameter are chosen to be independent
normal distributions whose standard deviations control the strength of the prior.
The variables that associate each point with one of the curves are taken as hidden
variables of the EM algorithm. One can thus derive an auxiliary function which can
be analytically (and cheaply) solved for the 6 free parameters O. Convergence of
the EM algorithm was typically obtained within 2 to 4 iterations (of maximization
of the auxiliary function).
3
AMAP
The recognition of handwritten characters from a pen trajectory on a digitizing
surface is often done in the time domain. Trajectories are normalized, and local
939
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Bengio, Le Cun, and Henderson
geometrical or dynamical features are sometimes extracted. The recognition is
performed using curve matching (Tappert 90), or other classification techniques such
as Neural Networks (Guyon et al 91). While, as stated earlier, these representations
have several advantages, their dependence on stroke ordering and individual writing
styles makes them difficult to use in high accuracy, writer independent systems that
integrate the segmentation with the recognition.
Since the intent of the writer is to produce a legible image, it seems natural to
preserve as much of the pictorial nature of the signal as possible, while at the same
time exploit the sequential information in the trajectory. We propose a representation scheme, called AMAP, where pen trajectories are represented by low-resolution
images in which each picture element contains information about the local properties of the trajectory. More generally, an AMAP can be viewed as a function in a
multidimensional space where each dimension is associated with a local property of
the trajectory, say the direction of motion e, the X position, and the Y position of
the pen. The value of the function at a particular location (e, X, Y) in the space
represents a smooth version of the "density" of features in the trajectory that have
values (e, X, Y) (in the spirit of the generalized Hough transform). An AMAP is a
multidimensional array (say 4x10x10) obtained by discretizing the feature density
space into "boxes". Each array element is assigned a value equal to the integral of
the feature density function over the corresponding box. In practice, an AMAP is
computed as follows. At each sample on the trajectory, one computes the position
of the pen (X, Y) and orientation of the motion () (and possibly other features, such
as the local curvature c). Each element in the AMAP is then incremented by the
amount of the integral over the corresponding box of a predetermined point-spread
function centered on the coordinates of the feature vector. The use of a smooth
point-spread function (say a Gaussian) ensures that smooth deformations of the
trajectory will correspond to smooth transformations of the AMAP. An AMAP can
be viewed as an "annotated image" in which each pixel is a feature vector.
A particularly useful feature of the AMAP representation is that it makes very few
assumptions about the nature of the input trajectory. It does not depend on stroke
ordering or writing speed, and it can be used with all types of handwriting (capital,
lower case, cursive, punctuation, symbols). Unlike many other representations (such
as global features), AMAPs can be computed for complete words without requiring
segmentation.
4
Convolutional Neural Networks
Image-like representations such as AMAPs are particularly well suited for use in
combination with Multi-Layer Convolutional Neural Networks (MLCNN) (Le Cun
89, Le Cun et al 90). MLCNNs are feed-forward neural networks whose architectures
are tailored for minimizing the sensitivity to translations, rotations, or distortions
of the input image. They are trained with a variation of the Back-Propagation
algorithm (Rumelhart et al 86, Le Cun 86).
The units in MCLNNs are only connected to a local neighborhood in the previous
layer. Each unit can be seen as a local feature detector whose function is determined
by the learning procedure. Insensitivity to local transformations is built into the
network architecture by constraining sets of units located at different places to use
identical weight vectors, thereby forcing them to detect the same feature on different
parts of the input. The outputs of the units at identical locations in different feature
maps can be collectively thought of as a local feature vector. Features of increasing
Globally Trained Handwritten Word Recognizer
complexity and globality are extracted by the neurons in the successive layers.
This weight-sharing technique has two interesting side effects. First, the number
of free parameters in the system is greatly reduced since a large number of units
share the same weights. Classically, MLCNNs are shown a single character at the
input, a.nd have a single set of outputs. However, an essential feature of MLCNNs
is that they can be scanned (replicated) over large input fields containing multiple
unsegmented characters (whole words) very economically by simply performing the
convolutions on larger inputs. Instead of producing a single output vector, SDNNs
produce a series of output vectors. The outputs detects and recognize characters at
different (and overlapping) locations on the input. These multiple-input, multipleoutput MLCNN are called Space Displacement Neural Networks (SDNN) (Matan
et al 92).
One of the best networks we found for character recognition has 5 layers arranged
as follows: layer 1: convolution with 8 kernels of size 3x3, layer 2: 2x2 subsampling,
layer 3: convolution with 25 kernels of size 5x5, layer 4 convolution with 84 kernels
of size 4x4, layer 5: 2x2 subsampling. The subsampling layers are essential to the
network's robustness to distortions. The output layer is one (single MLCNN) or
a series of (SDNN) 84-dimensional vectors. The target output configuration for
each character class was chosen to be a bitmap of the corresponding character in a
standard 7x12 (=84) pixel font. Such a code facilitates the correction of confusable
characters by the postprocessor.
5
Post-Processing
The convolutional neural network can be used to give scores associated to characters
when the network (or a piece of it corresponding to a single character output) has
an input field, called a segment, that covers a connected subset of the whole word
input. A segmentation is a sequence of such segments that covers the whole word
input. Because there are in general many possible segmentations, sophisticated
tools such as hidden Markov models and dynamic programming are used to search
for the best segmentation.
In this paper, we consider two approaches to the segmentation problem called INSEG (for input segmentation) and OUTSEG (for output segmentation). The postprocessor can be generally decomposed into two levels: 1) character level scores and
constraints obtained from the observations, 2) word level constraints (grammar,
dictionary). The INSEG and OUTSEG systems share the second level.
In an INSEG system, the network is applied to a large number of heuristically
segmented candidate characters. A cutter generates candidate cuts, which can potentially represent the boundary between two character segments. It also generates
definite cuts, which we assume that no segment can cross. Using these, a number
of candidate segments are constructed and the network is applied to each of them
separately. Finally, for each high enough character score in each of the segment, a
character hypothesis is generated, corresponding to a node in an observation graph .
The connectivity and transition probabilities on the arcs of the observation graph
represent segmentation and geometrical constraints (e.g., segments must not overlap and must cover the whole word, some transitions between characters are more
or less likely given the geometrical relations between their images).
In an OUTSEG system, all segmentation decisions are delayed until after the recog-
941
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Bengio, Le Cun, and Henderson
nition, as is often done in speech recognition. The AMAP of the entire word is
shown to an SDNN, which produces a sequence of output vectors equivalent to (but
obtained much more cheaply than) scanning the single-character network over all
possible pixel locations on the input. The Euclidean distances between each output
vector and the targets are interpreted as log-likelihoods of the output given a class .
To construct an observation graph, we use a set of character models (HMMs) . Each
character HMM models the sequence of network outputs observed for that character . We used three-state HMMs for each character, with a left and right state to
model transitions and a center state for the character itself. The observation graph
is obtained by connecting these character models , allowing any character to follow
any character.
On top of the constraints given in the observation graph , additional constraints that
are independent of the observations are given by what we call a gram mar graph,
which can embody lexical constraints. These constraints can be given in the form
of a dictionary or of a character-level grammar (with transition probabilities), such
as a trigram (in which we use the probability of observing a character in the context
of the two previous ones). The recognition finds the best path in the observation
graph that is compatible with the grammar graph. The INSEG and OUTSEG
architecture are depicted in Figure 2.
INSEG ARCHITECTURE
FOR WORD RECOGNITION
OUTSEG ARCHITECTURE
FOR WORD RECOGNITION
raw word
word
normalization
normalized
word ~--~'''''---"'''''1i
AMAP
computation
s~f
raw w0i"'r_d_""",,,_ __ ~
Mi.pf
': ":::",:, . ,. ': : :. : .:. t
~~':::
~:
:~:~:
~~~}
AMAP
t}
SDNN
graph
ofchar~a~c~e~r--'~----~
candi-'r-_ _f-_ _ """'II
dates
Character
~~~r~~~ ~,=.. ."""'_-_ <5~t'>r,ff:
. Cut hypotheses
I generation
segme~
n~""""_",,.._ _""""'1
graph
\r"!:~'1":"IWPII"""'~W
AMAP
HMMs
graph ~~_",-_ _-d! S....c .....r...... i....p .... t
of character
s....e.....n.....e.j ... o.T
cdaantedsi
5...... a... i ... u...... p.... .f
Lexical
constraints
word
Sec; p t
"Script"
h
Convolutional
Neural Network
~~~arliooa-cte-r----~
candl
dates" " - -+ --"""",!!
Lexical
constraints
d--""""",---J " Script "
wo r""'
Figure 2: INSEG and OUTSEG architectures for word recognition.
A crucial contribution of our system is the joint training of the neural network and
the post-processor with respect to a single criterion that approximates word-level
errors. We used the following discriminant criterion: minimize the total cost (sum
of negative log-likelihoods) along the "correct" paths (the ones that yield the correct
interpretations) , while minimizing the costs of all the paths (correct or not). The
discriminant nature of this criterion can be shown with the following example. If
Globally Trained Handwritten Word Recognizer
the cost of a path associated to the correct interpretation is much smaller than all
other paths, then the criterion is very close to 0 and no gradient is back-propagated.
On the other hand , if the lowest cost path yields an incorrect interpretation but differs from a path of correct interpretation on a sub-path, then very strong gradients
will be propagated along that sub-path , whereas the other parts of the sequence
will generate almost no gradient. \Vithin a probabilistic framework, this criterion
corresponds to the maximizing the mutual information (MMI) between the observations and the correct interpretation. During global training , it is optimized using
(enhanced) stochastic gradient descent with respect to all the parameters in the system, most notably the network weights. Experiments described in the next section
have shown important reductions in error rates when training with this word-level
criterion instead of just training the network separately for each character. Similar
combinations of neural networks with HMMs or dynamic programming have been
proposed in the past, for speech recognition problems (Bengio et al 92).
6
Experimental Results
In a first set of experiments, we evaluated the generalization ability of the neural
network classifier coupled with the word normalization preprocessing and AMAP
input representation. All results are in writer independent mode (different writers
in training and testing). Tests on a da tabase of isolated characters were performed
separately on four types of characters: upper case (2.99% error on 9122 patterns),
lower case (4.15% error on 8201 patterns), digits (1.4% error on 2938 patterns), and
punctuation (4.3% error on 881 patterns). Experiments were performed with the
network architecture described above.
The second and third set of experiments concerned the recognition of lower case
words (writer independent). The tests were performed on a database of 881 words.
First we evaluated the improvements brought by the word normalization to the
INSEG system. For the OUTSEG system we have to use a word normalization
since the network sees a whole word at a time. With the INSEG system, and
before doing any word-level training, we obtained without word normalization 7.3%
and 3.5% word and character errors (adding insertions, deletions and substitutions)
when the search was constrained within a 25461-word dictionary. When using the
word normalization preprocessing instead of a character level normalization, error
rates dropped to 4.6% and 2.0% for word and character errors respectively, i.e., a
relative drop of 37% and 43% in word and character error respectively.
In the third set of experiments, we measured the improvements obtained with the
joint training of the neural network and the post-processor with the word-level
criterion, in comparison to training based only on the errors performed at the character level. Training was performed with a database of 3500 lower case words. For
the OUTSEG system, without any dictionary constraints, the error rates dropped
from 38% and 12.4% word and character error to 26% and 8.2% respectively after
word-level training, i.e., a relative drop of 32% and 34%. For the INSEG system
and a slightly improved architecture, without any dictionary constraints, the error
rates dropped from 22.5% and 8.5% word and character error to 17% and 6.3%
respectively, i.e., a relative drop of 24.4% and 25.6%. With a 25461-word dictionary, errors dropped from 4.6% and 2.0% word and character errors to 3.2% and
1.4% respectively after word-level training, i.e., a relative drop of 30.4% and 30.0%.
Finally, some further improvements can be obtained by drastically reducing the size
of the dictionary to 350 words, yielding 1.6% and 0.94% word and character errors.
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Bengio, Le Cun, and Henderson
7
Conclusion
We have demonstrated a new approach to on-line handwritten word recognition
that uses word or sentence-level preprocessing and normalization, image-like representations, Convolutional neural networks, word models, and global training using
a highly discriminant word-level criterion. Excellent accuracy on various writer
independent tasks were obtained with this combination.
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7,047 | 82 | 82
SIMULATIONS SUGGEST
INFORMATION PROCESSING ROLES
FOR THE DIVERSE CURRENTS IN
HIPPOCAMPAL NEURONS
Lyle J. Borg-Graham
Harvard-MIT Division of Health Sciences and Technology and
Center for Biological Information Processing,
Massachusetts Institute of Technology, Cambridge, Massachusetts 02139
ABSTRACT
A computer model of the hippocampal pyramidal cell (HPC) is described
which integrates data from a variety of sources in order to develop a consistent description for this cell type. The model presently includes descriptions of eleven non-linear somatic currents of the HPC, and the electrotonic
structure of the neuron is modelled with a soma/short-cable approximation.
Model simulations qualitatively or quantitatively reproduce a wide range of
somatic electrical behavior i~ HPCs, and demonstrate possible roles for the
various currents in information processing.
1
The Computational Properties of Neurons
There are several substrates for neuronal computation, including connectivity, synapses, morphometries of dendritic trees, linear parameters of cell
membrane, as well as non-linear, time-varying membrane conductances, also
referred to as currents or channels. In the classical description of neuronal
function, the contribution of membrane channels is constrained to that of
generating the action potential, setting firing threshold, and establishing the
relationship between (steady-state) stimulus intensity and firing frequency.
However, it is becoming clear that the role of these channels may be much
more complex, resulting in a variety of novel "computational operators" that
reflect the information processing occurring in the biological neural net.
? American Institute of Physics 1988
83
2
Modelling Hippocampal Neurons
Over the past decade a wide variety of non-linear ion channels, have been
described for many excitable cells, in particular several kinds of neurons.
One such neuron is the hippocampal pyramidal cell (HPC). HPC channels are marked by their wide range of temporal, voltage-dependent, and
chemical-dependent characteristics, which results in very complex behavior
or responses of these stereotypical cortical integrating cells. For example,
some HPC channels are activated (opened) transiently and quickly, thus primarily affecting the action potential shape. Other channels have longer kinetics, modulating the response of HPCs over hundreds of milliseconds. The
measurement these channels is hampered by various technical constraints,
including the small size and extended electrotonic structure of HPCs and the
diverse preparations used in experiments. Modelling the electrical behavior
of HPCs with computer simulations is one method of integrating data from
a variety of sources in order to develop a consistent description for this cell
type.
In the model referred to here putative mechanisms for voltage-dependent
and calcium-dependent channel gating have been used to generate simulations of the somatic electrical behavior of HPCs, and to suggest mechanisms
for information processing at the single cell level. The model has also been
used to suggest experimental protocols designed to test the validity of simulation results. Model simulations qualitatively or quantitatively reproduce
a wide range of somatic electrical behavior in HPCs, and explicitly demonstrate possible functional roles for the various currents [1].
The model presently includes descriptions of eleven non-linear somatic
currents, including three putative N a+ currents - INa-trig, INa-rep, and
INa-tail; six K+ currents that have been reported in the literature - IDR
(Delayed Rectifier), lA, Ie, IAHP (After-hyperpolarization), 1M, and IQ;
and two Ca 2+ currents, also reported previously - lea and leas.
The electrotonic structure of the HPC is modelled with a soma/shortcable approximation, and the dendrites are assumed to be linear. While the
conditions for reducing the dendritic tree to a single cable are not met for
HPC (the so-called Rall conditions [3]), the Zin of the cable is close to that
of the tree. In addition, although HPC dendrites have non-linear membrane,
it assumed that as a first approximation the contribution of currents from
this membrane may be ignored in the somatic response to somatic stimulus.
Likewise, the model structure assumes that axon-soma current under these
conditions can be lumped into the soma circuit.
84
In part this paper will address the following question: if neural nets
are realizable using elements that have simple integrative all-or-nothing responses, connected to each other with regenerative conductors, then what
is the function for all the channels observed experimentally in real neurons?
The results of this HPC model study suggest some purpose for these complexities, and in this paper we shall investigate some of the possible roles of
non-linear channels in neuronal information processing. However, given the
speculative nature of many of the currents that we have presented in the
model, it is important to view results based on the interaction of the many
model elements as preliminary.
3
Defining Neural Information Coding is the First
Step in Describing Biological Computations
Determination of computational properties of neurons requires a priori assumptions as to how information is encoded in neuronal output. The classical description assumes that information is encoded as spike frequency.
However, a single output variable, proportional to firing frequency, ignores
other potentially information-rich degrees of freedom, including:
? Relative phase of concurrent inputs.
? Frequency modulation during single bursts.
? Cessation of firing due to intrinsic mechanisms.
? Spike shape.
Note that these variables apply to patterns of repetitive firingl. The
relative phase of different inputs to a single cell is very important at low
firing rates, but becomes less so as firing frequency approaches the time
constant of the postsynaptic membrane or some other rate-limiting process
in the synaptic transduction (e.g. neurotransmitter release or post synaptic channel activation/deactivation kinetics). Frequency modulation during
bursts/spike trains may be important in the interaction of a given axon's
output with other inputs at the target neuron. Cessation of firing due to
mechanisms intrinsic to the cell (as opposed to the end of input) may be
lSingle spikes may be considered as degenerate cases of repetitive firing responses.
85
important, for example, in that cell's transmission function. Finally, modulation of spike shape may have several consequences, which will be discussed
later.
4
Physiological Modulation of HPC Currents
In order for modulation of HPC currents to be considered as potential information processing mechanisms in vivo, it is necessary to identify physiological modulators. For several of the currents described here such factors
have been identified. For example, there is evidence that 1M is inhibited
by muscarinic (physiologically, cholinergic) agonists [2], that 1A is inhibited by acetylcholine [6], and that 1AHP is inhibited by noradrenaline [5].
In fact, the list of neurotransmitters which are active non-synaptically is
growing rapidly. It remains to be seen whether there are as yet undiscovered mechanisms for modulating other HPC currents, for example the three
N a+ currents proposed in the present model. Some possible consequences
of such mechanisms will be discussed later.
5
HPC Currents and Information Processing
The role of a given channel on the HPC electrical response depends on its
temporal characteristics as a function of voltage, intracellular messengers,
and other variables. This is complicated by the fact that the opening and
closing of channels is equivalent to varying conductances, allowing both linear and non-linear operations (e.g. [4] and [7]). In particular, a current
which is activated/deactivated over a period of hundreds of milliseconds
will, to a first approximation, act by slowly changing the time constant of
the membrane. At the other extreme, currents which activate/deactivate
with sub-millisecond time constants act by changing the trajectory of the
membrane voltage in complicated ways. The classic example of this is the
role of N a+ currents underlying the action potential.
To investigate how the different HPC currents may contribute to the
information processing of this neuron, we have looked at how each current
shapes the HPC response to a simple repertoire of inputs. At this stage
in our research the inputs have been very basic - short somatic current
steps that evoke single spikes, long lasting somatic current steps that evoke
spike trains, and current steps at the distal end of the dendritic cable. By
examining the response to these inputs the functional roles of the HPC
86
I Current" Spike Shape I Spike Threshold I Tm/Frequency-Intensity I
+++
++
-(+)
+
++
Ic
+
+
- (++)
++
+
+
IAHP
-
1M
-
++
+
INa-trig
INa-rep
ICa
IDR
IA
-
-
+++
+ (+++)
++
++
+++
+++
+
Table 1: Putative functional roles of HPC somatic currents. Entries in
parentheses indicate secondary role, e.g. Ca 2 + activation of J(+ current.
currents can be tentatively grouped into three (non-exclusive) categories:
? Modulation of spike shape.
? Modulation of firing threshold, both for single and repetitive spikes.
? Modulation of semi-steady-state membrane time constant.
? Modulation of repetitive firing, specifically the relationship between
strength of tonic input and frequency of initial burst and later "steady
state" spike train.
Table 1 summarizes speculative roles for some of the HPC currents as
suggested by the simulations. Note that while all four of the listed characteristics are interrelated, the last two are particularly so and are lumped
together in Table 1.
5.1
Possible Roles for Modulation of FI Characteristic
Again, it has been traditionally assumed that neural information is encoded
by (steady-state) frequency modulation, e.g. the number of spikes per second
over some time period encodes the output information of a neuron. For
example, muscle fiber contraction is approximately proportional to the spike
frequency of its motor neuron 2. If the physiological inhibition of a specific
2In fact, where action potential propagation is a stereotyped phenomena, such as in
long axons, then the timing of spikes is the only parameter that may be modulated.
87
. ..
-- -~
.........
'--;,
\
,
,
,
,
,
,
,
,
,
,
\
\
\
\
Stimulus Intensity (Constant Current)
Figure 1: Classical relation between total neuronal input (typically tonic
current stimulus) and spike firing frequency [solid line] and (qualitative)
biological relationships [dashed and dotted lines]. The dotted line applies
when INa-rep is blocked.
current changes the FI characteristic, this allows one way to modulate that
neuron's information processing by various agents.
Figure 1 contrasts the classical input-output relation of a neuron and
more biological input-output relations. The relationships have several features which can be potentially modulated either physiologically or pathologically, including saturation, threshold, and shape of the curves. Note in
particular the cessation of output with increased stimulation, as the depolarizing stimulus prevents the resetting of the transient inward currents.
For the HPC, simulations show (Figure 2 and Figure 3) that blocking the
putative INa-rep has the effect of causing the cell to "latch-up" in response
to tonic stimulus that would otherwise elicit stable spike trains. Both depolarizing currents and repolarizing currents playa role here. First, spike
upstroke is mediated by both INa-rep and the lower threshold INa-trig; at
high stimuli repolarization between spikes does not get low enough to reset
INa-trig' Second, spikes due to only one of these N a+ currents are weaker
and as a result do not activate the repolarizing [(+ currents as much as
normal because a) reduced time at depolarized levels activates the voltagedependent [(+ currents less and b) less Ca2+ influx with smaller spikes
reduces the Ca2+ -dependent activation of some [(+ currents. The net result is that repolarization between spikes is weaker and, again, does not reset
INa-trig.
Although the current being modulated here
(INa-rep)
is theoretical, the
88
Voltage (nV)
,~
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--
~~VL-,299.9
499.9
,899.9
2 nA Stinulus, Nornal
Vo leage (nV)
b~
Tine (sec) (x 1.ge-3)
699.9
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,899.9
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(x 1.ge-3)
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I
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~VVl/VvI/\/VV\/\/VVVVV1.,/VVVVV-~~
6 nA StiP'lulus, Nornal
Figure 2: Simulation of repetitive firing in response to constant current
injection into the soma. In this series, with the "normal" cell, a stimulus
of about 8 nA (not shown) will cause to cell to fire a short burst and then
cease firing.
possibility of selective blocking of INa-rep allows a mechanism for shifting
the saturation of the neuron's response to the left and, as can be seen by
comparing Figures 2 and 3, making the FI curve steeper over the response
range.
5.2
Possible Roles for Modulation of Spike Threshold
The somatic firing threshold determines the minimal input for eliciting a
spike, and in effect change the sensitivity of a cell. As a simple example,
blocking INa-trig in the HPe model raises threshold by about 10 millivolts.
This could cause the cell to ignore input patterns that would otherwise
generate action potentials.
There are two aspects of the firing "threshold" for a cell - static and
dynamic. Thus, the rate at which the soma membrane approaches threshold is important along with the magnitude of that threshold. In general
the threshold level rises with a slower depolarization for several reasons, including partial inactivation of inward currents (e.g. INa-trig) and partial
activation of outward currents (e.g. IA [8]) at subthreshold levels.
89
Tine (sec) (x 1.ge-3)
499.9
99.9
899.9
2 nA Stinulus,
4 nA Stinulus,
u~o
I-Na-Rep
u~o
I-Na-Rep
Tine (sec)
499 . 9
99.9
ex
1.ge-3}
899.9
-89.9
6 nA Stinulus,
u~o
I-Na-Rep
Figure 3: Blocking one of the putative N a+ currents (INa-rep) causes the
HPC repetitive firing response to fail at lower stimulus than "normal". This
corresponds to the leftward shift in the saturation of the response curve
shown in Figure 1.
Thus it is possible, for example, that IA helps to distinguish tonic dendritic distal synaptic input from proximal input. For input that eventually
will supply the same depolarizing current at the soma, dendritic input will
have a slower onset due to the cable properties of the dendrites. This slow
onset could allow IA to delay the onset of the spike or spikes. A similar depolarizing current applied more proximally would have a faster onset.
Sub-threshold activation of IA on the depolarizing phase would then be insufficient to delay the spike.
5.3
Possible Roles for Modulation of Somatic Spike Shape
How important is the shape of an individual spike generated at the soma?
First, we can assume that spike shape, in particular spike width, is unimportant at the soma spike-generating membrane - once the soma fires, it fires.
However, the effect of the spike beyond the soma mayor may not depend
on the spike shape, and this is dependent on both the degree which spike
propagation is linear and on the properties of the pre-synaptic membrane.
Axon transmission is both a linear and non-linear phenomena, and the
shorter the axon's electrotonic length, the more the shape of the somatic
90
action potential will be preserved at the distal pre-synaptic terminal. At
one extreme, an axon could transmit the spike a purely non-linear fashion
- once threshold was reached, the classic "all-or-nothing" response would
transmit a stereotyped action potential whose shape would be independent
of the post-threshold soma response. At the other extreme, i.e. if the axonal
membrane were purely linear, the propagation of the somatic event at any
point down the axon would be a linear convolution of the somatic signal
and the axon cable properties. It is likely that the situation in the brain lies
somewhere between these limits, and will depend on the wavelength of the
spike, the axon non-linearities and the axon length.
What role could be served by the somatic action potential shape modulating the pre-synaptic terminal signal? There are at least three possibilities.
First, it has been demonstrated that the release of transmitter at some presynaptic terminals is not an "all-or-nothing" event, and in fact is a function
of the pre-synaptic membrane voltage waveform. Thus, modulation of the
somatic spike width may determine how much transmitter is released down
the line, providing a mechanism for changing the effective strength of the
spike as seen by the target neuron. Modulation of somatic spike width could
be equivalent to a modulation ofthe "loudness" of a given neuron's message.
Second, pyramidal cell axons often project collateral branches back to the
originating soma, forming axo-somatic synapses which result in a feedback
loop. In this case, modulation of the somatic spike could affect this feedback
in complicated ways, particularly since the collaterals are typically short.
Finally, somatic spike shape may also playa role in the transmission of
spikes at axonal branch points. For example, consider a axonal branch point
with an impedance mismatch and two daughter branches, one thin and one
thick. Here a spike that is too narrow may not be able to depolarize the
thick branch sufficiently for transmission of the spike down that branch, with
the spike propagating only down the thin branch. Conversely, a wider spike
may be passed by both branches. Modulation of the somatic spike shape
could then be used to direct how a cell's output is broadcast, some times
allowing transmission to all the destinations of an HPC , and at other times
inhibiting transmission to a limited set of the target neurons.
For HPCs much evidence has been obtained which implicate the roles
of various HPC currents on modulating somatic spike shape, for example
the Ca 2 +-dependent K+ current Ie [9]. Simulations which demonstrate
the effect of Ie on the shape of individual action potentials are shown in
Figure 4.
91
Volt"9" (!'IV)
Tin" (~"c) (x 1.9,,-3)
Il 3.1l 4 . 9 S.1l
Volts" (nU)
Tin" (~"c) (x 1.9,,-3)
.9
Il 3.1l 4.1l 5.9
-81l.1l
-81l.9
"
,:"
1'\
, :
I"
\
Curr"nt (nA)"
..
1l.1l
....
-11l.1l
?.9
"
..
(x.1.1l,,-3)
3.'8- .'\.?.il"_".5.1l
Ti~,, : (~ec)
.Il
"
I
... ...
I-Na-Tris
-_. I-DR
" .. " I-C
Figure 4: Role of Ie during repolarization of spike. In the simulation on the
left, Ie is the largest repolarizing current. In the simulation on the right,
blocking Ie results in an wider spike.
6
The Assumption of Somatic Vs. Non-Somatic
Currents
In this research the somatic response of the HPC has been modelled under
the assumption that the data on HPC currents reflect activity of channels
localized at the soma. However, it must be considered that all channel proteins, regardless of their final functional destination, are manufactured at
the soma. Some of the so-called somatic channels may therefore be vestiges of channels intended for dendritic, axonal, or pre-synaptic membrane.
For example, if the spike-shaping channels are intended to be expressed for
pre-synaptic membrane, then modulation of these channels by endogenous
factors (e.g. ACh) takes place at target neuron. This may seem disadvantageous if a factor is to act selectively on some afferent tract. On the other
hand, in the dendritic field of a given neuron it is possible only some afferents have certain channels, thus allowing selective response to modulating
agents. These possibilities further expand the potential roles of membrane
channels for computation.
92
7
Other Possible Roles of Currents for Modulating HPC Response
There are many other potential ways that HPC currents may modulate the
HPC response. For example, the relationship between intracellular Ca2+
and the Ca2 +-dependent K+ currents, Ic and IAHP, may indicate possible
information processing mechanisms.
Intracellular Ca 2+ is an important second messenger for several intracellular processes, for example muscular contraction, but excessive [Ca 2+]in is
noxious. There are at least three negative feedback mechanisms for limiting
the flow of Ca2+ : voltage-dependent inactivation of Ca2+ currents; reduction of ECa (and thus the Ca2+ driving force) with Ca2+ influx; and the just
mentioned Ca2+ -mediation of repolarizing currents. A possible information
processing mechanism could be by modulation of IAHP, which plays an important role in limiting repetitive firing;. Simulations suggest that blocking
this current causes Ic to step in and eventually limit further repetitive firing, though after many more spikes in a train. Blocking both these currents
may allow other mechanisms to control repetitive firing, perhaps ones that
operate independently of [Ca 2+]in. Conceivably, this could put the neuron
into quite a differen t operating region.
8
Populations of Neurons V s. Single Cells: Implications for Graded Modulation of HPC Currents
In this paper we have considered the all-or-nothing contribution of the various channels, Le. the entire population of a given channel type is either
activated normally or all the channels are disabled/blocked. This description may be oversimplified in two ways. First, it is possible that a blocking
mechanism for a given channel may have a graded effect. For example, it is
possible that cholinergic input is not homogeneous over the soma membrane,
or that at a given time only a portion of these afferents are activated. In
either case it is possible that only a portion of the cholinergic receptors are
bound, thus inhibiting a portion of channels. Second, the result of channel
inhibition by neuromodulatory projections must consider both single cell
3The slowing down of the spike trains in Figure 2 and Figure 3 is mainly due to the
buildup of [Ca 2+];n, which progressively activates more IAHP.
93
response and population response, the size of the population depending on
the neuro-architecture of a cortical region and the afferents. For example,
activation of a cholinergic tract which terminates in a localized hippocampal
region may effect thousands of HPCs. Assuming that the 1M of individual
HPCs in the region may be either turned on or off completely with some
probability, the behavior of the population will be that of a graded response
of 1M inhibition. This graded response will in turn depend on the strength
of the cholinergic tract activity.
The key point is that the information processing properties of isolated
neurons may be reflected in the behavior of a population, and vica-versa.
While it is likely that removal of a single pyramidal cell from the hippocampus will have zero functional effect, no neuron is an island. Understanding the central nervous system begins with the spectrum of behavior in its
functional units, which may range from single channels, to specific areas of
a dendritic tree, to the single cell, to cortical or nuclear subfields, on up
through the main subsystems of CNS.
References
[1] L. Borg-Graham. Modelling the Somatic Electrical Behavior of Hippocampal Pyramidal Neurons. Master's thesis, Massachusetts Institute
of Technology, 1987.
[2] J. Halliwell and P. Adams. Voltage clamp analysis of muscarinic excitation in hippocampal neurons. Brain Research, 250:71-92, 1982.
[3] J. J. B. Jack, D. Noble, and R. W. Tsien. Electric Current Flow In
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| 82 |@word hippocampus:2 integrative:1 simulation:13 contraction:2 solid:1 biomathematics:1 reduction:1 initial:2 series:1 undiscovered:1 tine:5 past:1 current:65 comparing:1 nt:1 activation:6 yet:1 must:2 shape:18 eleven:2 motor:1 designed:1 progressively:1 v:1 nervous:1 slowing:1 short:4 contribute:1 burst:4 along:1 direct:1 borg:2 supply:1 qualitative:1 idr:2 upstroke:1 ica:1 behavior:9 growing:1 brain:2 terminal:3 oversimplified:1 rall:1 becomes:1 project:1 begin:1 underlying:1 linearity:1 circuit:1 inward:2 what:2 kind:1 depolarization:1 temporal:2 act:3 ti:1 control:2 normally:1 unit:1 timing:1 limit:2 consequence:2 receptor:1 oxford:1 establishing:1 firing:20 becoming:1 modulation:21 approximately:1 conversely:1 limited:1 range:5 subfields:1 lyle:1 block:1 vica:1 area:1 elicit:1 physiology:1 projection:1 pre:6 integrating:2 suggest:5 protein:1 get:1 close:1 subsystem:1 operator:1 put:1 equivalent:2 demonstrated:1 center:2 regardless:1 independently:1 stereotypical:1 nuclear:1 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7,048 | 820 | Temporal Difference Learning of
Position Evaluation in the Game of Go
Nicol N. Schraudolph
schraudo~salk.edu
Peter Dayan
dayan~salk.edu
Terrence J. Sejnowski
terry~salk.edu
Computational Neurobiology Laboratory
The Salk Institute for Biological Studies
San Diego, CA 92186-5800
Abstract
The game of Go has a high branching factor that defeats the tree
search approach used in computer chess, and long-range spatiotemporal interactions that make position evaluation extremely
difficult. Development of conventional Go programs is hampered
by their knowledge-intensive nature. We demonstrate a viable
alternative by training networks to evaluate Go positions via temporal difference (TD) learning.
Our approach is based on network architectures that reflect the
spatial organization of both input and reinforcement signals on
the Go board, and training protocols that provide exposure to
competent (though unlabelled) play. These techniques yield far
better performance than undifferentiated networks trained by selfplay alone. A network with less than 500 weights learned within
3,000 games of 9x9 Go a position evaluation function that enables
a primitive one-ply search to defeat a commercial Go program at
a low playing level.
1
INTRODUCTION
Go was developed three to four millenia ago in China; it is the oldest and one of the
most popular board games in the world. Like chess, it is a deterministic, perfect
information, zero-sum game of strategy between two players. They alternate in
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Schraudolph, Dayan, and Sejnowski
placing black and white stones on the intersections of a 19x19 grid (smaller for
beginners) with the objective of surrounding more board area (territory) with their
stones than the opponent. Adjacent stones of the same color form groups; an empty
intersection adjacent to a group is called a liberty of that group. A group is captured
and removed from the board when its last liberty is occupied by the opponent. To
prevent loops, it is illegal to make a move which recreates a prior board position. A
player may pass at any time; the game ends when both players pass in succession.
Unlike most other games of strategy, Go has remained an elusive skill for com puters
to acquire - indeed it has been recognized as a grand challenge" of Artificial
Intelligence (Rivest, 1993). The game tree search approach used extensively in
computer chess is infeasible: the game tree of Go has an average branching factor
of around 200, but even beginners may routinely look ahead up to 60 plies in
some situations. Humans appear to rely mostly on static evaluation of board
positions, aided by highly selective yet deep local lookahead. Conventional Go
programs are carefully (and protractedly) tuned expert systems (Fotland, 1993).
They are fundamentally limited by their need for human assistance in compiling
and integrating domain knowledge, and still play barely above the level of a
human beginner - a machine learning approach may thus offer considerable
advantages. (Brugmann, 1993) has shown that a knowledge-free optimization
approach to Go can work in principle: he obtained respectable (though inefficient)
play by selecting moves through simulated annealing (Kirkpatrick et al., 1983) over
possible continuations of the game.
/I
The pattern recognition component inherent in Go is amenable to connectionist
methods. Supervised backpropagation networks have been applied to the game
(Stoutamire, 1991; Enderton, 1991) but face a bottleneck in the scarcity of handlabelled training data. We propose an alternative approach based on the TD(A)
predictive learning algorithm (Sutton, 1984; Sutton, 1988; Barto et al., 1983), which
has been successfully applied to the game of backgammon by (Tesauro, 1992).
His TD-Gammon program uses a backpropagation network to map preselected
features of the board position to an output reflecting the probability that the player
to move would win. It was trained by TD(O) while playing only itself, yet learned
an evaluation function that - coupled with a full two-ply lookahead to pick the
estimated best move - made it competitive with the best human players in the
world (Robertie, 1992; Tesauro, 1994).
In an early experiment we investigated a straightforward adaptation of Tesauro's
approach to the Go domain. We trained a fully connected 82-40-1 backpropagation
network by randomized! self-play on a 9x9 Go board (a standard didactic size for
humans). The output learned to predict the margin of victory or defeat for black.
This undifferentiated network did learn to squeak past Wally, a weak public domain
program (Newman, 1988), but it took 659,000 games of training to do so. We have
found that the efficiency of learning can be vastly improved through appropriately
structured network architectures and training strategies, and these are the focus of
the next two sections.
1 Unlike backgammon, Go is a deterministic game, so we had to generate moves stochastically to ensure sufficient exploration of the state space. This was done by Gibbs sampling (Geman and Geman, 1984) over values obtained from single-ply search, annealing the
temperature parameter from random towards best-predicted play.
Temporal Difference Learning of Position Evaluation in the Game of Go
E evaluation
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Figure 1: A modular network architecture that takes advantage of board symmetries, translation invariance and localized reinforcement to evaluate Go positions.
Also shown is the planned connectivity prediction mechanism (see Discussion).
2 NETWORK ARCHITECTURE
One of the particular advantages of Go for predictive learning is that there is much
richer information available at the end of the game than just who won. Unlike
chess, checkers or backgammon, in which pieces are taken away from the board
until there are few or none left, Go stones generally remain where they are placed.
This makes the final state of the board richly informative with respect to the course
of play; indeed the game is scored by summing contributions from each point on
the board. We make this spatial credit assignment accessible to the network by
having it predict the fate of every point on the board rather than just the overall
score, and evaluate whole positions accordingly. This bears some similarity with
the Successor Representation (Dayan, 1993) which also integrates over vector rather
than scalar destinies. 2
Given the knowledge-based approach of existing Go programs, there is an embarrassment of input features that one might adopt for Go: Wally already uses about
30 of them, stronger programs disproportionately more. In order to demonstrate
reinforcement learning as a viable alternative to the conventional approach, however, we require our networks to learn whatever set of features they might need.
The complexity of this task can be significantly reduced by exploiting a number
2Sharing information within the network across multiple outputs restricts us to A = 0
for efficient implementation of TD( A). Note that although (Tesauro, 1992) did not have this
constraint, he nevertheless found A = 0 to be optimal.
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Schraudolph, Dayan, and Sejnowski
of constraints that hold a priori in this domain. Specifically, patterns of Go stones
retain their properties under color reversal, reflection and rotation of the board,
and - modulo the considerable influence of the board edges - translation. Each
of these invariances is reflected in our network architecture:
Color reversal invariance implies that changing the color of every stone in a Go
position, and the player whose tum it is to move, yields an equivalent position from
the other player's perspective. We build this constraint directly into our networks
by using antisymmetric input values (+1 for black, -1 for white) and squashing
functions throughout, and negating the bias input when it is white's tum to move.
Go positions are also invariant with respect to the eightfold (reflection x rotation)
symmetry of the square. We provided mechanisms for constraining the network
to obey this invariance by appropriate weight sharing and summing of derivatives
(Le Cun et al., 1989). Although this is clearly beneficial during the evaluation of
the network against its opponents, it appears to impede the course of learning. 3
To account for translation invariance we use convolution with a weight kernel
rather than multiplication by a weight matrix as the basic mapping operation in
our network, whose layers are thus feature maps produced by scanning a fixed
receptive field across the input. One particular advantage of this technique is the
easy transfer of learned weight kernels to different Go board sizes.
It must be noted, however, that Go is not translation-invariant: the edge of the board
not only affects local play but modulates other aspects of the game, and indeed
forms the basis of opening strategy. We currently account for this by allowing each
node in our network to have its own bias weight, giving it one degree of freedom
from its neighbors. This enables the network to encode absolute position at a
modest increse in the number of adjustable parameters. Furthermore, we provide
additional redundancy around the board edges by selective use of convolution
kernels twice as wide as the input.
Figure 1 illustrates the modular architecture suggested by these deliberations. In
the experiments described below we implement all the features shown except for
the connectivity map and lateral constraint satisfaction, which are the subject of
future work.
3 TRAINING STRATEGIES
Tern poral difference learning teaches the network to predict the consequences of
following particular strategies on the basis of the play they produce. The question
arises as to which strategies should be used to generate the large number of Go
games needed for training. We have identified three criteria by which we compare
alternative training strategies:
? the computational efficiency of move generation,
? the quality of generated play, and
? reasonable coverage of plausible Go positions.
3We are investigating possible causes and cures for this phenomenon.
Temporal Difference Learning of Position Evaluation in the Game of Go
Tesauro trained TD-Gammon by self-play - ie. the network's own position evaluation was used in training to pick both players' moves. This technique does not
require any external source of expertise beyond the rules of the game: the network
is its own teacher. Since Go is a deterministic game, we cannot always pick the
estimated best move when training by self-play without running the risk of trapping the network in some suboptimal fixed state. Theoretically, this should not
happen - the network playing white would be able to predict the idiosyncrasies
of the network playing black, take advantage of them thus changing the outcome,
and forcing black's predictions to change commensurately- but in practice it is a
concern. We therefore pick moves stochastically by Gibbs sampling (Geman and
Geman, 1984), in which the probability of a given move is exponentially related to
the predicted value of the position it leads to through a "temperature" parameter
that controls the degree of randomness.
We found self-play alone to be rather cumbersome for two reasons: firstly, the
single-ply search used to evaluate all legal moves is com putationally intensive and although we are investigating faster ways to accomplish it, we expect move
evaluation to remain a computational burden. Secondly, learning from self-play is
sluggish as the network must bootstrap itself out of ignorance without the benefit of
exposure to skilled opponents. However, there is nothing to keep us from training
the network on moves that are not based on its own predictions - for instance, it
can learn by playing against a conventional Go program, or even by just observing
games between human players.
We use three computer opponents to train our networks: a random move generator,
the public-domain program Wally (Newman, 1988), and the commercial program
The Many Faces of Go (Fotland, 1993). The random move generator naturally doesn't
play Go very we1l 4 , but it does have the advantages of high speed and ergodicity
- a few thousand games of random Go proved an effective way to prime our
networks at the start of training. The two conventional Go programs, by contrast,
are rather slow and deterministic, and thus not suitable generators of training data
when playing among themselves. However, they do make good opponents for the
network, which can provide the required variety of play through its Gibbs sam pIer.
When training on games played between such dissimilar players, we must match
their strength so as to prevent trivial predictions of the outcome. Against Many Faces
we use standard Go handicaps for this purpose; Wally we modified to intersperse its
play with random moves. The proportion of random moves is reduced adaptively
as the network improves, providing us with an on-line performance measure.
Since, in all cases, the strategies of both players are intimately intertwined in the
predictions, one would never expect them to be correct overall when the network
is playing a real opponent. This is a particular problem when the strategy for
choosing moves during learning is different from the policy adopted for 'optimal'
network play. (Samuel, 1959) found it inadvisable to let his checker program
learn from games which it won against an opponent, since its predictions might
otherwise reflect poor as well as good play. This is a particularly pernicious form
of over-fitting - the network can learn to predict one strategy in exquisite detail,
without being able to play well in general.
4In order to ensure a minimum of stability in the endgame, it does refuse to fill in its own
eyes - a particular, locally recognizable type of suicidal move.
821
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Schraudolph, Dayan, and Sejnowski
hl-+reinf
hO-+reinf
;0
archi tecture
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.
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board -+ reinf
turn -+ reinf
Figure 2: A small network that learned to play 9x9 Go. Boxes in the architecture
panel represent 9x9 layers of units, except for turn which is a single bias unit.
Arrows indicate convolutions with the corresponding weight kernels. Black disks
represent excitatory, white ones inhibitory weights; within each matrix, disk area
is proportional to weight magnitude.
4 RESULTS
In exploring this domain, we trained many networks by a variety of methods.
A small sample network that learned to beat Many Faces (at low playing level)
in 9x9 Go within 3,000 games of training is shown in Figure 2. This network
was grown during training by adding hidden layers one at a time; although it
was trained without the (reflection x rotation) symmetry constraint, many of the
weight kernels learned approximately symmetric features. The direct projection
from board to reinforcement layer has an interesting structure: the negative central
weight within a positive surround stems from the fact that a placed stone occupies
(thus loses) a point of territory even while securing nearby areas. Note that the wide
17x17 projections from the hidden layers have considerable fringes - ostensibly a
trick the network uses to incorporate edge effects, which are also prominent in the
bias projections from the turn unit.
We compared training this architecture by self-play versus play against Wally. The
initial rate of learning is similar, but soon the latter starts to outperform the former
(measured against both Wally and Many Faces), demonstrating the advantage of
having a skilled opponent. After about 2000 games, however, it starts to overfit to
Wally and consequently worsens against Many Faces. Switching training partner
to Many Faces at this point produced (after a further 1,000 games) a network that
could reliably beat this opponent. Although less capable, the self-play network did
manage to edge past Wally after 3,000 games; this compares very favorably with
Temporal Difference Learning of Position Evaluation in the Game of Go
the undifferentiated network described in the Introduction. Furthermore, we have
verified that weights learned from 9x9 Go offer a suitable basis for further training
on the full-size (19x19) board.
5 DISCUSSION
In general our networks appear more competent in the opening than further into
the game. This suggests that although reinforcement information is indeed propagating all the way back from the final position, it is hard for the network to capture
the multiplicity of mid-game situations and the complex combinatorics characteristic of the endgame. These strengths and weaknesses partially complement those
of symbolic systems, suggesting that hybrid approaches might be rewarding. We
plan to further improve network performance in a number of ways:
It is possible to augment the input representation of the network in such a way
that its task becomes fully translation-invariant. We intend to do this by adding an
extra input layer whose nodes are active when the corresponding points on the Go
board are empty, and inactive when they are occupied (regardless of color). Such
an explicit representation of liberties makes the three possible states of a point on
the board (black stone, white stone, or empty) linearly separable to the network,
and eliminates the need for special treatment of the board edges.
The use of limited receptive field sizes raises the problem of how to account for
long-ranging spatial interactions on the board. In Go, the distance at which groups
of stones interact is a function of their arrangement in context; an important subproblem of position evaluation is therefore to compute the connectivity of groups
of stones. We intend to model connectivity explicitly by training the network to
predict the correlation pattern of local reinforcement from a given position. This
information can then be used to control the lateral propagation of local features in
the hidden layer through a constraint satisfaction mechanism.
Finally, we can train networks on recorded games between human players, which
the Internet Go Server provides in steady quantities and machine-readable format.
We are only beginning to explore this promising supply of instantaneous (since
prerecorded), high-quality Go play for training. The main obstacle encountered so
far has been the human practice of abandoning the game once both players agree on
the outcome - typically well before a position that could be scored mechanically is
reached. We address this issue by eliminating early resignations from our training
set, and using Wally to bring the remaining games to completion.
We have shown that with sufficient attention to network architecture and training
procedures, a connectionist system trained by temporal difference learning alone
can achieve significant levels of performance in this knowledge-intensive domain.
Acknowledgements
We are grateful to Patrice Simard and Gerry Tesauro for helpful discussions, to Tim
Casey for the plethora of game records from the Internet Go Server, and to Geoff
Hinton for tniterations. Support was provided by the McDonnell-Pew Center for
Cognitive Neuroscience, SERC, NSERC and the Howard Hughes Medical Institute.
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7,049 | 821 | Backpropagation Convergence Via
Deterministic Nonmonotone Perturbed
Minimization
o.
L. Mangasarian & M. v. Solodov
Computer Sciences Department
University of Wisconsin
Madison, WI 53706
Email: [email protected], [email protected]
Abstract
The fundamental backpropagation (BP) algorithm for training artificial neural networks is cast as a deterministic nonmonotone perturbed gradient method. Under certain natural assumptions, such
as the series of learning rates diverging while the series of their
squares converging, it is established that every accumulation point
of the online BP iterates is a stationary point of the BP error function. The results presented cover serial and parallel online BP,
modified BP with a momentum term, and BP with weight decay.
1
INTRODUCTION
We regard training artificial neural networks as an unconstrained minimization
problem
N
min f(x) := ~ h(x)
xERn
~
(1)
j=l
where h : ~n --+ ~, j = 1, ... , N are continuously differentiable functions from the
n-dimensional real space ~n to the real numbers~. Each function Ii represents the
error associated with the j-th training example, and N is the number of examples
in the training set. The n-dimensional variable space here is that of the weights
associated with the arcs of the neural network and the thresholds of the hidden and
383
384
Mangasarian and Solodov
output units. For an explicit description of f(x) see (Mangasarian, 1993). We note
that our convergence results are equally applicable to any other form of the error
function, provided that it is smooth.
BP (Rumelhart,Hinton & Williams, 1986; Khanna, 1989) has long been successfully
used by the artificial intelligence community for training artificial neural networks.
Curiously, there seems to be no published deterministic convergence results for this
method. The primary reason for this is the nonmonotonic nature of the process.
Every iteration of online BP is a step in the direction of negative gradient of a partial
error function associated with a single training example, e.g. Ii (x) in (1). It is clear
that there is no guarantee that such a step will decrease the full objective function
f( x), which is the sum of the errors for all the training examples . Therefore a single
iteration of BP may, in fact, increase rather than decrease the objective function
f( x) we are trying to minimize. This difficulty makes convergence analysis of BP
a challenging problem that has currently attracted interest of many researchers
(Mangasarian & Solodov, 1994; Gaivoronski, 1994; Grippo, 1994; Luo & Tseng,
1994; White, 1989) .
By using stochastic approximation ideas (Kashyap,Blaydon & Fu, 1970; Ermoliev &
Wets, 1988), White (White, 1989) has shown that, under certain stochastic assumptions, the sequence of weights generated by BP either diverges or converges almost
surely to a point that is a stationary point of the error function. More recently,
Gaivoronski obtained stronger stochastic results (Gaivoronski, 1994). It is worth
noting that even if the data is assumed to be deterministic, the best that stochastic
analysis can do is to establish convergence of certain sequences with probability
one. This means that convergence is not guaranteed. Indeed, there may exist some
noise patterns for which the algorithm diverges, even though this event is claimed
to be unlikely.
By contrast, our approach is purely deterministic. In particular, we show that
online BP can be viewed as an ordinary perturbed nonmonotone gradient-type
algorithm for unconstrained optimization (Section 3) . We note in the passing, that
the term gradient descent which is widely used in the backpropagation and neural
networks literature is incorrect. From an optimization point of view, online BP
is not a descent method, because there is no guaranteed decrease in the objective
function at each step. We thus prefer to refer to it as a nonmonotone perturbed
gradient algorithm.
We give a convergence result for a serial (Algorithm 2.1), a parallel (Algorithm 2.2)
BP, a modified BP with a momentum term, and BP with weight decay. To the best
of our knowledge, there is no published convergence analysis, either stochastic or
deterministic, for the latter three versions of BP. The proposed parallel algorithm is
an attempt to accelerate convergence of BP which is generally known to be relatively
slow.
2
CONVERGENCE OF THE BACKPROPAGATION
ALGORITHM AND ITS MODIFICATIONS
We now turn our attention to the classical BP algorithm for training feedforward
artificial neural networks with one layer of hidden units (Rumelhart,Hinton &
Backpropagation Convergence via Deterministic Nonmonotone Perturbed Minimization
Williams, 1986; Khanna, 1989). Throughout our analysis the number of hidden
units is assumed to be fixed. The choice of network topology is a separate issue
that is not addressed in this work. For some methods for choosing the number of
hidden units see (Courrien, 1993; Arai, 1993).
We now summarize our notation.
N : Nonnegative integer denoting number of examples in the training set.
i = 1,2, ... : Index number of major iterations (epochs) of BP. Each major iteration consists of going through the entire set of error functions !1(x), ... , fN(X).
=
j
1, ... ,N : Index of minor iterations. Each minor iteration j consists of a step
in the direction of the negative gradient - \7 fmU)(zi,j) and a momentum step . Here
m(j) is an element of the permuted set {I, ... , N}, and zi,j is defined immediately
below. Note that if the training set is randomly permuted after every epoch, the
map m(?) depends on the index i. For simplicity, we skip this dependence in our
notation.
xi :
Iterate in ~n of major iteration (epoch) i = 1,2, ....
zi,; : Iterate in ~n of minor iteration j = 1, ... , N, within major iteration i
1,2, .... Iterates zi,j can be thought of as elements of a matrix with N columns and
infinite number of rows, with row i corresponding to the i-th epoch of BP.
By 1}i we shall denote the learning rate (the coefficient multiplying the gradient),
and by (ki the momentum rate (the coefficient multiplying the momentum term).
For simplicity we shall assume that the learning and momentum rates remain fixed
within each major iteration. In a manner similar to that of conjugate gradients
(Polyak, 1987) we reset the momentum term to zero periodically.
Algorithm 2.1. Serial Online BP with a Momentum Term.
Start with any xO E ~n. Having xi, stop if \7 f(x i )
0, else compute xi+l as
=
follows:
(2)
zi,j+l
= zi,j
- TJi \7 fmu)(i,j)
xi+l
+ aif1zi,j,
j
= 1, ... , N
= zi,N+l
(3)
(4)
where
if j = 1
otherwise
(5)
Remark 2.1. Note that the stopping criterion of this algorithm is typically that
used in first order optimization methods, and is not explicitly related to the ability of the neural network to generalize. However, since we are concerned with
convergence properties of BP as a numerical algorithm, this stopping criterion is
385
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Mangasarian and Solodov
justified. Another point related to the issue of generalization versus convergence is
the following. Our analysis allows the use of a weight decay term in the objective
function (Hinton, 1986; Weigend,Huberman & Rumelhart, 1990) which often yields
a network with better generalization properties. In the latter case the minimization
problem becomes
N
min I(x) := L~ hex)
xElRn
+ All x l1 2
(6)
i=l
where A is a small positive scaling factor.
= 0 reduces
Remark 2.2. The choice of C?i
without a momentum term.
Algorithm 2.1 to the original BP
Remark 2.3. We can easily handle the "mini-batch" methods (M!2l11er, 1992) by
merely redefining the meaning of the partial error function Ii to represent the error
associated with a subset of training examples. Thus "mini-batch" methods also fall
within our framework.
We next present a parallel modification of BP. Suppose we have k parallel processors, k 2: 1. We consider a partition of the set {l, ... , N} into the subsets
J" 1 1, ... ,k, so that each example is assigned to at least one processor. Let
N, be the cardinality of the corresponding set J,. In the parallel BP each processor
performs one (or more) cycles of serial BP on its set of training examples. Then a
synchronization step is performed that consists of averaging the iterates computed
by all the k processors. From the mathematical point of view this is equivalent to
each processor I E {I, ... , k} handling the partial error function I' (x) associated
with the corresponding set of training examples J , . In this setting we have
=
k
J'(x)=~Ii(x),
f(x)=~f'(x)
iEJ I
1=1
We note that in training a neural network it might be advantageous to assign
some training examples to more than one parallel processor. We thus allow for the
possibility of overlapping sets J,.
The notation for Algorithm 2.2 is similar to that for Algorithm 2.1, except for the
index 1 that is used to label the partial error function and minor iterates associated
with the l-th parallel processor. We now state the parallel BP with a momentum
term.
Algorithm 2.2. Parallel Online BP with a Momentum Term.
Start with any xO E ~n. Having xi, stop if x i+l = xi, else compute x i +l as follows
(i) Parallelization. For each parallel processor I E {I, ... , k} do
i,l
z,
z,i,i+l _- z,i,i where
'~f'
7], v m(j)
~zlili = {
0
= xi
(iIi)
z,
(7)
+ c?,uz"
. i,i
z;,i - z;,i-
A
=
l
J.
= 1, ... , N I
if j
1
otherwise
(8)
(9)
Backpropagation Convergence via Deterministic Nonmonotone Perturbed Minimization
o < TJi < 1,
O:s a i < 1
(ii) Synchronization
k
Xi+l =
~ L z;,Nr+l
(10)
1=1
We give below in Table 1 a flowchart of this algorithm.
/
i 1
Z 1'
..
-
x'.
Major iteration i :
..... ~
xi
.~
i1
.
z'I '.- x'
i1
.
z'k .'- x'
~
Serial BP on
examples in Jl
~
Serial BP on
examples in J,
Serial BP on
examples in Jk
~
~
i,Nr+l
z,
J
? ?IteratIOn
. z. + 1 : x ,'+1
M aJor
i,N,,+I
zk
/
i Nr+ 1
= k1 "k
L.....I=1 z,'
Table 1. Flowchart of the Parallel BP
Remark 2.4. It is well known that ordinary backpropagation is a relatively slow
algorithm. One appealing remedy is parallelization (Zhang,Mckenna,Mesirov &
Waltz, 1990). The proposed Algorithm 2.2 is a possible step in that direction.
Note that in Algorithm 2.2 all processors typically use the same program for their
computations. Thus load balancing is easily achieved.
Remark 2.5. We wish to point out that synchronization strategies other than
(10) are possible. For example, one may choose among the k sets of weights and
thresholds the one that best classifies the training data.
To the best of our knowledge there are no published deterministic convergence
387
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Mangasarian and Solodov
proofs for either of Algorithms 2.1,2.2. Using new convergence analysis for a class of
nonmonotone optimization methods with perturbations (Mangasarian & Solodov,
1994), we are able to derive deterministic convergence properties for online BP
and its modifications. Once again we emphasize the equivalence of either of those
methods to a deterministic nonmonotone perturbed gradient-type algorithm.
We now state our main convergence theorem. An important result used in the proof
is given in the Mathematical Appendix. We refer interested readers to (Mangasarian
& Solodov, 1994) for more details.
Theorem 2.1. If the learning and momentum rates are chosen such that
00
L =
l7i
i=O
00
00,
L 171 <
i=O
00
00,
L
O:'il7i
< 00,
(11)
i=O
then for any sequence {xi} generated by any of the Algorithms 2.1 or 2.2, it follows
that {/(xiH converges, {\7 !(xi)} - 0, and for each accumulation point x of the
sequence {x'}, \7 I( x) = O.
Remark 2.6. We note that conditions (11) imply that both the learning and
momentum rates asymptotically tend to zero. These conditions are similar to those
used in (White, 1989; Luo & Tseng, 1994) and seem to be the inevitable price paid
for rigorous convergence. For practical purposes the learning rate can be fixed or
adjusted to some small but finite number to obtain an approximate solution to the
minimization problem. For state-of-the-art techniques of computing the learning
rate see (Ie Cun, Simard & Pearlmutter, 1993).
Remark 2.7. We wish to point out that Theorem 2.1 covers BP with momentum
and/or decay terms for which there is no published convergence analysis of any
kind.
Remark 2.8. We note that the approach of perturbed minimization provides
theoretical justification to the well known properties of robustness and recovery
from damage for neural networks (Sejnowski & Rosenberg, 1987). In particular, this
approach shows that the net should recover from any reasonably small perturbation.
Remark 2.9. Establishing convergence to a stationary point seems to be the
best one can do for a first-order minimization method without any additional restrictive assumptions on the objective function. There have been some attempts
to achieve global descent in training, see for example, (Cetin,Burdick & Barhen,
1993). However, convergence to global minima was not proven rigorously in the
multidimensional case.
3
MATHEMATICAL APPENDIX: CONVERGENCE OF
ALGORITHMS WITH PERTURBATIONS
In this section we state a new result that enables us to establish convergence properties of BP. The full proof is nontrivial and is given in (Mangasarian & Solodov,
1994).
Backpropagation Convergence via Deterministic Nonmonotone Perturbed Minimization
Theorem 3.1. General Nonmonotonic Perturbed Gradient Convergence
(subsumes BP convergence).
Suppose that f(x) is bou?,!-ded below and that \1 f(x) is bounded and Lipschitz continuous on the sequence {x'} defined below. Consider the following perturbed gradient
algorithm. Start with any x O E ~n. Having xi, stop if \1 f(x i ) 0, else compute
=
(12)
where
di = -\1f(x i )
for some ei E ~n, TJi E~, TJi
00
L TJi =
;=0
L TJl <
i=O
(13)
> 0 and such that for some I > 0
00
00,
+ ei
00
00,
L TJdleili <
00,
Ileill ~ I
Vi
(14)
i=O
It follows that {f(x i)} converges, {\1 f(x i )} -+ 0, and for each accumulation point
x of the sequence {x'}, V' f(x) = O. If, in addition, the number of stationary points
of f(x) is finite, then the sequence {xi} converges to a stationary point of f(x).
Remark 3.1. The error function of BP is nonnegative, and thus the boundedness
condition on f(x) is satisfied automatically. There are a number of ways to ensure
that f(x) has Lipschitz continuous and bounded gradient on {xi} . In (Luo & Tseng,
1994) a simple projection onto a box is introduced which ensures that the iterates
remain in the box. In (Grippo, 1994) a regularization term as in (6) is added to the
error function so that the modified objective function has bounded level sets. We
note that the latter provides a mathematical justification for weight decay (Hinton,
1986; Weigend,Huberman & Rumelhart, 1990). In either case the iterates remain
in some compact set, and since f( x) is an infinitely smooth function, its gradient is
bounded and Lipschitz continuous on this set as desired.
Acknowledgements
This material is based on research supported by Air Force Office of Scientific
Research Grant F49620-94-1-0036 and National Science Foundation Grant CCR9101801.
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7,050 | 822 | Bounds on the complexity of recurrent
neural network implementations of finite
state machines
Bill G. Horne
NEC Research Institute
4 Independence Way
Princeton, NJ 08540
Don R. Hush
EECE Department
University of New Mexico
Albuquerque, NM 87131
Abstract
In this paper the efficiency of recurrent neural network implementations of m-state finite state machines will be explored. Specifically,
it will be shown that the node complexity for the unrestricted case
can be bounded above by 0 ( fo) . It will also be shown that the
node complexity is 0 (y'm log m) when the weights and thresholds
are restricted to the set {-I, I}, and 0 (m) when the fan-in is restricted to two. Matching lower bounds will be provided for each
of these upper bounds assuming that the state of the FSM can be
encoded in a subset of the nodes of size rlog m1.
1
Introduction
The topic of this paper is understanding how efficiently neural networks scale to
large problems. Although there are many ways to measure efficiency, we shall be
concerned with node complexity, which as its name implies, is a calculation of the
required number of nodes. Node complexity is a useful measure of efficiency since
the amount of resources required to implement or even simulate a recurrent neural
network is typically related to the number of nodes. Node complexity can also
be related to the efficiency of learning algorithms for these networks and perhaps
to their generalization ability as well. We shall focus on the node complexity of
recurrent neural network implementations of finite state machines (FSMs) when
the nodes of the network are restricted to threshold logic units.
359
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Home and Hush
In the 1960s it was shown that recurrent neural networks are capable of implementing arbitrary FSMs. The first result in this area was due to Minsky [7], who
showed that m-state FSMs can be implemented in a fully connected recurrent neural network. Although circuit complexity was not the focus of his investigation it
turns out that his construction, yields 0 (m) nodes. This construction was also
guaranteed to use weight values limited to the set {I, 2}. Since a recurrent neural
network with k hard-limiting nodes is capable of representing as many as 2k states,
one might wonder if an m-state FSM could be implemented by a network with
log m nodes. However, it was shown in [1] that the node complexity for a standard
((m
fully connected network is n
log m)1/3). They were also able to improve upon
Minsky's result by providing a construction which is guaranteed to yield no more
than 0 (m 3/ 4 ) nodes. In the same paper lower bounds on node complexity were investigated as the network was subject to restrictions on the possible range of weight
values and the fan-in and fan-out of the nodes in the network. Their investigation
was limited to fully connected recurrent neural networks and they discovered that
the node complexity for the case where the weights are restricted to a finite size set
is n (y'm log m) . Alternatively, if the nodes in the network were restricted to have a
constant fan-in then the node complexity becomes n (m) . However, they left open
the question of how tight these bounds are and if they apply to variations on the
basic architecture. Other recent work includes investigation of the node complexity
for networks with continuous valued nonlinearities [14]. However, it can also be
shown that when continuous nonlinearities are used, recurrent neural networks are
far more powerful than FSMs; in fact, they are Turing equivalent [13].
In this paper we improve the upper bound on the node complexity for the unrestricted case to 0 (yIm). We also provide upper bounds that match the lower
bounds above for various restrictions. Specifically, we show that a node complexity
of 0 ( y'm log m) can be achieved if the weights are restricted to the set {-I, I} , and
that the node complexity is 0 (m) for the case when the fan-in of each node in the
network is restricted to two. Finally, we explore the possibility that implementing
finite state machines in more complex models might yield a lower node complexity.
Specifically, we explore the node complexity of a general recurrent neural network
topology, that is capable of simulating a variety of popular recurrent neural network architectures. Except for the unrestricted case, we will show that the node
complexity is no different for this architecture than for the fully connected case if
the number of feedback variables is limited to rlog m1, i.e. if the state of the FSM
is encoded optimally in a subset of the nodes. We leave it as an open question if a
sparser encoding can lead to a more efficient implementation.
2
2.1
Background
Finite State Machines
FSMs may be defined in several ways. In this paper we shall be concerned with
Mealy machines, although our approach can easily be extended to other formulations
to yield equivalent results.
Bounds on the Complexity of Recurrent Neural Network Implementations
Definition 1 A Mealy machine is a quintuple M = (Q, qo, E, d, <1?, where Q is a
finite set of states; qo is the initial state ; E is the input alphabet; d is the output
alphabet; and <I> : Q x E Q x d is the combined transition and output function.
o
=
Throughout this paper both the input and output alphabets will be binary (i.e. E
d
{a, I}) . In general, the number of states, m
IQI, may be arbitrary. Since any
element of Q can be encoded as a binary vector whose minimum length is pog m 1,
the function <I> can be implemented as a boolean logic function of the form
=
=
<I> :
{a, l} pogm1+l _
{a, l} pogm1+l .
(1)
The number, N M , of different minimal FSMs with m states will be used to determine
lower bounds on the number of gates required to implement an arbitrary FSM in a
recurrent neural network. It can easily be shown that (2m)m :S NM [5]. However,
it will be convenient to reexpress N M in terms of n = flog m 1+ 1 as follows
(2)
2.2
Recurrent Neural Networks
The fundamental processing unit in the models we wish to consider is the perceptron,
which is a biased, linearly weighted sum of its inputs followed by a hard-limiting
nonlinearity whose output is zero if its input is negative and one otherwise. The
fan-in of the perceptron is defined to be the number of non-zero weights. When the
values of Xi are binary (as they are in this paper) , the perceptron is often referred
to as a threshold logic unit (TL U).
A count of the number of different partially specified threshold logic functions, which
are threshold logic functions whose values are only defined over v vertices of the
unit hypercube, will be needed to develop lower bounds on the node complexity
required to implement an arbitrary logic function . It has been shown that this
number, denoted L~, is [15]
L~:S
2v n
-,-.
n.
(3)
As pointed out in [10], many of the most popular discrete-time recurrent neural
network models can be implemented as a feedforward network whose outputs are
fed back recurrently through a set of unit time delays. In the most generic version of
this architecture, the feed forward section is lower triangular, meaning the [th node is
the only node in layer I and receives input from all nodes in previous layers (including
the input layer). A lower triangular network of k threshold logic elements is the
most general topology possible for a feedforward network since all other feedforward
networks can be viewed as a special case of this network with the appropriate weights
set equal to zero. The most direct implementation of this model is the architecture
proposed in [11] . However, many recurrent neural network architectures can be cast
into this framework. For example, fully connected networks [3] fit this model when
the the feedforward network is simply a single layer of nodes. Even models which
appear very different [2, 9] can be cast into this framework.
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Home and Hush
3
The unrestricted case
The unrestricted case is the most general, and thus explores the inherent power of
recurrent neural networks. The unrestricted case is also important because it serves
as a baseline from which one can evaluate the effect of various restrictions on the
node complexity.
In order to derive an upper bound on the node complexity of recurrent neural
network implementations of FSMs we shall utilize the following lemma, due to
Lupanov [6]. The proof of this lemma involves a construction that is extremely
complex and beyond the scope of this paper.
Lemma 1 (Lupanov, 1973) Arbitrary boolean logic functions with x inputs and
y outputs can be implemented in a network of perceptrons with a node complexity
of
o(
J ~~:g y) .
x
o
Theorem 1 Multilayer recurrent neural networks can implement FSMs having m
states with a node complexity of 0 (.Jffi) .
0
Proof: Since an m-state FSM can be implemented in a recurrent neural network
in which the multilayer network performs a mapping of the form in equation (1),
then using n = m = flog m1+ 1, and applying Lemma 1 gives an upper bound of
O(.Jffi).
Q.E.D.
Theorem 2 Multilayer recurrent neural networks can implement FSMs having m
states with a node complexity of n (fo) if the number of unit time delays is flog m1.
o
Proof: In order to prove the theorem we derive an expression for the maximum
number of functions that a k-node recurrent neural network can compute and compare that against the minimum number of finite state machines. Then we solve for
k in terms of the number of states of the FSM.
Specifically, we wish to manipulate the inequality
2(n-l)2 n -
2
< n!
-
( k- 1 )
n- 1
(a)
krr-l 2n(n+i~+1
.
(n + z)!
,=0
(b)
where the left hand side is given in equation (2), (a) represents the total number of
ways to choose the outputs and feedback variables of the network, and (b) represents the total number of logic functions computable by the feed forward section of
the network, which is lower triangular. Part (a) is found by simple combinatorial
arguments and noting that the last node in the network must be used as either
an output or feedback node. Part (b) is obtained by the following argument: If
the state is optimally encoded in flog m1 nodes, then only flog m1 variables need
Bounds on the Complexity of Recurrent Neural Network Implementations
to be fed back. Together with the external input this gives n = rlog m1 + 1 local
inputs to the feedforward network. Repeated application of (3) with v 2n yields
expression (b).
=
Following a series of algebraic manipulations it can easily be shown that there exists
a constant c such that
n2n < ck 2n.
Since n = flog ml
4
+ 1 it follows that k = f2 (fo).
Q.E.D.
Restriction on weights and thresholds
All threshold logic functions can be implemented with perceptrons whose weight
and threshold values are integers. It is well known that there are threshold logic
functions of n variables that require a perceptron with weights whose maximum
magnitude is f2(2n) and O( nn/2) [8]. This implies that if a perceptron is to be
implemented digitally, the number of bits required to represent each weight and
threshold in the worst case will be a super linear function of the fan-in. This is
generally undesirable ; it would be far better to require only a logarithmic number
of bits per weight, or even better, a constant number of bits per weight. We will be
primarily be interested in the most extreme case where the weights are limited to
values from the set {-I , I}.
In order to derive the node complexity for networks with weight restrictions, we
shall utilize the following lemma, proved in [4].
Lemma 2 Arbitrary boolean logic functions with x inputs and y outputs can be
implemented in a network ofperceptrons whose weights and thresholds are restricted
to the set {-I, I} with a node complexity of e (Jy2 x ) .
0
This lemma is not difficult to prove , however it is beyond the scope of this paper.
The basic idea involves using a decomposition of logic functions proposed in [12].
Specifically, a boolean function f may always be decomposed into a disjunction of
2 r terms of the form XIX2. ' . Xr fi(X r +1 , .. . , x n ) , one for each conjunction of the
first r variables, where Xj represents either a complemented or uncomplemented
version of the input variable Xj and each Ii is a logic function of the last n - r
variables. This expression can be implemented directly in a neural network. With
negligible number of additional nodes, the construction can be implemented in such
a way that all weights are either -lor 1. Finally, the variable r is optimized to
yield the minimum number of nodes in the network.
Theorem 3 Multilayer recurrent neural networks that have nodes whose weights
and thresholds are restricted to the set {-I , I} can implement FSMs having m
states with a node complexity of 0 (Jm log m) .
0
Proof: Since an m-state FSM can be implemented in a recurrent neural network
in which the multilayer network performs a mapping of the form in equation (1),
then using n = m = flog m1+ 1, and applying Lemma 2 gives an upper bound of
o (Jmlogm) .
Q.E.D.
363
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Home and Hush
Theorem 4 Multilayer recurrent neural networks that have nodes whose weights
and thresholds are restricted to a set of size IWI can implement FSMs having m
states with a node complexity of n (
if the number of unit time delays is
flogml.
0
Proof: The proof is similar to the proof of Theorem 2 which gave a lower bound
for the node complexity required in an arbitrary network of threshold logic units.
Here, the inequality we wish to manipulate is given by
k-l
k- 1
n-I
)
II IWln+i+
1.
i=O
(b)
(a)
where the left hand side and (a) are computed as before and (b) represents the
maximum number of ways to configure the nodes in the network when there are
only IWI choices for each weight and threshold. Following a series of algebraic
manipulations it can be shown that there exists a constant c such that
n2n ::; ck 2 log IWI.
Since n
= pog m1+ 1 it follows that k = n (
Clearly, for W
5
mlogm)
loglWI
.
Q.E.D.
= {-I, I} this lower bound matches the upper bound in Theorem 3.
Restriction on fan-in
A limit on the fan-in of a perceptron is another important practical restriction.
In the networks discussed so far each node has an unlimited fan-in. In fact, in
the constructions described above, many nodes receive inputs from a polynomial
number of nodes (in terms of m) in a previous layer. In practice it is not possible
to build devices that have such a large connectivity. Restricting the fan-in to 2, is
the most severe restriction, and will be of primary interest in this paper.
Once again, in order to derive the node complexity for restricted fan-in, we shall
utilize the following lemma, proved in [4].
Lemma 3 Arbitrary boolean logic functions with x inputs and y outputs can be
implemented in a network of perceptrons restricted to fan-in 2 with a node complexityof
y2X )
e ( x + logy .
o
This proof of this lemma is very similar to the proof of Lemma 2. Here Shannon's
decomposition is used with r = 2 to recursively decompose the logic function into
a set of trees, until each tree has depth d. Then, all possible functions of the last
n - d variables are implemented in an inverted tree-like structure, which feeds into
the bottom of the trees. Finally, d is optimized to yield the minimum number of
nodes.
Bounds on the Complexity of Recurrent Neural Network Implementations
Theorem 5 Multilayer recurrent neural networks that have nodes whose fan-in is
restricted to two can implement FSMs having m states with a node complexity of
Oem)
0
Proof: Since an m-state FSM can be implemented in a recurrent neural network
in which the multilayer network performs a mapping of the form in equation (1),
then using n = m = rlog m1+ 1, and applying Lemma 3 gives an upper bound of
o (m).
Q.E.D.
Theorem 6 Multilayer recurrent neural networks that have nodes whose fan-in is
restricted to two can implement FSMs having m states with a node complexity of
n (m) if the number of unit time delays is rlog m1.
0
Proof: Once again the proof is similar to Theorem 2, which gave a lower bound
for the node complexity required in an arbitrary network of threshold logic units.
Here, the inequality we need to solve for is given by
2(n-1)2'-'
:s n! ( ~:= ~
) D. 14 (
n
t
i )
,----_V~----A~----_V~----~
(a)
(b)
where the left hand side and (a) are computed as before and (b) represents the maximum number of ways to configure the nodes in the network. The term ( n
t
i )
is used since a node in the ith layer has n + i possible inputs from which two are
chosen. The constant 14 represents the fourteen possible threshold logic functions
of two variables. Following a series of algebraic manipulations it can be shown that
there exists a constant c such that
~
ck logk
Since n = rlog m1 + 1 it follows that k =
n (m) .
n2n
6
Q.E.D.
Summary
In summary, we provide new bounds on the node complexity of implementing FSMs
with recurrent neural networks. These upper bounds match lower bounds developed in [1] for fully connected recurrent networks when the size of the weight set
or the fan-in of each node is finite. Although one might speculate that more complex networks might yield more efficient constructions, we showed that these lower
bounds do not change for restrictions on weights or fan-in, at least when the state
of the FSM is encoded optimally in a subset of flog m1 nodes. When the network
is unrestricted, this lower bound matches our upper bound. We leave it as an open
question if a sparser encoding of the state variables can lead to a more efficient
implementation.
One interesting aspect of this study is that there is really not much difference
in efficiency when the network is totally unrestricted and when there are severe
restrictions placed on the weights. Assuming that our bounds are tight, then there
365
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Home and Hush
is only a y'log m penalty for restricting the weights to either -1 or 1. To get some
idea for how marginal this difference is consider that for a finite state machine with
18 x 10 18 states, y'log m is only eight!
m
=
A more detailed version of this paper can be found in [5].
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| 822 |@word version:3 polynomial:1 open:3 simulation:1 decomposition:2 recursively:1 initial:1 series:4 must:1 device:2 sys:2 ith:1 compo:1 node:70 lor:1 direct:1 prove:2 terminal:1 decomposed:1 jm:1 totally:1 becomes:1 provided:1 horne:3 bounded:1 circuit:3 flog:8 developed:1 nj:1 control:1 unit:10 appear:1 before:2 iqi:1 negligible:1 local:1 limit:1 acad:1 switching:1 encoding:2 might:4 limited:4 range:1 practical:1 tsoi:1 practice:1 implement:9 xr:1 area:1 universal:1 bell:1 matching:1 convenient:1 sycon:1 get:1 jffi:2 undesirable:1 prentice:1 applying:3 restriction:10 bill:1 equivalent:2 automaton:1 his:2 variation:1 limiting:2 construction:7 element:3 recognition:1 bottom:1 worst:1 cong:1 connected:6 digitally:1 complexity:41 dynamic:1 tight:2 upon:1 efficiency:5 f2:2 easily:3 hopfield:1 emergent:1 various:2 alphabet:3 disjunction:1 whose:11 encoded:5 valued:1 solve:2 otherwise:1 triangular:3 ability:2 net:1 rlog:6 y2x:1 leave:2 derive:4 recurrent:33 develop:1 alon:1 implemented:15 involves:2 implies:2 implementing:3 require:2 generalization:1 really:1 investigation:3 decompose:1 hall:1 scope:2 mapping:3 narendra:1 proc:2 combinatorial:1 krr:1 weighted:1 clearly:1 always:1 super:1 ck:3 conjunction:1 focus:2 tech:1 baseline:1 problemy:1 nn:1 typically:1 interested:1 denoted:1 special:1 marginal:1 equal:1 once:2 having:6 represents:6 muroga:1 report:3 inherent:1 primarily:1 minsky:3 interest:1 possibility:1 severe:2 extreme:1 configure:2 fsm:9 capable:3 tree:4 minimal:1 modeling:1 boolean:5 giles:1 ott:1 vertex:1 subset:3 wonder:1 delay:4 optimally:3 combined:1 fundamental:1 explores:1 ifip:1 together:1 synthesis:2 connectivity:1 again:2 ctr:1 nm:2 choose:1 fir:1 logy:1 external:1 winder:1 nonlinearities:2 speculate:1 coding:1 includes:1 iwi:3 who:1 efficiently:1 yield:8 identification:1 albuquerque:1 comp:1 synapsis:1 fo:3 definition:1 against:1 eece:5 proof:12 static:1 proved:2 popular:2 back:3 feed:3 formulation:1 anderson:1 until:1 hand:3 receives:1 qo:2 nonlinear:1 propagation:1 perhaps:1 name:1 effect:1 concept:1 elect:1 performs:3 meaning:1 fi:1 physical:1 fourteen:1 discussed:1 m1:13 pointed:1 nonlinearity:1 language:1 mealy:2 showed:2 recent:1 manipulation:3 inequality:3 binary:3 inverted:1 minimum:4 unrestricted:8 additional:1 determine:1 ii:2 technical:3 match:4 calculation:1 dewdney:1 manipulate:2 basic:2 fsms:14 multilayer:9 rutgers:1 represent:1 achieved:1 receive:1 background:1 biased:1 subject:1 integer:1 reexpress:1 noting:1 feedforward:5 concerned:2 variety:1 independence:1 fit:1 xj:2 gave:2 architecture:7 topology:2 idea:2 computable:1 expression:3 penalty:1 algebraic:3 sontag:2 speech:1 useful:1 generally:1 detailed:1 amount:1 per:2 discrete:1 shall:6 threshold:21 utilize:3 sum:1 turing:1 powerful:1 throughout:1 home:4 bit:3 layer:6 bound:30 guaranteed:2 followed:1 fan:17 unlimited:1 aspect:1 simulate:1 argument:2 extremely:1 quintuple:1 department:1 restricted:14 resource:1 equation:4 turn:1 count:1 needed:1 fed:2 serf:1 i1r:1 apply:1 eight:1 generic:1 appropriate:1 simulating:1 yim:1 gate:2 oem:1 unifying:1 siegelmann:2 build:1 hypercube:1 question:3 primary:1 fallside:1 n2n:3 sci:1 topic:1 evaluate:1 assuming:2 length:1 cont:1 providing:1 mexico:3 difficult:1 negative:1 implementation:11 collective:1 upper:10 finite:13 extended:1 discovered:1 arbitrary:9 cast:2 required:7 specified:1 optimized:2 hush:7 trans:2 robinson:1 able:1 beyond:2 dynamical:1 including:1 power:1 representing:1 improve:2 realizability:1 parthasarathy:1 understanding:1 fully:6 interesting:1 filtering:1 editor:1 summary:2 placed:1 last:3 side:3 perceptron:6 institute:1 feedback:3 depth:1 transition:1 world:1 forward:2 adaptive:1 far:3 ec:1 logic:20 ml:1 xi:1 alternatively:1 don:1 continuous:2 investigated:1 complex:3 linearly:1 repeated:1 referred:1 tl:1 wiley:1 wish:3 theorem:10 recurrently:1 explored:1 exists:3 restricting:2 logk:1 nec:1 magnitude:1 nat:1 sparser:2 logarithmic:1 simply:1 explore:2 jacm:1 partially:1 complemented:1 xix2:1 viewed:1 pog:2 hard:2 change:1 specifically:5 except:1 infinite:1 lemma:13 total:2 shannon:2 perceptrons:3 dept:2 princeton:1 |
7,051 | 823 | A Comparative Study Of A Modified
Bumptree Neural Network With Radial Basis
Function Networks and the Standard MultiLayer Perceptron.
Richard T .J. Bostock and Alan J. Harget
Department of Computer Science & Applied Mathematics
Aston University
Binningham
England
Abstract
Bumptrees are geometric data structures introduced by Omohundro
(1991) to provide efficient access to a collection of functions on a
Euclidean space of interest. We describe a modified bumptree structure
that has been employed as a neural network classifier, and compare its
performance on several classification tasks against that of radial basis
function networks and the standard mutIi-Iayer perceptron.
1
INTRODUCTION
A number of neural network studies have demonstrated the utility of the multi-layer
perceptron (MLP) and shown it to be a highly effective paradigm. Studies have also
shown, however, that the MLP is not without its problems, in particular it requires an
extensive training time, is susceptible to local minima problems and its perfonnance is
dependent upon its internal network architecture. In an attempt to improve upon the
generalisation performance and computational efficiency a number of studies have been
undertaken principally concerned with investigating the parametrisation of the MLP. It is
well known, for example, that the generalisation performance of the MLP is affected by
the number of hidden units in the network, which have to be determined empirically since
theory provides no guidance. A number of investigations have been conducted into the
possibility of automatically determining the number of hidden units during the training
phase (BostOCk, 1992). The results show that architectures can be attained which give
satisfactory, although generally sub-optimal, perfonnance.
Alternative network architectures such as the Radial Basis Function (RBF) network have
also been studied in an attempt to improve upon the performance of the MLP network.
The RBF network uses basis functions in which the weights are effective over only a
small portion of the input space. This is in contrast to the MLP network where the
weights are used in a more global fashion, thereby encoding the characteristics of the
training set in a more compact form. RBF networks can be rapidly trained thus making
240
Modified Bumptree Neural Network and Standard Multi-Layer Perceptron
them particularly suitable for situations where on-line incremental learning is required.
The RBF network has been successfully applied in a number of areas such as speech
recognition (Renals, 1992) and financial forecasting (Lowe, 1991). Studies indicate that
the RBF network provides a viable alternative to the MLP approach and thus offers
encouragement that networks employing local solutions are worthy of further
investigation.
In the past few years there has been an increasing interest in neural network architectures
based on tree structures. Important work in this area has been carried out by Omohundro
(1991) and Gentric and Withagen (1993). These studies seem to suggest that neural
networks employing a tree based structure should offer the same benefits of reduced
training time as that offered by the RBF network. The particular tree based architecture
examined in this study is the bumptree which provides efficient access to collections of
functions on a Euclidean space of interest. A bumptree can be viewed as a natural
generalisation of several other geometric data structures including oct-trees, k-d trees,
balltrees (Omohundro, 1987) and boxtrees (Omohundro, 1989).
In this paper we present the results of a comparative study of the performance of the three
types of neural networks described above over a wide range of classification problems.
The performance of the networks was assessed in terms of the percentage of correct
classifications on a test, or generalisation data set, and the time taken to train the
network. Before discussing the results obtained we shall give an outline of the
implementation of our bumptree neural network since this is more novel than the other
two networks.
2
THE BUMPTREE NEURAL NETWORK
Bumptree neural networks share many of the underlying principles of decision trees but
differ from them in the manner in which patterns are classified. Decision trees partition
the problem space into increasingly small areas. Classification is then achieved by
determining the lowest branch of the tree which contains a reference to the specified point.
The bumptree neural network described in this paper also employs a tree based structure to
partition the problem space, with each branch of the tree being based on multiple
dimensions. Once the problem space has been partitioned then each branch can be viewed
as an individual neural network modelling its own local area of the problem space, and
being able to deal with patterns from multiple output classes.
Bumptrees model the problem space by subdividing the space allowing each division to
be described by a separate function. Initial partitioning of the problem space is achieved
by randomly assigning values to the root level functions. A learning algorithm is applied
to determine the area of influence of each function and an associated error calculated. If
the error exceeds some threshold of acceptability then the area in question is further
subdivided by the addition of two functions; this process continues until satisfactory
performance is achieved. The bumptree employed in this study is essentially a binary tree
in which each leaf of the tree corresponds to a function of interest although the possibility
exists that one of the functions could effectively be redundant if it fails to attract any of
the patterns from its parent function.
A number of problems had to be resolved in the design and implementation of the
bumptree. Firstly, an appropriate procedure had to be adopted for partitioning the
241
242
Bostock and Harget
problem space. Secondly, consideration had to be given to the type of learning algorithm
to be employed. And finally, the mechanism for calculating the output of the network
had to be determined. A detailed discussion of these issues and the solutions adopted now
follows.
2.1
PARTITIONING THE PROBLEM SPACE
The bumptree used in this study employed gaussian functions to partition the problem
space, with two functions being added each time the space was partitioned. Patterns were
assigned to whichever of the functions had the higher activation level with the restriction
that the functions below the root level could only be active on patterns that activated their
parents. To calculate the activation of the gaussian function the following expression
was used:
(1)
where Afp is the activation of function f on pattern p over all the input dimensions, afi is
the radius of function f in input dimension i, Cfi is the centre of function f in input
dimension i, and Inpi is the ith dimension of the pth input vector.
It was found that the locations and radii of the functions had an important impact on the
performance of the network. In the original bumptree introduced by Omohundro every
function below the root level was required to be wholly enclosed by its parent function.
This restriction was found to degrade the performance of the bumptree particularly if a
function had a very small radius since this would produce very low levels of acti vation for
most patterns. In our studies we relaxed this constraint by assigning the radius of each
function to one, since the data presented to the bumptree was always normalised between
zero and one. This modification led to an improved performance.
A number of different techniques were examined in order to effectively position the
functions in the problem space. The first approach considered, and the simplest, involved
selecting two initial sets of centres for the root function with the centre in each dimension
being allocated a value between zero and one. The functions at the lower levels of the tree
were assigned in a similar manner with the requirement that their centres fell within the
area of the problem space for which their parent function was active. The use of nonhierarchical clustering techniques such as the Forgy method or the K-means clustering
technique developed by MacQueen provided other alternatives for positioning the
functions. The approach finally adopted for this study was the multiple-initial function
(MIF) technique.
In the MIF procedure ten sets of functions centres were initially defined by random
assignment and each pattern in the training set assigned to the function with the highest
activation level. A "goodness" measure was then determined for each function over all
patterns for which the function was active. The goodness measure was defined as the
square of the error between the calculated and observed values divided by the number of
active patterns. The function with the best value was retained and the remaining
functions that were active on one or more patterns had their centres averaged in each
dimension to provide a second function. The functions were then added to the network
structure and the patterns assigned to the function which gave the greater activation.
Modified Bumptree Neural Network and Standard Multi-Layer Perceptron
2.2
THE LEARNING ALGORITHM
A bumptree neural network comprises a number of functions each function having its
own individual weight and bias parameters and each function being responsive to different
characteristics in the training set. The bumptree employed a weighted value for every
input to output connection and a single bias value for each output unit. Several different
learning algorithms for determining the weight and bias values were considered together
with a genetic algorithm approach (Williams, 1993). A one-shot learning algorithm was
finally adopted since this gave good results and was computationally efficient. The
algorithm used a pseudo-matrix inversion technique to determine the weight and bias
parameters of each function after a single presentation of the relevant patterns in the
training set had been made. The output of any function for a given pattern p was
determined from
jmax
+ Piz
f.l.
j
GO ipz = ""
?..J a ijz * X (p)
(2)
j=l
where aoipz is the output of the zth output unit of the ith function on the pth pattern, j is
the input unit, jmax is the total number of input units, aijz is the weight that connects
the jth input unit to the zth output unit for the ith function, Xj(p) is the element of the
pth pattern concerned with the jth input dimension, and ~iz is the bias value for the zth
output unit.
The weight and bias parameters were determined by minimising the squared error given in
(3), where Ei is the error of the ith function across all output dimensions (zmax), for all
patterns upon which the function is active (pmax). The desired output for the zth output
dimension is tv pz " and aoipz is the actual output of the ith function on the zth
dimension of the pth pattern. The weight values are again represented by Ooijz and the bias
by ~iz'
(3)
After the derivatives of aijz and ~iz were determined it was a simple task to arrive at the
three matrices used to calculate the weight and bias values for the individual functions.
Problems were encountered in the matrix inversion when dealing with functions which
were only active on a few patterns and which were far removed from the root level of the
tree; this led to difficulties with singular matrices. It was found that the problem could be
overcome by using the Gauss-Jordan singular decomposition technique for the pseudoinversion of the matrices.
2.3
CALCULATION OF THE NETWORK OUTPUT
The difficulty in determining the output of the bumptree was that there were usually
functions at different levels of the tree that gave slightly different outputs for each active
pattern. Several different approaches were studied in order to resolve the difficulty
including using the normalised output of all the active functions in the tree irrespective of
their level in the structure. A technique which gave good results and was used in this
243
244
Bostock and Harget
study calculated the output for a pattern solely on the output of the lowest level active
function in the tree. The final output class of a pattern being given by the output unit
with the highest level of activation.
3
NETWORK PERFORMANCES
The perfonnance of the bumptree neural network was compared against that of the
standard MLP and RBF networks on a number of different problems. The bumptree used
the MIF placing technique in which the radius of each function was set to one. This
particular implementation of the bumptree will now be referred to as the MIF bumptree.
The MLP used the standard backpropagation algorithm (Rumelhart, 1986) with a
learning rate of 0.25 and a momentum value of 0.9. The initial weights and bias values
of the network were set to random values between -2 and +2. The number of hidden units
assigned to the network was determined empirically over several runs by varying the
number of hidden units until the best generalisation perfonnance was attained. The RBF
network used four different types of function, they were gaussian, multi-quadratic,
inverse multi-quadratic and thin plate splines. The RBF network placed the functions
using sample points within the problem space covered by the training set
3.1
INITIAL STUDIES
In the initial studies. a set of classical non-linear problems was used to compare the
perfonnance of the three types of networks. The set consisted of the XOR, Parity(6) and
Encoder(8) problems. The average results obtained over 10 runs for each of the data sets
are shown in Table 1 - the figures presented are the percentage of patterns correctly
classified in the training set together with the standard deviation.
Table 1. Percentage of Patterns Correctly Classified for the three Data Sets for each
Network type.
DATA SET
MLP
XOR
Parity(6)
Encoder(8)
100
100
100
RBF
100
92.1 ? 4.7
82.5 ? 16.8
MIF
100
98.3 ? 4.2
100
For the XOR problem the MLP network required an average of 222 iterations with an
architecture of 4 hidden units, for the parity problem an architecture of 10 hidden units and
an average of 1133 iterations. and finally for the encoder problem the network required an
average of 1900 iterations for an architecture consisting of three hidden units.
The RBF network correctly classified all the patterns of the XOR data set when four
multi-quadratic. inverse multi-quadratic or gaussian functions were used. For the parity(6)
problem the best result was achieved with a network employing between 60 and 64
inverse multi-quadratic functions. In the case of the encoder problem the best performance
was obtained using a network of 8 multi-quadratic functions.
The MIF bumptree required two functions to achieve perfect classification for the XOR
and encoder problems and an average of 40 functions in order to achieve the best
perfonnance on the parity problem. Thus in the case of the XOR and encoder problems
no further functions were required additional to the root functions.
Modif1ed Bumptree Neural Network and Standard Multi-Layer Perceptron
A comparison of the training times taken by each of the networks revealed considerable
differences. The MLP required the most extensive training time since it used the
backpropagation training algorithm which is an iterative procedure. The RBF network
required less training time than the MLP, but suffered from the fact that for all the
patterns in the training set the activity of all the functions had to be calculated in order to
arrive at the optimal weights. The bumptree proved to have the quickest training time for
the parity and encoder problems and a training time comparable to that taken by the RBF
network for the XOR problem. This superiority arose because the bumptree used a noniterative training procedure, and a function was only trained on those members of the
training set for which the function was active.
In considering the sensitivity of the different networks to the parameters chosen some
interesting results emerge. The performance of the MLP was found to be dependent on
the number of hidden units assigned to the network. When insufficient hidden units were
allocated the performance of the MLP degraded. The performance of the RBF network
was also found to be highly influenced by the values taken for various parameters, in
particular the number and type of functions employed by the network. The bumptree on
the other hand was assigned the same set of parameters for all the problems studied and
was found to be less sensitive than the other two networks to the parameter settings.
3.2
COMPARISON OF GENERALISATION PERFORMANCE
The performance of the three different networks was also measured for a set of four 'realworld' problems which allowed the generalisation performance of each network to be
determined. A summary of the results taken over 10 runs is given in Table 2.
Table 2 Performance of the Networks on the Training and Generalisation Data Sets of the
Test Problems.
DATA
NETWORK
FUNCTIONS
HIDDEN UNITS
TRAINING
TEST
Iris
? 0.6
? 0.0
? 0.4
MLP
RBF
MIF
4
75 gaussians
8
100
100
100
MLP
RBF
MIF
6
10 multi-quad
4
88.7
84.4
79.8
? 4.3
? 3.2
? 5.2
79.2 ? 1.7
80.3 ? 4.4
80.8 ? 1.9
MLP
RBF
MIF
20
50 Thin plate spl.
104
82.4
82.1
86.5
? 5.3
? 1.5
? 5.6
77.1 ? 6.6
77.8 ? 1.4
73.6 ? 4.6
MLP
RBF
MIF
16
25 Thin plate spl.
3
82.5 ? 2.7
76.0 ? 0.8
76.5 ? 1.2
95.7
96.0
97.5
Skin
Cancer
Vowel
Data
Diabetes
78.9
78.9
80.0
? 1.2
? 0.9
? 1.1
All three networks produce a comparable performance on the test problems, but in the
case of the bumptree this was achieved with a training time substantially less than that
required by the other networks. Inspection of the results also shows that the bumptree
required fewer functions in general than the RBF network.
245
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Bostock and Harget
The results shown above for the bumptree were obtained with the same set of parameters
used in the initial study which further confirms its lack of sensitivity to parameter
settings.
4.
CONCLUSION
A comparative study of the performance of three different types of networks, one of which
is novel, has been conducted on a wide range of problems. The results show that the
performance of the bumptree compared very favourably, both in terms of generalisation
and training times, with the more traditional MLP and RBF networks. In addition, the
performance of the bumptree proved to be less sensitive to the parameters settings than
the other networks. These results encourage us to continue further investigation of the
bumptree neural network and lead us to conclude that it has a valid place in the list of
current neural networks.
Acknowledgement
We gratefully acknowledge the assistance given by Richard Rohwer.
References
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constructive:1 |
7,052 | 824 | COMBINED NEURAL NETWORKS
FOR TIME SERIES ANALYSIS
Iris Ginzburg and David Horn
School of Physics and Astronomy
Raymond and Beverly Sackler Faculty of Exact Science
Tel-Aviv University
Tel-A viv 96678, Israel
Abstract
We propose a method for improving the performance of any network designed to predict the next value of a time series. Vve advocate analyzing the deviations of the network's predictions from the
data in the training set . This can be carried out by a secondary network trained on the time series of these residuals. The combined
system of the two networks is viewed as the new predictor. We
demonstrate the simplicity and success of this method, by applying it to the sunspots data. The small corrections of the secondary
network can be regarded as resulting from a Taylor expansion of
a complex network which includes the combined system. \\Te find
that the complex network is more difficult to train and performs
worse than the two-step procedure of the combined system.
1
INTRODUCTION
The use of neural networks for computational tasks is based on the idea that the
efficient way in which the nervous system handles memory and cognition is worth
immitating. Artificial implementations are often based on a single network of mathematical neurons. We note, however, that in biological systems one can find collections of consecutive networks, performing a complicated task in several stages, with
later stages refining the performance of earlier ones. Here we propose to follow this
strategy in artificial applications.
224
Combined Neural Networks for Time Series Analysis
We study the analysis of time series, where the problem is to predict the next element on the basis of previous elements of the series. One looks then for a functional
relation
Yn = f (Yn -1 , Yn - 2, ... , Yn - m) .
(1 )
This type of representation is particularly useful for the study of dynamical systems. These are characterized by a common continuous variable, time, and many
correlated degrees of freedom which combine into a set of differential equations.
Nonetheless, each variable can in principle be described by a lag-space representation of the type 1 . This is valid even if the Y = y(t) solution is unpredictable as in
chaotic phenomena.
Weigend Huberman and Rumelhart (1990) have studied the experimental series
of yearly averages of sunspots activity using this approach. They have realized
the lag-space representation on an (m, d, 1) network, where the notation implies a
hidden layer of d sigmoidal neurons and one linear output. Using m
12 and a
weight-elimination method which led to d = 3, they obtained results which compare
favorably with the leading statistical model (Tong and Lim, 1980). Both models do
well in predicting the next element of the sunspots series. Recently, Nowlan and
Hinton (1992) have shown that a significantly better network can be obtained if the
training procedure includes a complexity penalty term in which the distribution of
weights is modelled as a mixture of multiple gaussians whose parameters vary in an
adaptive manner as the system is being trained.
=
We propose an alternative method which is capable of improving the performance
of neural networks: train another network to predict the errors of the first one, to
uncover and remove systematic correlations that may be found in the solution given
by the trained network, thus correcting the original predictions. This is in agreement
with the general philosophy mentioned at the beginning, where we take from Nature
the idea that the task does not have to be performed by one complicated network; it
is advantageous to break it into stages of consecutive analysis steps. Starting with
a network which is trained on the sunspots data with back-propagation, we show
that the processed results improve considerably and we find solutions which match
the performance of Weigend et. al.
2
CONSTRUCTION OF THE PRIMARY NETWORK
Let us start with a simple application of back-propagation to the construction of
a neural network describing the sunspots data which are normalized to lie between
o and 1. The network is assumed to have one hidden layer of sigmoidal neurons,
hi i
1" . " d, which receives the input of the nth vector:
=
m
hi
= 0'(2: WijYn-j -
Oi)
(2)
j=l
The output of the network, Pn, is constructed linearly,
d
Pn =
2: Wi hi i=l
O.
(3)
225
226
Ginzburg and Hom
The error-function which we minimize is defined by
1
E
N
=2 L
(4)
(Pn - Yn)2
n=m+l
where we try to equate Pn, the prediction or output of the network, with Yn, the
nth value of the series. This is the appropriate formulation for a training set of N
data points which are viewed as N - m strings of length m used to predict the point
following each string.
We will work with two sets of data points. One will be labelled T and be used for
training the network, and the other P will be used for testing its predictive power.
Let us define the average error by
1
{s
= jjSfj 2:(Pn -
(5)
Yn)2
nES
where the set S is either Tor P. An alternative parameter was used by Weigend et.
al. ,in which the error is normalized by the standard deviation of the data. This
leads to an average relative variance (arv) which is related to the average error
through
(6)
=
Following Weigend et. al. we choose m
12 neurons in the first layer and
IITII 220 data points for the training set. The following IIPII 35 years are
used for testing the predictions of our network. We use three sigmoidal units in the
hidden layer and run with a slow convergence rate for 7000 periods. This is roughly
where cross-validation would indicate that a minimum is reached. The starting
parameters of our networks are chosen randomly. Five examples of such networks
are presented in Table 1.
=
3
=
THE SECONDARY NETWORK
Given the networks constructed above, we investigate their deviations from the
desired values
qn = Yn - Pn?
(7)
A standard statistical test for the quality of any predictor is the analysis of the
correlations between consecutive errors. If such correlations are found, the predictor
must be improved. The correlations reflect a systematic deviation of the primary
network from the true solution. We propose not to improve the primary network
by modifying its architecture but to add to it a secondary network which uses the
residuals qn as its new data. The latter is being trained only after the training
session of the primary network has been completed.
Clearly one may expect some general relation of the type
(8)
to exist. Looking for a structure of this kind enlarges considerably the original
space in which we searched for a solution to 1 . We wish the secondary network
Combined Neural Networks for Time Series Analysis
to do a modest task, therefore we assume that much can be gained by looking at
the interdependence of the residuals qn on themselves. This reduces the problem to
finding the best values of
Tn
= b(qn-l, qn-2,"', qn-I)
(9)
which would minimize the new error function
1
E2='2
N
L
(Tn-qn)2.
(10)
n=I+1
Alternatively, one may try to express the residual in terms of the functional values
Tn
=!2(Yn-1, Yn-2,"', Yn-I)
(11)
minimizing again the expression 10 .
When the secondary network completes its training, we propose to view
tn = Pn
+ Tn
(12)
as the new prediction of the combined system. We will demonstrate that a major
improvement can be obtained already with a linear perceptron. This means that
the linear regression
1
Tn =
L aIqn-i + /3
1
(13)
2
(14)
i=l
or
1
Tn =
L a;Yn-i + /3
i=l
is sufficient to account for a large fraction of the systematic deviations of the primary
networks from the true function that they were trained to represent.
4
NUMERICAL RESULTS
We present in Table 1 five examples of results of (12,5,1) networks, i.e. m = 12
inputs, a hidden layer of three sigmoidal neurons and a linear output neuron. These
five examples were chosen from 100 runs of simple back-propagation networks with
random initial conditions by selecting the networks with the smallest R values
(Ginzburg and Horn, 1992). This is a weak constraint which is based on letting
the network generate a large sequence of data by iterating its own predictions, and
selecting the networks whose distribution of function values is the closest to the
corresponding distribution of the training set.
The errors of the primary networks, in particular those of the prediction set ?p, are
quite higher than those quoted by Weigend et. al. who started out from a (12,8,1)
network and brought it down through a weight elimination technique to a (12,5,1)
structure. They have obtained the values ?T = 0.059 ?p = 0.06. We can reduce our
errors and reach the same range by activating a secondary network with I = 11 to
perform the linear regression (3.6) on the residuals of the predictions of the primary
network. The results are the primed errors quoted in the table. Characteristically
we observe a reduction of ?T by 3 - 4% and a reduction of ?p by more than 10%.
227
228
Ginzburg and Hom
#
fT
1
2
3
4
5
0.0614
0.0600
0.0611
0.0621
0.0616
{p
f'
T
0.0587
0.0585
0.0580
0.0594
0.0589
0.0716
0.0721
0.0715
0.0698
0.0681
{'
P
0.0620
0.0663
0.0621
0.0614
0.0604
Table 1
Error parameters of five networks. The unprimed errors are those of the primary
networks. The primed errors correspond to the combined system which includes
correction of the residuals by a linear perceptron with I 11 , which is an autoregressions of the residuals. Slightly better results for the short term predictions are
achieved by corrections based on regression of the residuals on the original input
vectors, when the regression length is 13 (Table 2).
=
#
{T
fT
fp
f'p
1
2
3
4
5
0.061
0.060
0.061
0.062
0.062
0.059
0.059
0.058
0.060
0.059
0.072
0.072
0.072
0.070
0.068
0.062
0.065
0.062
0.061
0.059
Table 2
Error parameters for the same five networks. The primed errors correspond to the
combined system which includes correction of the residuals by a linear perceptron
based on original input vectors with I 13.
=
5
LONG TERM PREDICTIONS
When short term prediction is performed, the output of the original network is
corrected by the error predicted by the secondary network. This can be easily generalized to perform long term predictions by feeding the corrected output produced
by the combined system of both networks back as input to the primary network. The
corrected residuals predicted by the secondary network are viewed as the residuals
needed as further inputs if the secondary network is the one performing autoregression of residuals. We run both systems based on regression on residuals and
regression on functional values to produce long term predictions.
In table 3 we present the results of this procedure for the case of a secondary
network performing regression on residuals. The errors of the long term predictions
are averaged over the test set P of the next 35 years. We see that the errors of
the primary networks are reduced by about 20%. The quality of these long term
predictions is within the range of results presented by Weigend et. al. Using the
regression on (predicted) functional values, as in Eq. 14 , the results are improved
by up to 15% as shown in Table 4.
Combined Neural Networks for Time Series Analysis
,
#
f2
fj
f5
f~
fll
f11
1
2
3
4
5
0.118
0.118
0.117
0.116
0.113
0.098
0.106
0.099
0.099
0.097
0.162
0.164
0.164
0.152
0.159
0.109
0.125
0.112
0.107
0.112
0.150
0.131
0.136
0.146
0.147
0.116
0.101
0.099
0.120
0.123
Table 3
Long term predictions into the future. fn denotes the average error of n time steps
predictions over the P set. The unprimed errors are those of the primary networks.
The primed errors correspond to the combined system which includes correction of
the residuals by a linear perceptron.
,
#
f2
f'2
f5
f'5
f11
f11
1
2
3
4
5
0.118
0.118
0.117
0.117
0.113
0.098
0.104
0.098
0.098
0.096
0.162
0.164
0.164
0.152
0.159
0.107
0.117
0.108
0.105
0.110
0.150
0.131
0.136
0.146
0.147
0.101
0.089
0.086
0.105
0.109
Table 4
Long term predictions into the future. The primed errors correspond to the combined system which includes correction of the residuals by a linear perceptron based
on the original inputs.
6
THE COMPLEX NETWORK
Since the corrections of the secondary network are much smaller than the characteristic weights of the primary network, the corrections can be regarded as resulting
from a Taylor expansion of a complex network which include's the combined system.
This can be simply implemented in the case of Eq. 14 which can be incorporated in
the complex network as direct linear connections from the input layer to the output
neuron, in addition to the non-linear hidden layer, i.e.,
tn
d
m
i=l
i=l
= L:: Wihi + L
viYn-i - () .
(15)
We train such a complex network on the same problem to see how it compares with
the two-step approach of the combined networks described in the previous chapters.
The results depend strongly on the training rates of the direct connections, as
compared with the training rates of the primary connections (i.e. those of the
primary network). When the direct connections are trained faster than the primary
ones, the result is a network that resembles a linear perceptron, with non-linear
229
230
Ginzburg and Hom
corrections. In this case, the assumption of the direct connections being small
corrections to the primary ones no longer holds. The training error and prediction
capability of such a network are worse than those of the primary network. On the
other hand, when the primary connections are trained using a faster training rate,
we expect the final network to be similar in nature to the combined system. Still,
the quality of training and prediction of these solutions is not as good as the quality
of the combined system, unless a big effort is made to find the correct rates. Typical
results of the various systems are presented in Table 5.
type of network
primary network
learning rate of linear weights = 0.1
learning rate of linear weights = 0.02
combined system
0.061
0.062
0.061
0.058
0.072
0.095
0.068
0.062
Table 5
Short term predictions of various networks. The learning rate of primary weights
is 0.04.
The performance of the complex network can be better than that of the primary
network by itself, but it is surpassed by the achievements of the combined system.
7
DISCUSSION
It is well known that increasing the complexity of a network is not the guaranteed
solution to better performance (Geman et. al. 1992). In this paper we propose an
alternative which increases very little the number of free parameters, and focuses on
the residual errors one wants to eliminate. Still one may raise the question whether
this cannot be achieved in one complex network. It can, provided we are allowed to
use different updating rates for different connections. In the extreme limit in which
one rate supersedes by far the other one, this is equivalent to a disjoint architecture
of a combined two-step system. This emphasizes the point that a solution of a
feedforward network to any given task depends on the architecture of the network
as well as on its training procedure.
The secondary network which we have used was linear, hence it defined a simple
regression of the residual on a series of residuals or a series of function values. In
both cases the minimum which the network looks for is unique. In the case in
which the residual is expressed as a regression on function values, the problem can
be recast in a complex architecture. However, the combined procedure guarantees
that the linear weights will be small, i.e. we look for a small linear correction to the
prediction of the primary network. If one trains all weights of the complex network
at the same rate this condition is not met, hence the worse results.
We advocate therefore the use of the two-step procedure of the combined set of
networks. We note that combined set of networks. We note that the secondary
networks perform well on all possible tests: they reduce the training errors, they
Combined Neural Networks for Time Series Analysis
improve short term predictions and they do better on long term predictions as well.
Since this approach is quite general and can be applied to any time-series forecasting
problem, we believe it should be always tried as a correction procedure.
REFERENCES
Geman, S., Bienenstock, E., & Doursat, R., 1992.
bias/variance dilemma. Neural Compo 4, 1-58.
Neural networks and the
Ginzburg, I. & Horn, D. 1992. Learning the rule of a time series. Int. Journal of
Neural Systems 3, 167-177.
Nowlan, S. J. & Hinton, G. E. 1992. Simplifying neural networks by soft weightsharing. Neural Compo 4, 473-493.
Tong, H., & Lim, K. S., 1980. Threshold autoregression, limit cycles and cyclical
data. J. R. Stat. Soc. B 42, 245.
Weigend, A. S., Huberman, B. A. & Rumelhart, D. E., 1990. Predicting the Future:
A Connectionist Approach, Int. Journal of Neural Systems 1, 193-209.
231
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7,053 | 825 | Fast Non-Linear Dimension Reduction
Nanda Kambhatla and Todd K. Leen
Department of Computer Science and Engineering
Oregon Graduate Institute of Science & Technology
P.O. Box 91000 Portland, OR 97291-1000
Abstract
We present a fast algorithm for non-linear dimension reduction.
The algorithm builds a local linear model of the data by merging
PCA with clustering based on a new distortion measure. Experiments with speech and image data indicate that the local linear
algorithm produces encodings with lower distortion than those built
by five layer auto-associative networks. The local linear algorithm
is also more than an order of magnitude faster to train.
1
Introduction
Feature sets can be more compact than the data they represent. Dimension reduction provides compact representations for storage, transmission, and classification.
Dimension reduction algorithms operate by identifying and eliminating statistical
redundancies in the data.
The optimal linear technique for dimension reduction is principal component analysis (PCA). PCA performs dimension reduction by projecting the original ndimensional data onto the m < n dimensional linear subspace spanned by the
leading eigenvectors of the data's covariance matrix. Thus PCA builds a global
linear model of the data (an m dimensional hyperplane). Since PCA is sensitive
only to correlations, it fails to detect higher-order statistical redundancies. One
expects non-linear techniques to provide better performance; i.e. more compact
representations with lower distortion.
This paper introduces a local linear technique for non-linear dimension reduction.
We demonstrate its superiority to a recently proposed global non-linear technique,
152
Fast Non-Linear Dimension Reduction
and show that both non-linear algorithms provide better performance than PCA
for speech and image data.
2
Global Non-Linear Dimension Reduction
Several researchers (e.g. Cottrell and Metcalfe 1991) have used layered feedforward
auto-associative networks with a bottle-neck middle layer to perform dimension
reduction. It is well known that auto-associative nets with a single hidden layer
cannot provide lower distortion than PCA (Bourlard and Kamp, 1988). Recent
work (e.g. Oja 1991) shows that five layer auto-associative networks can improve
on PCA. These networks have three hidden layers (see Figure l(a)). The first and
third hidden layers have non-linear response, and are referred to as the mapping
layers. The m < n nodes of the middle or representation layer provide the encoded
signal.
The first two layers of weights produce a projection from Rn to Rm. The last two
layers of weights produce an immersion from R minto R n. If these two maps are
well chosen, then the complete mapping from input to output will approximate the
identity for the training data. If the data requires the projection and immersion
to be non-linear to achieve a good fit, then the network can in principal find such
functions.
----,.
Low Dimensional
Encoding
J
Original High
?--- Dimensional
(a)
x
Representation
(b)
1
Figure 1: (a) A five layer feedforward auto-associative network. This network can
perform a non-linear dimension reduction from n to m dimensions. (b) Global
curvilinear coordinates built by a five layer network for data distributed on the
surface of a hemisphere. When the activations of the representation layer are swept,
the outputs trace out the curvilinear coordinates shown by the solid lines.
The activities of the nodes in the representation layer form global curvilinear coordinates on a submanifold of the input space (see Figure l(b)). We thus refer
to five layer auto-associative networks as a global, nonlinear dimension reduction
technique.
153
154
Kambhatla and Leen
3
Locally Linear Dimension Reduction
Five layer networks have drawbacks; they can be very slow to train and they are
prone to becoming trapped in poor local optima. Furthermore, it may not be
possible to accurately fit global, low dimensional, curvilinear coordinates to the
data. We propose an alternative that does not suffer from these problems.
Our algorithm pieces together local linear coordinate patches. The local regions are
defined by the partition of the input space induced by a vector quantizer (VQ). The
orientation of the local coordinates is determined by PCA (see Figure 2). In this
section, we present two ways to obtain the partition. First we describe an approach
that uses Euclidean distance, then we describe a new distortion measure which is
optimal for our task (local PCA).
-1
.5
-.5
o
.;.;.54r-----_ _~
r==---~~~~.......-
.25
o
-.25
1
Figure 2: Local coordinates built by our algorithm (dubbed VQPCA) for data distributed on the surface of a hemisphere. The solid lines represent the two principal
eigen-directions in each Voronoi cell. The region covered by one Voronoi cell is
shown shaded.
3.1
Euclidean partitioning
Here, we do a clustering (with Euclidean distance) followed by PCA in each of the
local regions. The hybrid algorithm, dubbed VQPCA, proceeds in three steps:
1. Using competitive learning, train a VQ (with Euclidean distance) with Q
reference vectors (weights) (rl' r2, ... ,rQ).
2. Perform a local PCA within each Voronoi cell of the VQ. For each cell,
compute the local covariance matrix for the data with respect to the corresponding reference vector (centroid) rc. Next compute the eigenvectors
(e 1, ... ,e~) of each covariance matrix.
3. Choose a target dimension m and project each data vector x onto
the leading m eigenvectors to obtain the local linear coordinates
z = (e 1. (x - r c), ... , e~ . (x - rc)).
Fast Non-Linear Dimension Reduction
The encoding of x consists of the index c of the reference cell closest (Euclidean
distance) to x, together with the m < n component vector z. The decoding is given
by
(1)
i=l
where r c is the reference vector (centroid) for the cell c, and ei are the leading
eigenvectors of the covariance matrix of the cell c. The mean squared reconstruction
error incurred by VQPCA is
m
(2)
i=l
where E[?] denotes an expectation with respect to x, and
x is defined in
(1).
Training the VQ and performing the local PCA are very fast relative to training a
five layer network. The training time is dominated by the distance computations
for the competitive learning. This computation can be speeded up significantly by
using a multi-stage architecture for the VQ (Gray 1984).
3.2
Projection partitioning
The VQPCA algorithm as described above is not optimal because the clustering is
done independently of the PCA projection. The goal is to minimize the expected
error in reconstruction (2). We can realize this by using the expected reconstruction
error as the distortion measure for the design of the VQ.
The reconstruction error for VQPCA (Erecon defined in (2)) can be written in matrix
form as
Erecon = E[ (x - ref P; Pc(X - rc)] ,
(3)
where Pc is an m x n matrix whose rows are the orthonormal trailing eigenvectors
of the covariance matrix for the cell c. This is the mean squared Euclidean distance
between the data and the local hyperplane.
The expression for the VQPCA error in (2) suggests the distortion measure
d(x, rc) = (x - rc)T P; Pc(x - rc) .
(4)
We call this the reconstruction distance. The reconstruction distance is the error
incurred in approximating x using only m local PCA coefficients. It is the squared
projection of the difference vector x - r c on the trailing eigenvectors of the covariance
matrix for the cell c. Clustering with respect to the reconstruction distance directly
minimizes the expected reconstruction error Erecon.
The modified VQPCA algorithm is:
1. Partition the input space using a VQ with the reconstruction distance measure 1 in (4) .
2. Perform a local PCA (same as in steps 2 and 3 of the algorithm as described
in section 3.1).
IThe VQ is trained using the (batch mode) generalized Lloyd's algorithm (Gersho and
Gray, 1992) rather than an on-line competitive learning. This avoids recomputing the
matrix Pc (which depends on Tc) for each input vector.
155
156
Kambhatla and Leen
4
Experimental Results
We apply PCA, five layer networks (5LNs), and VQPCA to dimension reduction
of speech and images. We compare the algorithms using two performance criteria:
training time and the distortion in the reconstructed signal. The distortion measure
is the normalized reconstruction error:
?recon
?norm
4.1
E[
IIx1l 2 ]
E[llx-xI12]
E [ IIxll 2 ]
Model Construction
The 5LNs were trained using three optimization techniques: conjugate gradient
descent (CGD), the BFGS algorithm (a quasi-Newton method (Press et al1987)),
and stochastic gradient descent (SGD). In order to limit the space of architectures,
the 5LNs have the same number of nodes in both of the mapping (second and
fourth) layers.
For the VQPCA with Euclidean distance, clustering was implemented using standard VQ (VQPCA-Eucl) and multistage quantization (VQPCA-MS-E). The multistage architecture reduces the number of distance calculations and hence the training time for VQPCA (Gray 1984).
4.2
Dimension Reduction of Speech
We used examples of the twelve monothongal vowels extracted from continuous
speech drawn from the TIMIT database (Fisher and Doddington 1986). Each input
vector consists of 32 DFT coefficients (spanning the frequency range 0-4kHz), timeaveraged over the central third of the utterance. We divided the data set into a
training set containing 1200 vectors, a validation set containing 408 vectors and
a test set containing 408 vectors. The validation set was used for architecture
selection (e.g the number of nodes in the mapping layers for the five layer nets).
The test set utterances are from speakers not represented in the training set or the
validation set. Motivated by the desire to capture formant structure in the vowel
encodings, we reduced the data from 32 to 2 dimensions. (Experiments on reduction
to 3 dimensions gave similar results to those reported here (Kambhatla and Leen
1993).)
Table 1 gives the test set reconstruction errors and the training times. The VQPCA
encodings have significantly lower reconstruction error than the global PCA or five
layer nets. The best 5LNs have slightly lower reconstruction error than PC A, but
are very slow to train. Using the multistage search, VQPCA trains more than
two orders of magnitude faster than the best 5LN, and achieves an error about 0.7
times as great. The modified VQPCA algorithm (with the reconstruction distance
measure used for clustering) provides the least reconstruction error among all the
architectures tried.
Fast Non-Linear Dimension Reduction
Table 1: Speech data test set reconstruction errors and training times. Architectures represented here are from experiments with the lowest validation set error
over the parameter ranges explored. The numbers in the parentheses are the values
of the free parameters for the algorithm represented (e.g 5LN-CGD (5) indicates a
network with 5 nodes in both the mapping (2nd and 4th) layers, while VQPCA-Eucl
(50) indicates a clustering into 50 Voronoi cells).
ALGORITHM
i norm
PCA
5LN-CGD (5)
5LN-BFGS (30)
5LN-SGD (25)
VQPCA-Eucl (50)
VQPCA-MS-E (9x9)
VQPCA-Recon (45)
0.0060
0.0069
0.0057
0.0055
0.0037
0.0036
0.0031
TRAINING TIME
(in seconds)
11
956
28,391
94,903
1,454
142
931
Table 2: Reconstruction errors and training times for a 50 to 5 dimension reduction
of images. Architectures represented here are from experiments with the lowest
validation set error over the parameter ranges explored.
4.3
ALGORITHM
i norm
PCA
5LN-CGD (40)
5LN-BFGS (20)
5LN-SGD (25)
VQPCA-Eucl (20)
VQPCA-MS-E (8x8)
VQPCA-Recon (25)
0.458
0.298
0.052
0.350
0.140
0.176
0.099
TRAINING TIME
(in seconds)
5
3,141
10,389
15,486
163
118
108
Dimension Reduction of Images
The data consists of 160 images of the faces of 20 people. Each is a 64x64, 8-bit/pixel
grayscale image. We extracted the first 50 principal components of each image and
use these as our experimental data. This is the same data and preparation that
DeMers and Cottrell used in their study of dimension reduction with five layer
auto-associative nets (DeMers and Cottrell 1993). They trained auto-associators to
reduce the 50 principal components to 5 dimensions.
We divided the data into a training set containing 120 images, a validation set (for
architecture selection) containing 20 images and a test set containing 20 images.
We reduced the images to 5 dimensions using PCA, 5LNs 2 and VQPCA. Table 2
2We used 5LNs with a configuration of 50-n-5-n-50, n varying from 10 to 40 in increments of 5. The BFGS algorithm posed prohibitive memory and time requirements for
n > 20 for this task.
157
158
Kambhatla and Leen
Table 3: Reconstruction errors and training times for a SO to S dimension reduction
of images (training with all the data). Architectures represented here are from
experiments with the lowest error over the parameter ranges explored.
ALGORITHM
[norm
PCA
SLN-SGD (30)
SLN-SGD (40)
VQPCA-Eucl (SO)
VQPCA-Recon (SO)
0.40S4
0.1034
0.0729
0.0009
0.0017
TRAINING TIME
(in seconds)
7
2S,306
31,980
90S
216
summarizes the results. We notice that a five layer net obtains the encoding with
the least error for this data, but it takes a long time to train. Presumably more
training data would improve the best VQPCA results.
~.-
.
'~'-??f-???:
.,..
...-
~. >':~ .. ~.:'
- -.?.
.
_:.I
~1-' T
,., .
'
~
~ ..~
.. '(;'.... '-".'.'- -
Figure 3: Two representative images: Left to right - Original SO-PC image, reconstruction from S-D encodings: PCA, SLN-SGD(40), VQPCA(lO), and VQPCA(SO).
For comparison with DeMers and Cottrell's (DeMers and Cottrell 1993) work, we
also conducted experiments training with all the data. The results are summarized 3
in Table 3 and Figure 3 shows two sample faces. Both non-linear techniques produce
encodings with lower error than PCA, indicating significant non-linear structure in
the data. With the same data, and with a SLN with 30 nodes in each mapping layer,
DeMers (DeMers and Cottrell 1993) obtains a reconstruction error [norm 0.13174 .
We note that the VQPCA algorithms achieve an order of magnitude improvement
over five layer nets both in terms of speed of training and the accuracy of encodings.
3For 5LNs, we only show results with SGD in order to compare with the experimental
results of DeMers. For this data, 5LN-CGD gave encodings with a higher error and 5LNBFGS posed prohibitive memory and computational requirements.
4DeMers reports half the MSE per output node, E = (1/2) * (1/50) * MSE = 0.00l.
This corresponds to [norm = 0.1317
Fast Non-Linear Dimension Reduction
5
Summary
We have presented a local linear algorithm for dimension reduction. We propose
a new distance measure which is optimal for the task of local PCA. Our results
with speech and image data indicate that the nonlinear techniques provide more
accurate encodings than PCA. Our local linear algorithm produces more accurate
encodings (except for one simulation with image data), and trains much faster than
five layer auto-associative networks.
Acknowledgments
This work was supported by grants from the Air Force Office of Scientific Research
(F49620-93-1-0253) and Electric Power Research Institute (RP8015-2). The authors
are grateful to Gary Cottrell and David DeMers for providing their image database
and clarifying their experimental results. We also thank our colleagues in the Center
for Spoken Language Understanding at OGI for providing speech data.
References
H. Bourlard and Y. Kamp. (1988) Auto-association by multilayer perceptrons and
singular value decomposition. Biological Cybernetics, 59:291-294.
G. Cottrell and J. Metcalfe. (1991) EMPATH: Face, emotion, and gender recognition using holons. In R. Lippmann, John Moody and D. Touretzky, editors,
Advances in Neural Information Processing Systems 3, pages 564-571. Morgan
Kauffmann.
D. DeMers and G. Cottrell. (1993) Non-linear dimensionality reduction. In Giles,
Hanson, and Cowan, editors, Advances in Neural Information Processing Systems
5. San Mateo, CA: Morgan Kaufmann.
W. M. Fisher and G. R. Doddington. (1986) The DARPA speech recognition research database: specification and status. In Proceedings of the DARPA Speech
Recognition Workshop, pages 93-99, Palo Alto, CA.
A. Gersho and R. M. Gray. (1992) Vector Quantization and Signal Compression.
Kluwer academic publishers.
R. M. Gray. (1984) Vector quantization. IEEE ASSP Magazine, pages 4-29.
N. Kambhatla and T. K. Leen. (1993) Fast non-linear dimension reduction. In IEEE
International Conference on Neural Networks, Vol. 3, pages 1213-1218. IEEE.
E. Oja. (1991) Data compression, feature extraction, and autoassociation in feedforward neural networks. In Artificial Neural Networks, pages 737-745. Elsevier
Science Publishers B. V. (N orth-Holland) .
W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling. (1987) Numerical Recipes - the Art of Scientific Computing. Cambridge University Press,
Cambridge/New York.
159
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7,054 | 826 | Decoding Cursive Scripts
Yoram Singer
and
Naftali Tishby
Institute of Computer Science and
Center for Neural Computation
Hebrew University, Jerusalem 91904, Israel
Abstract
Online cursive handwriting recognition is currently one of the most
intriguing challenges in pattern recognition. This study presents a
novel approach to this problem which is composed of two complementary phases. The first is dynamic encoding of the writing trajectory into a compact sequence of discrete motor control symbols.
In this compact representation we largely remove the redundancy of
the script, while preserving most of its intelligible components. In
the second phase these control sequences are used to train adaptive
probabilistic acyclic automata (PAA) for the important ingredients
of the writing trajectories, e.g. letters. We present a new and efficient learning algorithm for such stochastic automata, and demonstrate its utility for spotting and segmentation of cursive scripts.
Our experiments show that over 90% of the letters are correctly
spotted and identified, prior to any higher level language model.
Moreover, both the training and recognition algorithms are very
efficient compared to other modeling methods, and the models are
'on-line' adaptable to other writers and styles.
1
Introduction
While the emerging technology of pen-computing is already available on the world's
markets, there is an on growing gap between the state of the hardware and the
quality of the available online handwriting recognition algorithms. Clearly, the
critical requirement for the success of this technology is the availability of reliable
and robust cursive handwriting recognition methods.
833
834
Singer and Tishby
We have previously proposed a dynamic encoding scheme for cursive handwriting
based on an oscillatory model of handwriting [8, 9] and demonstrated its power
mainly through analysis by synthesis . Here we continue with this paradigm and use
the dynamic encoding scheme as the front-end for a complete stochastic model of
cursive script.
The accumulated experience in temporal pattern recognition in the past 30 years
has yielded some important lessons relevant to handwriting. The first is that one
can not predefine the basic 'units' of such temporal patterns due to the strong interaction, or 'coarticulation ' , between such units. Any reasonable model must allow for
the large variability of the basic handwriting components in different contexts and
by different writers. Thus true adaptability is a key ingredient of a good stochastic model of handwriting. Most, if not all, currently used models of handwriting
and speech are hard to adapt and require vast amounts of training data for some
robustness in performance. In this paper we propose a simpler stochastic modeling
scheme , which we call Probabilistic Acyclic Automata (PAA), with the important
feature of being adaptive. The training algorithm modifies the architecture and
dimensionality of the model while optimizing its predictive power. This is achieved
through the minimization of the "description length" of the model and training
sequences, following the minimum description length (MDL) principle. Another
interesting feature of our algorithm is that precisely the same procedure is used in
both training and recognition phases, which enables continuous adaptation.
The structure of the paper is as follows. In section 2 we review our dynamic encoding method, used as the front-end to the stochastic modeling phase. We briefly
describe the estimation and quantization process, and show how the discrete motor
control sequences are estimated and used , in section 3. Section 4 deals with our
stochastic modeling approach and the PAA learning algorithm. The algorithm is
demonstrated by the modeling of handwritten letters. Sections 5 and 6 deal with
preliminary applications of our approach to segmentation and recognition of cursi ve
handwriting.
2
Dynamic encoding of cursive handwriting
Motivated by the oscillatory motion model of handwriting, as described e.g. by
Hollerbach in 1981 [2], we developed a parameter estimation and regularization
method which serves for the analysis, synthesis and coding of cursive handwriting .
This regularization technique results in a compact and efficient discrete representation of handwriting.
Handwriting is generated by the human muscular motor system, which can be simplified as spring muscles near a mechanical equilibrium state. When the movements
are small it is justified to assume that the spring muscles operate in the linear
regime , so the basic movements are simple harmonic oscillations, superimposed by
a simple linear drift. Movements are excited by selecting a pair of agonist-antagonist
muscles that are modeled by the spring pair. In a restricted form this simple motion
is described by the following two equations ,
Vx(t) = x(t) = acos(wxt + f/;) + c Vy(t) = yet) = bcos(wyt) ,
(1)
where Vx(t) and Vy(t) are the horizontal and vertical pen velocities respectively, Wx
and
Wy
are the angular velocities,
a,
b are the velocity amplitudes, ? is the relative
Decoding Cursive Scripts
phase lag , and c is the horizontal drift velocity. Assuming that these describe
the true trajectory, the horizontal drift, c, is estimated as the average horizontal
velocity, c = Jv 2:[:1 Vx(i). For fixed values of the parameters a, b,w and 1; these
equations describe a cycloidal trajectory.
Our main assumption is that the cycloidal trajectory is the natural (free) pen motion, which is modified only at the velocity zero crossings. Thus changes in the
dynamical parameters occur only at t he zero crossings and preserve the continuity
of the velocity field. This assumption implies that the angular velocities W x , Wy
and amplitudes a, b can be considered constant between consecutive zero crossings.
Denoting by tf and t; , the i'th zero crossing locations of the horizontal and vertical
velocities , and by Li and L; , the horizontal and vertical progression during the i 'th
interval, then the estimated amplitudes are, a = 2(tf~ =tX) , b = 2(J~ :t Y )' Those
.+1 ?
.+1 ?
amplitudes define the vertical and horizontal scales of the written letters.
Examination of the vertical velocity dynamics reveals the following : (a) There is
a virtual center of the vertical movement and velocity trajectory is approximately
symmetric around this center. (b) The vertical velocity zero crossings occur while
the pen is at almost fixed vertical levels which correspond to high, normal and low
modulation values, yielding altogether 5 quantized levels. The actual pen levels
achieved at the vertical velocity zero crossings vary around the quantized values,
with approximately normal distribution. Let the indicator, It (It E {I , . . . , 5}),
be the most probable quantized level when the pen is at the position obtained at
the t'th zero crossing. \Ve need to estimate concurrently the 5 quantized levels
H 1, ... , H 5, their variance (J' (assumed the same for all levels), and the indicators
It. In this model the observed data is the sequence of actual pen levels L(t), while
the complete data is the sequence of levels and indicators {It , L(t)} . The task of
estimating the parameters {Hi , (J'} is performed via maximum likelihood estimation
from incomplete data, commonly done by the EM algorithm[l] and described in [9].
The horizontal amplitude is similarly quantized to 3 levels.
After performing slant equalization of the handwriting, namely, orthogonalizing the
x and y motions , the velocities Vx(t) , "~(t) become approximately uncorrelated.
When Wx ~ w y , the two velocities are uncorrelated if there is a ?90 0 phase-lag
between Vx and Vy . There are also locations of total halt in both velocities (no pen
movement) which we take as a zero phase lag . Considering the vertical oscillations
as a 'master clock', the horizontal oscillations can be viewed as a 'slave clock ' whose
phase and amplitude vary around the 'master clock'. For English cursive writing,
the frequency ratio between the two clocks is limited to the set {~, 1,2}, thus Vy
induces a grid for the possible Vx zero crossings. The phase-lag of the horizontal
oscillation is therefore restricted to the values 00, ?90 0 at the zero crossings of
Vy . The most likely phase-lag trajectory is determined by dynamic programming
over the entire grid. At the end of this process the horizontal oscillations are fully
determined by the vertical oscillations and the pen trajectory 's description greatly
simplified.
The variations in the vertical angular velocity for a given writer are small, except
in short intervals where the writer hesitates or stops. The only information that
should be preserved is the typical vertical angular velocity, denoted by w. The
835
836
Singer and Tishby
normalized discretized equations of motion now become,
{~
sin(wt + <Pi) + 1
hsin(wt)
ai
ai
E {AI, Ai, A3} <Pj E {-90?, 0?, 90?}
hE {H1 2
-
Hil
11::; 11 ,/2
::;
5} .
(2)
We used analysis by synthesis technique in order to verify our assumptions and
estimation scheme. The final result of the whole process is depicted in Fig. 1,
where the original handwriting is plotted together with its reconstruction from the
discrete representation.
Figure 1: The original and the fully quantized cursive scripts.
3
Discrete control sequences
The process described in the previous section results in a many to one mapping
from the continuous velocity field, Vx(t), Vy(t), to a discrete set of symbols. This
set is composed of the cartesian product of the quantized vertical and horizontal
amplitudes and the phase-lags between these velocities . We treat this discrete control sequence as a cartesian product time series . Using the value (0' to indicate
that the corresponding oscillation continues with the same dynamics , a change in
the phase lag can be encoded by setting the code to zero for one dimension, while
switching to a new value in the other dimension. A zero in both dimensions indicates no activity. In this way we can model 'pen ups' intervals and incorporate
auxiliary symbols like 'dashes', 'dots', and 'crosses', that play an important role in
resolving disambiguations between letters. These auxiliary are modeled as a separate channel and are ordered according to their X coordinate . We encode the
control levels by numbers from 1 to 5 , for the 5 levels of vertical positions. The
quantized horizontal amplitudes are coded by 5 values as well: 2 for positive amplitudes (small and large), 2 for negative amplitudes, and one for zero amplitude.
Below is an example of our discrete representation for the handwriting depicted in
Fig. 1. The upper and lower lines encode the vertical and horizontal oscillations
respectively, and the auxiliary channel is omitted. In this example there is only one
location where both symbols are (0', indicating a pen-up at the end of the word.
240204204001005002040202204020402424204020500204020402400440240220
104034030410420320401050010502425305010502041032403050033105001000
4
Stochastic modeling of the motor control sequences
Existing stochastic modeling methods, such as Hidden Markov Models (HMM) [3],
suffer from several serious drawbacks. They suffer from the need to 'fix' a-priory the
Decoding Cursive Scripts
architecture of the model; they require large amounts of segmented training data;
and they are very hard to adapt to new data. The stochastic model presented here
is an on-line learning algorithm whose important property is its simple adaptability
to new examples. We begin with a brief introduction to probabilistic automata ,
leaving the theoretical issues and some of the more technical details to another
place.
A probabilistic automaton is a 6-tuple (Q , ~ , T", qs, qe), where Q is a finite set
of n states, ~ is an alphabet of size k, T : Q x ~ --+ Q is the state transition
function, , : ~ x Q --+ [0,1] is the transition (output) probability where for every
q E Q, LaE~ ,( O'lq) = l. qs E Q is a start state, and qe E Q is an end state. A
probabilistic automaton is called acyclic if it contains no cycles. We denote such
automata by PAA. This type of automaton is also known as a Markov process with
a single source and a single absorbing state. The rest of the states are all transient
states . Such automata induce non-zero probabilities on a finite set of strings . Given
an input string a = (0'1, .. . , 0' n) if at the of end its 'run' the automaton entered the
final state qe, the probability of a string a is defined to be, pea) n{:l ,(O'ilqi-l)
where qo = qs, qi = T(qi-1, O'i) . On the other hand , if qN f. qe then pea) = O.
=
The inference of the P AA structure from data can be viewed as a communication
problem. Suppose that one wants to transmit an ensemble of strings, all created
by the same PAA. If both sides know the structure and probabilities of the PAA
then the transmitter can optimally encode the strings by using the PAA transition
probabilities. If only the transmitter knows the structure and the receiver has
to discover it while receiving new strings, each time a new transition occurs , the
transmitter has to send the next state index as well . Since the automaton is acyclic,
the possible next states are limited to those which do not form a cycle when the
new edge is added to the automaton. Let k~ be the number of legal next states
from a state q known to the receiver at time t. Then the encoding of the next
state index requires at least log2(k~ + 1) bits. The receiver also needs to estimate
the state transition probability from the previously received strings. Let n(O'lq) be
the number of times the symbol 0' has been observed by the receiver while being in
state q. Then the transition probability is estimated by Laplace 's rule of succession ,
?(O'lq) = L n(alq )~\ 1 I' In sum, if q is the current state and ktq the number of
I
(7
EE
n(al
q
+~
possible next states known to the receiver , the number of bits required to encode the
next symbol 0' (assuming optimal coding scheme) is given by: (a) if the transition
T(q, 0') has already been observed: -log2(P(0'Iq)) ; (b) if the transition T(q, 0') has
never occurred before : -log2(.P(0'Iq)) + log2(k~ + 1).
In training such a model from empirical observations it is necessary to infer the
structure of the PAA as well its parameters . We can thus use the above coding
scheme to find a minimal description length (MDL) of the data , provided that our
model assumption is correct. Since the true PAA is not known to us, we need to
imitate the role of the receiver in order to find the optimal coding of a message. This
can be done efficiently via dynamic programming for each individual string. After
the optimal coding for a single string has been found , the new states are added , the
transition probabilities ?(O'lq) are updated and the number of legal next states kg
is recalculated. An exan~ple of the learning algorithm is given in Fig. 2, with the
estimated probabilities P, written on the graph edges.
837
838
Singer and Tishby
(b)
(d)
Figure 2: Demonstration of the PAA learning algorithm . Figure (a) shows the
original automaton from which the examples were created. Figures (b )-( d) are the
intermediate automata built by the algorithm. Edges drawn with bold , dashed, and
grey lines correspond to transitions with the symbols '0', '1', and the terminating
symbol , respectively.
5
Automatic segmentation of cursive scripts
Since the learning algorithm of a PAA is an on-line scheme, only a small number
of segmented examples is needed in order to built an initial model. For cursive
handwriting we manually collected and segmented about 10 examples, for each
lower case cursive letter , and built 26 initial models. At this stage the models are
small and do not capture the full variability of the control sequences. Yet this set
of initial automata was sufficient to gradually segment cursive scripts into letters
and update the models from these segments. Segmented words with high likelihood
are fed back into the learning algorithm and the models are further refined. The
process is iterated until all the training data is segmented with high likelihood.
The likelihood of new data might not be defined due the incompleteness of the
automata, hence the learning algorithm is again applied in order to induce probabilities. Let Pi~j be the probability that a model 5 (which represents a cursive
letter) generates the control symbols Si, ... , Sj -1 (j > i). The log-likelihood of a
proposed segmentation (i1, i 2 , ... , iN+d of a word 5 1 ,52 , ... , 5 N is,
N
L ((i1, . . . , iN+1)1(51, ... , 5N) , (Sl, . . . , sL)) = log(II Pi~~iJ+J =
j=l
N
L log(Pi~~iJ+l)
j=l
The segmentation is calculated efficiently by maintaining a layers graph and using
dynamic programming to compute recursively the most likely segmentation. Formally, let M L( n, k) be the highest likelihood segmentation of the word up to the
Decoding Cursive Scripts
n'th control symbol and the k'th letter in the word. Then,
M L(n, k)
= . ma~
tk-l~t~n
{M L(i, k - 1) + log
(Pi:~)}
The best segmentation is obtained by tracking the most likely path from M(N, L)
back to M(l, 1) . The result of such a segmentation is depicted in Fig. 3.
Figure 3: Temporal segmentation of the word impossible. The segmentation is
performed by applying the automata of the letters contained in the word, and
finding the Maximum-Likelihood sequence of models via dynamic programming.
6
Inducing probabilities for unlabeled words
Using this scheme we automatically segmented a database which contained about
1200 frequent english words , by three different writers. After adding the segmented
letters to the training set the resulting automata were general enough, yet very
compact. Thus inducing probabilities and recognition of unlabeled data could be
performed efficiently. The probability of locating letters in certain locations in new
unlabeled words (i.e. words whose transcription is not given) can be evaluated by
the automata. These probabilities are calculated by applying the various models
on each sub-string of the control sequence, in parallel. Since the automata can
accommodate different lengths of observations, the log-likelihood should be divided
by the length of the sequence. This normalized log-likelihood is an approximation
of the entropy induced by the models, and measures the uncertainty in determining
the transcription of a word. The score which measures the uncertainty of the occurrence of a letter S in place n in the a word is, Score(nIS)
maxI 10g(P:'n+l_d.
The result of applying several automata to a new word is shown in Fig. 4. High
probability of a given automaton indicates a beginning of a letter with the corresponding model. The probabilities for the letters k, a, e, b are plotted top to
bottom. The correspondence between high likelihood points and the relevant locations in the words are shown with dashed lines. These locations occur near the
'true' occurrence of the letter and indicate that these probabilities can be used for
recognition and spotting of cursive handwriting. There are other locations where
the automata obtain high scores. These correspond to words with high similarity to
the model letter and can be resolved by higher level models, similar to techniques
used in speech.
=
7
t
Conclusions and future research
In this paper we present a novel stochastic modeling approach for the analysis,
spotting, and recognition of online cursive handwriting. Our scheme is based on a
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Singer and Tishby
Figure 4: The normalized log-likelihood scores induced by the automata for the
letters k, a, e, and b (top to bottom). Locations with high score are marked with
dashed lines and indicate the relative positions of the letters in the word.
discrete dynamic representation of the handwriting trajectory, followed by training
adaptive probabilistic automata for frequent writing sequences. These automata
are easy to train and provide simple adaptation mechanism with sufficient power
to capture the high variability of cursively written words . Preliminary experiments
show that over 90% of the single letters are correctly identified and located, without
any additional higher level language model. Methods for higher level statistical
language models are also being investigated [6], and will be incorporated into a
complete recognition system.
Acknowledgments
We would like to thank Dana Ron for useful discussions and Lee Giles for providing us
with the software for plotting finite state machines. Y.S. would like to thank the Clore
foundation for its support.
References
[1] A. Dempster, N. Laird, and D. Rubin. Maximum likelihood estimation from
incomplete data via the EM algorithm. 1. Roy. Statist. Soc., 39(B):1-38, 1977.
[2] J .M. Hollerbach. An oscillation theory of handwriting. Bio. Cyb., 39, 1981.
[3] L.R. Rabiner. A tutorial on hidden markov models and selected applications in
speech recognition. Proc. IEEE, pages 257-286, Feb. 1989.
[4] J . Rissanen. Modeling by shortest data description. Automaiica, 14, 1978.
[5] J. Rissanen. Stochastic complexity and modeling. Annals of Stat., 14(3), 1986.
[6] D. Ron, Y. Singer, and N. Tishby. The power of amnesia. In this volume.
[7] D.E. Rumelhart. Theory to practice: a case study - recognizing cursive handwriting. In Proc. of 1992 NEC Conf. on Computation and Cognition.
[8] Y. Singer and N. Tishby. Dynamical encoding of cursive handwriting. In IEEE
Conference on Computer Vision and Pattern Recognition, 1993.
[9] Y. Singer and N. Tishby. Dynamical encoding of cursive handwriting. Technical
Report CS93-4, The Hebrew University of Jerusalem, 1993.
PART VII
IMPLEMENTATIONS
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hence:1 symmetric:1 deal:2 sin:1 during:1 naftali:1 qe:4 antagonist:1 complete:3 demonstrate:1 motion:5 harmonic:1 novel:2 absorbing:1 volume:1 he:2 occurred:1 slant:1 ai:4 automatic:1 grid:2 similarly:1 language:3 dot:1 similarity:1 feb:1 wxt:1 optimizing:1 certain:1 success:1 continue:1 muscle:3 preserving:1 minimum:1 additional:1 paradigm:1 shortest:1 dashed:3 ii:1 resolving:1 full:1 infer:1 segmented:7 technical:2 adapt:2 cross:1 divided:1 spotted:1 coded:1 halt:1 qi:2 basic:3 vision:1 achieved:2 justified:1 preserved:1 want:1 interval:3 leaving:1 source:1 operate:1 rest:1 induced:2 call:1 ee:1 near:2 intermediate:1 enough:1 easy:1 architecture:2 identified:2 motivated:1 utility:1 suffer:2 locating:1 speech:3 useful:1 cursive:23 amount:2 statist:1 hardware:1 induces:1 sl:2 vy:6 tutorial:1 estimated:5 correctly:2 discrete:9 redundancy:1 key:1 rissanen:2 acos:1 drawn:1 jv:1 pj:1 vast:1 graph:2 year:1 sum:1 run:1 letter:20 master:2 uncertainty:2 cursively:1 place:2 almost:1 reasonable:1 oscillation:9 disambiguation:1 incompleteness:1 bit:2 layer:1 hi:1 dash:1 followed:1 correspondence:1 yielded:1 activity:1 occur:3 precisely:1 software:1 generates:1 spring:3 performing:1 according:1 em:2 restricted:2 gradually:1 legal:2 equation:3 previously:2 mechanism:1 singer:8 know:2 needed:1 fed:1 end:6 serf:1 available:2 predefine:1 progression:1 occurrence:2 robustness:1 altogether:1 original:3 top:2 log2:4 maintaining:1 yoram:1 already:2 added:2 occurs:1 separate:1 thank:2 hmm:1 collected:1 priory:1 assuming:2 length:5 code:1 modeled:2 index:2 ratio:1 demonstration:1 hebrew:2 providing:1 negative:1 implementation:1 upper:1 vertical:16 observation:2 markov:3 finite:3 variability:3 communication:1 incorporated:1 drift:3 pair:2 mechanical:1 namely:1 required:1 spotting:3 wy:2 pattern:4 dynamical:3 below:1 regime:1 hollerbach:2 challenge:1 built:3 reliable:1 power:4 critical:1 natural:1 examination:1 indicator:3 scheme:9 technology:2 brief:1 created:2 prior:1 review:1 determining:1 relative:2 lae:1 fully:2 interesting:1 acyclic:4 dana:1 ingredient:2 foundation:1 sufficient:2 rubin:1 principle:1 plotting:1 uncorrelated:2 pi:5 free:1 english:2 side:1 allow:1 institute:1 dimension:3 calculated:2 world:1 transition:10 qn:1 commonly:1 adaptive:3 simplified:2 ple:1 sj:1 compact:4 transcription:2 reveals:1 receiver:6 assumed:1 continuous:2 pen:11 channel:2 robust:1 investigated:1 main:1 intelligible:1 whole:1 complementary:1 fig:5 sub:1 position:3 slave:1 lq:4 symbol:10 maxi:1 a3:1 quantization:1 adding:1 orthogonalizing:1 nec:1 cartesian:2 gap:1 vii:1 entropy:1 depicted:3 likely:3 ordered:1 contained:2 tracking:1 aa:1 ma:1 viewed:2 marked:1 hard:2 change:2 muscular:1 determined:2 except:1 typical:1 wt:2 total:1 called:1 indicating:1 formally:1 support:1 incorporate:1 |
7,055 | 827 | Connectionist Models for
A uditory Scene Analysis
Richard
o. Duda
Department of Electrical Engineering
San Jose State University
San Jose, CA 95192
Abstract
Although the visual and auditory systems share the same basic
tasks of informing an organism about its environment, most connectionist work on hearing to date has been devoted to the very
different problem of speech recognition . VVe believe that the most
fundamental task of the auditory system is the analysis of acoustic
signals into components corresponding to individual sound sources,
which Bregman has called auditory scene analysis . Computational
and connectionist work on auditory scene analysis is reviewed, and
the outline of a general model that includes these approaches is
described.
1
INTRODUCTION
The primary task of any perceptual system is to tell us about the external world.
The primary problem is that the sensory inputs provide too much data and too little
information. A perceptual system must glean from the flood of incomplete, noisy,
redundant and constantly changing streams of data those invariant properties that
reveal important objects and events in the environment. For humans, the perceptual
systems with the widest bandwidths are the visual system and the auditory system.
There are many obvious similarities and differences between these modalities, and
in addition to using them to perceive different aspects of the physical world, we also
use them in quite different ways to communicate with one another.
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The earliest neural-network models for vision and hearing addressed problems in
pattern recognition, with optical character recognition and isolated word recognition among the first engineering applications. However, about twenty years ago
the research goals in vision and hearing began to diverge. In particular, the need
for computers to perceive the external environment motivated vision researchers to
seek the principles and procedures for recovering information about the physical
world from visual data (Marr, 1982; Ballard and Brown, 1982). By contrast, the
vast majority of work on machine audition remained focused on the communication problem of speech recognition (Morgan and Scofield, 1991; Rabiner and Juang,
1993). While this focus has produced considerable progress, the resulting systems
are still not very robust, and perform poorly in uncontrolled environments. Furthermore, as Richards (1988) has noted, " ... Speech, like writing and reading, is
a specialized skill of advanced animals, and understanding speech need not be the
best route to understanding how we interpret the patterns of natural sounds that
comprise most of the acoustic spectrum about us."
In recent years, some researchers concerned with modeling audition have begun to
shift their attention from speech understanding to sound understanding. The inspiration for much of this activity has come from the work of Bregman, whose book
on auditory scene analysis documents experimental evidence for important gestalt
principles that summarize the ways that people group elementary events in frequency /time into sound objects or streams (Bregman, 1990). In this survey paper,
we briefly review this activity and consider its implications for the development of
connectionist models for auditory scene analysis.
2
AUDITORY SCENE ANALYSIS
In vision, Marr (1982) emphasized the importance of identifying the tasks of the
visual system and developing a computational theory that is distinct from particular algorithms or implementations. The computational theory had to specify the
problems to be solved, the sensory data that is available, and the additional knowledge or assumptions required to solve the problems. Among the various tasks of
the visual system, Marr believed that the recovery of the three-dimensional shapes
of the surfaces of objects from the sensory image data was the most fundamental.
The auditory system also has basic tasks that are more primitive than the recognition of speech. These include (1) the separation of different sound sources, (2)
the localization of the sources in space (3) the suppression of echoes and reverberation, (4) the decoupling of sources from the environment, (5) the characterization
of the sources, and (6) the characterization of the environment. Unfortunately, the
relation between physical sound sources and perceived sound streams is not a simple one-to-one correspondence. Distributed sound sources, echoes, and synthetic
sounds can easily confuse auditory perception. Nevertheless, humans still do much
better at these six basic tasks than any machine hearing system that exists today.
From the standpoint of physics, the raw data available for performing these tasks
is the pair of acoustic signals arriving at the two ears. From the standpoint of
neurophysiology, the raw data is the activity on the auditory nerve. The nonlinear,
mechallo-neural spectral analysis performed by the cochlea converts sound pressure fluctuations into auditory nerve firings. For better or for worse , the cochlea
Connectionist Models for Auditory Scene Analysis
decomposes the signal into many frequency components, transforming it into a frequency /time (or, more accurately, a place/time) spectrogram-like representation.
The auditory system must find the underlying order in this dynamic flow of data.
For a specific case, consider a simple musical mixture of several periodic signals.
\Vithin its limits of resolution, the cochlea decomposes each individual signal into
its discrete harmonic components. Yet, under ordinary circumstances, we do not
hear these components as separate sounds, but rather we fuse them into a single
sound having, as musicians say, its particular timbre or tone color. However, if
there is something distinctive about the different signals (such as different pitch or
different modulation), we do not fuse all of the sounds together, but rather hear the
separate sources, each with its own timbre.
What information is available to group the spectral components into sound streams?
Hartmann (1988) identifies the following factors that influence grouping: (1) common onset/offset, (2) common harmonic relations, (3) common modulation, (4)
common spatial origin, (5) continuity of spectral envelope, (6) duration, (7) sound
pressure level, and (8) context. These properties are easier to name than to precisely specify, and it is not surprising that no current model incorporates them all.
However, several auditory scene analysis systems have been built that exploit some
subset of these cues (''''eintraub, 1985; Cooke, 1993; Mellinger, 1991; Brown, 1992;
Brown and Cooke, 1993; Ellis, 1993). Although these are computational rather
than connectionist models, most of them at least find inspiration in the structure
of the mammalian auditory system.
3
NEURAL AND CONNECTIONIST MODELS
The neural pathways from the cochlea through the brainstem nuclei to the auditory
cortex are complex, but have been extensively investigated. Although this system
is far from completely understood, neurons in the brainstem nuclei are known to be
sensitive to various acoustic features - onsets, offsets and modulation in the dorsal
cochlear nucleus, interaural time differences (lTD's) in the medial superior olive
(MSO), interaural intensity differences (IID's) in the lateral superior olive (LSO),
and spatial location maps in the inferior colliculus (Pickles, 1988).
Both functional and connectionist models have been developed for all of these functions. Because it is both important and relatively well understood, the cochlea
has received by far the most attention (Allen, 1985). As a result of this work, we
now have real-time implementations for some of these models as analog VLSI chips
(Lyon and Mead, 1988; Lazzaro et al., 1993). Connectionist models for sound localization have also been extensively explored. Indeed, one of the earliest of all neural
network models was Jeffress's classic crosscorrelation model (Jeffress, 1948), which
was hypothesized forty years before neural crosscorrelation structures were actually
found in the barn owl (Carr and Konishi, 1988). Models have subsequently been
proposed for both the LSO (Reed and Blum, 1990) and the TvISO (Han and Colburn, 1991). Mathematically, both the lTD and IID cues for binaural localization
are exposed by crosscorrelation. Lyon showed that cross correlation can also be used
to separate as well as localize the signals (Lyon, 1983). VLSI cross correlation chips
can provide this information in real time (Lazzaro and Mead, 1989; Bhadkamkar
and Fowler, 1993).
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While interaural crosscorrelation can determine the azimuth to a sound source,
full three-dimensional localization also requires the determination of elevation and
range. Because of a lack of symmetry in the orientation of its ears, the barn owl can
actually determine azimuth from the lTD and elevation from the IID. This at least
in part explains why it has been such a popular choice for connectionist modeling
(Spence et al., 1990; Moiseff et al., 1991; Palmieri et al., 1991; Rosen, Rumelhart
and Knudsen, 1993) . Unfortunately, the localization mechanisms used by humans
are more complicated.
It is well known that humans use monaural, spectral shape cues to estimate elevation
in the median sagittal plane (Blauert, 1983; Middlebrooks and Green, 1991), and
source localization models based on this approach have been developed (Neti, Young
and Schneider, 1992; Zakarauskas and Cynander, 1993). The author has shown that
there are strong binaural cues for elevation at short distances away from the median
plane, and has used statistical methods to estimate both azimuth and elevation
accurately from IID data alone (Duda, 1994). In addition, backprop models have
been developed that can estimate azimuth and elevation from IID and lTD inputs
jointly (Backman and Karjalainen 93; Anderson, Gilkey and Janko, 1994).
Finally, psychologists have long been aware of an important reverberationsuppression phenomenon known as the precedence effect or the law of the first
wavefront (Zurek , 1987). It is usually summarized by saying that echoes of a sound
source have little effect on its localization, and are not even consciously heard if
they are not delayed more than the so-called echo threshold, which ranges from
5-10 ms for sharp clicks to more than 50 ms for music. It is generally believed
that the precedence effect can be accounted for by contralateral inhibition in the
crosscorrelation process, and Lindemann has accounted for many of the phenomena
by a conceptually simple connectionist model (Lindemann, 1986).
However, Clifton and her colleagues have found that the echoes are indeed heard
if the timing of the echoes suddenly changes, as might happen when one moves
from one acoustic environment into another one (Clifton 1987; Freyman, Clifton
and Litovsky, 1991). Clifton conjectures that the auditory system is continually
analyzing echo patterns to model the acoustic environment, and that the resulting
expectations modify the echo threshold . This suggests that simple crosscorrelation
models will not be adequate when the listener is moving, and thus that even the
localization problem is still unsolved.
4
ARCHITECTURE OF AN AUDITORY MODEL
If we look back at the six basic tasks for the auditory system, we see that only one
(source localization) ha.s received much attention from connectionist researchers,
and its solution is incomplete. In particular, current localization models cannot
handle multiple sources and cannot cope with significant room echoes and reverberation. The common problem for all of the basic tasks is that of source separation,
which only the a.uditory scene analysis systems have addressed.
Fig. 1 shows a functional block diagram for a hypothetical auditory model that
combines the computational and connectionist models and has the potential of
coping with a multisource environment . The inputs to the model are the left-ear
Connectionist Models for Auditory Scene Analysis
and right-ear signals, and the main output is a dynamic set of streams. The system
is primarily data driven, although low-bandwidth efferent paths could be added for
tasks such as automatic gain control.
Data flow on the left half of the diagram is monaural, and dataflow of the right
half is binaural. The binaural processing is based on crosscorrelation analysis of
the cochlear outputs. The author has shown that interaural differences not only
effective in determining azimuth, but can also be used to determine elevation as well
(Duda, 1994). V\'e have chosen to follow Slaney and Lyon (Slaney and Lyon, 1993)
in basing the monaural analysis on autocorrelation analysis. Originally proposed
by Licklider (1951) to explain pitch phenomena, autocorrelation is a biologically
plausible operation that supports the common onset, modulation and harmonicity
analysis needed for stream formation (Duda, Lyon and Slaney, 1990; Brown and
Cooke, 1993).
While the processes at lower levels of this diagram are relatively well understood,
the process of stream formation is problematic. Bregman (1990) has posed this
problem in terms of grouping the components of the "neural spectrogram" in both
frequency and time. He has identified two principles that seem to be employed
in stream formation: exclusive allocation (a component may not be used in more
than one description at a time) and accounting (all incoming components must be
assigned to some source). The various auditory scene analysis systems that we
mentioned earlier provide different mechanisms for exploiting these principles to
form auditory streams. Unfortunately, the principles admit of exceptions, and the
existing implementations seem rather ad hoc and arbitrary. The development of a
biologically plausible model for stream formation is the central unsolved problem
for connectionist research in audition.
Short? Term
Memory
Stream Formation
Aud~ory
Monaural Maps
Auto-Correlatlon
Analysis
Cross-Correiation
Analysis
Spectral Analysis
(Cochlear Model)
Spectral Analysis
(Cochlear Model)
Left Input
Right Input
I
Figure 1: Block diagram for a basic auditory model
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Acknowledgements
This work was supported by the National Science Foundation under NSF Grant
No. IRI-9214233. This paper could not have been written without the many discussions on these topics with Al Bregman, Dick Lyon, David Mellinger, Bernard
MontReynaud, John R. Pierce, Malcolm Slaney and J. Martin Tenenbaum, and
from the stimulating CCRMA Hearing Seminar at Stanford University that Bernard
initiated and that Malcolm has maintained and invigorated.
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7,056 | 828 | Connectionist Modeling and
Parallel Architectures
Joachim Diederich
Ah Chung Tsoi
Neurocomputing Research Centre
Department of Electrical and
Computer Engineering
School of Computing Science
Queensland University of Technology
University of Queensland
St Lucia, Queensland 4072, Australia
Brisbane Q 400 1 Australia
The introduction of specialized hardware platforms for connectionist modeling
("connectionist supercomputer") has created a number of research topics. Some of
these issues are controversial, e.g. the efficient implementation of incremental learning techniques, the need for the dynamic reconfiguration of networks and possible
programming environments for these machines.
Joachim Diederich, Queensland University of Technology (Brisbane), started with
a brief introduction to connectionist modeling and parallel machines. Neural network
modeling can be done on various levels of abstraction. On a low level of abstraction,
a simulator can support the definition and simulation of "compartmental models,"
chemical synapses, dendritic trees etc., i.e. explicit computational models of single
neurons. These models have been built by use of SPICE (DC Berkeley) and Genesis
(Caltech). On a higher level of abstraction, the Rochester Connectionist Simulator
(RCS~ University of Rochester) and ICSIM (lCSI Berkeley) allow the definition of
unit types and complex connectivity patterns. On a very high level of abstraction,
simulators like tleam (UCSD) allow the easy realization of pre-defined network architectures (feedforward networks) and leaming algorithms such as backpropagation.
Ben Gomes, International Computer Science Institute (Berkeley) introduced the
Connectionist Supercomputer 1. The CNS-l is a multiprocessor system designed for
moderate precision fixed point operations used extensively in connectionist network
calculations. Custom VLSI digital processors employ an on-chip vector coprocessor
unit tailored for neural network calculations and controlled by RISC scalar CPU. One
processor and associated commercial DRAM comprise a node, which is connected in a
mesh topology with other nodes to establish a MIMD array. One edge of the communications mesh is reserved for attaching various 110 devices, which connect via a custom network adaptor chip. The CNS-l operates as a compute server and one 110 port
is used for connecting to a host workstation.
Users with mainstream connectionist applications can use CNSim, an object-oriented,
graphical high-level interface to the CNS-l environment. Those with more complicated applications can use one of several high-level programming languages (C. C++.
1178
Connectionist Modeling and Parallel Architectures
Sather}, and access a complete set of hand-coded assembler subroutine libraries for
connectionist applications. Simulation, debugging and profiling tools will be available to aid both types of users. Additional tools are available for the systems programmer to code at a low level for maximum perfonnance. Access to the 1I0-level
processor and network functions are provided, along with the evaluation tools needed
to complement the process.
Urs Muller, Swiss Federal Institute of Technology (Zurich) introduced MUSIC: A
high performance neural network simulation tool on a multiprocessor. MUSIC
(Multiprocessor System with Intelligent Communication), a 64 processor system,
runs backpropagation at a speed of 247 million connection updates per second using
32 bit floating-point precision. TIlUS the system reaches supercomputer speed (3.8
gflops peak), it still can be used as a personal desk-top computer at a researchers own
disposal: The complete system consumes less than 800 Watt and fits into a 19 inch
rack.
Fin Martin, Intel Corporation, introduced INiI000," an REF processor which accepts 40,000 patterns per second. Input patterns of 256 dimensions by 5 bits are
transferred from the host to the NilO00 and compared with the chip's "memory" of
1024 stored reference patterns, in parallel. A custom 16 bit on-chip microcontroller
runs at 20 MIPS and controls all the programming and algorithm functions. RBF's
are considered an advancement over traditional template matching algorithms and back
propagation.
Paul Murtagh and Ah Chung Tsoi, University of Queensland (St. Lucia) described a reconfigurable VLSI Systolic Array for artificial neural networks. After a
brief review of some of the most common neural network architectures, e.g., multilayer perceptron, Hopfield net, Boltzmann machine, Ah Chung Tsoi showed that the
training algorithms of these networks can be written in a unified manner. This unified
training algoritlml is then shown to be implementable in a systolic array fashion. The
individual processor can be designed accordingly. Each processor incorporates functionality reconfiguration to allow a number of neural network models to be implemented. The architecture also incorporates reconfiguration for fault tolerance and processor arrangement. Each processor occupies very little silicon area with 16 processors being able to fit onto a lOx 10 nm12 die.
GUnther Palm and Franz Kurfess introduced "Neural Associative Memories."
Despite having processing elements which are thousands of times faster than the neurons in the brain, modem computers still cannot match quite a few processing capabilities of the brain, many of which we even consider trivial (such as recognizing
faces or voices, or following a conversation). A common principle for those capabilities lies in the use of correlations between patterns in order to identify patterns which
are similar. Looking at the brain as an information processing mechanism with -probably among others -- associative processing capabilities together with the converse view of associative memories as certain types of artificial neural networks initiated a number of interesting results. These range from theoretical considerations to insights in the functioning of neurons, as well as parallel hardware implementations of
neural associative memories. The talk discussed some implementation aspects and
presented a few applications.
Finally, Ernst Niebur, California Institute of Technology (pasadena) presented his
work on biologically realistic modeling on SIMD machines (No abstract available).
1179
| 828 |@word coprocessor:1 implemented:1 establish:1 functioning:1 chemical:1 arrangement:1 functionality:1 simulation:3 queensland:5 occupies:1 australia:2 traditional:1 programmer:1 die:1 topic:1 dendritic:1 complete:2 trivial:1 interface:1 systolic:2 code:1 considered:1 mimd:1 consideration:1 written:1 common:2 mesh:2 realistic:1 specialized:1 designed:2 update:1 million:1 discussed:1 dram:1 implementation:3 boltzmann:1 device:1 advancement:1 silicon:1 accordingly:1 neuron:3 modem:1 fin:1 implementable:1 tool:4 federal:1 centre:1 communication:2 node:2 language:1 looking:1 genesis:1 dc:1 ucsd:1 access:2 mainstream:1 along:1 etc:1 introduced:4 own:1 showed:1 complement:1 joachim:2 moderate:1 connection:1 manner:1 certain:1 server:1 california:1 accepts:1 fault:1 simulator:3 brain:3 abstraction:4 multiprocessor:3 i0:1 muller:1 caltech:1 additional:1 able:1 pattern:6 cpu:1 little:1 pasadena:1 vlsi:2 subroutine:1 provided:1 built:1 memory:4 issue:1 among:1 faster:1 match:1 platform:1 calculation:2 profiling:1 rcs:1 unified:2 comprise:1 simd:1 corporation:1 having:1 host:2 coded:1 controlled:1 berkeley:3 technology:4 brief:2 library:1 created:1 multilayer:1 started:1 lox:1 connectionist:10 others:1 intelligent:1 control:1 unit:2 converse:1 employ:1 few:2 oriented:1 tailored:1 review:1 neurocomputing:1 engineering:1 individual:1 floating:1 brisbane:2 despite:1 cns:3 interesting:1 initiated:1 probably:1 digital:1 custom:3 evaluation:1 incorporates:2 controversial:1 port:1 principle:1 feedforward:1 range:1 easy:1 mips:1 fit:2 tsoi:3 edge:1 architecture:5 topology:1 swiss:1 backpropagation:2 sather:1 perfonnance:1 allow:3 tree:1 perceptron:1 institute:3 template:1 area:1 face:1 attaching:1 tolerance:1 theoretical:1 matching:1 dimension:1 pre:1 assembler:1 modeling:6 onto:1 cannot:1 franz:1 recognizing:1 desk:1 extensively:1 hardware:2 stored:1 risc:1 connect:1 gomes:1 spice:1 insight:1 st:2 array:3 international:1 peak:1 per:2 his:1 connecting:1 together:1 commercial:1 connectivity:1 user:2 programming:3 gunther:1 complex:1 element:1 chung:3 paul:1 ref:1 run:2 electrical:1 intel:1 thousand:1 fashion:1 connected:1 aid:1 precision:2 view:1 consumes:1 microcontroller:1 explicit:1 lie:1 environment:2 bit:3 parallel:5 complicated:1 capability:3 dynamic:1 personal:1 rochester:2 reconfigurable:1 reserved:1 lucia:2 identify:1 inch:1 hopfield:1 chip:4 speed:2 various:2 aspect:1 talk:1 music:2 niebur:1 martin:1 researcher:1 transferred:1 processor:10 ah:3 department:1 artificial:2 palm:1 synapsis:1 reach:1 debugging:1 watt:1 diederich:2 quite:1 definition:2 ur:1 compartmental:1 biologically:1 scalar:1 associated:1 workstation:1 associative:4 zurich:1 conversation:1 net:1 murtagh:1 mechanism:1 needed:1 rbf:1 leaming:1 back:1 realization:1 disposal:1 higher:1 available:3 operation:1 operates:1 ernst:1 reconfiguration:3 done:1 correlation:1 voice:1 hand:1 supercomputer:3 top:1 incremental:1 support:1 ben:1 object:1 propagation:1 rack:1 graphical:1 gflops:1 adaptor:1 school:1 |
7,057 | 829 | Neurobiology, Psychophysics, and
Computational Models of Visual
Attention
Ernst Niebur
Computation and Neural Systems
California Institute of Technology
Pasadena, CA 91125, USA
Bruno A. Olshausen
Department of Anatomy and Neurobiology
Washington University School of Medicine
St. Louis, MO 63110
The purpose of this workshop was to discuss both recent experimental findings and
computational models of the neurobiological implementation of selective attention.
Recent experimental results were presented in two of the four presentations given
(C.E. Connor, Washington University and B.C. Motter, SUNY and V.A. Medical
Center, Syracuse), while the other two talks were devoted to computational models
(E. Niebur, Caltech, and B. Olshausen, Washington University).
Connor presented the results of an experiment in which the receptive field profiles of
V 4 neurons were mapped during different states of attention in an awake, behaving
monkey. The attentional focus was manipulated in this experiment by altering the
position of a behaviorally relevant ring-shaped stimulus. The animal's task was to
judge the size of the ring when compared to a reference ring (i.e., same or different).
In order to map the receptive field profile, a behaviorally irrelevant bar stimulus
was flashed at one of several positions inside and outside the classical receptive field
(CRF). It was found that shifts of attention produced alterations in receptive field
profiles for over half the cells studied. In most cases the receptive field center of
gravity translated towards attentional foci in or near the CRF. Changes in width
and shape of the receptive field profile were also observed, but responsive regions
were not typically limited to the location of the attended ring stimulus. Attentionrelated effects often included enhanced responses at certain locations as well as
diminished responses at other locations.
Motter studied the basic mechanisms of visual search as manifested in the single
unit activity of rhesus monkeys. The animals were trained to select a bar stimulus
among others based on the color or luminance of the target stimulus. The majority
of neurons were selectively activated when the color or luminance of the stimulus in
the receptive field matched the color or luminance of the cue, whereas the activity
was attenuated when there was no match. Since a cell responds differently to the
same stimulus depending on the color or luminance of the cue (which is given far
away from the stimulus by the color or luminance of the fixation spot), the activity
of the neurons reflect a selection based on the cued feature and not simply the
physical color or luminance of the receptive field stimulus. Motter showed that the
1167
1168
Niebur and Olshausen
selection can also be based on memory by switching off the cue in the course of
the experiment. The monkey could then perform the task only by relying on his
memory and the pattern of V4 activity. In the memory-based task as well as in
the experiments with the stimulus continuously present, the differential activation
was independent of spatial location and offers therefore a physiological correlate
to psychophysical studies suggesting that stimuli can be preferentially selected in
parallel across the visual field.
Niebur presented a model for the neuronal implementation of selective visual attention based on temporal correlation among groups of neurons. In the model, neurons
in primary visual cortex respond to visual stimuli with a Poisson distributed spike
train with an appropriate, stimulus-dependent mean firing rate. The spike trains
of neurons whose receptive fields do not overlap with the "focus of attention" are
distributed according to homogeneous (time-independent) Poisson process with no
correlation between action potentials of different neurons. In contrast, spike trains
of neurons with receptive fields within the focus of attention are distributed according to non-homogeneous (time-dependent) Poisson processes. Since the short-term
average spike rates of all neurons with receptive fields in the focus of attention covary, correlations between these spike trains are introduced which are detected by
inhibitory interneurons in V 4. These cells, modeled as modified integrate-and-fire
neurons, function as coincidence detectors and suppress the response of V 4 cells
associated with non-attended visual stimuli. The model reproduces quantitatively
experimental data obtained in cortical area V4 of monkey.
The model presented by Olshausen proposed that attentional gating takes place via
an explicit control process, without relying on temporal correlation. This model is
designed to serve as a possible explanation for how the visual cortex forms position
and scale invariant representations of objects. Control neurons dynamically modify
the synaptic strengths of intracortical connections so that information from a windowed region of primary visual cortex is selectively routed to higher cortical areas,
preserving spatial relationships. The control signals for setting the position and size
ofthe attentional window are hypothesized to originate from neurons in the pulvinar
and in the deep layers of visual cortex. The dynamics of these control neurons are
governed by simple differential equations that can be realized by neurobiologically
plausible circuits. In pre-attentive mode, the control neurons receive their input
from a low-level "saliency map" representing potentially interesting regions of a
scene. During the pattern recognition phase, control neurons are driven by the interaction between top-down (memory) and bottom-up (retinal input) sources. The
model predicts that the receptive fields of cortical neurons should shift with attention, as found in Connor's experiments, although the predicted shifts are somewhat
larger than those found to date.
Acknowledgement
The work of EN and BAO was supported by the Office of Naval Research. EN was
additionally supported by the National Science Foundation.
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7,058 | 83 | 72
ANALYSIS AND COMPARISON OF DIFFERENT LEARNING
ALGORITHMS FOR PATTERN ASSOCIATION PROBLEMS
J. Bernasconi
Brown Boveri Research Center
CH-S40S Baden, Switzerland
ABSTRACT
We investigate the behavior of different learning algorithms
for networks of neuron-like units. As test cases we use simple pattern association problems, such as the XOR-problem and symmetry detection problems. The algorithms considered are either versions of
the Boltzmann machine learning rule or based on the backpropagation
of errors. We also propose and analyze a generalized delta rule for
linear threshold units. We find that the performance of a given
learning algorithm depends strongly on the type of units used. In
particular, we observe that networks with ?1 units quite generally
exhibit a significantly better learning behavior than the corresponding 0,1 versions. We also demonstrate that an adaption of the
weight-structure to the symmetries of the problem can lead to a
drastic increase in learning speed.
INTRODUCTION
In the past few years, a number of learning procedures for
neural network models with hidden units have been proposed 1 ,2. They
can all be considered as strategies to minimize a suitably chosen
error measure. Most of these strategies represent local optimization
procedures (e.g. gradient descent) and therefore suffer from all the
problems with local m1n1ma or cycles. The corresponding learning
rates, moreover, are usually very slow.
The performance of a given learning scheme may depend criticallyon a number of parameters and implementation details. General
analytical results concerning these dependences, however, are practically non-existent. As a first step, we have therefore attempted
to study empirically the influence of some factors that could have a
significant effect on the learning behavior of neural network systems.
Our preliminary investigations are restricted to very small
networks and to a few simple examples. Nevertheless, we have made
some interesting observations which appear to be rather general and
which can thus be expected to remain valid also for much larger and
more complex systems.
NEURAL NETWORK MODELS FOR PATTERN ASSOCIATION
An artificial neural network consists of a set of interconnected units (formal neurons). The state of the i-th unit is described
by a variable S. which can be discrete (e.g. S. = 0,1 or S. = ?1) or
continuous (e.l. 0 < S. < 1 or -1 < S. < +ll, and each ~onnection
j-7i carries a weight- W.1. which can be 1positive, zero, or negative.
1J
? American Institute of Physics 1988
73
The dynamics of the network is determined by a local update
rule,
S.(t+l)
1
= HIj
W. . S . (t))
1J
(1)
J
where f is a nonlinear activation function, specifically a threshold
function in the case of discrete units and a sigmoid-type function,
e.g.
(2)
or
(3)
respectively, in the case of continuous units. The individual units
can be given different thresholds by introducing an extra unit which
always has a value of 1.
If the network is supposed to perform a pattern association
task, it is convenient to divide its units into input units, output
units, and hidden units. Learning then consists in adjusting the
weights in such a way that, for a given input pattern, the network
relaxes (under the prescribed dynamics) to a state in which the
output units represent the desired output pattern.
Neural networks learn from examples (input/output pairs) which
are presented many times, and a typical learning procedure can be
viewed as a strategy to minimize a suitably defined error function
F. In most cases, this strategy is a (stochastic) gradient descent
method: To a clamped input pattern, randomly chosen from the learning examples, the network produces an output pattern {O . }. This is
compared with the desired output, say {T . }, and the erfor F( {O. },
{T . }) is calculated . Subsequently, each 1weight is changed by ~an
am~unt proportional to the respective gradient of F,
b.W ..
~J
of
= -r} oW ..
(4)
~J
and the procedure is repeated for a new learning example until F is
minimized to a satisfactory level.
In our investigations, we shall consider two different types of
learning schemes. The first is a deterministic version of the Boltzmann machine learning rule! and has been proposed by Yann Le Cun 2 ?
It applies to networks with symmetric weights, W.. = W.. , so that an
~J
J~
energy
E(~) == - I
W.. S. S .
(i ,j) ~J
~
J
(5)
can be associated with each state S = {S.}. If X refers to the net1
work state when only the input units are clamped and Y to the state
when both the input and output units are clamped, the error function
74
is defined as
= E c:~)
F
- E QO
(6)
and the gradients are simply given by
of- = Y.
-oW. .
1
1J
Y.J
x. X.
1
J
(7)
The second scheme, called backpropagation or generalized delta
rule 1 ,3, probably represents the most widely used learning algorithm.
In its original form, it applies to networks with feedforward connections only, and it uses gradient descent to minimize the mean squared
error of the output signal,
F
= -21 L. (T1
. -1
0.)2
(8)
1
For a weight W.. from an (input or hidden) unit j to an output
unit i, we simply ha~
(9 )
where f' is the derivative of the nonlinear activation function
introduced in Eq. (1), and for weights which do not connect to an
output unit, the gradients can successively be determined by applying the chain rule of differentiation.
In the case of discrete units, f is a threshold function, so
that the backpropagation algorithm described above cannot be applied.
We remark, however, that the perceptron learning rUle 4 ,
~W ..
1J
= ?(T.1
- O.)S.
1
J
(10)
is nothing else than Eq. (9) with f' replaced by a constant ?.
Therefore, we propose that a generalized delta rule for linear
threshold units can be obtained if f' is replaced by a constant ? in
all the backpropagation expressions for of/oW ... This generalization
of the perceptron rule is, of course, not u1dque. In layered networks, e.g., the value of the constant which replaces f' need not be
the same for the different layers.
ANALYSIS OF LEARNING ALGORITHMS
The proposed learning algorithms suffer from all the problems
of gradient descent on a complicated landscape. If we use small
weight changes, learning becomes prohibitively slow, while large
weight changes inevitably lead to oscillations which prevent the
algorithm from converging to a good solution. The error surface,
moreover, may contain many local minima, so that gradient descent is
not guaranteed to find a global minimum.
75
There are several ways to improve a stochastic gradient descent
procedure. The weight changes may, e.g., be accumulated over a
number of learning examples before the weights are actually changed.
Another often used method consists in smoothing the weight changes
by overrelaxation,
of
~W ..
(k+1) = -~ ~W + a ~W .. (k)
a ..
1J
1J
1J
(11)
where ~W .. (k) refers to the weight change after the presentation of
the k-th 1 1earning example (or group of learning examples, respectively). The use of a weight decay term,
~W ..
1J
of
= -11 a~W
..
1J
- BW ..
1J
(12)
prevents the algorithm from generating very large weights which may
create such high barriers that a solution cannot be found in reasonable time.
Such smoothing methods suppress the occurrence of oscillations,
at least to a certain extent, and thus allow us to use higher learning rates. They cannot prevent, however, that the algorithm may
become trapped in bad local minimum. An obvious way to deal with the
problem of local minima is to restart the algorithm with different
initial weights or, equivalently, to randomize the weights with a
certain probability p during the learning procedure. More sophisticated approaches involve, e.g., the use of hill-climbing methods.
The properties of the error-surface over the weight space not
only depend on the choice of the error function F, but also on the
network architecture, on the type of units used, and on possible
restrictions concerning the values which the weights are allowed to
assume.
The performance of a learning algorithm thus depends on many
factors and parameters. These dependences are conveniently analyzed
in terms of the behavior of an appropriately defined learning curve.
For our small examples, where the learning set always consists of
all input/output cases, we have chosen to represent the performance
of a learning procedure by the fraction of networks that are
"perfect" after the presentation of N input patterns. (Perfect networks are networks which for every input pattern produce the correct
output). Such learning curves give us much more detailed information
about the behavior of the system than, e. g., averaged quantities
like the mean learning time.
RESULTS
In the following, we shall present and discuss some representative results of our empirical study. All learning curves refer to
a set of 100 networks that have been exposed to the same learning
procedure, where we have varied the initial weights, or the sequence
76
of learning examples, or both. With one exception (Figure 4), the
sequences of learning examples are always random.
A prototype pattern association problem is the exclusive-or
(XOR) problem. Corresponding networks have two input units and one
output unit. Let us first consider an XOR-network with only one
hidden unit, but in which the input units also have direct connections to the output unit. The weights are symmetric, and we use the
deterministic version of the Boltzmann learning rule (see Eqs. (5)
to (7)). Figure 1 shows results for the case of tabula rasa initial
conditions, i.e. the initial weights are all set equal to zero. If
the weights are changed after every learning example, about 2/3 of
the networks learn the problem with less than 25 presentations per
pattern (which corresponds to a total number of 4 x 25 = 100 presentations). The remaining networks (about 1/3), however, never learn
to solve the XOR-problem, no matter how many input/output cases are
presented. This can be understood by analyzing the corresponding
evolution-tree in weight-space which contains an attractor consisting of 14 "non-perfect" weight-configurations. The probability to
become trapped by this attractor is exactly 1/3. If the weight
changes are accumulated over 4 learning examples, no such attractor
100
en
~
a::
80
I
.-w
z
.-u
60
??? ? ?000?
I-
-
?
??
?
?
-
w
a::
40
Q.
20 .....
lL.
i
ij
0
0
0
0
0
0
0
0
0
? ? ? ? ? ? ? ? ?
0
-
00
0
00
-
00
0
W
~
I
I
-
0
~
I
0
0
-
?0
0
~
0
.o.~.
0
I
I
I
I
20
40
60
80
100
#: PRESENTATIONS /PATTERN
Fig. 1. Learning curves for an XOR-network with one hidden unit
(deterministic Boltzmann learning, discrete ?I units, initial
weights zero). Full circles: weights changed after every learning
example; open circles: weight changes accumulated over 4 learning
examples.
77
seems to exist (see Fig. 1), but for some networks learning at least
takes an extremely long time . The same saturation effect is observed
with random initial weights (uniformly distributed between -1 and
+1), see Fig. 2.
Figure 2 also exhibits the difference in learning behavior
between networks with ?1 units and such with 0,1 units. In both
cases, weight randomization leads to a considerably improved learning behavior. A weight decay term, by the way, has the same effect.
The most striking observation, however, is that ?1 networks learn
much faster than 0,1 networks (the respective average learning times
differ by about a factor of 5). In this connection, we should mention
that ~ = 0.1 is about optimal for 0,1 units and that for ?1 networks
the learning behavior is practically independent of the value of ~.
It therefore seems that ?1 units lead to a much more well-behaved
error-surface than 0,1 units. One can argue, of course, that a
discrete 0,1 model can always be translated into a ?1 model, but
this would lead to an energy function which has a considerably more
complicated weight dependence than Eq. (5).
100
en
~
a:::
80
0
3:
....w
z
60
....
u
w
lL.
40
a:::
w
a..
~
0
20
0
2
5
#
10 20
50 100 200
1000
PRESENTATIONS / PATTERN
Fig. 2. Learning curves for an XOR-network with one hidden unit
(deterministic Boltzmann learning, initial weights random, weight
changes accumulated over 5 learning examples). Circles: discrete ?1
units, ~ = 1; triangles: discrete 0,1 units, ~ = 0.1; broken curves:
without weight randomization; solid curves: with weight randomization (p
0.025).
=
78
Figures 3 and 4 refer to a feedforward XOR-network with 3
hidden units, and to backpropagation or generalized delta rule
learning. In all cases we have included an overrelaxation (or momentum) term with a
0.9 (see Eq. (11?. For the networks with continuous units we have used the activation functions given by Eqs. (2)
and (3), respectively, and a network was considered "perfect" if for
all input/output cases the error was smaller than 0.1 in the 0,1
case, or smaller than 0.2 in the ?1 case, respectively.
In Figure 3, the weights have been changed after every learning
example, and all curves refer to an optimal choice of the only
remaining parameter, ?. or ", respectively. For discrete as well as
for continuous units, the ?1 networks again perform much better than
their 0,1 counterparts. In the continuous case, the average learning
times differ by about a factor of 7, and in the discrete case the
discrepancy is even more pronounced. In addition, we observe that in
?1 networks learning with the generalized delta rule for discrete
units is about twice as fast as with the backpropagation algorithm
for continuous units.
=
100~--~--~----~----~~~~--~--~
en
~
a::
80
0
~
I-
w
60
Z
I0
w
a::
w
a.
lL.
~
40
20
0
O~----~--~------~----~----~~~~--~
10
5
20
50
100
200
500 1000
:# PRESENTATIONS / PATTERN
Fig. 3. Learning curves for an XOR-network with three hidden units
(backpropagation/generalized delta rule, initial weights random,
weights changed after every learning example). Open circles: discrete ?1 units, ?. = 0.05; open triangles: discrete 0,1 units, ?. = 0.025;
full circles: continuous ?1 units, "
0.125; full triangles; continuous 0,1 units, "
0.25.
=
=
79
In Figure 4, the weight changes are accumulated over all 4
input/output cases, and only networks with continuous units are
considered. Also in this case, the ?1 units lead to an improved
learning behavior (the optimal Il-values are about 2.5 and 5.0,
respectively). They not only lead to significantly smaller learning
times, but ?1 networks also appear to be less sensitive with respect
to a variation of 11 than the corresponding 0,1 versions.
The better performance of the ?1 models with continuous units
can partly be attributed to the steeper slope of the chosen activation function, Eq. (3). A comparison with activation functions that
have the same slope, however, shows that the networks with ?1 units
still perform significantly better than those with 0,1 units. If the
weights are updated after every learning example, e.g., the reduction in learning time remains as large as a factor of 5. In the case
of backpropagation learning, the main reason for the better performance of ?1 units thus seems to be related to the fact that the
algorithm does not modify weights which emerge from a unit with
value zero. Similar observations have been made by Stornetta and
Huberman,s who further find that the discrepancies become even more
pronounced if the network size is increased.
100
"1
CI)
~
a:
=5.0
80
0
~
I-
w
z
60
I-
u
w
lL.
40
a:
w
a..
~
0
20
0
0
50
#
100
150
200
250
PRESENTATIONS I PATTERN
Fig. 4. Learning curves for an XOR-network with three hidden units
(backpropagation, initial weights random, weight changes accumulated
over all 4 input/output cases). Circles: continuous ?1 units;
triangles: continuous 0,1 units.
80
In Figure 5, finally, we present results for a network that
learns to detect mirror symmetry in the input pattern. The network
consists of one output, one hidden, and four input units which are '
also directly connected to the output unit. We use the deterministic
version of Boltzmann learning and change the weights after every
presentation of a learning pattern . If the weights are allowed to
assume arbitrary values, learning is rather slow and on average
requires almost 700 presentations per pattern. We have observed,
however, that the algorithm preferably seems to converge to solutions in which geometrically symmetric weights are opposite in sign
and almost equal in magnitude (see also Ref. 3). This means that the
symmetric input patterns are automatically treated as equivalent, as
their net input to the hidden as well as to the output unit is zero.
We have therefore investigated what happens if the weights are
forced to be antisymmetric from the beginning. (The learning procedure, of course, has to be adjusted such that it preserves this
antisymmetry). Figure 5 shows that such a problem-adapted weightstructure leads to a dramatic decrease in learning time.
100
en
~
?
??
?
?
??
?
80
a::
0
3:
I-
w
z
60
(,)
w
a::
0
0
0
0
0
0
?
?
???
?
lLL.
0
0
40
lLI
0
0
0
0
a..
~
0
20
0
2
0
?
??
10 20
5
#
0
0
0
0
50 100 200
500
2000
PRESENTATIONS I PATTERN
Fig. 5. Learning curves for a symmetry detection network with 4
input units and one hidden unit (deterministic Boltzmann learning,
11
1, discrete ?1 units, initial weights random, weights changed
after every learning example).
Full circles: symmetry-adapted
weights; open circles: arbitrary weights, weight randomization
(p
0.015).
=
=
81
CONCLUSIONS
The main results of our empirical study can be summarized as
follows:
- Networks with ?1 units quite generally exhibit a significantly
faster learning than the corresponding 0,1 versions.
- In addition, ?1 networks are often less sensitive to parameter variations than 0,1 networks.
- An adaptation of the weight-structure to the symmetries of the
problem can lead to a drastic improvement of the learning behavior.
Our qualitative interpretations seem to indicate that the observed effects should not be restricted to the small examples considered in this paper. It would be very valuable, however, to have
corresponding analytical results.
REFERENCES
1. "Parallel Distributed Processing: Explorations in the Microstructure of Cognition", vol. 1: "Foundations", ed. by D.E. Rumelhart
and J.L. McClelland (MIT Press, Cambridge), 1986, Chapters 7 & 8.
2. Y. Ie Cun, in "Disordered Systems and Biological Organization",
ed . by E. Bienenstock, F. Fogelman Soulie, and G. Weisbuch (Springer, Berlin), 1986, pp. 233-240.
3. D.E. Rumelhart, G.E. Hinton, and R.J. Williams, Nature 323, 533
(1986).
4. M.L. Minsky and S. Papert, "Perceptrons" (MIT Press, Cambridge),
1969.
5. W.S. Stornetta and B.A. Huberman, IEEE Conference on "Neural Networks", San Diego, California, 21-24 June 1987.
| 83 |@word version:7 seems:4 suitably:2 open:4 dramatic:1 mention:1 solid:1 carry:1 reduction:1 initial:10 configuration:1 contains:1 past:1 activation:5 net1:1 update:1 beginning:1 direct:1 become:3 qualitative:1 consists:5 expected:1 behavior:10 automatically:1 lll:1 becomes:1 moreover:2 what:1 weisbuch:1 differentiation:1 every:8 preferably:1 exactly:1 prohibitively:1 unit:66 appear:2 positive:1 t1:1 before:1 local:6 understood:1 modify:1 analyzing:1 twice:1 averaged:1 backpropagation:9 procedure:9 empirical:2 significantly:4 convenient:1 refers:2 cannot:3 layered:1 influence:1 applying:1 restriction:1 equivalent:1 deterministic:6 center:1 williams:1 rule:13 variation:2 updated:1 diego:1 us:1 rumelhart:2 observed:3 cycle:1 connected:1 decrease:1 valuable:1 broken:1 dynamic:2 existent:1 depend:2 unt:1 exposed:1 triangle:4 translated:1 chapter:1 forced:1 fast:1 artificial:1 quite:2 larger:1 widely:1 solve:1 say:1 sequence:2 analytical:2 net:1 propose:2 interconnected:1 adaptation:1 supposed:1 pronounced:2 produce:2 generating:1 perfect:4 ij:1 eq:7 indicate:1 differ:2 switzerland:1 correct:1 stochastic:2 subsequently:1 exploration:1 disordered:1 microstructure:1 generalization:1 preliminary:1 investigation:2 randomization:4 biological:1 adjusted:1 practically:2 onnection:1 considered:5 cognition:1 sensitive:2 create:1 mit:2 always:4 rather:2 june:1 improvement:1 am:1 detect:1 accumulated:6 i0:1 hidden:12 bienenstock:1 fogelman:1 smoothing:2 equal:2 never:1 represents:1 discrepancy:2 minimized:1 few:2 randomly:1 preserve:1 individual:1 replaced:2 minsky:1 consisting:1 bw:1 attractor:3 detection:2 organization:1 investigate:1 analyzed:1 chain:1 respective:2 tree:1 divide:1 desired:2 circle:8 increased:1 introducing:1 connect:1 considerably:2 ie:1 physic:1 squared:1 again:1 baden:1 successively:1 american:1 derivative:1 summarized:1 matter:1 depends:2 analyze:1 steeper:1 complicated:2 parallel:1 slope:2 minimize:3 il:1 xor:9 who:1 landscape:1 climbing:1 lli:1 ed:2 energy:2 pp:1 obvious:1 associated:1 attributed:1 adjusting:1 sophisticated:1 actually:1 higher:1 improved:2 strongly:1 until:1 qo:1 nonlinear:2 behaved:1 effect:4 brown:1 contain:1 counterpart:1 evolution:1 symmetric:4 satisfactory:1 deal:1 ll:5 during:1 generalized:6 hill:1 demonstrate:1 sigmoid:1 empirically:1 association:5 interpretation:1 significant:1 refer:3 cambridge:2 rasa:1 surface:3 certain:2 minimum:4 tabula:1 converge:1 signal:1 full:4 faster:2 long:1 concerning:2 converging:1 represent:3 addition:2 else:1 appropriately:1 extra:1 probably:1 seem:1 feedforward:2 relaxes:1 architecture:1 opposite:1 prototype:1 expression:1 suffer:2 remark:1 generally:2 detailed:1 involve:1 mcclelland:1 exist:1 sign:1 delta:6 trapped:2 per:2 discrete:13 shall:2 vol:1 group:1 four:1 threshold:5 nevertheless:1 prevent:2 overrelaxation:2 geometrically:1 fraction:1 year:1 striking:1 almost:2 reasonable:1 yann:1 oscillation:2 bernasconi:1 earning:1 layer:1 guaranteed:1 replaces:1 adapted:2 speed:1 prescribed:1 extremely:1 remain:1 smaller:3 cun:2 happens:1 restricted:2 remains:1 discus:1 drastic:2 observe:2 occurrence:1 original:1 remaining:2 quantity:1 strategy:4 randomize:1 exclusive:1 dependence:3 exhibit:3 gradient:9 ow:3 berlin:1 restart:1 argue:1 extent:1 reason:1 equivalently:1 hij:1 negative:1 suppress:1 implementation:1 boltzmann:7 perform:3 neuron:2 observation:3 descent:6 inevitably:1 hinton:1 antisymmetry:1 varied:1 arbitrary:2 introduced:1 pair:1 connection:3 california:1 usually:1 pattern:21 saturation:1 treated:1 scheme:3 improve:1 interesting:1 proportional:1 foundation:1 course:3 changed:7 formal:1 allow:1 perceptron:2 institute:1 barrier:1 emerge:1 distributed:2 curve:11 calculated:1 soulie:1 valid:1 made:2 san:1 global:1 continuous:12 learn:4 nature:1 symmetry:6 investigated:1 complex:1 antisymmetric:1 main:2 nothing:1 stornetta:2 repeated:1 allowed:2 ref:1 fig:7 representative:1 en:4 slow:3 papert:1 momentum:1 clamped:3 learns:1 bad:1 decay:2 ci:1 mirror:1 magnitude:1 simply:2 conveniently:1 prevents:1 applies:2 springer:1 ch:1 corresponds:1 adaption:1 viewed:1 presentation:11 change:11 included:1 determined:2 specifically:1 typical:1 uniformly:1 huberman:2 called:1 total:1 partly:1 attempted:1 perceptrons:1 exception:1 |
7,059 | 830 | A Computational Model
for Cursive Handwriting
Based on the Minimization Principle
Yasuhiro Wada
*
Yasuharu Koike
Eric Vatikiotis-Bateson
Mitsuo Kawato
ATR Human Infonnation Processing Research Laboratories
2-2 Hikaridai, Seika-cho, Soraku-gun, Kyoto 619-02, Japan
ABSTRACT
We propose a trajectory planning and control theory for continuous
movements such as connected cursive handwriting and continuous
natural speech. Its hardware is based on our previously proposed
forward-inverse-relaxation neural network (Wada & Kawato, 1993).
Computationally, its optimization principle is the minimum torquechange criterion. Regarding the representation level, hard constraints
satisfied by a trajectory are represented as a set of via-points extracted
from a handwritten character. Accordingly, we propose a via-point
estimation algorithm that estimates via-points by repeating the
trajectory formation of a character and the via-point extraction from the
character. In experiments, good quantitative agreement is found
between human handwriting data and the trajectories generated by the
theory. Finally, we propose a recognition schema based on the
movement generation. We show a result in which the recognition
schema is applied to the handwritten character recognition and can be
extended to the phoneme timing estimation of natural speech.
1 INTRODUCTION
In reaching movements, trajectory formation is an ill-posed problem because the hand
can move along an infinite number of possible trajectories from the starting to the target
point. However, humans move an arm between two targets along consistent one of an
>II
Present Address: Systems Lab., Kawasaki Steel Corporation,
Makuhari Techno Garden, 1-3.Nakase, Mihama-ku, Chiba 261, Japan
727
728
Wada, Koike, Vatikiotis-Bateson, and Kawato
infinite number of trajectories. Therefore, the brain should be able to compute a unique
solution by imposing an appropriate criterion to the ill-posed problem. Especially, a
smoothness performance index was intensively studied in this context.
Flash & Hogan (1985) proposed a mathematical model, the minimum-jerk model. Their
model is based on the kinematics of movement, independent of the dynamics of the
musculoskeletal system. On the other hand, based on the idea that the objective function
must be related to dynamics, Uno, Kawato & Suzuki (1989) proposed the minimum
torque-change criterion which accounts for the desired trajectory determination. The
criterion is based on the theory that the trajectory of the human arm is determined so as to
minimize the time integral of the square of the rate of torque change. They proposed the
following quadratic measure of performance. Where -r j is the torque generated by the jth actuator of M actuators, and ljis the movement time.
CT =
0)2 dt
rJo L( -d-r'
dt
"
M
(1)
j=l
Handwriting production is an attractive subject in human motor control studies. In
cursive handwriting, a symbol must be transformed into a motor command stream. This
transformation process raises several questions. How can the central nervous system
(eNS) represent a character symbol for producing a handwritten letter? By what principle
can motor planning be made or a motor command be produced? In this paper we propose
a handwriting model whose computational theory and representation are the same as the
model in reaching movements. Our proposed computational model for cursive
handwriting is assumed to generate a trajectory that passes through many via-points. The
computational theory is based on the minimum torque-change criterion, and a
representation of a character is assumed to be expressed as a set of via-points extracted
from a handwritten character. In reaching movement, the boundary condition is given by
the visual information, such as the location of a cup, and the trajectory formation is based
on the minimum torque-change criterion, which is completely the same as the model of
handwriting (Fig. 1). However, it is quite difficult to determine the via-points in order to
reproduce a cursive handwritten character. We propose an algorithm that can determine
the via-points of the handwritten character, based only on the same minimization
principle and which does not use any other ad hoc information such as zero-crossing
velocity (Hollerbach, 1981).
Reaching
(reach to
the object)
Representation
Computational
n================nTheory
Location
.of the object
Handwriting - .
(write a character)
t
Visual Information
Via-Point
(representation
of character)
Via-poitt Estimation
Algorithm
Hardware
l-,
: '!H;11~ ~
... r- -~-~--0r;1::""=""'=~="":"'~"
- jk l~t~C(
..
Figure 1: A handwriting model.
A Computational Model for Cursive Handwriting Based on the Minimization Principle
2 PREVIOUS WORK ON THE HANDWRITING MODEL
Several handwriting models (Hollerbach, 1981; Morasso & Mussa-Ivaldi, 1982; Edleman
& Flash, 1987) have been proposed. Hollerbach proposed a handwriting model based on
oscillation theory. The model basically used a vertical oscillator and a horizontal
oscillator. Morasso & Mussa-Ivaldi proposed a trajectory formation model using a spline
function, and realized a handwritten character using the formation model.
Edleman & Flash (1987) proposed a handwriting model based on snap (fourth derivative
of position) minimization. The representation of a character was four basic strokes and a
handwritten character was regenerated by a combination of several strokes. However,
their model was different from their theory for reaching movement. Flash & Hogan
(1985) have proposed the minimum jerk criterion in the reaching movement.
3
A HANDWRITING MODEL
3.1 Trajectory formation neural network:
Forward-Inverse Relaxation Model (FIRM)
First, we explain the trajectory formation neural network. Because the dynamics of the
human arm are nonlinear, finding a unique trajectory based on the minimum torquechange criterion is a nonlinear optimization problem. Moreover, it is rather difficult.
There are several criticisms of previous proposed neural networks based on the minimum
torque-change criterion: (1) their spatial representation of time, (2) back propagation is
essential, and (3) much time is required. Therefore, we have proposed a new neural
network, FIRM(Forward-Inverse Relaxation Model) for trajectory formation (Wada &
Kawato, 1993). This network can be implemented as a biologically plausible neural
network and resolve the above criticisms.
3.2 Via-point estimation model
Edelman & Flash (1987) have pointed out the difficulty of finding the via-points in a
handwritten character. They have argued two points: (1) the number of via-points, (2) a
reason for the choice of every via-point locus. It is clear in approximation theory that a
character can be regenerated perfectly if the number of extracted via-pOints is large.
Appropriate via-points can not be assigned according to a regular sampling rule if the
sample duration is constant and long. Therefore, there is an infinite number of
combinations of numbers and via-point positions in the problem of extracting via-points
from a given trajectory, and a unique solution can not be found if a trajectory reformation
theory is not identified. That is, it is an ill-posed problem.
The algorithm for assigning the via-points finds the via-points by iteratively activating
both the trajectory formation module (FIRM) and the via-point extraction module (Fig.
2). The trajectory formation module generates a trajectory based on the minimum torquechange criterion using the via-points which are extracted by the via-point extraction
module. The via-point extraction module assigns the via-points so as to minimize the
square error between the given trajectory and the trajectory generated by the trajectory
formation module. The via-point extraction algorithm will stop when the error between
the given trajectory and the trajectory generated from the extracted via-points reaches a
threshold.
729
730
Wada, Koike, Vatikiotis-Bateson, and Kawato
Via-Points Extraction Module Minimum TorqueChange Trajectory
f'IM
o j~l((J1.(I) -9~ta(t)5dl
- - . . Min
Via-points assignment to
decrease the above trajectory
error
--....
..-
Via-Point
Information
(Position . Time)
Trajectory Formation Module
(FIRM)
f'r~
o (~y<h
dI
j=1
?
Min
Trajectory generation
based on minimum torquechange criterion
Figure 2: Via-point estimation model. 9~ta(t) is the given trajectory of the j-th joint angle
and ei (I) represents the generated trajectory.
3.2.1 Algorithm of via-point extraction
There are a via-point extraction procedure and a trajectory production procedure in the
via-point extraction module. and they are iteratively computed. Trajectory production in
the module is based on the minimum-jerk model (Flash & Hogan 1985) on a joint angle
space. which is equivalent to the minimum torque-change model when arm dynamics are
approximated as in the following dynamic equation:
",i = [i Oi
(j= 1..... M)
(2)
where Ii and iji are the inertia of the link and the acceleration of the j-th joint angle.
respectively.
The algorithm for via-point extraction is illustrated in Fig. 3. The procedural sequence is
as follows:
(Step 1) A trajectory between a starting point and a final point is generated by using the
minimum torque-change principle of the linear dynamics model.
(Step 2) The point with the maximum square error value between the given trajectory and
the generated trajectory is selected as a via-point candidate.
(Step 3) If the maximum value of the square error is less than the preassigned threshold.
the procedure described above is finished. If the maximum value of the square error is
greater than the threshold. the via-point candidate is assigned as via-point i and a
trajectory is generated from the starting point through the via-point i to the final point.
This generated trajectory is added to the trajectory that has already been generated. The
time of the start point of the generated trajectory is a via-point located just before the
assigned via-point i. and the time of the final point of the generated trajectory is a viapoint located just after the assigned via-point i. The position error of the start point and
the final point equal O. since the compensation for the error has already been made. Thus,
the boundary conditions of the generated trajectory at the start and final point become O.
The velocity and acceleration constraints at the start and final point are set to O.
(Step 4) By repeating Steps 2 and 3, a set of via-points is found.
The j-th actuator velocity constraint 9!ia and acceleration constraint O!ia at the via-point i
are set by minimizing the following equation.
J(8!ia,O~a) [p{ J,r:!" (lP)2 dt + J,r:}... (8'i)2 dt} ~ Min
=
O
(3)
A Computational Model for Cursive Handwriting Based on the Minimization Principle
I Step 3 1
~Ory
time
..
by Step3
Figure 3: An algorithm for extracting via-points.
Finally, the via-points are fed to the FIRM, and the minimum torque change trajectory is
produced. This trajectory and the given trajectory are then compared again. If the value
of the square error does not reach the threshold, the procedure above is repeated.
It can be mathematically shown that a given trajectory is perfectly approximated with this
method (completeness), and furthermore that the number of extracted via-points for a
threshold is the minimum (optimality). (Wada & Kawato, 1994)
4 PERFORMANCE OF THE VIA-POINT ESTIMATION MODEL
4.1 Performance of single via-point movement
First, we examine the performance of our proposed via-point estimation model. A result
of via-point estimation in a movement with a via-point is shown in Fig 4. Two
movements (T3-PI-T5 and T3-P2-T5) are examined. The white circle and the solid lines
show the target points and measured trajectories, respectively. PI and P2 show target
via-points. The black circle shows the via-points estimated by the algorithm. The
estimated via-points were close to the target via-points. Thus, our proposed via-point
estimation algorithm can find a via-point on the given trajectory.
0.65
0.60
? Estimated Via-Point
0 TargetPoint
0
PI
0.55
T5
]: 0.50
>0 0.45
0.40
0.35
0 .30 '--.-----,..--...,....-~---r--r-__.-0.3 -0.2 -0.1
0.0
0.1
0.2
0.3
X[m]
Figure 4: A result of via-point estimation in a movement with a via-point.
4.2 Performance of the handwriting model
Fig. 5 shows the case of cursive connected handwritten characters. The handwriting
model can generate trajectories and velocity curves of cursive handwritten characters that
are almost identical to human data. The estimated via-points are classified into two
groups. The via-points in one group are extracted near the minimum points of the
731
732
Wada, Koike, Vatikiotis-Bateson, and Kawato
0.$2
?
Eatimar.cd Via-Point
????? Trajeclary by IIICIdoI
~.10
0.00
0.10
X(ID)
(a)
(b)
Figure 5: Estimated via-points in cursive handwriting. (a) and (b) show the trajectory and
tangential velocity profile, respectively. The via-point estimation algorithm extracts a viapoint (segmentation point) between characters.
velocity profile. The via-points of the other group are assigned to positions that are
independent of the above points. Generally, the minimums of the velocity are considered
to be the feature points of the movement. However, we confirmed that a given trajectory
can not be reproduced by using only the first group of via-points. This finding shows that
the second group of via-points is important. Our proposed algorithm based on the
minimization principle can estimate points that can not be selected by any kinematic
criterion. Funhermore, it is important in handwritten character recognition that the viapoint estimation algorithm extracts via-points between characters, that is, their
segmentation points.
5
FROM FORMATION TO RECOGNITION
5.1 A recognition model
Next, we propose a recognition system using the trajectory formation model and the viapoint estimation model. There are several reports in the literature of psychology which
suggest that the formation process is related to the recognition process. (Liberman &
Mattingly, 1985; Freyd, 1983)
Here, we present a pattern recognition model that strongly depends on the handwriting
model and the via-point estimation model (Fig.6). (1) The features of the handwritten
character are extracted by the via-point estimation algorithm. (2) Some of via-points are
segmented and normalized in space and time. Then, (3) a trajectory is regenerated by
using the normalized via-points. (4) A symbol is identified by comparing the regenerated
trajectory with the template trajectory.
....
QJ
~
E
''=
~
-E
.5.:
o.~
c.."""
IQ
.!~
;>
Recognizer
~ (Reformation
& Comparison)
~Ymb'
Figure 6: Movement pattern recognition using extracted via-points obtained through
movement pattern generator
A Computational Model for Cursive Handwriting Based on the Minimization Principle
rItwz-
1 :BAD : (0,17) (18,35) (36,52)
2 :BAD : (0,18) (18,35) (36,52)
3 :BAD : (0,17) (18,35) (35,52)
1 :DEAR : (0,8) (9,18) (19,31) (30,51)
2 :DEAR : (0,8) (9,18) (19,31) (30,50)
3 :DEAR : (0,8) (9,18) (19,30) (30,51)
Figure 7: Results of character recognition
5.2 Performance of the character recognition model
Fig. 7 shows a result of character recognition. The right-hand side shows the recognition
results for the left-hand side. The best three candidates for recognition are listed.
Numerals in parentheses show the number of starting via-points and the final via-point
for the recognized character.
5.3 Performance of the estimation of timing of phonemes in real speech
Fig. 8 shows the acoustic waveform, the spectrogram, and the articulation movement
when the sentence" Sam sat on top of the potato cooker... " is spoken. The phonemes are
identified, and the vertical lines denote phoneme midpoints. White circles show the viapoints estimated by our proposed algorithm. Rather good agreement is found between the
estimated via-points and the phonemes.
From this experiment, we can point out two important possibilities for the estimation
model of phoneme timing. The first possibility concerns speech recognition, and the
second concerns speech data compression. It seems possible to extend the via-point
estimation algorithm to speech recognition if a mapping from acoustic to articulator
motion is identified (Shirai & Kobayashi, 1991, Papcun et al., 1992). Furthermore, with
training of a forward mapping from articulator motion to acoustic data (Hirayama et al.,
1993), the via-point estimation model can be used for speech data compression.
6 SUMMARY
We have proposed a new handwriting model. In experiments, good qualitative and
quantitative agreement is found between human handwriting data and the trajectories
generated by the model. Our model is unique in that the same optimization principle and
hard constraints used for reaching are also used for cursive handwriting. Also, as
opposed to previous handwriting models, determination of via-points is based on the
optimization principle and does not use a priori knowledge.
We have demonstrated two areas of recognition, connected cursive handwritten character
recognition and the estimation of phoneme timing. We incorporated the formation model
into the recognition model and realized the recognition model suggested by Freyd (1983)
and Liberman and Mattingly(1985). The most important point shown by the models is
that the human recognition process can be realized by specifying the human formation
process.
REFERENCES
S. Edelman & T. Flash (1987) A Model of Handwriting. Bioi. Cybern. ,57,25-36.
733
734
Wada, Koike, Vatikiotis-Bateson, and Kawato
... n.~"'~fl>
cooker...
Figure 8: Estimation result of phoneme time. Temporal acoustics and vertical
positions of the tongue blade (TBY),tongue tip (TTY), jaw (lY), and lower lip (LLY)
are shown with overlaid via-point trajectories. Vertical lines correspond to acoustic
segment centers; 0 denotes via-points.
T. Flash, & N. Hogan (1985) The coordination of arm movements; An experimentally
confirmed mathematical model. Journal of Neuroscience, 5, 1688-1703.
J. J. Freyd (1983) Representing the dynamics of a static fonn. Memory & Cognition, 11,
342-346.
M. Hirayama, E. Vatikiotis-Bateson, K. Honda, Y. Koike, & M. Kawato (1993)
Physiologically based speech synthesis. In Giles, C. L., Hanson, S. J., and Cowan, J. D.
(eds) Advances in Neural Information Processing Systems 5,658-665. San Mateo, CA:
Morgan Kaufmann Publishers.
1. M. Hollerbach (1981) An oscillation theory of handwriting. Bioi. Cybern., 39,139-156.
A. M. Liberman & 1. G. Mattingly (1985) The motor theory of speech perception
revised. Cognition, 21, 1-36.
P. Morasso, & F. A. Mussa-Ivaldi (1982) Trajectory formation and handwriting: A
computational model. Bioi. Cybern. ,45, 131-142.
J. Papcun, J. Hochberg, T. R. Thomas, T. Laroche, J. Zacks, & S. Levy (1992) Inferring
articulation and recognition gestures from acoustics with a neural network trained on xray microbeam data. Journal of Acoustical Society of America, 92 (2) Pt. 1.
K. Shirai, & T. Kobayashi (1991) Estimation of articulatory motion using neural
networks. Journal of Phonetics, 19, 379-385.
Y. Uno, M. Kawato, & R. Suzuki (1989) Formation and control of optimal trajectory in
human arm movement - minimum torque-change model. BioI. Cybern. 61, 89-101.
Y. Wada, & M. Kawato (1993) A neural network model for arm trajectory formation
using forward and inverse dynamics models. Neural Networks, 6(7),919-932.
Y. Wada, & M. Kawato (1994) Long version of this paper, in preparation.
PART VI
ApPLICATIONS
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7,060 | 831 | Learning Classification with Unlabeled Data
Virginia R. de Sa
[email protected]
Department of Computer Science
University of Rochester
Rochester, NY 14627
Abstract
One of the advantages of supervised learning is that the final error metric is available during training. For classifiers, the algorithm can directly
reduce the number of misclassifications on the training set. Unfortunately, when modeling human learning or constructing classifiers for autonomous robots, supervisory labels are often not available or too expensive. In this paper we show that we can substitute for the labels by
making use of structure between the pattern distributions to different sensory modalities. We show that minimizing the disagreement between the
outputs of networks processing patterns from these different modalities is
a sensible approximation to minimizing the number of misclassifications
in each modality, and leads to similar results. Using the Peterson-Barney
vowel dataset we show that the algorithm performs well in finding appropriate placement for the codebook vectors particularly when the confuseable classes are different for the two modalities.
1
INTRODUCTION
This paper addresses the question of how a human or autonomous robot can learn to classify
new objects without experience with previous labeled examples. We represent objects
with n-dimensional pattern vectors and consider piecewise-linear classifiers consisting of
a collection of (labeled) codebook vectors in the space of the input patterns (See Figure 1).
The classification boundaries are gi ven by the voronoi tessellation of the codebook vectors.
Patterns are said to belong to the class (given by the label) of the codebook vector to which
they are closest.
112
Learning Classification with Unlabeled Data
?
0
XB
?
0
o
o
o
? XB
o
Figure 1: A piecewise-linear classifier in a 2-Dimensional input space. The circles represent data
samples from two classes (filled (A) and not filled (B)). The X's represent codebook vectors (They
are labeled according to their class A and B). Future patterns are classified according to the label of
the closest codebook vector.
In [de Sa and Ballard, 1993] we showed that the supervised algorithm LVQ2.1[Kohonen,
1990] moves the codebook vectors to minimize the number of misclassified patterns. The
power of this algorithm lies in the fact that it directly minimizes its final error measure (on
the training set). The positions of the codebook vectors are placed not to approximate the
probability distributions but to decrease the number of misclassifications .
Unfortunately in many situations labeled training patterns are either unavailable or expensive. The classifier can not measure its classification performance while learning (and
hence not directly maximize it). One such unsupervised algorithm, Competitive Learning[Grossberg, 1976; Kohonen, 1982; Rumelhart and Zipser, 1986], has unlabeled codebook vectors that move to minimize a measure of the reconstruction cost. Even with subsequent labeling of the codebook vectors, they are not well suited for classification because
they have not been positioned to induce optimal borders.
Supervised
- implausible label
Unsupervised
-limited power
"COW"
Self-Supervised
- derives label from a
co-occuring input to
another modality
~
Target
\I
??
o
?
O{}OO
~
O~ 0
000
000
o
?
??
O{}OO
~
o
??
?
O{}OO
~
0 600
o
??
?
O{}OO
Input 2
moo
Figure 2: The idea behind the algorithm
This paper presents a new measure for piecewise-linear classifiers receiving unlabeled patterns from two or more sensory modalities. Minimizing the new measure is an approximation to minimizing the number of misclassifications directly. It takes advantage of the
structure available in natural environments which results in sensations to different sensory
modalities (and sub-modalities) that are correlated. For example, hearing "mooing" and
113
114
de Sa
p
0.5
0.4
0.3
p
0.5
1\
I \
0.4
0.3
P(CB)P(,,*~)
0.2
I I
I
I
\
\
\
\
Figure 3: This figure shows an example world as sensed by two different modalities. If modality A
receives a pattern from its Class A distribution, modality 2 receives a pattern from its own class A
distribution (and the same for Class B). Without receiving information about which class the patterns
came from, they must try to determine appropriate placement of the boundaries b l and b2 ? P(C;) is
the prior probability of Class i and p(xjIC;) is the conditional density of Class i for modality j
seeing cows tend to occur together. So, although the sight of a cow does not come with an
internal homuncular "cow" label it does co-occur with an instance of a "moo". The key
is to process the "moo" sound to obtain a self-supervised label for the network processing
the visual image of the cow and vice-versa. See Figure 2.
2 USING MULTI-MODALITY INFORMATION
One way to make use of the cross-modality structure is to derive labels for the codebook
vectors (after they have been positioned either by random initialization or an unsupervised
algorithm). The labels can be learnt with a competitive learning algorithm using a network
such as that shown in Figure 4. In this network the hidden layer competitive neurons represent the codebook vectors. Their weights from the input neurons represent their positions
in the respective input spaces. Presentation of the paired patterns results in activation of
the closest codebook vectors in each modality (and D's elsewhere). Co-occurring codebook vectors will then increase their weights to the same competitive output neuron. After
several iterations the codebook vectors are given the (arbitrary) label of the output neuron
to which they have the strongest weight. We will refer to this as the "labeling algorithm".
2.1
MINIMIZING DISAGREEMENT
A more powerful use of the extra information is for better placement of the codebook
vectors themselves.
In [de Sa, 1994] we derive an algorithm that minimizes l the disagreement between the
outputs of two modalities. The algorithm is originally derived not as a piecewise-linear
classifier but as a method of moving boundaries for the case of two classes and an agent
with two I-Dimensional sensing modalities as shown in Figure 3.
Each class has a particular pro babili ty distri buti on for the sensation received by each modality. If modality 1 experiences a sensation from its pattern A distribution, modality 2 experiences a sensation from its own pattern A distribution. That is, the world presents patterns
Ithe goal is actually to find a non-trivial local minimum (for details see [de Sa, 1994])
Learning Classification with Unlabeled Data
Output (Class)
000
Hidden Layer
Code book
Vectors (W)
Input (X)
ModaiitylNetwork 1
ModalitylNetwork 2
Figure 4: This figure shows a network for learning the labels of the codebook vectors. The weight
vectors of the hidden layer neurons represent the codebook vectors while the weight vectors of the
connections from the hidden layer neuron!; to the output neurons represent the output class that each
codebook vector currently represents. In this example there are 3 output classes and two modalities
each of which has 2-D input patterns and 5 codebook vectors.
from the 2-D joint distribution shown in Figure 5a) but each modality can only sample its
1-D marginal distribution (shown in Figure 3 and Figure 5a).
We show [de Sa, 1994] that minimizing the disagreement error of patterns for which the two modalities output different labels E(b), b2) = Pr{x) < b) & X2 > bJ}
the proportion of pairs
+ Pr{x) > b) & X2 < b2}
(1)
(2)
(where f(x). X2) = P(CA)p(xtICA)P(X2ICA) + P(CB)p(x1ICB)p(x2ICB) is the joint probability
density for the two modalities) in the above problem results in an algorithm that corresponds
to the optimal supervised algorithm except that the "label" for each modality's pattern is
the hypothesized output of the other modality.
Consider the example illustrated in Figure 5. In the supervised case (Figure 5a?) the labels
are given allowing sampling of the actual marginal distributions. For each modality, the
number of misclassifications can be minimized by setting the boundaries for each modality
at the crossing points of their marginal distributions.
However in the self-supervised system, the labels are not available. Instead we are given
the output of the other modality. Consider the system from the point of view of modality
2. Its patterns are labeled according to the outputs of modality 1. This labels the patterns
in Class A as shown in Figure 5b). Thus from the actual Class A patterns, the second
modality sees the "labeled" distributions shown. Letting a be the fraction of misclassified
patterns from Class A, the resulting distributions are given by (1 - a)P(CA)P(X2ICA) and
(a)P(CA)P(X2ICA).
Similarly Figure 5c) shows the effect on the patterns in class B. Letting b be the fraction of Class B patterns misclassified, the distributions are given by (1 - b)P( CB)P(X2ICB)
115
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de Sa
and (b)P( CB)p(X2ICB). Combining the effects on both classes results in the "labeled"
distributions shown in Figure 5d). The "apparent Class ~' distribution is given by
(1 - a)P(CA)P(X2ICA) + (b)P(CB)p(X2ICB and the "apparent Class B" distribution by
(a)P(CA)P(X2ICA) + (1-b)P(CB)p(x2ICB). Notice that even though the approximated distributions may be discrepant, if a:::: b, the crossing point will be close.
Simultaneously the second modality is labeling the patterns to the first modality. At each
iteration of the algorithm both borders move according to the samples from the "apparent"
marginal distributions.
- P(CA)p(x1ICA)
- P(CB)p(x1ICB)
- (a)P(CA}p(x2ICA)
- (1-a)P(CA)p(x2ICA)
a)
Figure 5: This figure shows an example of the joint and marginal distributions (For better visualization the scale of the joint distribution is twice that of the marginal distributions) for the example
problem introduced in Figure 3. The darker gray represents patterns labeled "N', while the lighter
gray are labeled "B". The dark and light curves are the corresponding marginal distributions with
bold and regular labels respectively. a) shows the labeling for the supervised case. b),c) and d) reflect
the labels given by modality 1 and the corresponding marginal distributions seen by modality 2. See
text for more details
2.2
Self-Supervised Piecewise-Linear Classifier
The above ideas have been extended[de Sa, 1994] to rules for moving the codebook vectors
in a piecewise-linear classifier. Codebook vectors are initially chosen randomly from the
data patterns. In order to complete the algorithm idea, the codebook vectors need to be
given initial labels (The derivation assumes that the current labels are correct). In LVQ2.1
Learning Classification with Unlabeled Data
the initial codebook vectors are chosen from among the data patterns that are consistent
with their neighbours (according to a k-nearest neighbour algorithm); their labels are then
taken as the labels of the data patterns. In order to keep our algorithm unsupervised the
"labeling algorithm" mentioned earlier is used to derive labels for the initial codebook
vectors.
Also due to the fact that the codebook vectors may cross borders or may not be accurately
labeled in the initialization stage, they are updated throughout the algorithm by increasing the weight to the output class hypothesized by the other modality, from the neuron
representing the closest codebook vector. The final algorithm is given in Figure 6
1. Randomly choose initial codebook vectors from data vectors
2. Initialize labels of codebook vectors using the labeling algorithm
described in text
3 . Repeat for each presentation of input patterns XI(n) and X 2(n) to their
respective modalities
? find the two nearest codebook vectors in modality 1 -- wl.i; , Wl.i;, and
modality 2 -- W2,k;, W2,k; to the respective input patterns
? Find the hypothesized output class (CA , CB ) in each modality (as
given by the label of the closest codebook vector)
? For each modality update the weights according to the following
rules (Only the rules for modality 1 are given)
If neither or both Wli',
WI;' have the same label as w2,k' or XI(n) does
, 1
' 2
1
not lie within c(n) of the border between them no updates are done,
otherwise
*(
1)
)(XI(n)-wv(n-l))
n+a(n IIX I (n)-wV(n-I)1I
()
wi,i' n
=WI,i
WIJ* (n)
= wi/n -
*
(X I (n)-wIJ,(n-I))
1) - a(n) IIXI (n) _ w~J(n -1)11
where WI ,i' is the codebook vector wi th the same label, and WIJ' is
the codebook vector with another label.
? update the labeling weights
Figure 6: The Self-Supervised piecewise-linear classifier algorithm
3 EXPERIMENTS
The following experiments were all performed using the Peterson and Barney vowel formant data 2. The dataset consists of the first and second formants for ten vowels in a /h Vd/
context from 75 speakers (32 males, 28 females, 15 children) who repeated each vowel
twice 3.
To enable performance comparisons, each modality received patterns from the same
dataset. This is because the final classification performance within a modality depends
20 btained
from Steven Nowlan
33 speakers were missing one vowel and the raw data was linearly transformed to have zero mean
and fall within the range [-3, 3] in both components
117
118
de Sa
Table 1: Tabulation of performance figures (mean percent correct and sample standard deviation
over 60 trials and 2 modalities). The heading i - j refers to performance measured after the lh step
during the ilh iteration. (Note Step 1 is not repeated during the multi-iteration runs).
same-paired vowels
random pairing
not only on the difficulty of the measured modality but also on that of the other "labeling"
modality. Accuracy was measured individually (on the training set) for both modalities
and averaged. These results were then averaged over 60 runs. The results described below
are also tabulated in Table 1
In the first experiment, the classes were paired so that the modalities received patterns
from the same vowel class. If modality 1 received an [a] vowel, so did modality 2 and
likewise for all the vowel classes (i.e. p(xt!Cj ) = p(x2ICj) for all j). After the labeling
algorithm stage, the accuracy was 60?5% as the initial random placement of the codebook
vectors does not induce a good classifier. After application of the third step in Figure 6 (the
minimizing-disagreement part of the algorithm) the accuracy was 75 ?4%. At this point the
codebook vectors are much better suited to defining appropriate classification boundaries.
It was discovered that all stages of the algorithm tended to produce better results on the
runs that started with better random initial configurations. Thus, for each run, steps 2 and
3 were repeated with the final codebook vectors. Average performance improved (73?4%
after step 2 and 76?4% after step 3). Steps 2 and 3 were repeated several more times with
no further significant increase in performance.
The power of using the cross-modality information to move the codebook vectors can be
seen by comparing these results to those obtained with unsupervised competitive learning within modalities followed by an optimal supervised labeling algorithm which gave a
performance of 72 %.
One of the features of multi-modality information is that classes that are easily confuseable
in one modality may be well separated in another. This should improve the performance of
the algorithm as the "labeling" signal for separating the overlapping classes will be more
reliable. In order to demonstrate this, more tests were conducted with random pairing of
the vowels for each run. For example presentation of [a] vowels to one modality would be
paired with presentation of [i] vowels to the other. That is p(xIICj ) = p(x2ICaj) for a random
permutation aI, a2 .. alO. For the labeling stage the performance was as before (60 ? 4%)
as the difficulty within each modality has not changed. However after the minimizingdisagreement algorithm the results were better as expected. After 1 and 2 iterations of the
algorithm, 77 ? 3% and 79 ? 2% were classified correctly. These results are close to those
obtained with the related supervised algorithm LVQ2.1 of 80%.
4
DISCUSSION
In summary, appropriate classification borders can be learnt without an explicit external
labeling or supervisory signal. For the particular vowel recognition problem, the performance of this "self-supervised" algorithm is almost as good as that achieved with super-
Learning Classification with Unlabeled Data
vised algorithms. This algorithm would be ideal for tasks in which signals for two or more
modalities are available, but labels are either not available or expensive to obtain.
One specific task is learning to classify speech sounds from images of the lips and the
acoustic signal. Stork et. al. [1992] performed this task with a supervised algorithm
but one of the main limitations for data collection was the manual labeling of the patterns
[David Stork, personal communication, 1993]. This task also has the feature that the speech
sounds that are confuseable are not confuseable visually and vice-versa [Stork et ai., 1992].
This complementarity helps the performance of this classifier as the other modality provides
more reliable labeling where it is needed most.
The algorithm could also be used for learning to classify signals to a single modality where
the signal to the other "modality" is a temporally close sample. As the world changes
slowly over time, signals close in time are likely from the same class. This approach
should be more powerful than that of [FOldiak, 1991] as signals close in time need not be
mapped to the same codebook vector but the closest codebook vector of the same class.
Acknowledgements
I would like to thank Steve Nowlan for making the vowel formant data available to me.
Many thanks also to Dana Ballard, Geoff Hinton and Jeff Schneider for their helpful conversations and suggestions. A preliminary version of parts of this work appears in greater
depth in [de Sa, 1994].
References
[de Sa, 1994] Virginia R. de Sa, "Minimizing disagreement for self-supervised classification," In
M.C. Mozer, P. Smolensky, D.S. Touretzky, J.L. Elman, and A.S. Weigend, editors, Proceedings
of the 1993 Connectionist Models Summer School, pages 300-307. Erlbaum Associates, 1994.
[de Sa and Ballard, 1993] Virginia R. de Sa and Dana H. Ballard, "a note on learning vector quantization," In c.L. Giles, SJ.Hanson, and J.D. Cowan, editors, Advances in Neural Information
Processing Systems 5, pages 220-227. Morgan Kaufmann, 1993.
[Foldiak, 1991] Peter FOldiak, "Learning Invariance from Transformation Sequences," Neural Computation, 3(2):194-200, 1991.
[Grossberg, 1976] Stephen Grossberg, "Adaptive Pattern Classification and Universal Recoding: I.
Parallel Development and Coding of Neural Feature Detectors," Biological Cybernetics, 23: 121134, 1976.
[Kohonen, 1982] Teuvo Kohonen, "Self-Organized Formation of Topologically Correct Feature
Maps," Biological Cybernetics, 43:59-69, 1982.
[Kohonen, 1990] Teuvo Kohonen, "Improved Versions of Learning Vector Quantization," In IJCNN
International Joint Conference on Neural Networks, volume 1, pages 1-545-1-550, 1990.
[Rumelhart and Zipser, 1986] D. E. Rumelhart and D. Zipser, "Feature Discovery by Competitive
Learning," In David E. Rumelhart, James L. McClelland, and the PDP Research Group, editors,
Parallel Distributed Processing: Explorations in the Microstructure of Cognition, volume 2,
pages 151-193. MIT Press, 1986.
[Stork et at., 1992] David G. Stork, Greg Wolff, and Earl Levine, "Neural network lipreading system
for improved speech recognition," In IJCNN International Joint Conference on Neural Networks,
volume 2, pages 11-286-11-295, 1992.
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7,061 | 832 | Classifying Hand Gestures with a View-based
Distributed Representation
Trevor J. Darrell
Perceptual Computing Group
MIT Media Lab
Alex P. Pentland
Perceptual Computing Group
MIT Media Lab
Abstract
We present a method for learning, tracking, and recognizing human hand
gestures recorded by a conventional CCD camera without any special
gloves or other sensors. A view-based representation is used to model
aspects of the hand relevant to the trained gestures, and is found using an
unsupervised clustering technique. We use normalized correlation networks, with dynamic time warping in the temporal domain, as a distance
function for unsupervised clustering. Views are computed separably for
space and time dimensions; the distributed response of the combination
of these units characterizes the input data with a low dimensional representation. A supervised classification stage uses labeled outputs of the
spatio-temporal units as training data. Our system can correctly classify
gestures in real time with a low-cost image processing accelerator.
1
INTRODUCTION
Gesture recognition is an important aspect of human interaction, either interpersonally or
in the context of man-machine interfaces. In general, there are many facets to the "gesture
recognition" problem. Gestures can be made by hands, faces, or one's entire body; they
can be static or dynamic, person-specific or cross-cultural. Here we focus on a subset of
the general task, and develop a method for interpreting dynamic hand gestures generated
by a specific user. We pose the problem as one of spotting instances of a set of known
(previously trained) gestures. In this context, a gesture can be thought of as a set of hand
views observed over time, or simply as a sequence of images of hands over time. These
images may occur at different temporal rates, and the hand may have different spatial
945
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Darrell and Pentland
offset or gross illumination condition. We would like to achieve real- or near real-time
performance with our system, so that it can be used interactively by users.
To achieve this level of performance, we take advantage of the principle of using only
as much "representation" as needed to perform the task. Hands are complex, 3D articulated structures, whose kinematics and dynamics are difficult to fully model. Instead of
performing explicit model-based reconstruction, and attempting to extract these 3D model
parameters (for example see [4, 5, 6]), we use a simpler approach which uses a set of 2D
views to represent the object. Using this approach we can perform recognition on objects
which are either too difficult to model or for which a model recovery method is not feasible.
As we shall see below, the view-based approach affords several advantages, such as the
ability to form a sparse representation that only models the poses of the hands that are
relevant to the desired recognition tasks, and the ability to learn the relevant model directly
from the data using unsupervised clustering.
2
VIEW-BASED REPRESENTATION
Our task is to recognize spatio-temporal sequences of hand images. To reduce the dimensionality of the matching involved, we find a set of view images and a matching function
such that the set of match scores of a new image with the view images is adequate for recognition. The matching function we use is the normalized correlation between the image and
the set of learned spatial views.
Each view represents a different pose of the object being tracked or recognized. We
construct a set of views that "spans" the set of images seen in the training sequences, in
the sense that at least one view matches every frame in the sequence (given a distance
metric and threshold value). We can then use the view with the maximum score (minimum
distance) to localize the position of the object during gesture performance, and use the
ensemble response of the view units (at the location of maximal response) to characterize
the actual pose of the object. Each model is based on one or more example images of a
view of an object, from which mean and variance statistics about each pixel in the view are
computed.
The general idea of view-based representation has been advocated by Ullman [12] and
Poggio [9] for representing 3-D objects by interpolating between a small set of 2-D views.
Recognition using views was analyzed by Breuel, who established bounds on the number
of views needed for a given error rate [3]. However the view-based models used in these
approaches rely on a feature-based representation of an image, in which a "view" is the
list of vertex locations of semantically relevant features. The automatic extraction of these
features is not a fully solved problem. (See [2] for a nearly automated system of finding
corresponding points and extracting views.)
Most similar to our work is that of Murase and Nayar[8] and Turk[11] which use loworder eigenvectors to reduce the dimensionality of the signal and perform recognition. Our
work differs from theirs in that we use normalized-correlation model images instead of
eigenfunctions and can thus localize the hand position more directly, and we extend into
the temporal domain, recognizing image sequences of gestures rather than static poses.
A particular view model will have a range of parameter values of a given transformation
(e.g., rotation, scale, articulation) for which the correlation score shows a roughly convex
"tuning curve". If we have a set of view models which sample the transformation parameter
Classifying Hand Gestures with a View-Based Distributed Representation
(a)
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Figure 1: (a) Three views of an eyeball: +30, O. and -30 of gaze angle. (a) Normalized
correlation scores of the +30 degree view model when tracking a eyeball rotating from
approximately -30 to +30 degrees of gaze angle. (b) Score for 0 degree view model. (c)
Score for - 30 degree model.
finely enough, it is possible to infer the actual transform parameters for new views by
examining the set of model correlation scores. For example, Figure la shows three views
of an eyeball that could be used for gaze tracking; one looking 30 degrees left, one looking
center-on, and one looking 30 degrees to the right. The three views span a ?30 degree
subspace of the gaze direction parameter. Figure I (b,c,d) shows the normalized correlation
score for each view model when tracking a rotating eyeball. Since the tuning curves
produced by these models are fairly broad with respect to gaze angle, one could interpolate
from their responses to obtain a good estimate of the true angle.
When objects are non-rigid, either constructed out of flexible materials or an articulated
collection of rigid parts (like a hand), then the dimensionality of the space of possible
views becomes much larger. Full coverage of the view space in these cases is usually
not possible since enumerating it even with very coarse sampling would be prohibitively
expensive in terms of storage and search computation required. However, many parts of a
high dimensional view space may never be encountered when processing real sequences,
due to unforeseen additional constraints. These may be physical (some joints may not
be completely independent), or behavioral (some views may never be used in the actual
communication between user and machine). A major advantage of our adaptive scheme is
that it has no difficulty with sparse view spaces, and derives from the data which regions of
the space are full.
947
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Darrell and Pentland
(
Figure 2: (a) Models automatically acquired from a sequence of images of a rotating box.
(b) Normalized correlation scores for each model as a function of image sequence frame
number.
3
UNSUPERVISED LEARNING OF VIEW UNITS
To derive a set of new view models, we use a simple form of unsupervised clustering
in which the first example forms a new view, and subsequent examples that are below a
distance threshold are merged into the nearest existing view. A new view is created when
an example is below the threshold distance for all views in the current set, but is above a
base threshold which establishes that the object is still (roughly) being tracked. Over time,
this "follow-the-Ieader" algorithm results in a family of view models that sample the space
of object poses in the training data. This method is similar to those commonly used in
vector quantization [7]. Variance statistics are updated for each model pixel, and can be
used to exclude unreliable points from the correlation computation.
For simple objects and transformations, this adaptive scheme can build a model which
adequately covers the entire space of possible views. For example, for a convex rigid body
undergoing aID rotation with fixed relative illumination, a relatively small number of view
models can suffice to track and interpolate the position of the object at any rotation. Figures
2 illustrates this with a simple example of a rotating box. The adaptive tracking scheme
was run with a camera viewing a box rotating about a fixed axis. Figure 2a shows the view
models in use when the algorithm converged, and all possible rotations were matched with
score greater than 0\. To demonstrate the tuning properties of each model under rotation,
Figure 2b shows the correlation scores for each model plotted as a function of input frame
Classifying Hand Gestures with a View-Based Distributed Representation
Figure 3: Four spatial views found by unsupervised clustering method on sequence containing two hand-waving gestures: side-to-side and up-down.
I
I
IT]
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views
. . . c:::::J
temporal
views
Figure4: Overview of unsupervised clustering stage to learn spatial and temporal views. An
input image sequence is reduced to sequence of feature vectors which record the maximum
value in a normalized correlation network corresponding to each spatial view. A similar
process using temporal views reduces the spatial feature vectors to a single spatia-temporal
feature vector.
number of a demonstration sequence. In this sequence the box was held fixed at its initial
position for the first 5 frames, and then rotated continuously from 0 to 340 degrees. The
responses of each model are broadly tuned as a function of object angle, with a small
number of models sufficing to represent/interpolate the object at all rotations (at least about
a single axis).
We ran our spatial clustering method on images of hands performing two different "waving"
gestures. One gesture was a side-to-side wave, with the fingers rigid, and the other was
an up-down wave, with the wrist held fixed and the fingers bending towards the camera
in synchrony. Running instances of both through our view learning method, with a base
threshold of Bo=0.6 and a "new model" threshold of BI = 0.7, the clustering method found
4 four spatial templates to span all of the images in the both sequences Figure 3 shows the
pixel values for these four models.
949
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Darrell and Pentland
Figure 5: Surface plot of temporal templates found by unsupervised clustering method on
sequences of two hand-waving gestures. Vertical axis is score, horizontal axis is time, and
depth axis is spatial view index.
3.1
TEMPORAL VIEWS
The previous sections provide a method for finding spatial views to reduce the dimensionality in a tracking task. The same method can be applied in the temporal domain as
well, using a set of "temporal views". Figure 4 shows an overview of these two stages.
We construct temporal views using a similar method to that used for spatial views, but
with temporal segmentation cues provided by the user. Sequences of spatial-feature vector
outputs (the normalized correlation scores of the spatial views) are passed as input to the
unsupervised clustering method, yielding a set of temporal views. To find the distance
between two sequences, we again use a normalized correlation metric, with Dynamic Time
Warping (DlW) method [1, 10]. This allows the time course of a gesture to vary, as long
as the same series of spatial poses is present.
In this way a set of temporal views acting on spatial views which in turn act on image
intensities, is created. The responses of these composi te views yield a single spatio-temporal
stimulus vector which describes spatial and temporal properties of the input signal. As an
example, for the "hand-waving" example shown above, two temporal views were found by
the clustering method. These are shown as surface plots in Figure 5. Empirically we have
found that the spatio-temporal units capture the salient aspects of the spatial and temporal
variation of the hand gestures in a low-dimensional representation, so efficient classification
is possible. The response of these temporal view units on an input sequence containing
three instances of each gesture is shown in Figure 6.
4
CLASSIFICATION OF GESTURES
The spatio-temporal units obtained by the unsupervised procedure described above are used
as inputs to a supervised learning/classification stage (Figure 7(a)). We have implemented
two different classification strategies, a traditional Diagonal Gaussian Classifier, and a
multi-layer perceptron.
Classifying Hand Gestures with a View-Based Distributed Representation
(a)
(b)
Figure 6: (a) surface plot of spatial view responses on input sequence containing three
instances of each hand-waving gesture. (b) final spatio-temporal view unit response: the
time-warped, normalized correlation score of temporal views on spatial view feature vectors.
As an experiment, we collected 42 examples of a "hello" gesture, 26 examples of "goodbye" and 10 examples of other gestures intended to generate false alarms in the classifier.
All gestures were performed by a single user under similar imaging conditions. For each
trial we randomly selected half of the target gestures to train the classifier, and tested on the
remaining half. (All of the conflictor gestures were used in both training and testing sets
since they were few in number.)
Figure 7(b) summarizes the results for the different classification strategies. The Gaussian
classifier (DG) achieved an hit rate of 67%, with zero false alarms. The multi-layer
perceptron (MLP) was more powerful but less conservative, with a hit rate of 86% and a
false alarm rate of 5%. We found the results of the MLP classifier to be quite variable;
on many of the trials the classifier was stuck in a local minima and failed to converge on
the test set. Additionally there was considerable dependence on the number of units in
the hidden layer; empirically we found 12 gave best performance. Nonetheless, the MLP
classifier provided good performance. When we excluded the trials on which the classifier
failed to converge on the training set, the performance increased to 91 % hit rate, 2% false
alarm rate.
5
CONCLUSION
We have demonstrated a system for tracking and recognition of simple hand gestures. Our
entire recognition system, including time-warping and classification, runs in real time (over
10Hz). This is made possible through the use of a special purpose normalized correlation
search co-processor. Since the dimensionality of the feature space is low, the dynamic time
warping and classifications steps can be implemented on conventional workstations and
still achieve real-time performance. Because of this real-time performance, our system is
951
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Darrell and Pentland
hello II
"bye"
II
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CLASSIFIER -
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ST unit
outputs
Figure 7: Overview of supervised classification stage and results obtained for different
types of classifiers.
directly applicable to interactive "glove-free" gestural user interfaces.
References
[1] Bellman, R E., (1957) Dynamic Programming. Princeton, NJ: Princeton Univ. Press.
[2] Beymer, D., Shashua, A., and Poggio, T., (1993) ''Example Based Image Analysis
and Synthesis", MIT AI Lab Memo No. 1431
[3] Breuel, T., (1992) "View-based Recognition", IAPR Workshop on Machine Vision
Applications.
[4] Cipolla, R, Okamotot, Y., and Kuno, Y., (1992) "Qualitative visual interpretation
of 3D hand gestures using motion parallax", IAPR Workshop on Machine Vision
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[5] Fukumoto, M., Mase, K., and Suenaga, Y., (1992) "Real-Time Detection of Pointing
Actions for a Glove-Free Interface", IAPR Workshop on Machine Vision Applications.
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using Pipe-line Image Processor", (1992) IEEE Workshop on Robot and Human
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Appearance", Proc. IEEE Qualitative Vision Workshop, New York City, pp. 39-49.
[9] Poggio, T., and Edelman, S., (1990) "A Network that Learns to Recognize Three
Dimensional Objects," Nature, Vol. 343, No. 6255, pp. 263-266.
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word recognition", IEEE Trans. ASSP, Vol. 26, pp. 623-625.
[11] Turk, M., and Pentland, A. P., (1991) "Eigenfaces for Recognition", Journal of
Cognitive Neuroscience, vol. 3, pp. 71-89.
[12] Ullman, S., and Basri, R, (1991)"Recognition by Linear Combinations of Models,"
IEEE PAMI, Vol. 13, No. 10, pp. 992-1007.
| 832 |@word trial:3 gish:1 initial:1 series:1 score:14 tuned:1 existing:1 current:1 subsequent:1 shape:1 plot:3 cue:1 selected:1 half:2 record:1 coarse:1 loworder:1 location:2 simpler:1 constructed:1 qualitative:2 edelman:1 behavioral:1 parallax:1 sakoe:1 acquired:1 roughly:2 multi:2 bellman:1 automatically:1 actual:3 becomes:1 provided:2 matched:1 cultural:1 suffice:1 medium:2 spoken:1 finding:2 transformation:3 nj:1 temporal:26 every:1 act:1 interactive:1 prohibitively:1 classifier:10 hit:3 unit:10 local:1 approximately:1 pami:1 co:1 range:1 bi:1 camera:3 wrist:1 testing:1 differs:1 procedure:1 thought:1 matching:3 word:1 gestural:1 storage:1 context:2 conventional:2 demonstrated:1 center:1 convex:2 recovery:1 variation:1 updated:1 target:1 user:6 programming:2 us:2 recognition:15 expensive:1 labeled:1 observed:1 solved:1 capture:1 region:1 ran:1 gross:1 dynamic:8 trained:2 iapr:3 completely:1 joint:1 finger:2 articulated:2 train:1 univ:1 whose:1 quite:1 larger:1 ability:2 statistic:2 transform:1 final:1 sequence:19 advantage:3 breuel:2 reconstruction:1 interaction:1 maximal:1 relevant:4 achieve:3 darrell:5 rotated:1 object:16 derive:1 develop:1 pose:7 eyeball:4 nearest:1 advocated:1 coverage:1 implemented:2 murase:2 direction:1 merged:1 human:3 viewing:1 material:1 ao:1 pointing:1 major:1 vary:1 purpose:1 proc:2 applicable:1 establishes:1 city:1 mit:3 sensor:1 gaussian:2 hello:2 rather:1 focus:1 sense:1 rigid:4 entire:3 hidden:1 pixel:3 classification:9 flexible:1 figure4:1 spatial:20 special:2 fairly:1 construct:2 never:2 extraction:1 sampling:1 represents:1 broad:1 unsupervised:10 nearly:1 stimulus:1 few:1 randomly:1 dg:1 recognize:2 interpolate:3 intended:1 detection:1 mlp:3 analyzed:1 yielding:1 held:2 poggio:3 rotating:5 desired:1 plotted:1 instance:4 classify:1 increased:1 facet:1 cover:1 cost:1 vertex:1 subset:1 recognizing:2 examining:1 too:1 characterize:1 person:1 st:1 gaze:5 synthesis:1 unforeseen:1 continuously:1 again:1 recorded:1 interactively:1 containing:3 cognitive:1 warped:1 ullman:2 exclude:1 coding:1 performed:1 view:76 lab:3 characterizes:1 shashua:1 wave:2 waving:5 synchrony:1 variance:2 who:1 ensemble:1 yield:1 produced:1 processor:2 converged:1 trevor:1 nonetheless:1 pp:7 involved:1 turk:2 static:2 workstation:1 dlw:1 dimensionality:5 segmentation:1 supervised:3 follow:1 response:9 box:4 stage:5 correlation:15 hand:25 horizontal:1 normalized:11 true:1 adequately:1 excluded:1 during:1 demonstrate:1 motion:1 interface:3 interpreting:1 image:21 rotation:6 physical:1 tracked:2 overview:3 empirically:2 extend:1 interpretation:1 theirs:1 ai:1 automatic:1 tuning:3 robot:1 surface:3 base:2 spatia:1 seen:1 minimum:2 additional:1 greater:1 recognized:1 converge:2 signal:2 ii:2 full:2 infer:1 reduces:1 match:2 gesture:31 cross:1 long:1 vision:4 metric:2 represent:2 achieved:1 finely:1 eigenfunctions:1 hz:1 extracting:1 near:1 enough:1 automated:1 gave:1 reduce:3 idea:1 enumerating:1 kishino:1 passed:1 speech:1 york:1 action:1 adequate:1 eigenvectors:1 reduced:1 generate:1 affords:1 neuroscience:1 correctly:1 track:1 broadly:1 shall:1 vol:5 group:2 four:3 salient:1 threshold:6 localize:2 imaging:1 run:2 angle:5 powerful:1 family:1 summarizes:1 bound:1 layer:3 encountered:1 occur:1 constraint:1 alex:1 aspect:3 span:3 performing:2 attempting:1 relatively:1 combination:2 makhoul:1 describes:1 previously:1 turn:1 kinematics:1 needed:2 clustering:11 running:1 remaining:1 ccd:1 composi:1 build:1 warping:4 strategy:2 dependence:1 traditional:1 diagonal:1 subspace:1 distance:6 separably:1 collected:1 index:1 demonstration:1 difficult:2 memo:1 perform:3 vertical:1 pentland:6 looking:3 communication:2 assp:1 frame:4 intensity:1 required:1 pipe:1 learned:1 established:1 trans:1 spotting:1 below:3 usually:1 articulation:1 including:1 difficulty:1 rely:1 representing:1 scheme:3 axis:5 created:2 extract:1 mase:1 bending:1 relative:1 fully:2 takemura:1 accelerator:1 degree:8 principle:1 classifying:4 course:1 bye:1 free:2 side:4 perceptron:2 template:2 face:1 eigenfaces:1 sparse:2 distributed:5 curve:2 dimension:1 depth:1 chiba:1 stuck:1 made:2 collection:1 adaptive:3 commonly:1 roucos:1 basri:1 unreliable:1 spatio:6 search:2 additionally:1 learn:2 nature:1 interpolating:1 complex:1 domain:3 alarm:4 body:2 aid:1 position:4 explicit:1 perceptual:2 learns:1 down:2 specific:2 undergoing:1 offset:1 list:1 derives:1 workshop:5 quantization:2 false:4 te:1 illumination:2 illustrates:1 simply:1 appearance:1 beymer:1 visual:1 failed:2 tracking:7 bo:1 cipolla:1 towards:1 man:1 feasible:1 considerable:1 glove:3 semantically:1 acting:1 conservative:1 la:1 kuno:1 princeton:2 tested:1 nayar:2 |
7,062 | 833 | Backpropagation without Multiplication
Patrice Y. Simard
AT &T Bell Laboratories
Holmdel, NJ 07733
Hans Peter Graf
AT&T Bell Laboratories
Holmdel, NJ 07733
Abstract
The back propagation algorithm has been modified to work without any multiplications and to tolerate comput.ations with a low
resolution, which makes it. more attractive for a hardware implementatioll. Numbers are represented in float.ing point format with
1 bit mantissa and 3 bits in the exponent for the states, and 1 bit
mantissa and 5 bit exponent. for the gradients, while the weights are
16 bit fixed-point numbers. In this way, all the computations can
be executed with shift and add operations . Large nehvorks with
over 100,000 weights were t.rained and demonstrat.ed the same performance as networks comput.ed with full precision. An estimate of
a circuit implementatioll shows that a large network can be placed
on a single chip , reaching more t.han 1 billion weight updat.es pel'
second. A speedup is also obtained on any machine where a multiplication is slower than a shift op erat.ioJl.
1
INTRODUCTION
One of the main problems for implement.ing the backpropagation algorithm in hardware is the large number of multiplications t.hat. have to be executed. Fast multipliers for operands wit.h a high resolution l'eqllire a large area. Hence the multipliers
are the elements dominating t.he area of a circuit. i\'Iany researchers have tried to
reduce the size of a circuit by limit.ing the resolution of the computation. Typically,
this is done by simply reducing the number of bits utilized for the computation. For
a forward pass a reduction tOjllst a few , 4 to 6. bits, often degl'ades the performance
very little, but. learning requires considerably more resolution. Requirements ranging anywhere from 8 bits to more than 16 bits were report.ed to be necessary to make
learning converge relia.bly (Sakaue et al., 1993; Asanovic, I\'Iorgan and \Va.wrzYllek,
1993; Reyneri and Filippi, 1991). But t.here is no general theory, how much resolution is enough, and it depends on several factors, such as the size and architecture
of the network as \-vell as on the t.ype of problem to be solved .
232
Backpropagation without Multiplication
Several researchers have tried to tl'ain networks where the weights are limited to
powers of two (Kwan and Tang, 1993; White and Elmasry, 1992; l'vlarchesi et. al.,
1993). In this way all the multiplications can be reduced to shift operations, an
operation that can be implemented with much less hardware than a multiplication.
But restricting the weight values severely impacts the performance of a network, and
it is tricky to make t.he learning procedure converge. III fact , some researchers keep
weights with a full resolution off-line and update t.hese weights in the backward pass,
while the weights with reduced resolution are used in the forward pass (Marchesi
et al., 1993) . Similar tricks are usually used when networks implemented in analog
hardware are trained. Weight.s with a high resolution are stored in an external,
digital memory while the analog net.work with its limited resolution is used in the
forward pass. If a high resolution copy is not stored, the weight update process
needs to be modified. This is typically done by using a stochastic update technique,
such as weight dithering (Vincent and l\lyers, 19~)2), or weight perturbation (.J abri
and Flower, 1992).
We present here an algorithm that instead of reducing the resolut.ion of the weights,
reduces the resolution of all t.he other values, namely those of the states, gradients
and learning rates, to powers of two. This eliminates multiplications without affecting the learning capabilities of t.he network. Therefore we ohtain the benefit of
a much compacter circuit without any compromises on the learning performance.
Simulations of large net.works with over 100,000 weights show that this algorithm
perf?r.ms as well as standal'd backpwpagation computed with 32 bit floating point
preCIsIon.
2
THE ALGORITHM
The forward propagat.ion for each unit i. is given by th e pquation:
Xj
= j~(L
wji.t' il
( 1)
where f is the unit functjoll, Wji is the weight from unit i to unit j, and Xi is the
activation of unit i. The backpwpagation algorithm is wbust with regard to the
unit function as long as the function is nonlinear, monotonically increasing, and a
derivative exists (the most commonly used function is depicted in Figure 1, left.
A saturated ramp function (see Figure 1, middle), for instance, performs as well
as the differentiable sigmoid. The binary threshold function, however, is too much
of a simplification and results in poor performance . The choice of OUl' function is
dictated by the fact that we would like t.o have only powers of two for the unit values.
This function is depicted ill Figure 1, right. It gives performances comparable to
the sigmoid or the saturated ramp . Its values can be represented by a 1 bit mantissa
(the sign) with a. 2 or 3 bit exponent. (negative powers of t.wo).
The derivative of this funct.ion is a. SlIm of Dirac delta functions, but we take instead
the derivative of a piecewise linear ramp funct.ion (see Figure 1) . 0\1(" could actually
consider this a low pass filtered version of the real derivat.ive . After the gradients
of all the units have been computed using the equation.
[h
=
If (s 11 In i ) L
U'j i [lj
(2)
j
we will discretize the values to be a power of two (wit h sign) . This introduces noise
into the gradient and its effect, on th e learning has to be considered carefully. This
233
234
Simard and Graf
Piecewise linear
Sigmoid
.. ,
Functlon
F\Jflctl.On
F'Unction
1.,
1..
Power of two
...
..,
- o. S
-,
-,
-l.$_I:-.--:_,:--:.?~_":",~_.~.?~.-=.--=.?~,""7,--=-.,~,
-'1----1
-l.~_':-,-:_,:'":
.?~_":",-:_,~.,~,-:,--=. ,~,""7,"":
.? -:,
FunctIon derlvatl.Ve
Functlon derl.Vatlve
-:",'.,-!,
"'"
- ) . s_':-,-:
_,-=.,~_.,..,-:_.-=
. ,--::-.-,.'"'".,-.-,
FunctIon derl.vatl.ve (apprOXlmationJ
1..
1..
..,
-o.S
-o.S
-O.!
-,
-,
-,
?.:-,"':,--;"
..:-,~,
-1 ? '_':-,~_,~
??:-':"_,-_-:?? :-."'=.~
-1.
S_I:-,--:_,:--:.,-_":",~_.~.,~.-=,'"'".,~,""7,'"',--:,
".
-1.
'-'="2--:""1.':-':"-1--":""
?.:-,"'=o~
?. :-,-:"'--:"'1.:-,-:,
Figure 1: Left: sigmoid function with its derivative. ]\>'1iddle: piecewise linear
function with its derivative. Right.: Sat.urated power of two function with a power
of two approximation of its derivative (ident.ical to t.he piecewise linear derivative).
problem will be discussed in section 4. The backpropagat.ion algorithm can now be
implemented with addit.ions (or subtract.iolls) and shifting only. The weight update
is given by the equa.tion:
D.1.Vji
Since both 9j and
and shifts.
3
Xi
=
-119jXi
(3)
are powers of two, the weight update also reduces to additions
RESULTS
A large structured network wit.h five layers and overall 11l00'e t.han 100,000 weights
was used to test this algorithm. The applicat.ion analyzed is recognizing handwrit.ten
character images. A database of 800 digits was used for training and 2000 handwritten digits were used for test.ing. A description of this network can be found in
(Le Cun et aI., 1990). Figure 2 shows the learning curves on t.he test set for various
unit functions and discretization processes.
First, it should be noted that t.he results given by the sigmoid function and the
saturated ramp with full precision on unit values, gradients, and weights are very
similar. This is actually a well known behavior. The surprising result comes from
the fact that reducing the precision of the unit values and the gradients to a 1 bit
mantissa does not reduce the classification accuracy and does not even slow down the
learning process. During these tests the learning process was interrupted at various
stages to check that both the unit values (including the input layer, but excluding
the output layer) and t.he gradient.s (all gradients) were restricted to powers of two.
It was further confirmed that. ollly 2 bits wet'e sufficient. for the exponent of the unit
Backpropagation without Multiplication
Training
error
100
Testing
error
100
? sigmoid
o piecewise lin
o power of 2
go
eo
? sigmoid
piecewise lin
o power of 2
90
D
80
70
70
60
60
50
50
40
40
30
30
20
20
IlohU:e::a.n:u
10
10
~
0
0
2
4
6
8
10 12 14 18 18 20 22 24
age (in 1000)
0
0
2
4
6
8
10 12 14 16 16 20 22 24
age (in 1000)
Figure 2: Training and testing error during leaming. The filled squares (resp.
empty squared) represent the points obtained with the vanilla backpropagation and
a sigmoid function (resp. piecewise linear function) used a<; an activation function.
The circles represent the same experiment done wit.h a power of t.wo function used
as the activation function, and wit.h all lInit. gradients discretized to the nearest
power of two.
values (from 2? to 2- 3 ) and 4 bit.s were sufficient for the exponent. of the gradients
(from 2? to 2- 15 ).
To test whether there was any asymptot.ic limit on performance, we ran a long
term experiment (several days) with our largest network (17,000 free parameters)
for handwritten character recognition. The training set (60,000 patterns) was made
out 30,000 patterns of the original NIST t.raining set (easy) and 30,000 patterns
out of the original NIST testing set (hard). Using the most basic backpropagation
algorithm (with a guessed constant learning rate) we got the training raw error rate
down to under 1% in 20 epochs which is comparable to our standard learning time.
Performance on the test set was not as good with the discrete network (it took twice
as long to reach equal performance with the discrete network). This was attributed
to the unnecessary discretization of the output units 1.
These results show that gradients and unit activations can be discretized to powers
of two with negligible loss in pel"formance and convergence speed! The next section
will present theoretical explanations for why this is at. all possible and why it is
generally the case.
lSince the output units are not multiplied by anything, t.here is no need to use a
discrete activation funct.ion. As a matter of fact the continuous sigmoid function can be
implemented by just changing the target. values (using the inverse sigmoid function) and
by using no activation function for the output units. This modificat.ion was not introduced
but we believe it would improves the performance on t.he t.est. set. especially when fancy
decision rules (with confidencE' evaluatioll) are used, since t.hey require high precision on
the output units.
235
236
Simard and Graf
~:s:.ogram
2000
lROO
1600
1"00
1200
1000
aDD
600
'00
200
histogram
Best case: Noise
is uncorrelated and
a" weights are equal
Worse case: NOise
is correlated or the
weights are unequal
Figure 3: Top left: histogram of t.he gradients of one output unit after more than
20 epochs of learning over a training set of GO,OOO pallel'lIs . Bottom left: same
histogram assuming that the distt'ibutioll is constant between powers of two. Right:
simplified network architectlll'es fOl' noise effect. analysis .
4
DISCUSSION
Discretizing the gradient is potentially very dangerous. Convergence may no longer
be guaranteed, learning may hecome prohibitively slow, and final performance after
learning may be be too poor to be interesting, "Ve will now explain why these
problems do not arise for our choice of discret.ization.
Let gi(p) be the error gradient at weight i and pattern p. Let 1'.,: and Ui be the mean
and standard deviation of gi(p) over the set of patterns. The mean Pi is what is
driving the weights to their final values, the standard deviation Ui represents the
amplitudes of the variations of the gradients from pattern to pattern. In batch
learning, only Pi is used for the weight upda.te, while in stochastic gradient descent,
each gi(p) is used for the weight update. If the learning rate is small enough the
effects of the noise (measured by u;) of the stochastic variable Ui (p) are negligible,
but the frequent weight updates in stochastic gradient descent result. in important
speedups,
To explain why the discretization of the gradient to a power of two has negligible
effect on the pel'formance, consider that in stochastic gradient descent, the noise on
the gradient is already so large that it is minimally affected by a rounding (of the
gradient) to the nearest power of two. Indeed asymptotically, t.he gradient a.verage
(Pi) tends to be negligible compared to it.s standard deviation (ui), meaning that
from pattern to pattern the gradient can undergo sign reversals, Rounding to the
nearest power of two in comparison is a change of at. most 33%, but never a change
in sign. This additional noise can therefore be compensated by a slight. decrease in
the learning rate which will hardly affect the leal'l1ing process .
Backpropagation without Multiplication
The histogram of gi(p) after learning in the last experiment described in the result
section, is shown in Figure 3 (over the training set of 60,000 patterns). It is easy to
see in the figure that J.li is small wit.h respect to (7i (in this experiment J.li was one to
two orders of magnitude smaller than (7i depending on the layer). vVe can also see
that rounding each gradient to the nearest power of two will not affect significantly
the variance (7i and therefore the learning rate will not need to be decreased to
achieve the same performance.
We will now try to evaluate the rounding to the nearest power of two effect more
quantitatively. The standard deviation of the gradient for any weight can be written
as
'I
1~
')
l~
')
')
l~
2
(4)
(7; = N ~(gi(P) - pi)- = N ~ gi(p)- - J.l- ~ N ~ gi(p)
p
p
p
This approximation is very good asymptotically (after a few epochs of learning).
For instance if lJ.li I < (7;/ 10, the above formula gives the standard deviation to 1%.
Rounding the gradient gi to the nearest power of two (while keeping the sign) can
be viewed as the effect of a multiplicative noise 11i in the equation
g/ = 2k = ad 1 + nd
for some k
(5)
where g/ is the nearest power of two from gj. It can be easily verified that this
implies that 11.i ranges from -1/3 and 1/3 . From now on , we will view Hi as a
random variable which models as noise the effect of discretization . To simplifY the
computation we will assume that 11j has uniform distribution. The effect of this
assumption is depicted in figure :3, where the bottom histogram has been assumed
constant between any two powers of t.wo.
To evaluate the effect of the noise ni in a multilayer network , let 7Ili be the multiplicative noise introduced at layer I (l = 1 for the output, and I = L for the first
layer above the input) for weight i. Let's further assume that there is only one unit
per layer (a simplified diagra.m of the network architecture is shown on figure 3.
This is the worst case analysis. If there are several units per layer, the gradients
will be summed to units in a lower layer. The gradients within a layer are correlated from unit to unit (they all originate from the same desired values), but the
noise introduced by the discretization can only be less correlated, not more . The
summation of the gradient in a lower layer can therefore only decrease the effect of
the discretization . The worst case analysis is t.herefore when there is only one unit.
per layer as depicted in figure :3, extreme right. \Ve will further assume that the
noise introduced by the discretizat.ion ill one layer is independent from the Iloise
introduced in the next layer . This is not ~'eally true but it greatly simplifies the
derivation.
Let J.l~ and (7i be the mean and standard deviation of Oi (p)'. Since nli has a zero
mean, J.l~ = J.li and J1~ is negligible with respect to gd}J)? In the worst case, when the
~radient has to be backpropagated all the way to t.he input , the standard deviation
IS:
L
1L
N
p
gi(p)2
II (3-2 j1/3 (1 +
1
-1/3
11 Ii
)2d7l/i
)-
/1 2 ~
(1 )L
(7; 1 + -.
27
(6)
237
238
Simard and Graf
As learning progresses, the minimum average distance of each weight to the weight
corresponding to a local minimum becomes proportional to the variance of the noise
on that weight, divided by the learning rate. Therefore, asymptotically (which is
where most of the time is spent), for a given convergence speed, the learning rate
should be inversely proportional to the variance of the noise in the gradient. This
means that to compensClte the effect of the discretization. the learning rate should
be divided by
(1"
I
(11+ ;7)
L
'" 1.02?
(7)
Even for a 10 layer network this value is only 1.2, (u~ is 20 % larger than ud.
The assumption that the noise is independent from layer to layer tends to slightly
underestimate this number while the assumption that the noise from unit to unit
in the same layer is completely correlated tends to overestimate it.
All things considered, we do not expect that the learning rate should be decrea'Sed
by more than 10 to 20% for any practical application. In all our simulations it was
actually left unchanged!
5
HARDWARE
This algorithm is well suited for integrating a large network on a single chip . The
weights are implemented with a resolution of 16 bits, while the states need only 1
bit in the mantissa and 3 bits in the exponent, the gradient 1 bit in the mantissa
and 5 bits in the exponent, and for the learning rate 1 bits mantissa and 4 bits
exponent suffice. In this way, all the multiplications of weights with states, and of
gradients with learning rates and st.at.t's. reduce to add operations of the exponents.
For the forward pass the weights are multiplied with the states and then summed.
The mUltiplication is executed as a shift operation of the weight values. For summing two products their mantissae have to be aligned, again a shift operation, and
then they can be added. The partial sums are kept at full resolution until the end of
the summing process. This is necessary to avoid losing the influence of many small
products. Once the sum is computed, it is then quantized simply by checking the
most significant bit in the mantissa. For the backward propagation the computation
runs in the same way, except t.hat now the gradient is propagated through the net,
and the learning rate has to be taken into account..
The only operations required for this algorithm are 'shift' and 'add'. An ALU
implementing these operations with the resolution ment.ioned can be built with
less than 1,000 transistors. In order to execut.e a network fast, its weights have to
be stored on-chip. Ot.herwise, t.he time required to t.ransfer weight!; from external
memory onto the chip boundary makes the high compute power all but useless. If
storage is provided for 10,000 weights plus 2,000 states, this requires less than 256
kbit of memory. Together with 256 ALUs and circuitry for routing the data, this
leads to a circuit with about 1.7 million transistors, where over 80% of them are
contained in the memory. This assumes that the memory is implemented with static
cells, if dynamic memory is used instead the transistor count drops considerably..
An operating speed of 40 MHz resnlts in a compute rate of 10 billion operat.ions
per second. \-\lith such a chip a network may be trained at a speed of more than 1
billion weight updates per second.
Backpropagation without Multiplication
This algorithm has been optimized for an implementation on a chip, but it can
also provide a considerable speed up when executed on a standard computer. Due
to the small resolution of the numbers, several states can be packed into a 32 bit
number and hence many more fit int.o a chache. Moreover on a machine without
a hardware multiplier, where the multiplication is executed with microcode, shift
operations may be much faster than multiplications. Hence a suhstancial speedup
may be observed.
References
Asanovic, K., Morgan, N., and \Vawrzynek, J. (1993). Using Simulations of Reduced Precision Arithmetic t.o Design a Neura- Microprocessor. 1. FLSI Signal
Processing, 6(1):33-44.
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networks. IEEE Trans. Neural Networks, 3(3):154-157.
Kwan, H. and Tang, C. (1993). Multipyerless Multilayer Feedforward Neural Network Desi~n Suitable for Continuous Input-Output Mapping. Elecironic Lei-
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Le Cun, Y., Boser, B., Denker, J. S., Henderson, D., Howard, R. E., Hubbard,
W., and Jackel, L. D. (1990). Handwritten Digit Recognition with a BackPropagation Network . In Touretzky, D., editor, Neural Injo1'lnaiio71 Processing
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Reyneri, L. and Filippi, E. (1991). An analysis on the Performance of Silicon Implementations of Backpropagation Algorithms for Artificial Nemal Networks.
IEEE Trans. Computer's, 40( 12): 1380-1389.
Sakaue, S., Kohda, T., Yamamoto, II., l\'laruno, S., and Shimeki, Y. (1993). Reduction of Required Precision Bits for Back-Propagation Applied to Pattern
Recognition. IEEE Tralls. Neural Neiworks, 4(2):270-275.
Vincent, J. and Myers, D. (1992). Weight dithering and \Vordlength Selection for
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7,063 | 834 | Event-Driven Simulation of Networks of
Spiking l'Ieurons
Lloyd Watts
Synaptics Inc.
2698 Orchard Parkway
San Jose CA 95134
11oydGsynaptics.com
Abstract
A fast event-driven software simulator has been developed for simulating large networks of spiking neurons and synapses. The primitive network elements are designed to exhibit biologically realistic behaviors, such as spiking, refractoriness, adaptation, axonal
delays, summation of post-synaptic current pulses, and tonic current inputs. The efficient event-driven representation allows large
networks to be simulated in a fraction of the time that would be
required for a full compartmental-model simulation. Corresponding analog CMOS VLSI circuit primitives have been designed and
characterized, so that large-scale circuits may be simulated prior
to fabrication.
1
Introduction
Artificial neural networks typically use an abstraction of real neuron behaviour,
in which the continuously varying mean firing rate of the neuron is presumed to
carry the information about the neuron's time-varying state of excitation [1]. This
useful simplification allows the neuron's state to be represented as a time-varying
continuous-amplitude quantity. However, spike timing is known to be important in
many biological systems. For example, in nearly all vertebrate auditory systems,
spiral ganglion cells from the cochlea are known to phase lock to pure-tone stimuli
for all but the highest perceptible frequencies [2]. The barn owl uses axonal delays
to compute azimuthal spatial localization [3]. Studies in the cat [4] have shown that
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relative timing of spikes is preserved even at the highest cortical levels. Studies in
the visual system of the blowfly [5] have shown that the information contained in
just three spikes is enough for the fly to make a decision to turn, if the visual input
IS sparse.
Thus, it is apparent that biological neural systems exploit the spiking and timedependent behavior of the neurons and synapses to perform system-level computation. To investigate this type of computation, we need a simulator that includes
detailed neural behavior, yet uses a signal representation efficient enough to allow
simulation of large networks in a reasonable time.
2
Spike: Event-Driven Simulation
Spike is a fast event-driven simulator optimized for simulating networks of spiking
neurons and synapses. The key simplifying assumption in Spike is that all currents
injected into a cell are composed of piecewise-constant pulses (i.e., boxcar pulses),
and therefore all integrated membrane voltage trajectories are piecewise linear in
time. This very simple representation is capable of surprisingly complex and realistic
behaviors, and is well suited for investigating system-level questions that rely on
detailed spiking behavior.
The simulator operates by maintaining a queue of scheduled events. The occurrence
of one event (i.e., a neuron spike) usually causes later events to be scheduled in the
queue (Le., end of refractory period, end of post-synaptic current pulse). The total
current injected into a cell is integrated into the future to predict the time of firing,
at which time a neuron spike event is scheduled. If any of the current components
being injected into the cell subsequently change, the spike event is rescheduled. The
simulator runs until the queue is empty or until the desired run-time has elapsed.
A similar event-driven neural simulator was developed by Pratt [6].
The simulator output may be plotted by a number of commercially available plotting
programs, including Gnuplot, Mathematica, Xvgr, and Cview.
3
N euraLOG: Neural Schematic Capture
NeuraLOG is a schematic entry tool, which allows the convenient entry of "neural"
circuit diagrams, consisting of neurons, synapses, test inputs, and custom symbols.
NeuraLOG is a customization of the program AnaLOG, by John Lazzaro and Dave
Gillespie.
The parameters of the neural elements are entered directly on the schematic diagram; these parameters include the neuron refractory period, duration and intensity
of the post-synaptic current pulse following an action potential, saturation value of
summating post-synaptic currents, tonic input currents, axonal delays, etc. Custom symbols can be defined, so that arbitrarily complex hierarchical designs may
be made. It is common to create a complex "neuron" containing many neuron and
synapse primitive elements. Spiking inputs may be supplied as external stimuli for
the circuit in a number of different formats, including single spikes, periodic spike
trains, periodic bursts, poisson random spike trains, and gaussian-jittered periodic
Event-Driven Simulation of Networks of Spiking Neurons
spike trains. Textual input to Spike is also supported, to allow simulation of circuit
topologies that would be too time-consuming to enter graphically.
4
A Simple Example
A simple example of a neural circuit is shown in Figure 1. This circuit consists of
two neurons (the large disks), several synapses (the large triangles), and two tonic
inputs (the small arrows). The text strings associated with each symbol define
that symbol's parameters: neuron parameters are identifier labels (Le., ni) and
refractory period in milliseconds (ms); synapse parameters are the value of the postsynaptic current in nA, and the duration of the current pulse in ms, and an optional
saturation parameter, which indicates how many post-synaptic current pulses may
be superposed before saturation; the tonic input parameter is the injected current
in nA. Filled symbols (tonic inputs and synapses) indicate inhibitory behavior.
-.8108>
-815>
6.4>
.132.5>
-.001
>
Figure 1: Graphical input representation of a simple neural circuit, as entered in NeuraLOG.
The simulated behavior of the circuit is shown in Figure 2. The neuron ni exhibits
an adapting bursting behavior, as seen in the top trace of the plot.
The excitatory tonic current input to neuron ni causes ni to fire repeatedly. The
weakly excitatory synapse from ni to neuron n2 causes n2 to fire after many spikes
from n1. The synaptic current in the synapse from ni to n2 is plotted in the trace
labeled snin2. The strongly inhibitory synapse from n2 to ni causes ni to stop
firing after n2 fires a spike. The synaptic current in the synapse from n2 to ni is
plotted in the trace labeled sn2ni. The combination of the excitatory tonic input
to ni and the inhibitory feedback from n2 to ni causes the bursting behavior.
The adapting behavior is caused by the self-inhibitory accumulating feedback from
neuron ni to itself, via the summating inhibitory synapse in the top left of the
diagram. Each spike on ni causes a slightly increased inhibitory current into ni,
which gradually slows the rate of firing with each successive pulse. The synaptic
current in this inhibitory synapse is plotted in the trace snini; it is similar to the
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n1
_mJJJD"'----___
JillJJU"'----___OOJJU~
J~
J
J
n2
sn1n1
sn2n1
o
10
20
30
40
50
60
70
Time (ms)
Figure 2: Simulation results for the circuit of Figure 1, showing adapting
bursting behavior.
current that would be generated by a calcium-dependent potassium channel.
This simple example demonstrates that the summating synapse primitive can be
used to model a behavior that is not strictly synaptic in origin; it can be thought
of as a general time-dependent state variable. This example also illustrates the
principle that proper network topology (summating synapse in a negative feedback
loop) can lead to realistic system-level behavior (gradual adaptation), even though
the basic circuit elements may be rather primitive (boxcar current pulses).
5
Applications of the Simulation Tools
NeuraLOG and Spike have been used by the author to model spiking associative
memories, adaptive structures that learn to predict a time delay, and chaotic spiking
circuits. Researchers at Caltech [7, 8] and the Salk Institute have used the tools in
their studies of locust central pattern generators (CPGs) and cortical oscillations.
The cortical oscillation circuits contain a few hundred neurons and a few thousand
synapses. A CPG circuit, developed by Sylvie Ryckebusch, is shown in Figure 3;
the corresponding simulation output is shown in Figure 4.
NeuraLOG and Spike are distributed at no charge under the GNU licence. They
are currently supported on HP and Sun workstations. The tools are supplied with
a user's manual and working tutorial examples.
Event-Driven Simulation of Networks of Spiking Neurons
in)>--~.
~)>--~.
pill
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i11)
~
levi)
<l
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pill)
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pill)
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ItIVr )
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<l
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IwI)
Figure 3: Sylvie Ryckebusch's locust CPG circuit. For clarity, the synapse
parameters have been omitted.
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I
In
dfl
dsl
i11
levi
pirl
dfr
dsr
j
11r
t
levr
j
pirr
J J J J
-L t -I 1 -I
j
J j .J j
0
j
20
40
60
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L -L t -I
j
j
j
j
80
J
?
100
Time (ms)
Figure 4: Simulation results for Sylvie Ryckebusch's locust CPG circuit.
6
The Link to Analog VLSI
Analog VLSI circuit primitives that can be modelled by Spike have been designed
and tested. The circuits are shown in Figure 5, and have been described previously
[9, 10]. These circuits have been used by workers at Caltech to implement VLSI
models of central pattern generators. The software simulation tools allow simulation
of complex neural circuits prior to fabrication, to improve the likelihood of success
on first silicon, and to allow optimization of shared parameters (bias wires).
7
Conclusion
NeuraLOG and Spike fill a need for a fast neural simulator that can model large
networks of biologically realistic spiking neurons. The simple computational primitives within Spike can be used to create complex and realistic neural behaviors
in arbitrarily complex hierarchical designs. The tools are publicly available at no
charge. NeuraLOG and Spike have been used by a number of research labs for
detailed modeling of biological systems.
Acknowledgements
NeuraLOG is a customization of the program AnaLOG, which was written by John
Lazzaro and David Gillespie. Lloyd Watts gratefully acknowledges helpful discussions with Carver Mead, Sylvie Ryckebusch, Misha Mahowald, John Lazzaro,
Event-Driven Simulation of Networks of Spiking Neurons
.......................?... ....... ..???....... ....................?.....?.............. ......?......................
~
Ie I
m
IKJ
I
.
~ ............ ~y.f!~.P.s.~ .............. I~~.i~ .. L............................... ~~':Ir~!"............................ .
Figure 5: CMOS Analog VLSI circuit primitives. The neuron circuit models
a voltage-gated sodium channel and a delayed rectifier potassium channel
to produce a spiking mechanism. The tonic circuit allows constant currents
The synapse circuit
to be injected onto the membrane capacitance
creates a boxcar current pulse in response to a spike input.
em.
David Gillespie, Mike Vanier, Brad Minch, Rahul Sarpeshkar, Kwabena Boahen,
John Platt, and Steve Nowlan. Thanks to Sylvie Ryckebusch for permission to use
her CPG circuit example.
References
[1] J. Hertz, A. Krogh and R. Palmer, Introduction to the Theory of Neural Computation, Addison-Wesley, 1991.
[2] N. Y-S. Kiang, T. Watanabe, E. C. Thomas, L. F. Clark, "Discharge Patterns
of Single Fibers in the Cat's Auditory Nerve", MIT Res. Monograph No. 35,
(MIT, Cambridge, MA).
[3] M. Konishi, T.T. Takahashi, H. Wagner, W.E. Sullivan, C.E. Carr, "Neurophysiological and Anatomical Substrates of Sound Localization in the Owl",
In Auditory Function, G.M. Edelman, W.E. Gall, and W.M. Cowan, eds., pp.
721-745, Wiley, New York.
[4] D. P. Phillips and S. E. Hall, "Response Timing Constraints on the Cortical
Representation of Sound Time Structure" , Journal of the Acoustical Society of
America, 88 (3), pp. 1403-1411, 1990.
[5] R.R. de Ruyter van Steveninck and W. Bialek, "Real-time Performance of a
movement-sensitive neuron in the blowfly visual system: Coding and infor-
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[6]
[7]
[8]
[9]
[10]
mation transfer in short spike sequences", Proceedings of the Royal Society of
London, Series B, 234, 379-414.
G. A. Pratt, Pulse Computation, Ph.D. Thesis, Massachusetts Institute of
Technology, 1989.
M. Wehr, S. Ryckebusch and G. Laurent, Western Nerve Net Conference, Seattle, Washington, 1993.
S. Ryckebusch, M. Wehr, and G. Laurent, "Distinct rhythmic locomotor patterns can be generated by a simple adaptive neural circuit: biology, simulation,
and VLSI implementation", in review, Journal of Computational Neuroscience.
R. Sarpeshkar, L. Watts, C.A. Mead, "Refractory Neuron Circuits", Internal Memorandum, Physics of Computation Laboratory, California Institute of
Technology, 1992.
L. Watts, "Designing Networks of Spiking Silicon Neurons and Synapses", Proceedings of Computation and Neural Systems Meeting CNS*92, San Francisco,
CA,1992.
PART VIII
VISUAL PROCESSING
| 834 |@word disk:1 pulse:11 azimuthal:1 simulation:15 simplifying:1 gradual:1 carry:1 series:1 current:22 com:1 nowlan:1 yet:1 written:1 john:4 realistic:5 designed:3 plot:1 tone:1 short:1 dfl:1 successive:1 cpg:4 burst:1 edelman:1 consists:1 presumed:1 behavior:14 simulator:8 chi:1 vertebrate:1 circuit:27 kiang:1 string:1 developed:3 charge:2 demonstrates:1 platt:1 before:1 timing:3 mead:2 laurent:2 firing:4 bursting:3 palmer:1 steveninck:1 locust:3 timedependent:1 implement:1 chaotic:1 sullivan:1 adapting:3 thought:1 convenient:1 onto:1 superposed:1 accumulating:1 primitive:8 graphically:1 duration:2 pure:1 fill:1 konishi:1 memorandum:1 discharge:1 user:1 substrate:1 us:2 gall:1 designing:1 origin:1 element:4 labeled:2 mike:1 fly:1 capture:1 thousand:1 sun:1 movement:1 highest:2 monograph:1 boahen:1 weakly:1 localization:2 creates:1 triangle:1 pill:3 represented:1 cat:2 sarpeshkar:2 fiber:1 america:1 train:3 distinct:1 fast:3 london:1 artificial:1 apparent:1 compartmental:1 itself:1 associative:1 sequence:1 net:1 adaptation:2 loop:1 entered:2 seattle:1 potassium:2 empty:1 produce:1 cmos:2 krogh:1 indicate:1 subsequently:1 owl:2 behaviour:1 licence:1 biological:3 summation:1 strictly:1 hall:1 barn:1 predict:2 omitted:1 label:1 currently:1 sensitive:1 create:2 tool:6 mit:2 gaussian:1 mation:1 rather:1 varying:3 voltage:2 indicates:1 likelihood:1 helpful:1 abstraction:1 dependent:2 typically:1 integrated:2 her:1 vlsi:6 infor:1 spatial:1 washington:1 kwabena:1 biology:1 nearly:1 future:1 commercially:1 stimulus:2 piecewise:2 few:2 composed:1 delayed:1 phase:1 consisting:1 cns:1 fire:3 n1:2 investigate:1 custom:2 misha:1 capable:1 worker:1 filled:1 carver:1 desired:1 plotted:4 re:1 increased:1 modeling:1 mahowald:1 entry:2 hundred:1 delay:4 fabrication:2 too:1 periodic:3 jittered:1 minch:1 thanks:1 ie:1 physic:1 continuously:1 na:2 thesis:1 central:2 containing:1 external:1 summating:4 takahashi:1 potential:1 de:1 lloyd:2 coding:1 includes:1 inc:1 caused:1 later:1 lab:1 iwi:2 dll:1 ni:14 publicly:1 ir:1 dsl:1 modelled:1 trajectory:1 researcher:1 dave:1 synapsis:8 manual:1 synaptic:9 ed:1 frequency:1 mathematica:1 pp:2 associated:1 workstation:1 stop:1 auditory:3 massachusetts:1 amplitude:1 nerve:2 wesley:1 steve:1 response:2 rahul:1 synapse:12 refractoriness:1 strongly:1 though:1 just:1 until:2 working:1 western:1 scheduled:3 contain:1 laboratory:1 self:1 excitation:1 m:4 carr:1 common:1 spiking:15 refractory:4 analog:6 silicon:2 cambridge:1 enter:1 phillips:1 boxcar:3 hp:1 gratefully:1 synaptics:1 locomotor:1 etc:1 driven:9 pirl:1 arbitrarily:2 success:1 meeting:1 caltech:2 seen:1 period:3 signal:1 full:1 sound:2 characterized:1 post:5 schematic:3 basic:1 poisson:1 cochlea:1 cell:4 preserved:1 diagram:3 cowan:1 axonal:3 spiral:1 enough:2 pratt:2 topology:2 sylvie:5 queue:3 york:1 cause:6 lazzaro:3 action:1 repeatedly:1 dar:2 useful:1 detailed:3 ph:1 supplied:2 millisecond:1 inhibitory:7 tutorial:1 neuroscience:1 anatomical:1 key:1 levi:5 clarity:1 fraction:1 run:2 jose:1 injected:5 reasonable:1 oscillation:2 decision:1 gnu:1 simplification:1 constraint:1 software:2 format:1 orchard:1 watt:8 combination:1 membrane:2 hertz:1 slightly:1 em:1 postsynaptic:1 perceptible:1 biologically:2 ikj:1 gradually:1 previously:1 turn:1 mechanism:1 addison:1 end:2 available:2 i1r:2 hierarchical:2 blowfly:2 simulating:2 occurrence:1 permission:1 thomas:1 top:2 include:1 graphical:1 lock:1 maintaining:1 exploit:1 society:2 capacitance:1 question:1 quantity:1 spike:25 ryckebusch:7 bialek:1 exhibit:2 link:1 simulated:3 acoustical:1 viii:1 trace:4 slows:1 negative:1 vanier:1 design:2 implementation:1 calcium:1 proper:1 perform:1 gated:1 neuron:28 wire:1 optional:1 tonic:8 intensity:1 david:2 required:1 optimized:1 california:1 elapsed:1 textual:1 usually:1 pattern:4 dsr:1 program:3 saturation:3 including:2 memory:1 royal:1 gillespie:3 event:14 rely:1 sodium:1 improve:1 technology:2 acknowledges:1 text:1 prior:2 review:1 acknowledgement:1 relative:1 generator:2 clark:1 plotting:1 principle:1 excitatory:3 surprisingly:1 supported:2 cpgs:1 dfr:1 bias:1 allow:4 institute:3 wagner:1 rhythmic:1 sparse:1 distributed:1 van:1 feedback:3 cortical:4 author:1 made:1 adaptive:2 san:2 investigating:1 parkway:1 francisco:1 consuming:1 continuous:1 channel:3 learn:1 ruyter:1 ca:2 transfer:1 complex:6 wehr:2 arrow:1 n2:8 identifier:1 salk:1 wiley:1 watanabe:1 rectifier:1 showing:1 symbol:5 i11:3 illustrates:1 suited:1 customization:2 ganglion:1 neurophysiological:1 visual:4 contained:1 brad:1 ma:1 rescheduled:1 shared:1 change:1 operates:1 total:1 internal:1 tested:1 |
7,064 | 835 | Statistics of Natural Images:
Scaling in the Woods
Daniel L. Ruderman* and William Bialek
NEe Research Institute
4 Independence Way
Princeton, N.J. 08540
Abstract
In order to best understand a visual system one should attempt
to characterize the natural images it processes. We gather images
from the woods and find that these scenes possess an ensemble scale
invariance. Further, they are highly non-Gaussian, and this nonGaussian character cannot be removed through local linear filtering. We find that including a simple "gain control" nonlinearity in
the filtering process makes the filter output quite Gaussian, meaning information is maximized at fixed channel variance. Finally, we
use the measured power spectrum to place an upper bound on the
information conveyed about natural scenes by an array of receptors.
1
Introduction
Natural stimuli are playing an increasingly important role in our understanding of
sensory processing. This is because a sensory system's ability to perform a task is a
statistical quantity which depends on the signal and noise characteristics. Recently
several approaches have explored visual processing as it relates to natural images
(Atick & Redlich '90, Bialek et al '91, van Hateren '92, Laughlin '81, Srinivasan
et al '82) . However, a good characterization of natural scenes is sorely lacking. In
this paper we analyze images from the woods in an effort to close this gap. We
? Current address:
CB2 3EG, England.
The Physiological Laboratory,
Downing Street,
Cambridge
551
552
Ruderman and Bialek
further attempt to understand how a biological visual system should best encode
these images.
2
The Images
Our images consist of 256 x 256 pixels 1(x) which are calibrated against luminance
(see Appendix). We define the image contrast logarithmically as
cf;(x) = In(I(x)/10),
where 10 is a reference intensity defined for each image. We choose this constant
such that Ex cf;(x) = 0; that is, the average contrast for each image is zero. Our
analysis is of the contrast data cf;( x).
3
Scaling
Recent measurements (Field '87, Burton & Moorhead '87) suggest that ensembles
of natural scenes are scale-invariant. This means that and any quantity defined on
a given scale has statistics which are invariant to any change in that scale. This
seems sensible in light of the fact that the images are composed of objects at all
distances, and so no particular angular scale should stand out. (Note that this does
not imply that any particular image is fractal! Rather, the ensemble of scenes has
statistics which are invariant to scale.)
3.1
Distribution of Contrasts
We can test this scaling hypothesis directly by seeing how the statistics of various
quantities change with scale. We define the contrast averaged over a box of size
N x N (pixels) to be
cf;N =
~2
N
L
cf;( i, j).
i,j=l
We now ask: "How does the probability P( cf;N) change with N?"
=
In the left graph of figure 1 we plot log(P( cf;N / cf;~MS)) for N
1,2,4,8,16,32 along
with the parabola corresponding to a Gaussian of the same variance. By dividing
out the RMS value we simply plot all the graphs on the same contrast scale. The
graphs all lie atop one another, which means the contrast scales-the distribution's
shape is invariant to a change in angular scale. Note that the probability is far from
Gaussian, as the graphs have linear, and not parabolic, tails. Even after averaging
nearly 1000 pixels (in the case of 32x32), it remains non-Gaussian. This breakdown
of the central limit theorem implies that the pixels are correlated over very long
distances. This is analogous to the physics of a thermodynamic system at a critical
point.
3.2
Distribution of Gradients
As another example of scaling, we consider the probability distribution of image
gradients. We define the magnitude of the gradient by a discrete approximation
Statistics of Natural Images: Scaling in the Woods
.,
? 15
-2.5
?35
.,.,' - - -............
-2
.,
-~--'---~-----'-----'
Figure 1: Left: Semi-log plot of P(</JN/(VJMS ) for N
1,2,4,8,16,32 with a
Gaussia~ of the same variance for comparison (solid line). Right: Semi-log plot of
P(GN/G N) for same set of N's with a Rayleigh distribution for comparison (solid
line) .
such that
G(x)
= IG(x)1 ~ 1'V</J (x) I?
We examine this quantity over different scales by first rescaling the images as above
and then evaluating the gradient at the new scale. We plot log( P( G N / GN )) for N
1,2,4,8,16,32 in the right graph of figure 1, along with the Rayleigh distribution,
P ~ G exp( -aG 2 ). If the images had Gaussian statistics, local gradients would be
Rayleigh distributed. Note once again scaling of the distribution.
=
3.3
Power Spectrum
Scaling can also be demonstrated at the level of the power spectrum. If the ensemble
is scale-invariant, then the spectrum should be of the form
A
S(k)
= k 2 -'7'
where k is measured in cycles/degree, and S is the power spectrum averaged over
orientations.
The spectrum is shown in figure 2 on log-log axes. It displays overlapping data from
the two focal lengths, and shows that the spectrum scales over about 2.5 decades in
spatial frequency. We determine the parameters as A = (6.47?0.13) x 1O- 3 deg.(O.19)
and 1J = 0.19 ? 0.01. The integrated power spectrum up to 60 cycles/degree (the
human resolution limit) gives an RMS contrast of about 30%.
4
Local Filtering
The early stages of vision consist of neurons which respond to local patches of
images. What do the statistics of these local processing units look like? We convolve
images with the filter shown in the left of figure 3, and plot the histogram of its
output on a semi-log scale on the right of the figure.
553
554
Ruderman and Bialek
~
<
-1
"?
?
'"?
~
'tl
-2
.,
~
.,"
~
~
-3
~
0
~
~
?
)
-4
0
.':
0
rl
'"
...,0
-5
??
-6
-1. 5
-1
-0 . 5
0
0.5
1
LoglO[Spatial Frequency (cycles/degree?)
1.5
Figure 2: Power spectrum of the contrast of natural scenes (log-log plot).
The distribution is quite exponential over nearly 4 decades in probability. In fact,
almost any local linear filter which passes no DC has this property, including centersurround receptive fields. Information theory tells us that it is best to send signals
with Gaussian statistics down channels which have power constraints. It is of interest, then, to find some type of filtering which transforms the exponential distributions we find into Gaussian quantities.
Music, as it turns out, has some similar properties. An amplitude histogram from 5
minutes of "The Blue Danube" is shown on the left of figure 4. It is almost precisely
exponential over 4 decades in probability. We can guess what causes the excesses
over a Gaussian distribution at the peak and the tails; it's the dynamics. When
a quiet passage is played the amplitudes lie only near zero, and create the excess
in the peak. When the music is loud the fluctuations are large, thus creating the
?0.
?1
+
-
.,.
-
+
?2'
.
?2
Figure 3: Left: 2 X 2 local filter. Right: Semi-log plot of histogram of its output
when filtering natural scenes.
Statistics of Natural Images: Scaling in the Woods
-<15
.,
.
....
.
...
..
".
.
...
.....
..
".""." .????? ".,1, ???????????? ," ??
-.
-15
"
I
-2
I
I
I
"
"
I
:::::::1:::::::0:::::::1:::::::
/
?25
-.
i
I
;
...
I
.
.
...
I I . . . . . . . . 111,.,., II ?? I ??? I I . I ??? ,1111,.
I
!
/
i
-4
-2
Figure 4: Left: Semi-log histogram of "The Blue Danube" with a Gaussian for
comparison (dashed). Right: 5 x 5 center-surround filter region.
tails. Most importantly, these quiet and loud passages extend coherently in time;
so to remove the peak and tails, we can simply slowly adjust a "volume knob" to
normalize the fluctuations. The images are made of objects which have coherent
structure over space, and a similar localized dynamic occurs. To remove it, we need
some sort of gain control.
To do this, we pass the images through a local filter and then normalize by the local
standard deviation of the image (analogous to the volume of a sound passage):
./,( ) = ?(x) -
?(x)
O'(x)'
'f/ X
Here ?(x) is the mean image contrast in the N x N region surrounding x, and O'(x)
is the standard deviation within the same region (see the right of figure 4) .
.-",-"
/
-1
>
-1
/'
,
~
!
~
I
-, ,
-J
,
,
/
I
--,\
!
i
i
I
I
-1
,,
\
,/'
;
~
~
\
i
\,
\
I
3
\\
"''\''
-, ,
-J
\
\\
:'
?
Contr .... t
\'~"
-2
\\
i
"""" " ."~,
,
.J
,
-.
c
0
,
1 5
Gradlen t
:l
(U rHtl
2.5
of Me.n1
)
S
Figure 5: Left: Semi-log plot of histogram of 1/J, with Gaussian for comparison
(dashed). Right: Semi-log plot of histogram of gradients of 1/J, with Rayleigh distribution shown for comparison (dashed).
We find that for a value N = 5 (ratio of the negative surround to the positive
center), the histograms of 1/J are the closest to Gaussian (see the left of figure 5) .
Further, the histogram of gradients of 1/J is very nearly Rayleigh (see the right of
555
556
Ruderman and Bialek
figure 5). These are both signatures of a Gaussian distribution. Functionally, this
"variance normalization" procedure is similar to contrast gain control found in the
retina and LGN (Benardete et ai, '92). Could its role be in "Gaussianizing" the
image statistics?
5
Information in the Retina
From the measured statistics we can place an upper bound on the amount of information an array of photo receptors conveys about natural images. We make the
following assumptions:
? Images are Gaussian with the measured power spectrum. This places an
upper bound on the entropy of natural scenes, and thus an upper bound
on the information represented.
? The receptors sample images in a hexagonal array with diffraction-limited
optics. There is no aliasing.
? Noise is additive, Gaussian, white, and independent of the image.
The output of the
nth
receptor is thus given by
Yn
=
J
d2x ?(x) M(x - xn)
+
'f/n,
where Xn is the location of the receptor, M(x) is the point-spread function of the
optics, and 'f/n is the noise. For diffraction-limited optics,
M(k) ~ 1 -
Ikl/kc,
where kc is the cutoff frequency of 60 cycles/degree.
In the limit of an infinite lattice, Fourier components are independent, and the total
information is the sum of the information in each component:
+= 47J"
Ac fkCdkklog[1+A1 2 IM (k)1 2 S(k)].
u
Jo
c
Here I is the information per receptor, Ac is the area of the unit cell in the lattice,
and u 2 is the variance of the noise.
We take S(k) = A/k 2 - fJ , with A and 'f/ taking their measured values, and express
the noise level in terms of the signal-to-noise ratio in the receptor. In figure 6 we
plot the information per receptor as a function of SN R along with the information
capacity (per receptor) of the photoreceptor lattice at that SN R, which is
C=
1
2 log [1 + SN R] .
=
The information conveyed is less than 2 bits per receptor per image, even at SN R
1000. The redundancy of this representation is quite high, as seen by the gap
between the curves; at least as much of the information capacity is being wasted as
is being used .
Statistics of Natural Images: Scaling in the Woods
I
(bits)
5
4
0.5
1
1.5
2
2.5
3 LoglO[SNR)
Figure 6: Information per receptor per image (in bits) as a function of 10g(SN R)
(lower line). Information capacity per receptor ( upper line).
6
Conclusions
We have shown that images from the forest have scale-invariant, highly nonGaussian statistics. This is evidenced by the scaling of the non-Gaussian histograms
and the power-law form of the power spectrum. Local linear filtering produces values with quite exponential probability distributions. In order to "Gaussianize," we
must use a nonlinear filter which acts as a gain control. This is analogous to contrast
gain control, which is seen in the mammalian retina. Finally, an array of receptors
which encodes these natural images only conveys at most a few bits per receptor
per image of information, even at high SN R. At an image rate of 50 per second,
this places an information requirement of less than about 100 bits per second on a
foveal ganglion cell.
Appendix
Snapshots were gathered using a Sony Mavica MVC-5500 still video camera
equipped with a 9.5-123.5mm zoom lens. The red, green, and blue signals were
combined according to the standard CIE formula Y = 0.59 G + 0.30 R + 0.11 B
to produce a grayscale value at each pixel. The quantity Y was calibrated against
incident luminance to produce the image intensity I(x). The images were cropped
to the central 256 x 256 region.
The dataset consists of 45 images taken at a 15mm focal length (images subtend
15 0 of visual angle) and 25 images at an 80mm focal length (3 0 of visual angle) . All
images were of distant objects to avoid problems of focus. Images were chosen by
placing the camera at a random point along a path and rotating the field of view
until no nearby objects appeared in the frame. The camera was tilted by less than
10 0 up or down in an effort to avoid sky and ground. The forested environment
(woods in New Jersey in springtime) consisted mainly of trees, rocks, hillside, and
a stream.
557
558
Ruderman and Bialek
Acknowledgements
We thank H. B. Barlow, B. Gianulis, A. J. Libchaber, M. Potters, R. R. de Ruyter
van Stevenink, and A. Schweitzer. Work was supported in part by a fellowship from
the Fannie and John Hertz Foundation (to D.L.R.).
References
J .J. Atick and N. Redlich. Towards a theory of early visual processing Neural
Computation, 2:308, 1990.
E. A. Benardete, E. Kaplan, and B. W. Knight. Contrast gain control in the primate
retina: P cells are not X-like, some M-cells are. Vis. Neuosci., 8:483-486, 1992.
W. Bialek, D. L. Ruderman, and A. Zee. The optimal sampling of natural images: a design principle for the visual system?, in Advances in Neural Information
Processing systems, 3, R. P. Lippman, J. E. Moody and D. S. Touretzky, eds., 1991.
G. J. Burton and I. R. Moorhead. Color and spatial structure in natural scenes.
Applied Optics, 26:157-170, 1987.
D. J. Field. Relations between the statistics of natural images and the response
properties of cortical cells. I. Opt. Soc. Am. A, 4:2379, 1987.
J. H. van Hateren. Theoretical predictions of spatiotemporal receptive fields of fly
LMCs, and experimental validation. I. Compo Physiol. A, 171:157-170, 1992.
S. B. Laughlin. A simple coding procedure enhances a neuron's information capacity. Z. Naturforsh., 36c:910-912, 1981.
M. V. Srinivasan, S. B. Laughlin, and A. Dubs. Predictive coding: a fresh view of
inhibition in the retina. Proc. R. Soc. Lond. B, 216:427-459, 1982.
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7,065 | 836 | When Will a Genetic Algorithm
Outperform Hill Climbing?
Melanie Mitchell
Santa Fe Institute
1660 Old Pecos Trail, Suite A
Santa Fe, NM 87501
John H. HoUand
Dept. of Psychology
University of Michigan
Ann Arbor, MI 48109
Stephanie Forrest
Dept. of Computer Science
University of New Mexico
Albuquerque, NM 87131
Abstract
We analyze a simple hill-climbing algorithm (RMHC) that was previously shown to outperform a genetic algorithm (GA) on a simple
"Royal Road" function. We then analyze an "idealized" genetic
algorithm (IGA) that is significantly faster than RMHC and that
gives a lower bound for GA speed. We identify the features of the
IGA that give rise to this speedup, and discuss how these features
can be incorporated into a real GA.
1
INTRODUCTION
Our goal is to understand the class of problems for which genetic algorithms (GA)
are most suited, and in particular, for which they will outperform other search
algorithms. Several studies have empirically compared GAs with other search and
optimization methods such as simple hill-climbing (e.g., Davis, 1991), simulated
annealing (e.g., Ingber & Rosen, 1992), linear, nonlinear, and integer programming
techniques, and other traditional optimization techniques (e.g., De Jong, 1975).
However, such comparisons typically compare one version of the GA with a second
algorithm on a single problem or set of problems, often using performance criteria
which may not be appropriate. These comparisons typically do not identify the
features that led to better performance by one or the other algorithm, making it
hard to distill general principles from these isolated results. In this paper we look in
depth at one simple hill-climbing method and an idealized form of the GA, in order
to identify some general principles about when and why a GA will outperform hill
climbing.
51
52
Mitchell, Holland, and Forrest
81 = 11111111??????????????????????????????????????????????.......... j C1 =8
82 = ????????11111111??????????????????????????????????????????...... j C2 = 8
83 = ????????????????11111111??????????????????????????????.......... j C3 =8
84 = ????????????????????????11111111??????????????????????.......... ; C4 =8
85 = ????????????????????????????????11111111????????????????........ ; Cs = 8
86 = ????????????????????????????????????????11111111??????.......... ; C6 =8
87
????????????????????????????????????????????????11111111?
.......; Cs
C7 =
8S =
= ......................................................
??11111111;
=8
8
8~t=1111111111111111111111111111111111111111111111111111111111111111
Figure 1: Royal Road function Rl.
In previous work we have developed a class of fitness landscapes (the "Royal Road"
functions; Mitchell, Forrest, & Holland, 1992; Forrest & Mitchell, 1993) designed to
be the simplest class containing the features that are most relevant to the performance of the GA. One of our purposes in developing these landscapes is to carry
out systematic comparisons with other search methods.
A simple Royal Road function, R l , is shown in Figure 1. Rl consists of a list of
partially specified bit strings (schemas) Si in which '*' denotes a wild card (either
o or 1). Each schema is given with a coefficient Ci. The order of a schema is
the number of defined (non-'*') bits. A bit string x is said to be an instance of a
schema 8, x E 8, if x matches s in the defined positions. The fitness Rl(X) of a bit
string x is defined as follows:
8,
Rl(X)
{I
=~
~ CiOi(X), where o,(x) = 0
,
E
if x Si
otherwise.
For example, if x is an instance of exactly two of the order-8 schemas, Rl (x)
Likewise, Rl (111 ... 1) = 64.
= 16.
The Building Block Hypothesis (Holland, 1975/1992) states that the GA works well
when instances of low-order, short schemas ("building blocks") that confer high fitness can be recombined to form instances of larger schemas that confer even higher
fitness. Given this hypothesis, we initially expected that the building-block structure of Rl would layout a "royal road" for the GA to follow to the optimal string.
We also expected that simple hill-climbing schemes would perform poorly since a
large number of bit positions must be optimized simultaneously in order to move
from an instance of a lower-order schema (e.g., 11111111** ... *) to an instance of a
higher-order intermediate schema (e.g., 11111111*****?*?11111111**... *). However both these expectations were overturned (Forrest & Mitchell, 1993). In our
experiments, a simple GA (using fitness-proportionate selection with sigma scaling,
single-point crossover, and point mutation) optimized Rl quite slowly, at least in
part because of "hitchhiking": once an instance of a higher-order schema is discovered, its high fitness allows the schema to spread quickly in the population, with Os
in other positions in the string hitchhiking along with the Is in the schema's defined
positions. This slows down the discovery of schemas in the other positions, especially those that are close to the highly fit schema's defined positions. Hitchhiking
can in general be a serious bottleneck for the GA, and we observed similar effects
When Will a Genetic Algorithm Outperform Hill Climbing?
Table 1: Mean and median number of function evaluations to find the optimum
string over 200 runs of the GA and of various hill-climbing algorithms on R 1 . The
standard error is given in parentheses.
in several variations of our original GA.
Our other expectation-that the GA would outperform simple hill-climbing on
these functions-was also proved wrong. Forrest and Mitchell (1993) compared
the GA's performance on a variation of Rl with three different hill-climbing methods: steepest ascent hill-climbing (SAHC), next-ascent hill-climbing (NAHC), and a
zero-temperature Monte Carlo method, which Forrest and Mitchell called ''random
mutation hill-climbing" (RMHC). In RMHC, a string is chosen at random and its
fitness is evaluated. The string is then mutated at a randomly chosen single locus,
and the new fitness is evaluated. If the mutation leads to an equal or higher fitness,
the new string replaces the old string. This procedure is iterated until the optimum
has been found or a maximum number of function evaluations has been performed.
Here we have repeated these experiments for R 1 . The results (similar to those given
for R2 in Forrest & Mitchell, 1993) are given in Table 1. We compare the mean
and median number of function evaluations to find the optimum string rather than
mean and median absolute run time, because in almost all GA applications (e.g.,
evolving neural-network architectures), the time to perform a function evaluation
vastly dominates the time required to execute other parts of the algorithm. For this
reason, we consider all parts of the algorithm excluding the function evaluations to
take negligible time.
The results on SAHC and NAHC were as expected-while the GA found the optimum on RI in an average of 61,334 function evaluations, neither SAHC nor NAHC
ever found the optimum within the maximum of 256,000 function evaluations. However, RMH C found the optimum on Rl in an average of 6179 function evaluationsnearly a factor often faster than the GA. This striking difference on landscapes originally designed to be "royal roads" for the GA underscores the need for a rigorous
answer to the question posed earlier: "Under what conditions will a GA outperform
other search algorithms, such as hill climbing?"
2
ANALYSIS OF RMHC AND AN IDEALIZED GA
To begin to answer this question, we analyzed the RMHC algorithm with respect to
R 1 ? Suppose the fitness function c,onsists of N adjacent blocks of K Is each (in RI,
N = 8 and K = 8). What is the expected time (number of function evaluations)
E(K, N) to find the optimum string of allIs? We can first ask a simpler question:
what is the expected time E(K, 1) to find a single block of K Is? A Markov-chain
analysis (not given here) yields E(K, 1) slightly larger than 2K , converging slowly
to 2K from above as K -+ 00 (Richard Palmer, personal communication). For
S3
54
Mitchell, Holland, and Forrest
example, for K
=8, E(K, 1) = 301.2.
Now suppose we want RMHC to discover a string with N blocks of K Is. The
time to discover a first block of K Is is E(K, 1), but, once it has been found, the
time to discover a second block is longer, since many of the function evaluations are
"wasted" on testing mutations inside the first block. The proportion of non-wasted
mutations is (K N - K) / K N; this is the proportion of mutations that occur in the
KN - K positions outside the first block. The expected time E(K, 2) to find a
second block is E(K, 1) + E(K, l)[KN/(KN - K)]. Similarly, the total expected
time is:
E(K,N)
N
N
= E(K, 1) + E(K, 1) N _ 1 + ... + E(K, 1) N _ (N _ 1)
[ 1+ 31+ ... + 1]
E(K,l)N 1 + "2
N
.
(1)
(The actual value may be a bit larger, since E(K,l) is the expected time to the first
block, whereas E(K, N) depends on the worst time for the N blocks.) Expression
(1) is approximately E(K, l)N(logN + r), where r is Euler's constant. For K
8, N
8, the value of expression (1) is 6549. When we ran RMHC on the Rl
function 200 times, the average number of function evaluations to the optimum was
6179, which agrees reasonably well with the expected value.
=
=
Could a GA ever do better than this? There are three reasons why we might expect
a GA to perform well on Rl. First, at least theoretically the GA is fast because
of implicit parallelism (Holland, 1975/1992): each string in the population is an
instance of many different schemas, and if the population is large enough and is
initially chosen at random, a large number of different schemas-many more than
the number of strings in the population-are being sampled in parallel. This should
result in a quick search for short, low-order schemas that confer high fitness. Second,
fitness-proportionate reproduction under the GA should conserve instances of such
schemas. Third, a high crossover rate should quickly combine instances oflow-order
schemas on different strings to create instances of longer schemas that confer even
higher fitness. Our previous experiments (Forrest & Mitchell, 1993) showed that
the simple GA departed from this "in principle" behavior. One major impediment
was hitchhiking, which limited implicit parallelism by fixing certain schema regions
sub optimally. But if the GA worked exactly as described above, how quickly could
it find the optimal string of Rl?
To answer this question we consider an "idealized genetic algorithm" (IGA) that
explicitly has the features described above. The IGA knows ahead of time what the
desired schemas are, and a "function evaluation" is the determination of whether a
given string contains one or more of them. In the IGA, at each time step a single
string is chosen at random, with uniform probability for each bit. The string is
"evaluated" by determining whether it is an instance of one or more of the desired
schemas. The first time such a string is found, it is sequestered. At each subsequent
discovery of an instance of one or more not-yet-discovered schemas the new string
is instantaneously crossed over with the sequestered string so that the sequestered
string contains all the desired schemas that have been discovered so far.
This procedure is unusable in practice, since it requires knowing a priori which
schemas are relevant, whereas in general an algorithm such as the GA or RMHC
When Will a Genetic Algorithm Outperform Hill Climbing?
directly measures the fitness of a string, and does not know ahead of time which
schemas contribute to high fitness. However, the idea behind the GA is to do
implicitly what the IGA is able to do explicitly. This idea will be elaborated below.
Suppose again that our desired schemas consist of N blocks of K 1s each. What is
the expected time (number of function evaluations) until the saved string contains
all the desired schemas? Solutions have been suggested by G. Huber (personal communication), and A. Shevoroskin (personal communication), and a detailed solution
is given in (Holland, 1993). The main idea is to note that the probability of finding
a single desired block 8 on a random string is p = 1/2K, and the probability of
finding s by time t is 1 - (1 - p)t. Then the probability PN(t) that all N blocks
have been found by time tis:
PN(t) = (1 - (1 - p)t)N,
and the probability PN(t) that all N blocks are found at exactly time tis:
PN(t)
=[1- (1- p)t]N -
[1- (1- p)t-l]N.
The expected time is then
00
EN =
2:t ([1- (1- p)t]N -
[1- (1- p)t-l]N).
1
This sum can be expanded and simplified, and with some work, along with the
approximation (1- p)n ~ 1- np for small p, we obtain the following approximation:
EN ~ (lip)
N 1
I:; ~ 2K(logN + 1)?
n=l
The major point is that the IGA gives an expected time that is on the order of
2K log N, where RMHC gives an expected time that is on the order of 2K N log N,
a factor of N slower. This kind of analysis can help us predict how and when the
G A will outperform hill climbing.
What makes the IGA faster than RMHC? A primary reason is that the IGA perfectly implements implicit parallelism: each new string is completely independent
of the previous one, so new samples are given independently to each schema region.
In contrast, RMHC moves in the space of strings by single-bit mutations from an
original string, so each new sample has all but one of the same bits as the previous sample. Thus each new string gives a new sample to only one schema region.
The IGA spends more time than RMHC constructing new samples, but since we
are counting only function evaluations, we ignore the construction time. The IGA
"cheats" on each function evaluation, since it knows exactly the desired schemas,
but in this way it gives a lower bound on the number of function evaluations that
the GA will need on this problem.
Independent sampling allows for a speed-up in the IGA in two ways: it allows for
the possibility of more than one desirable schema appearing simultaneously on a
given sample, and it also means that there are no wasted samples as there are
in RMHC. Although the comparison we have made is with RMHC, the IGA will
also be significantly faster on Rl (and similar landscapes) than any hill-climbing
55
56
Mitchell, Holland, and Forrest
Levell:
Level 2:
Level 3:
Level 4:
81 82
83 8,
85 8S
81 8a
89 810
811 812
813 8H
815 81S
(81 82) (83 8,) (85 8S) (81 8a) (89 810) (811 812) (813 81') (815 81S)
(81 82
(81 82
83 8,) (85 8S
83 8,
85 8S
81 8a) (89 810
81 8a) (89 810
811 812) (813 8H
811 812
813 8H
815 81S)
815 81S)
Figure 2: Royal Road Function R4.
method that works by mutating single bits (or a small number of bits) to obtain
new samples.
The hitchhiking effects described earlier also result in a loss of independent samples
for the real GA. The goal is to have the real GA, as much as possible, approximate
the IGA. Of course, the IGA works because it explicitly knows what the desired
schemas are; the real GA does not have this information and can only estimate
what the desired schemas are by an implicit sampling procedure. But it is possible
for the real GA to approximate a number of the features of the IGA. Independent
samples: The population size has to be large enough, the selection process has to
be slow enough, and the mutation rate has to be sufficient to make sure that no
single locus is fixed at a single value in every (or even a large majority) of strings in
the population. Sequestering desired schemas: Selection has to be strong enough to
preserve desired schemas that have been discovered, but it also has to be slow enough
(or, equivalently, the relative fitness of the non-overlapping desirable schemas has
to be small enough) to prevent significant hitchhiking on some highly fit schemas,
which can crowd out desired schemas in other parts of the string. Instantaneous
crossover: The crossover rate has to be such that the time for a crossover to occur
that combines two desired schemas is small with respect to the discovery time for
the desired schemas. Speed-up over RMHC: The string length (a function of N) has
to be large enough to make the N speed-up factor significant.
These mechanisms are not all mutually compatible (e.g., high mutation works
against sequestering schemas), and thus must be carefully balanced against one
another. A discussion of how such a balance might be achieved is given in Holland
(1993).
3
RESULTS OF EXPERIMENTS
As a first step in exploring these balances, we designed R3, a variant of our previous
function R2 (Forrest & Mitchell, 1993), based on some of the features described
above. In R3 the desired schemas are 81-88 (shown in Fig. 1) and combinations
of them, just as in R2. However, in R3 the lowest-level order-8 schemas are each
separated by "introns" (bit positions that do not contribute to fitness-see Forrest
& Mitchell, 1993; Levenick, 1991) of length 24.
In R3, a string that is not an instance of any desired schema receives fitness 1.0.
Every time a new level is reached-i.e., a string is found that is an instance of one
or more schemas at that level-a small increment u is added to the fitness. Thus
strings at level 1 (that are instances of at least one level-l schema) have fitness
1 + u, strings at level 2 have fitness 1 + 2u, etc. For our experiments we set u 0.2.
=
When Will a Genetic Algorithm Outperfonn Hill Climbing?
Table 2: R4: Mean function evaluations (over 37 runs) to attain each level for
the GA and for RMHC. In the GA runs, the number of function evaluations is
sampled every 500 evaluations, so each value is actually an upper bound for an
interval of length 500. The standard errors are in parentheses. The percentage of
runs which reached each level is shown next to the heading "% runs." Only runs
which successfully reached a given level were included in the function evaluation
calculations for that level.
The purpose of the introns was to help maintain independent samples in each schema
position by preventing linkage between schema positions. The independence of
samples was also helped by using a larger population (2000) and the much slower
selection scheme given by the function. In preliminary experiments on R3 (not
shown) hitchhiking in the GA was reduced significantly, and the population was
able to maintain instances of all the lowest-level schemas throughout each run.
Next, we studied R4 (illustrated in Figure 2). R4 is identical to R3, except that it
does not have introns. Further, R4 is defined over 128-bit strings, thus doubling the
size of the problem. In preliminary runs on R4, we used a population size of 500,
a mutation rate of 0.005 (mutation always flips a bit), and multipoint crossover,
where the number of crossover points for each pair of parents was selected from a
Poisson distribution with mean 2.816.
Table 2 gives the mean number of evaluations to reach levels 1, 2, and 3 (neither
algorithm reached level 4 within the maximum of 10 6 function evaluations). As
can be seen, the time to reach level one is comparable for the two algorithms, but
the GA is much faster at reaching levels 2 and 3. Further, the GA discovers level
3 approximately twice as often as RMHC. As was said above, it is necessary to
balance the maintenance of independent samples with the sequestering of desired
schemas. These preliminary results suggest that R4 does a better job of maintaining
this balance than the earlier Royal Road functions. Working out these balances in
greater detail is a topic of future work.
4
CONCLUSION
We have presented analyses of two algorithms, RMHC and the IGA, and have used
the analyses to identify some general principles of when and how a genetic algorithm
will outperform hill climbing. We then presented some preliminary experimental
results comparing the GA and RMHC on a modified Royal Road landscape. These
analyses and results are a further step in achieving our original goals-to design the
simplest class of fitness landscapes that will distinguish the GA from other search
methods, and to characterize rigorously the general features of a fitness landscape
that make it suitable for a GA.
57
S8
Mitchell, Holland, and Forrest
Our modified Royal Road landscape R4, like Rl, is not meant to be a realistic
example of a problem to which one might apply a GA. Rather, it is meant to be
an idealized problem in which certain features most relevant to GAs are explicit,
so that the GA's performance can be studied in detail. Our claim is that in order
to understand how the GA works in general and where it will be most useful, we
must first understand how it works and where it will be most useful on simple yet
carefully designed landscapes such as these. The work reported here is a further
step in this direction.
Acknowledgments
We thank R. Palmer for suggesting the RMHC algorithm and for sharing his careful
analysis with us, and G. Huber for his assistance on the analysis of the IGA. We
also thank E. Baum, L. Booker, T. Jones, and R. Riolo for helpful comments and
discussions regarding this work. We gratefully acknowledge the support of the Santa
Fe Institute's Adaptive Computation Program, the Alfred P. Sloan Foundation
(grant B1992-46), and the National Science Foundation (grants IRI-9157644 and
IRI-9224912).
References
L. D. Davis (1991). Bit-climbing, representational bias, and test suite design. In R.
K. Belew and L. B. Booker (eds.), Proceedings of the Fourth International Conference on Genetic Algorithms, 18-23. San Mateo, CA: Morgan Kaufmann.
K. A. De Jong (1975). An Analysis of the Behavior of a Class of Genetic Adaptive
Systems. Unpublished doctoral dissertation. University of Michigan, Ann Arbor,
MI.
S. Forrest and M. Mitchell (1993). Relative building-block fitness and the buildingblock hypothesis. In D. Whitley (ed.), Foundations of Genetic Algorithms 2, 109126. San Mateo, CA: Morgan Kaufmann.
J. H. Holland (1975/1992). Adaptation in Natural and Artificial Systems. Cambridge, MA: MIT Press. (First edition 1975, Ann Arbor: University of Michigan
Press.)
J. H. Holland (1993). Innovation in complex adaptive systems: Some mathematical
sketches. Working Paper 93-10-062, Santa Fe Institute, Santa Fe, NM.
L. Ingber and B. Rosen (1992). Genetic algorithms and very fast simulated reannealing: A comparison. Mathematical Computer Modelling, 16 (11),87-100.
J. R. Levenick (1991). Inserting introns improves genetic algorithm success rate:
Taking a cue from biology. In R. K. Belew and L. B. Booker (eds.), Proceedings of
the Fourth International Conference on Genetic Algorithms, 123-127. San Mateo,
CA: Morgan Kaufmann.
M. Mitchell, S. Forrest, and J. H. Holland (1992). The royal road for genetic algorithms: Fitness landscapes and GA performance. In F. J. Varela and P. Bourgine
(eds.), Proceedings of the First European Conference on Artificial Life, 245-254.
Cambridge, MA: MIT Press.
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7,066 | 837 | Agnostic PAC-Learning of Functions on
Analog Neural Nets
(Extended Abstract)
Wolfgang Maass
Institute for Theoretical Computer Science
Technische Universitaet Graz
Klosterwiesgasse 32/2
A-BOlO Graz, Austria
e-mail: [email protected]
Abstract:
There exist a number of negative results ([J), [BR), [KV]) about
learning on neural nets in Valiant's model [V) for probably approximately correct learning ("PAC-learning"). These negative results
are based on an asymptotic analysis where one lets the number of
nodes in the neural net go to infinit.y. Hence this analysis is less adequate for the investigation of learning on a small fixed neural net.
with relatively few analog inputs (e.g. the principal components of
some sensory data). The latter type of learning problem gives rise
to a different kind of asymptotic question: Can the true error of the
neural net be brought arbitrarily close to that of a neural net with
"optimal" weights through sufficiently long training? In this paper
we employ some new arguments ill order to give a positive answer
to this question in Haussler's rather realistic refinement of Valiant's
model for PAC-learning ([H), [KSS)). In this more realistic model
no a-priori assumptions are required about the "learning target" ,
noise is permitted in the training data, and the inputs and outputs
are not restricted to boolean values. As a special case our result
implies one of the first positive results about learning on multi-layer
neural net.s in Valiant's original PAC-learning model. At the end
of this paper we will describe an efficient parallel implementation
of this new learning algorit.hm.
311
312
Maass
We consider multi-layer high order feedforward neural nets N with arbitrary piecewise polynomial activation functions . Each node g of fan-in m > 0 in N is
called a computation node. It is labelled by some polynomial Q9(Yl, ... , Ym)
and some piecewise polynomial activation funetion
R --+ R. We assume
that
consists of finitely many polynomial pieces and that its definition involves only rational parameters. The computation node g computes the function
(Yl, ... ,Ym) t-+
(Q9 (Yl, ... , Ym)) from R minto R. The nodes of fan-in 0 in N
("input nodes") are labelled by variables Xl, ... , Xk. The nodes g of fan-out 0 in
N ("output nodes") are labelled by 1, ... , I. We assume that the range B of their
activation functions
is bounded. Any parameters that occur in the definitions of
the
are referred to as architectural parameters of N.
,9 :
,9
,9
,9
,9
The coefficient.s of all the polynomials Q9 are called the programmable parameters
(or weights) of N. Let w be the number of programmable parameters of N. For any
assignment a E R W to the programmable parameters of N the network computes
a function from Rk into RI which we will denote by N!!...
We write Q n for the set of rational numbers that can be written as quotients of
integers with bit-length::; n. For;,. = (Zl, .. . ,ZI) E RI we write
11;,.lh
I
for
E Iz;l.
;=1
Let F : Rk --+ RI be some arbitrary function, which we will view as a "prediction
rule". For any given instance (~, 1/) E R k X Rl we measure the error of F by
"F(~) - 111 II? For any distribution A over some subset of R k x Rl we measure the
true error of F with regard to A by E(?,Y)EA [IIF(~) -lllll]' i.e. the expected value
of the error of F with respect to distribution A.
Theorelll 1: Let N be some arbitrary high order feedforward neural net with piecewise polynomial activation functions. Let tv be the number of programmable parameters of N (we assume that w = 0(1)). Then one can construct from N some
first order feedforward neural net jj with piecewise linear activation functions and
the quadratic activation function ,(x) = x2, which has the following property:
There exists a polynomial m(:,
and a learning algorithm LEARN such that for
any given ?, 6, E (0,1) and s, n E N and any distribution A over Q~ x (Qn n B)l
the following holds:
For any sample (
({Xi, Yi) )i=l, ... ,m of m ~ m(:,
points that are independently
drawn according to A the algorithm LEARN computes in polynomially in m, s, n
computation steps an assignment ii of rational numb~rs to the programmable parameters of jj such that with probability ~ 1 - 6:
i)
=
or in other words:
The true error of jjli with regard to A is within
that can be achieved by any N!!.. with a E Q:.
i)
?
of the least possible true error
Remarks
a) One can easily see (see [M 93b] for details) that Theorem 1 provides a
positive learning result in Haussler's extension of Valiant's model for PAClearning ([H], [KSS]). The "touchstone class" (see [KSS)) is defined as the
Agnostic PAC-Learning of Functions on Analog Neural Nets
class of function f : Rk -+ Rl that are computable on N with programmable parameters from Q.
This fact is of some general interest, since so far only very few positive
results are known for any learning problem in this rather realistic (but
quite demanding) learning model.
b) Consider the special case where the distribution A over Q~ x (Qn
of the form
n B)l is
D(~)
ADIO'T(~' y) = {
0
otherwise
for some arbitrary distribution D over the domain Q~ and some arbitrary
Q: T E Q~. Then the term
inf EC~IY}EA[IINQ.(~)
a
EQw
3
-lllhl
is equal to O. Hence the preceding theorem states that with learning algorithm LEARN the "learning network" jj can "learn" with arbitrarily small
true error any target function NQT that is computable on N with rational
"weights" aT' Thus by choosing N sufficiently large, one can guarantee
that the associated "learning network" jj can learn any target-function
that might arise in the context of a specific learning problem.
In addition the theorem also applies to the more realistic situation where
the learner receives examples (~, y) of the form (~, NQT (~)+ noise), or even
if there exists no "target function" NQT that would "explain" the actual
distribution A of examples (~, ll) ("agnostic learning").
The proof of Theorem 1 is mathematically quite involved, and we can give here
only an outline. It consists of three steps:
(1) Construction of the auxiliary neural net fl .
(2) Reducing the optimization of weights in jj for a given distribution A to a
finite nonlinear optimization problem.
(3) Reducing the resulting finite nonlinear optimization problem to a family of
finite linear optimization problems.
,9
Details to step (1): If the activation functions
in N are piecewise linear and
all computation nodes in N have fan-out::; 1 (this occurs for example if N has just
one hidden layer and only one output) then one can set fI := N. If the
are
piecewise linear but not all computation nodes in N have fan-out::; lone defines
jj as the tree of the same depth as N, where sub circuits of computation nodes with
fan-out m > 1 are duplicated 111 times. The activation functions remain unchanged
in this case.
,9
,9
If the activation functions
are piecewise polynomial but not piecewise linear,
one has to apply a rather complex construction which is described in detail in the
Journal version of [M 93a]. In any case if has the property that all functions that
313
314
Maass
are computable on N can also be computed on N, the depth of N is bounded by a
constant, and the size of N is bounded by a polynomial in the size of N (provided
that the depth and order of N, as well as the number and degrees of the polynomial
pieces of the "'(9 are bounded by a constant).
Details to step (2): Since the VC-dimension of a neural net is only defined
for neural nets with boolean output, one has to consider here instead the pseudodimension of the function class F that is defined by N.
Definition: (see Haussler (H]).
Let X be some arbitrary domain, and let F be an arbitrary class of functions from
X into R. Then the pseudo-dimension of F is defined by
dimp(F) := max{ISI: S ~ X and 3h : S --+ R such that
Vb E {O, l}s 31 E F Vx E S (I(x) ~ hex) ~ b(x) = I)}.
Note that in the special case where F is a concept class (i.e. all 1 E Fare
?-
1
valued) the pseudo-dimension dimp(F) coincides with the VC-dimension of F. The
pseudo-dimension of the function class associated with network architectures N with
piecewise polynomial activation functions can be bounded with the help of Milnor's
Theorem [Mi] in the same way as the VC-dimension for the case of boolean network
output (see [GJ)):
Theorenl 2: Consider arbitrary network architectures N of order v with k input
nodes, I output nodes, and w programmable parameters. Assume that each gate in
N employs as activation function some piecewise polynomial (or piecewise rational)
function of degree ~ d with at most q pieces. For some arbitrary p E {I, 2, ...}
we define F
{ 1 : R k+1 --+ R : 30: E R W Vx E Rk V1!. E Rl(l(~,1!.)
IINQ'.(.~) -1!.lIp)}? Then one has dimp(F)
0(w 2 10gq) if v, d, 1= 0(1).
?
=
With the help of the pseudo-dimension one can carry out the desired reduction of
the optimization of weights in N (with regard to an arbitrary given distribution A
of examples (~, 11.) to a finite optimization problem. Fix some interval [b 1 , b2 ] ~ R
such that B ~ [b 1 , b2], b1 < b2, and such that the ranges of the activation functions
of the output gates of N are contained in [b 1 , b2]. We define b := I? (b 2 - bt) , and
F:= {f :RkX[b 1 ,b 2]I--+[0,b]: 30:ERwV~ERkV1!.E[bl,b2F(f(~,1!.)=
IINQ'.(~) - YIII)}? Assume now that parameters c, 6 E (0,1) with c ~ band s, n E N
have been -fixed. For convenience we assume that s is sufficiently large so that
all architectural parameters in N are from Qs (we assume that all architectural
parameters in Ai are rational). We define
257?b 2 ( .
33eb
771 ?'"8 := c 2
2? dllnp(F) .Inc - + In"8 .
( 11)
8)
By Corollary 2 of Theorem 7 in Haussler [H) one has for 771 ~ 771(:, i), I< := y~57 E
(2,3), and any distribution A over Q~ x (Qn n [b 1 ,b2))1
1 ~
c
(1)
P7'(EAm[{31 E F: 1(771 L...J /(!1.,1!.?) - E(~,.~)EA[f(!1.'1!.)]I > I<}] < 6,
(~,~)E(
Agnostic PAC-Learning of Functions on Analog Neural Nets
where
E(~.!!)EA [f(~, u)]
is the expectation of
f(~, u)
with regard to distribution A.
We design an algorithm LEARN that computes for any mEN, any sample
(= ((Xi,yi))iE{l ?..? m} E (Q~ x (Qn
n [b 1 ,b 2])I)m,
and any given sEN in polynomially in m, s, n computation steps an assignment
a of rational numbers to the parameters in j\( such that the function it that is
computed by j\(!i. satisfies
m
1 m _
2
inf
~ ~ IIN?(xd
- ydh?
(2)
Tn
Ilh(xd - ydh ~ (1 - ]{)e +
w m~
-i=l
a E Q"
i=l
This suffices for the proof of Theorem 1, since (1) and (2) together imply that, for
any distribution A over Q~ x (Qn n [b 1 , b2])1 and any m ~ m( 1, i), with probability
~ 1 - 6 (with respect to the random drawing of ( E Am) the algorithm LEARN
outputs for inputs ( and s an assignment a of rational numbers to the parameters
in j\( such that
L
E(~'1!:)EA[IIN!i.(~)
-ulld ~ c +
inf
a E Q~
E(!:.Y)EA[IIN?(~)
-
-ulh]?
Details to step (3): The computation of weights a that satisfy (2) is nontrivial,
since this amounts t.o solving a nonlinear optimization problem. This holds even if
each activation function in N is piecewise linear, because weights from successive
layers are multiplied with each other.
We employ a method from [M 93a] that allows us to replace the nonlinear conditions
on the programmable parameters a of N by linear conditions for a transformed set
.?, f3 of parameters. We simulate j\(? by another network architecture N[?]~ (which
one may view as a "normal form" for j\(?) that uses the same graph (V, E) as
N, but different activation functions and different values f3 for its programmable
parameters. The activation functions of N[.?] depend on IVI new architectural
parameters .? E RI vI, which we call scaling parameters in the following. Whereas
the architectural parameters of a network architecture are usually kept fixed, we
will be forced to change the scaling parameters of N along with its programmable
parameters f3. Although this new network architecture has the disadvantage that
it requires IVI additional parameters .?, it has the advantage that we can choose in
N[?] all weights on edges between computation nodes to be from {-I,O, I}. Hence
we can treat them as constants with at most 3 possible values in the system of
inequalities that describes computations of N[?]. Thereby we can achieve that all
variables that appear in the inqualities that describe computations of N[?J for fixed
network inputs (the variables for weights of gates on levell, the variables for the
biases of gates on all levels, and the new variables for the scaling parameters .?)
appear only linearly in those inqualities.
We briefly indicate the construction of N in the case where each activation function
"I in N is piecewise linear. For any c > we consider the associated piecewise linear
activation function "I c with
T;f x E R( "I c (c . x) = c . "I ( x ) ).
?
315
316
Maass
Assume that fr is some arbitrary given assignment to the programmable parameters
in jj. We transform jjsr through a recursive process into a "normal form" N(?]t
in which all weights on edges between computation nodes are from {-I, 0, I}, such
that \:fll. E R k (jjsr(ll.) = N(?]t(ll.?) .
q
Assume that an output gate gout of jjsr receives as input
L: aiYi + ao,
where
i=l
al, ... , a q , ao are the weights and the bias of gout (under the assignment a) and
Yl, ... ,Yq are the (real valued) outputs of the immediate predecessors g1, ... ,gq of
g. For each i E {I, ... , q} with 0i =/:- 0 such that gi is not an input node we replace
the activation function "fi of gi by "f!a,l, and we multiply the weights and the bias
of gate gi with lail. Finally we replace the weight ai of gate gout by sgn( ad, where
sgn(ad := 1 ifai > 0 and sgn(ai) := -1 ifai < o. This operation has the effect
that the multiplication with IOj I is carried out before the gate gi (rather than after
gj, as done in jjsr), but that the considered output gate gout still receives the same
input as before. If aj = 0 we want to "freeze" that weight at O. This can be done
by deleting gi and all gates below gi from N.
The analogous operations are recursively carried out for the predecessors gi of gout
(note however that the weights of gj are no longer the original ones from jjsr, since
they have been changed in the preceding step). We exploit here the assumption
that each gate in jj has fan-out::; 1.
Let f3 consist of the new weights on edges adjacent to input nodes and of the
resulting biases of all gates in N. Let f consist of the resulting scaling parameters
at the gates of N. Then we have \:f~ E Rk (jjsr(~) = N[.?]t(~?). Furthermore c > 0
for all scaling parameters c in f.
At the end of this proof we will also need the fact that the previously described parameter transformation can be inverted, i.e. one can compute from Q, f3 an equivalent
weight assignment a for jj (with the original activation functions "f).
We now describe how the algorithm LEARN computes for any given sample
(= ({Xi,Yi)i=l ..... m E (Q~ x (Q" n[b l ,b 2 W)m and any given sEN with the
help of linear programming a new assignment .?, ~ to the parameters in N such that
the function It that is computed by N@]i satisfies (2). For that purpose we describe
the computations of N for the fixed inputs Xi from the sample ( = ((Xi, Yi) )i=l .. ..,m
by polynomially in m many systems L l , . .. , Lp(m) that each consist of Oem) linear
inequalities with the transformed parameters Q, f3 as variables. Each system Lj reflects one possibility for employing specific linear pieces of the activation functions in
N for specific network inputs Xl, ... , X m , and for employing different combinations
of weights from {-I, 0, I} for edges between computation nodes.
One can show that it suffices to consider only polynomially in Tn many systems
of inequalities L j by exploiting that all inequalities are linear, and that the input
space for N has bounded dimension k.
Agnostic PAC-Learning of Functions on Analog Neural Nets
We now expand each of the systems Lj (which has only 0(1) variables) into a
linear programming problem LPj with Oem) variables. We add to Lj for each of
the I output nodes IJ of N 2m new variables
for i = 1, ... , m, and the 4m
inequalities
ur, vr
tj(xd :S (Y;)II + ui - vi,
tj(xd
~ (Ydll
+ ui - vi,
ui
~ 0,
vi
~ 0,
where ((Xi, Yi) )i=l , .. . ,m is the fixed sample ( and (Yi)1I is that coordinate of Yj which
corresponds to the output node IJ of N. In these inequalities the symbol tj(xd denotes the term (which is by construction linear in the variables f, (3) that represents
the output of gate IJ for network input Xi in this system Lj. One-should note that
these terms tj( Xi) will in general be different for different j, since different linear
pieces of the activation functions at preceding gates may be used in the computation
of N for the same network input Xi. We expand the system Lj of linear inequalities
to a linear programming problemLPj in canonical form by adding the optimization
requirement
m
mmlmlze
i=l
IJ
output node
The algorithm LEARN employs an efficient algorithm for linear programming (e.g.
the ellipsoid algorithm, see [PS]) in order to compute in altogether polynomially
in m, sand n many steps an optimal solution for each of the linear programming
problems LP1 , ... , LPp(m). We write h j for the function from Rk into Rl that is
computed by N[f]~ for the optimal solution ?, (3 of LPj. The algorithm LEARN
m
computes
~
' " Ilhj(xj) mL...J
i=l
Yilll for j
-
= 1, . .. ,p(m).
Let] be that index for which
this expression has a minimal value . Let f, ~ be the associated optimal solution of
LPl (i.e. N@)l computes hl). LEARN employs the previously mentioned backwards transformation from f, j3 into values Ii for the programmable parameters of
jj such that 'V~ E Rk (jjQ.(~)
the algorithm LEARN.
= N[f.]l(~)).
These values a are given as output of
We refer to [M 93b] for the verification that this weight assignment a has the
property that is claimed in Theorem 1. We also refer to [M 93b] for the proof in the
more general case where the activation functions of N are piecewise polynomial. ?
Reillark: The algorithm LEARN can be speeded up substantially on a parallel machine. Furthermore if the individual processors of the parallel machine are allowed
to use random bits, hardly any global control is required for this parallel computation. We use polynomially in m many processors. Each processor picks at random
one of the systems Lj of linear inequalit.ies and solves the corresponding linear programming problem LPj . Then the parallel machine compares in a "competitive
m
phase" the costs
L: Ilhj(Xi) - ydh
i=l
-
-
of the solutions hj that have been computed by
the individual processors. It outputs the weights
a for
jj that correspond to the
317
318
Maass
best ones of these solutions hj . If one views the number w of weights in N no longer
as a c.onstant, one sees that the number of processores that are needed is simply
exponential in w, but that the parallel computation time is polynomial in m and
w.
Acknowledgements
I would like to thank Peter Auer, Phil Long and Hal White for their helpful com-
ments.
References
[BR]
A. Blum, R. L. Rivest, "Training a 3-node neural network is NPcomplete", Proc. of the 1988 Workshop on Computational Learning
Theory, Morgan Kaufmann (San Mateo, 1988), 9 - 18
[GJ]
P. Goldberg, M. Jerrum, "Bounding the Vapnik-Chervonenkis dimension of concept classes parameterized by real numbers", Proc. of the
6th Annual A CM Conference on Computational Learning Theory, 361
[H]
[J]
[KV]
[KSS]
[M 93a]
[M 93b]
[Mi]
[PS]
[V]
- 369.
D. Haussler, "Decision theoretic generalizations of the PAC model
for neural nets and other learning applications", Information and
Computation, vol. 100, 1992, 78 - 150
J. S. Judd, "Neural Network Design and the Complexity of Learning" ,
MIT-Press (Cambridge, 1990)
M. Kearns, L. Valiant, "Cryptographic limitations on learning
boolean formulae and finite automata", Proc. of the 21st ACM Symposium on Theory of Computing, 1989,433 - 444
M. J. Kearns, R. E. Schapire, L. M. Sellie, "Toward efficient agnostic
learning", Proc. of the 5th A CM Workshop on Computational Learning Theory, 1992, 341 - 352
W. Maass, "Bounds for t.he c.omputational power and learning c.omplexity of analog neural nets" (extended abstract), Proc. of the 25th
ACM Symposium on Theory of Computing, 1993,335 - 344. Journal
version submitted for publication
W. Maass, "Agnostic PAC-learning of functions on analog neural
nets" (journal version), to appear in Neural Computation.
.J. Milnor, "On the Betti numbers ofreal varieties", Proc. of the American Math. Soc., vol. 15, 1964, 275 - 280
C. H. Papadimitrioll, K. Steiglitz, "Combinatorial Optimization: Algorithms and Complexity" , Prent.ice Hall (Englewood Cliffs, 1982)
L. G. Valiant, "A theory of the learnable", Comm. of the ACM, vol.
27, 1984, 1134 - 1142
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7,067 | 838 | Clustering with a Domain-Specific
Distance Measure
Steven Gold, Eric Mjolsness and Anand Rangarajan
Department of Computer Science
Yale University
New Haven, CT 06520-8285
Abstract
With a point matching distance measure which is invariant under
translation, rotation and permutation, we learn 2-D point-set objects, by clustering noisy point-set images. Unlike traditional clustering methods which use distance measures that operate on feature
vectors - a representation common to most problem domains - this
object-based clustering technique employs a distance measure specific to a type of object within a problem domain. Formulating
the clustering problem as two nested objective functions, we derive
optimization dynamics similar to the Expectation-Maximization
algorithm used in mixture models.
1
Introduction
Clustering and related unsupervised learning techniques such as competitive learning and self-organizing maps have traditionally relied on measures of distance, like
Euclidean or Mahalanobis distance, which are generic across most problem domains.
Consequently, when working in complex domains like vision, extensive preprocessing is required to produce feature sets which reflect properties critical to the domain,
such as invariance to translation and rotation. Not only does such preprocessing
increase the architectural complexity of these systems but it may fail to preserve
some properties inherent in the domain. For example in vision, while Fourier decomposition may be adequate to handle reconstructions invariant under translation
and rotation, it is unlikely that distortion invariance will be as amenable to this
technique (von der Malsburg, 1988).
96
Clustering with a Domain-Specific Distance Measure
These problems may be avoided with the help of more powerful, domain-specific
distance measures, including some which have been applied successfully to visual
recognition tasks (Simard, Le Cun, and Denker, 1993; Huttenlocher et ai., 1993).
Such measures can contain domain critical properties; for example, the distance
measure used here to cluster 2-D point images is invariant under translation, rotation and labeling permutation. Moreover, new distance measures may constructed,
as this was, using Bayesian inference on a model of the visual domain given by a
probabilistic grammar (Mjolsness, 1992). Distortion invariant or graph matching
measures, so formulated, can then be applied to other domains which may not be
amenable to description in terms of features.
Objective functions can describe the distance measures constructed from a probabilistic grammar, as well as learning problems that use them. The clustering problem in the present paper is formulated as two nested objective functions: the inner
objective computes the distance measures and the outer objective computes the
cluster centers and cluster memberships. A clocked objective function is used, with
separate optimizations occurring in distinct clock phases (Mjolsness and Miranker,
1993). The optimization is carried out with coordinate ascent/descent and deterministic annealing and the resulting dynamics is a generalization of the ExpectationMaximization (EM) algorithm commonly used in mixture models.
2
2.1
Theory
The Distance Measure
Our distance measure quantifies the degree of similarity between two unlabeled
2-D point images, irrespective of their position and orientation. It is calculated
with an objective that can be used in an image registration problem. Given two
sets of points {Xj} and {Yk }, one can minimize the following objective to find the
translation, rotation and permutation which best maps Y onto X :
Ereg(m, t, 0) =
L mjkllXj -
t -
R(0) . Yk l1 2
jk
with constraints: 'Vj L:k mjk
=1 ,
'Vk
L:j
mjk
= l.
Such a registration permits the matching of two sparse feature images in the presence
of noise (Lu and Mjolsness, 1994). In the above objective, m is a permutation matrix
which matches one point in one image with a corresponding point in the other image.
The constraints on m ensure that each point in each image corresponds to one and
only one point in the other image (though note later remarks regarding fuzziness).
Then given two sets of points {Xj} and {Yk } the distance between them is defined
as:
D({Xj}, {Yk})
min(Ereg(m,t,0) I constraints on m) .
(1)
= m,t,e
This measure is an example of a more general image distance measure derived in
(Mjolsness, 1992):
d(x, y) = mind(x, T(y)) E [0,00)
T
where T is a set of transformation parameters introduced by a visual grammar. In
(1) translation, rotation and permutation are the transformations, however scaling
97
98
Gold, Mjolsness, and Rangarajan
or distortion could also have been included, with consequent changes in the objective
function.
The constraints are enforced by applying the Potts glass mean field theory approximations (Peterson and Soderberg,1989) and then using an equivalent form of
the resulting objective, which employs Lagrange multipliers and an x log x barrier
function (as in Yuille and Kosowsky, 1991):
Ereg(m, t, 8)
L: mjkllXj -
t - R(8) ? YkW
jk
+ f31 L: mjk(logmjk -1)
jk
+L:J.tj(L:mjk-1)+L:vk(L:mjk-1).
j
k
k
(2)
j
In this objective we are looking for a saddle point. (2) is minimized with respect to
m, t, and 8, which are the correspondence matrix, translation,and rotation, and is
maximized with respect to J.t and v, the Lagrange multipliers that enforce the row
and column constraints for m.
2.2
The Clustering Objective
The learning problem is formulated as follows: Given a set of I images, {Xd, with
each image consisting of J points, find a set of A cluster centers {Ya } and match
variables {Mia} defined as
if Xi is in Ya's cluster
otherwise,
M. - {I
la 0
such that each image is in only one cluster, and the total distance of all the images
from their respective cluster centers is minimized. To find {Ya} and {Mia} minimize
the cost function,
Ec/U8ter(Y, M)
MiaD(Xi, Ya) ,
ia
with the constraint that 'Vi l:a Mia = 1. D(Xi, Y a), the distance function, is
defined by (1).
= L:
The constraints on M are enforced in a manner similar to that described for the
distance measure, except that now only the rows of the matrix M need to add to
one, instead of both the rows and the columns. The Potts glass mean field theory
method is applied and an equivalent form of the resulting objective is used:
1
Ec/u8ter(Y, M) = ~ MiaD(Xi, Ya) + f3 ~ Mia (log Mia - 1) + ~ Ai(L: Mia -1)
ta
za
z
a
(3)
Replacing the distance measure by (2), we derive:
L:
Ec/u8ter(Y, M, t, 8, m) = L:Mia
miajkllXij - tia - R(8ia) . Ya k11 2 +
ia
jk
~[f3~ ~k miajk(logmiajk za
J
1) + ~ J.tiaj(L:
k miajk - 1) +
J
L:Viak(L:miajk -1)]+ -;- L:Mia(logMia -1)+
k
j
M ia
L: Ai(L:a Mia -1)
i
Clustering with a Domain-Specific Distance Measure
A saddle point is required. The objective is minimized with respect to Y, M, m,
t, 0, which are respectively the cluster centers, the cluster membership matrix, the
correspondence matrices, the rotations, and the translations. It is maximized with
respect to A, which enforces the row constraint for M, and J..l and v which enforce
the column and row constraints for m. M is a cluster membership matrix indicating
for each image i, which cluster a it falls within, and mia is a permutation matrix
which assigns to each point in cluster center Ya a corresponding point in image Xi.
0ia gives the rotation between image i and cluster center a. Both M and mare
fuzzy, so a given image may partially fall within several clusters, with the degree of
fuzziness depending upon 13m and 13M.
Therefore, given a set of images, X, we construct Ecltuter and upon finding the
appropriate saddle point of that objective, we will have Y, their cluster centers,
and M, their cluster memberships.
3
The Algorithm
3.1
Overview - A Clocked Objective Function
The algorithm to minimize the above objective consists of two loops - an inner
loop to minimize the distance measure objective (2) and an outer loop to minimize
the clustering objective (3). Using coordinate descent in the outer loop results
in dynamics similar to the EM algorithm for clustering (Hathaway, 1986). (The
EM algorithm has been similarly used in supervised learning [Jordan and Jacobs,
1993].) All variables occurring in the distance measure objective are held fixed
during this phase. The inner loop uses coordinate ascent/descent which results in
repeated row and column projections for m. The minimization of m, t and 0 occurs
in an incremental fashion, that is their values are saved after each inner loop call
from within the outer loop and are then used as initial values for the next call to
the inner loop. This tracking of the values of m, t, and 0 in the inner loop is
essential to the efficiency of the algorithm since it greatly speeds up each inner loop
optimization. Each coordinate ascent/descent phase can be computed analytically,
further speeding up the algorithm. Local minima are avoided, by deterministic
annealing in both the outer and inner loops.
The resulting dynamics can be concisely expressed by formulating the objective as
a clocked objective function, which is optimized over distinct sets of variables in
phases,
Ecloc1ced
= Ecl'luter( (((J..l, m)A , (v, m)A)$' 0 A , t A)$, (A, M)A, yA)$
with this special notation employed recursively:
E{x, Y)$ : coordinate descent on x, then y, iterated (if necessary)
xA
:
use analytic solution for x phase
The algorithm can be expressed less concisely in English, as follows:
Initialize t, 0 to zero, Y to random values
Begin Outer Loop
Begin Inner Loop
Initialize t, 0 with previous values
99
100
Gold, Mjolsness, and Rangarajan
Find m, t, e for each ia pair:
Find m by softmax, projecting across j, then k, iteratively
Find e by coordinate descent
Find t by coordinate descent
End Inner Loop
If first time through outer loop i 13m and repeat inner loop
Find M ,Y using fixed values of m, t, e determined in inner loop:
Find M by soft max, across i
Find Y by coordinate descent
i 13M, 13m
End Outer Loop
When the distances are calculated for all the X - Y pairs the first time time through
the outer loop, annealing is needed to minimize the objectives accurately. However
on each succeeding iteration, since good initial estimates are available for t and e
(the values from the previous iteration of the outer loop) annealing is unnecessary
and the minimization is much faster.
The speed of the above algorithm is increased by not recalculating the X - Y distance
for a given ia pair when its Mia membership variable drops below a threshold.
3.2
Inner Loop
The inner loop proceeds in three phases. In phase one, while t and e are held fixed,
m is initialized with the softmax function and then iteratively projected across its
rows and columns until the procedure converges. In phases two and three, t and e
are updated using coordinate descent. Then 13m is increased and the loop repeats.
In phase one m is updated with softmax:
miajk
=
exp( -13m "Xij Lk' exp( -13m IIXij
tia -
-
tia
R(e ia ) . Yak 112)
- R(eia) . Yak/112)
Then m is iteratively normalized across j and k until
miajk
=
miajk
=-~-?
'1\'.,
L.JJ
Using coordinate descent
m?
.I
,aJ k
e is calculated in phase two:
And t in phase three:
Finally
13m
Ljk t:t.miajk
is increased and the loop repeats.
<
f :
Clustering with a Domain-Specific Distance Measure
v2
By setting the partial derivatives of (2) to zero and initializing I-lJ and
to zero,
the algorithm for phase one may be derived. Phases two and three may be derived
by taking the partial derivative of (2) with respect to 0, setting it to zero, solving
for 0, and then solving for the fixed point of the vector (tl, t2).
Beginning with a small 13m allows minimization over a fuzzy correspondence matrix
m, for which a global minimum is easier to find. Raising 13m drives the m's closer
to 0 or 1, as the algorithm approaches a saddle point.
3.3
Outer Loop
The outer loop also proceeds in three phases: (1) distances are calculated by calling
the inner loop, (2) M is projected across a using the softmaxfunction, (3) coordinate
descent is used to update Y .
Therefore, using softmax M is updated in phase two:
Mia =
exp( -13M Ljk miajkllXij - tia - R(0ia) . Yak112)
~----------~------~----------~~~----~7
La' exp( -13M Ljk mia' jk IIXij - t ia , - R(0 ia ,) . Y a, k 112)
Y, in phase three is calculated using coordinate descent:
Li Mia Lj miajk( cos 0 ia (Xij 1
-
tiad
+ sin 0ia(Xij2 -
tia2))
Li Mia Lj miaj k
Y ak2
Then
4
13M
Li Mia Lj miajk( - sin 0ia(Xi jl - tiad
Li Mia
Ej
+ cos 0ia(Xij2 -
tia2))
miajk
is increased and the loop repeats.
Methods and Experimental Results
In two experiments (Figures la and Ib) 16 and 100 randomly generated images of
15 and 20 points each are clustered into 4 and 10 clusters, respectively.
A stochastic model, formulated with essentially the same visual grammar used to
derive the clustering algorithm (Mjolsness, 1992), generated the experimental data.
That model begins with the cluster centers and then applies probabilistic transformations according to the rules laid out in the grammar to produce the images.
These transformations are then inverted to recover cluster centers from a starting
set of images. Therefore, to test the algorithm, the same transformations are applied to produce a set of images, and then the algorithm is run in order to see if it
can recover the set of cluster centers, from which the images were produced.
First, n = 10 points are selected using a uniform distribution across a normalized
square. For each of the n = 10 points a model prototype (cluster center) is created
by generating a set of k = 20 points uniformly distributed across a normalized
square centered at each orginal point. Then, m = 10 new images consisting of
k = 20 points each are generated from each model prototype by displacing all k
model points by a random global translation, rotating all k points by a random
global rotation within a 54? arc, and then adding independent noise to each of the
translated and rotated points with a Gaussian distribution of variance (1"2.
101
102
Gold, Mjolsness, and Rangarajan
j
t t
,.
I
10
t j
t
10
0.2
0.'
0.6
0.1
1.2
1.'
Figure 1: (a): 16 images, 15 points each (b):100 images, 20 points each
The p = n x m = 100 images so generated is the input to the algorithm. The
algorithm, which is initially ignorant of cluster membership information, computes
n = 10 cluster centers as well as n x p = 1000 match variables determining the
cluster membership of each point image. u is varied and for each u the average
distance of the computed cluster centers to the theoretical cluster centers (i.e. the
original n = 10 model prototypes) is plotted.
Data (Figure 1a) is generated with 20 random seeds with constants of n = 4, k =
15, m = 4, p = 16, varying u from .02 to .14 by increments of .02 for each seed.
This produces 80 model prototype-computed cluster center distances for each value
of u which are then averaged and plotted, along with an error bar representing
the standard deviation of each set. 15 random seeds (Figure 1b) with constants
of n = 10, k
20, m
10, p = 100, u varied from .02 to .16 by increments of
.02 for each seed, produce 150 model prototype-computed cluster center distances
for each value of u. The straight line plotted on each graph shows the expected
model prototype-cluster center distances, b = ku /
which would be obtained if
there were no translation or rotation for each generated image, and if the cluster
memberships were known. It can be considered a lower bound for the reconstruction
performance of our algorithm. Figures 1a and 1b together summarize the results of
280 separate clustering experiments.
=
=
vn,
For each set of images the algorithm was run four times, varying the initial randomly
selected starting cluster centers each time and then selecting the run with the lowest
energy for the results. The annealing rate for 13M and 13m was a constant factor of
1.031. Each run of the algorithm averaged ten minutes on an Indigo SGI workstation
for the 16 image test, and four hours for the 100 image test. The running time of
the algorithm is O(pnk2). Parallelization, as well as hierarchical and attentional
mechanisms, all currently under investigation, can reduce these times.
5
Summary
By incorporating a domain-specific distance measure instead of the typical generic
distance measures, the new method of unsupervised learning substantially reduces
the amount of ad-hoc pre-processing required in conventional techniques. Critical
features of a domain (such as invariance under translation, rotation, and permu-
Clustering with a Domain-Specific Distance Measure
tation) are captured within the clustering procedure, rather than reflected in the
properties of feature sets created prior to clustering. The distance measure and
learning problem are formally described as nested objective functions. We derive
an efficient algorithm by using optimization techniques that allow us to divide up
the objective function into parts which may be minimized in distinct phases. The
algorithm has accurately recreated 10 prototypes from a randomly generated sample
database of 100 images consisting of 20 points each in 120 experiments. Finally, by
incorporating permutation invariance in our distance measure, we have a technique
that we may be able to apply to the clustering of graphs. Our goal is to develop
measures which will enable the learning of objects with shape or structure.
Acknowledgements
This work has been supported by AFOSR grant F49620-92-J-0465 and
ONR/DARPA grant N00014-92-J-4048.
References
R. Hathaway. (1986) Another interpretation of the EM algorithm for mixture
distributions. Statistics and Probability Letters 4:53:56.
D. Huttenlocher, G. Klanderman and W. Rucklidge. (1993) Comparing images using the Hausdorff Distance . Pattern Analysis and Machine Intelligence
15(9):850:863.
A. L. Yuille and J.J. Kosowsky. (1992) . Statistical physics algorithms that converge.
Technical Report 92-7, Harvard Robotics Laboratory.
M.l. Jordan and R.A. Jacobs. (1993). Hierarchical mixtures of experts and the
EM algorithm. Technical Report 9301, MIT Computational Cognitive Science.
C. P. Lu and E. Mjolsness. (1994). Two-dimensional object localization by coarseto-fine correlation matching. In this volume, NIPS 6 .
C. von der Malsburg . (1988) . Pattern recognition by labeled graph matching.
Neural Networks,1:141:148 .
E. Mjolsness and W. Miranker. (1993). Greedy Lagrangians for neural networks:
three levels of optimization in relaxation dynamics. Technical Report 945, Yale
University, Department of Computer Science.
E. Mjolsness. Visual grammars and their neural networks . (1992) SPIE Conference
on the Science of Artificial Neural Networks, 1710:63:85.
C. Peterson and B. Soderberg. A new method for mapping optimization problems
onto neural networks. (1989) International Journal of Neural Systems,I(1):3:22.
P. Simard, Y. Le Cun, and J. Denker. Efficient pattern recognition using a new
transformation distance. (1993). In S. Hanson, J . Cowan, and C. Giles, (eds.),
NIPS 5 . Morgan Kaufmann, San Mateo CA.
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7,068 | 839 | Discontinuous Generalization in Large
Committee Machines
H. Schwarze
Dept. of Theoretical Physics
Lund University
Solvegatan 14A
223 62 Lund
Sweden
J. Hertz
Nordita
Blegdamsvej 17
2100 Copenhagen 0
Denmark
Abstract
The problem of learning from examples in multilayer networks is
studied within the framework of statistical mechanics. Using the
replica formalism we calculate the average generalization error of a
fully connected committee machine in the limit of a large number
of hidden units. If the number of training examples is proportional
to the number of inputs in the network, the generalization error
as a function of the training set size approaches a finite value. If
the number of training examples is proportional to the number of
weights in the network we find first-order phase transitions with a
discontinuous drop in the generalization error for both binary and
continuous weights.
1
INTRODUCTION
Feedforward neural networks are widely used as nonlinear, parametric models for the
solution of classification tasks and function approximation. Trained from examples
of a given task, they are able to generalize, i.e. to compute the correct output for
new, unknown inputs. Since the seminal work of Gardner (Gardner, 1988) much
effort has been made to study the properties of feedforward networks within the
framework of statistical mechanics; for reviews see e.g. (Hertz et al., 1989; Watkin et
al., 1993). Most of this work has concentrated on the simplest feedforward network,
the simple perceptron with only one layer of weights connecting the inputs with a
399
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Schwarze and Hertz
single output. However, most applications have to utilize architectures with hidden
layers, for which only a few general theoretical results are known, e.g. (Levin et al.,
1989; Krogh and Hertz, 1992; Seung et al., 1992).
As an example of a two-layer network we study the committee machine (Nilsson,
1965). This architecture has only one layer of adjustable weights, while the hiddento-output weights are fixed to + 1 so as to implement a majority decision of the
hidden units. For binary weights this may already be regarded as the most general
two-layer architecture, because any other combination of hidden-output weights can
be gauged to + 1 by flipping the signs of the corresponding input-hidden weights.
Previous work has been concerned with some restricted versions of this model, such
as learning geometrical tasks in machines with local input-to-hidden connectivity
(Sompolinsky and Tishby, 1990) and learning in committee machines with nonoverlapping receptive fields (Schwarze and Hertz, 1992; Mato and Parga, 1992). In
this tree-like architecture there are no correlations between hidden units and its
behavior was found to be qualitatively similar to the simple perceptron.
Recently, learning in fully connected committee machines has been studied within
the annealed approximation (Schwarze and Hertz, 1993a,b; Kang et aI, 1993), revealing properties which are qualitatively different from the tree model. However,
the annealed approximation (AA) is only valid at high temperatures, and a correct
description of learning at low temperatures requires the solution of the quenched
theory. The purpose of this paper is to extend previous work towards a better
understanding of the learning properties of multilayer networks. We present results
for the average generalization error of a fully connected committee machine within
the replica formalism and compare them to results obtained within the AA. In particular we consider a large-net limit in which both the number of inputs Nand
the number of hidden units K go to infinity but with K ~ N. The target rule is
defined by another fully connected committee machine and is therefore realizable
by the learning network.
2
THE MODEL
We consider a network with N inputs, K hidden units and a single output unit (j.
Each hidden unit (jl, I E {I, ... , K}, is connected to the inputs 8 = (81 , .?? , 8N)
through the weight vector W, and performs the mapping
(j1(W I , 8) = sign
(Jw
W, . 8).
(1)
The hidden units may be regarded as outputs of simple perceptrons and will be
referred to as students. The factor N- 1 / 2 in (1) is included for convenience; it
ensures that in the limit N -+ 00 and for iid inputs the argument of the sign
function is of order 1. The overall network output is defined as the majority vote
of the student committee, given by
(2)
Discontinuous Generalization in Large Committee Machines
=
This network is trained from P
aK N input-output examples ({", T({")), J.I. E
ofthe training inputs
{1, ... , P}, ofthe desired mapping T, where the components
are independently drawn from a distribution with zero mean and unit variance. We
study a realizable task defined by another committee machine with weight vectors
L (the teachers), hidden units Tz and an overall output T(S) of the form (2). We
will discuss both the binary version of this model with W" L E {? l}N and the
continuous version in which the W,'s and L's are normalized to VN.
{r
The goal of learning is to find a network that performs well on unknown examples,
which are not included in the training set. The network quality can be measured
by the generalization error
?({W,}) = (0[-(T({~},S) T(S)])~,
(3)
the probability that a randomly chosen input is misclassified.
Following the statistical mechanics approach we consider a stochastic learning algorithm that for long training times yields a Gibbs distribution of networks with
the corresponding partition function
Z =
J
dpo({W, }) e- f1Et ({W,}) ,
(4)
where
(5)
is the training error, {3 = liT is a"formal temperature parameter, and po( {W,})
includes a priori constraints on the weights. The average generalization and training errors at thermal equilibrium, averaged over all representations of the training
examples, are given by
(( (?({W,}))T))
1
P (( (Et({~}))T )),
(6)
where (( ... )) denotes a quenched average over the training examples and ( ... )T a
thermal average. These quantities may be obtained from the average free energy
F = - T (( In Z )), which can be calculated within the standard replica formalism
(Gardner, 1988; Gyorgyi and Tishby, 1990).
Following this approach, we introduce order parameters and make symmetry assumptions for their values at the saddle point of the free energy; for details of the
calculation see (Schwarze, 1993). We assume replica symmetry (RS) and a partial committee symmetry allowing for a specialization of the hidden units on their
respective teachers. Furthermore, a self-consistent solution of the saddle-point
equations requires scaling assumptions for the order parameters. Hence, we are left
with the ansatz
1
R'k = N (( ( ~)T . V k ))
1
D ,k = N(((W,)T,(((Wk)T))
1
C'k= N(((W"Wk)T))
(7)
401
402
Schwarze and Hertz
where p, ~, d, q and c are of order 1. For ~ = q = 0 this solution is symmetric
under permutations of hidden units in the student network, while nonvanishing ~
and q indicate a specialization of hidden units that breaks this symmetry. The
values of the order parameters at the saddle point of the replica free energy finally
allow the calculation of the average generalization and training errors.
3
THEORETICAL RESULTS
In the limit of small training set sizes, Q ' " 0(1/ K), we find a committee-symmetric
solution where each student weight vector has the same overlap to all the teacher
vectors, corresponding to ~ = q = O. For both binary and continuous weights
the generalization error of this solution approaches a nonvanishing residual value as
shown in figure 1. Note that the asymptotic generalization ability of the committeesymmetric solution improves with increasing noise level.
0.50
0.30
DAD
0.25
0.30
...-...
E-<
w
'-'
0
? ? ? ?" ?"
" " " "
0.20
0.10
w
0.20
0.15
,,
0.10
0.05
0.00
0
10
20
C(
30
= PiN
40
50
,,
,,
,,
,
,,
,
-.- .-- ..
,-
--
~~~---
_....... --
Et
I
0.00
a)
Eg
b)
0.0
--
,I
,,
,
I
0.5
1.0
1.5
2.0
T
Figure 1: a) Generalization (upper curve) and training (lower curve) error as functions of 0
K Q. The results of Monte Carlo simulations for the generalization
(open symbols) and training (closed symbols) errors are shown for K
5 (circles)
and K = 15 (triangles) with T = 0.5 and N = 99. The vertical lines indicate the
predictions of the large- K theory for the location of the phase transition Oc = K Q c
in the binary model for K = 5 and K = 15, respectively.
b) Temperature dependence of the asymptotic generalization and training errors for
the committee-symmetric solution.
=
=
Only if the number of training examples is sufficiently large, Q ' " 0(1), can the
committee symmetry be broken in favor of a specialization of hidden units. We find
first-order phase transitions to solutions with ~,q > 0 in both the continuous and
the binary model. While in the binary model the transition is accompanied by a
perfect alignment of the hidden-unit weight vectors with their respective teachers
(~
1), this is not possible in a continuous model. Instead, we find a close approach
of each student vector to one of the teachers in the continuous model: At a critical
value Q" (T) of the load parameter a second minimum of the free energy appears,
corresponding to the specialized solution with ~, q > O. This solution becomes the
=
Discontinuous Generalization in Large Committee Machines
global minimum at Ckc(T) > Ck.(T), and its generalization error decays algebraically.
In both models the symmetric, poorly generalizing state remains metastable for
arbitrarily large Ck. For increasing system sizes it will take exponentially long times
for a stochastic training algorithm to escape from this local minimum (see figure
1a). Figure 2 shows the qualitative behavior of the generalization error for the
continuous model, and the phase diagrams in figure 3 show the location of the
transitions for both models.
1/2
?o(T)
a.
ac
--------------------=--'=-----+f--.,...I---..---i
i
I
f
I
j
~
~----------------~/~/_---------------
a", O(l/K)
'"
~ -
p
KN
a'" 0(1)
Figure 2: Schematic behavior of the generalization error in the large- K committee
machine with continuous weights.
In the binary model a region of negative thermodynamic entropy (below the dashed
line in figure 3a) suggests that replica symmetry has to be broken to correctly
describe the metastable, symmetric solution at large Ck.
A comparison of the RS solution with the results previously obtained within the
AA (Schwarze and Hertz, 1993a,b) shows that the AA gives a qualitatively correct
description of the main features of the learning curve. However, it fails to predict the
temperature dependence of the residual generalization error (figure 1b) and gives an
incorrect description of the approach to this value. Furthermore, the quantitative
predictions for the locations of the phase transitions differ considerably (figure 3).
4
SIMULATIONS
We have performed Monte Carlo simulations to check our analytical findings for the
binary model (see figure 1a). The influence of the metastable, poorly generalizing
state is reflected by the fact that at low temperatures the simulations do not follow
the predicted phase transition but get trapped in the metastable state. Only at
higher temperatures do the simulations follow the first order transition (Schwarze,
1993). Furthermore, the deviation of the training error from the theoretical result
indicates the existence of replica symmetry breaking for finite Q. However, the generalization error of the symmetric state is in good quantitative agreement with the
403
404
Schwarze and Hertz
0.8
0.6
E-<
1.0
;
.I
;,l'
0.4
,.I
i
;
;
;
;
,l
;
;
;
0.8
/
/ .......?.?./
0.6
... -- ..
0.4
i
i
0.2
j
j
i
O.O~~~~~~!~~~~~~~~
5
10
15
0:
r
0.2
,i
a)
// .....
l./????????????????????
I
20
25
O.O~~~~~~~~~!~-u~~
30
b)
1.0
1.5
= P/KN
2.0
0:
2.5
3.0
3.5
4.0
= P/KN
Figure 3: Phase diagrams of the large-K committee machine.
a) continuous weights: The two left lines show the RS results for the spinodal
line (--), where the specialized solution appears, and the location of the phase
transition (-). These results are compared to the predictions of the AA for the
spinodal line (- . -) and the phase transition ( ... ).
b) binary weights: The RS result for the location of the phase transition ( - ) and
its zero-entropy line (--) are compared to the prediction of the AA for the phase
transition ( ... ) and its zero-entropy line (- . -).
theoretical results.
In order to investigate whether our analytical results for a Gibbs ensemble of committee machines carries over to other learning scenarios we have studied a variation
of this model allowing the use of backpropagation. We have considered a 'softcommittee' whose output is given by
q( {W,}. S) = tanh
(t.
tanh (J?, . S?.
(8)
The first-layer weights W, of this network were trained on examples (el', r(el'?,
J.? E {l, ... , P}, defined by another soft-committee with weight vectors V, using
on-line backpropagation with the error function
?(S)
= (1/2)[0'({~}, S) -
r(S)]2.
(9)
In general this procedure is not guaranteed to yield a Gibbs distribution of weights
(Hansen et al., 1993) and therefore the above analysis does not apply to this case.
However, the generalization error for a network with N = 45 inputs and K =
3 hidden units, averaged over 50 independent runs, shows the same qualitative
behavior as predicted for the Gibbs ensemble of committee machines (see figure 4).
After an initial approach to a nonvanishing value, the average generalization error
decreases rather smoothly to zero. This smooth decrease of the average error is
due to the fact that some runs got trapped in a poorly-generalizing, committeesymmetric solution while others found a specialized solution with a close approach
to the teacher.
Discontinuous Generalization in Large Committee Machines
0.18 r--.....,.----r--.....,.---r--.....,.------r'1
0.16
0.1.
i
0.12
0.06
0.0.
0.02
200
600
800
1000
1200
P
=
Figure 4: Generalization error and training error of the 'soft-committee' with N
45 and K
3. We have used standard on-line backpropagation for the first-layer
weights with a learning rate 11 = 0.01 for 1000 epochs. the results are averaged over
50 runs with different teacher networks and different training sets.
=
5
CONCLUSION
We have presented the results of a calculation of the generalization error of a multilayer network within the statistical mechanics approach. We have found nontrivial
behavior for networks with both continuous and binary weights. In both models, phase transitions from a symmetric, poorly-generalizing solution to one with
specialized hidden units occur, accompanied by a discontinuous drop of the generalization error. However, the existence of a metastable, poorly generalizing solution
beyond the phase transition implies the possibility of getting trapped in a local
minimum during the training process. Although these results were obtained for a
Gibbs distribution of networks, numerical experiments indicate that some of the
general results carryover to other learning scenarios.
Acknowledgements
The authors would like to thank M. Biehl and S. Solla for fruitful discussions. HS
acknowledges support from the EC under the SCIENCE programme (under grant
number B/SCl * /915125) and by the Danish Natural Science Council and the Danish
Technical Research Council through CONNECT.
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T. Watkin, A. Rau, and M. Biehl (1993), Rev. Mod. Phys. 65, 499.
| 839 |@word h:1 version:3 nd:1 open:1 r:4 simulation:5 carry:1 initial:1 numerical:1 partition:1 j1:1 drop:2 location:5 qualitative:2 incorrect:1 introduce:1 behavior:5 mechanic:4 increasing:2 becomes:1 finding:1 quantitative:2 unit:17 grant:1 local:3 limit:4 ak:1 studied:3 mateo:3 suggests:1 palmer:1 averaged:3 implement:1 backpropagation:3 procedure:1 got:1 revealing:1 quenched:2 get:1 convenience:1 close:2 influence:1 seminal:1 fruitful:1 annealed:2 go:1 independently:1 rule:1 regarded:2 oh:1 variation:1 target:1 agreement:1 preprint:1 calculate:1 region:1 ensures:1 connected:5 sompolinsky:2 solla:2 decrease:2 edited:1 broken:2 seung:2 trained:3 triangle:1 po:1 describe:1 monte:2 europhys:2 whose:1 widely:1 biehl:2 ability:1 favor:1 net:1 analytical:2 poorly:5 description:3 getting:1 perfect:1 ac:1 measured:1 krogh:3 predicted:2 indicate:3 implies:1 differ:1 discontinuous:6 correct:3 stochastic:2 generalization:26 sufficiently:1 considered:1 equilibrium:1 mapping:2 predict:1 purpose:1 proc:1 tanh:2 hansen:2 council:2 city:1 ck:3 rather:1 hiddento:1 check:1 indicates:1 realizable:2 glass:1 el:2 nand:1 hidden:19 misclassified:1 overall:2 classification:1 spinodal:2 priori:1 field:1 lit:1 park:1 others:1 carryover:1 escape:1 few:1 kwon:1 randomly:1 phase:13 investigate:1 possibility:1 alignment:1 partial:1 respective:2 sweden:1 ckc:1 korea:1 tree:2 iv:1 desired:1 circle:1 theoretical:5 formalism:3 soft:2 deviation:1 levin:2 tishby:6 kn:3 connect:1 teacher:7 considerably:1 physic:1 ansatz:1 connecting:1 moody:1 nonvanishing:3 connectivity:1 watkin:2 tz:1 nonoverlapping:1 accompanied:2 student:5 wk:2 includes:1 performed:1 break:1 dad:1 closed:1 spin:1 solvegatan:1 variance:1 kaufmann:3 ensemble:2 yield:2 ofthe:2 generalize:1 parga:2 iid:1 carlo:2 mato:2 phys:7 ed:1 danish:2 energy:4 improves:1 appears:2 salamon:1 wesley:1 higher:1 follow:2 reflected:1 jw:1 furthermore:3 correlation:1 nonlinear:1 schwarze:14 quality:1 scientific:1 normalized:1 hence:1 symmetric:7 eg:1 during:1 self:1 oc:1 hill:1 performs:2 temperature:7 geometrical:1 recently:1 specialized:4 exponentially:1 jl:1 extend:1 rau:1 gibbs:5 ai:1 scenario:2 binary:11 arbitrarily:1 morgan:3 minimum:4 algebraically:1 dashed:1 thermodynamic:1 smooth:1 technical:1 calculation:3 long:2 schematic:1 prediction:4 multilayer:3 diagram:2 mod:1 feedforward:3 concerned:1 architecture:4 whether:1 specialization:3 effort:1 york:1 gyorgyi:2 concentrated:1 simplest:1 singapore:1 sign:3 trapped:3 correctly:1 nordita:1 pohang:1 drawn:1 replica:7 utilize:1 run:3 vn:1 decision:1 scaling:1 layer:7 guaranteed:1 nontrivial:1 occur:1 infinity:1 constraint:1 argument:1 metastable:5 combination:1 hertz:15 rev:2 nilsson:2 restricted:1 equation:1 remains:1 previously:1 discus:1 pin:1 committee:22 addison:1 scl:1 apply:1 existence:2 denotes:1 already:1 quantity:1 flipping:1 parametric:1 receptive:1 dependence:2 thank:1 blegdamsvej:1 majority:2 denmark:1 negative:1 unknown:2 adjustable:1 allowing:2 upper:1 vertical:1 finite:2 thermal:2 redwood:1 copenhagen:1 hanson:1 kang:2 able:1 beyond:1 below:1 lund:2 overlap:1 critical:1 natural:1 residual:2 technology:1 gardner:4 acknowledges:1 koberle:1 review:1 understanding:1 epoch:1 acknowledgement:1 asymptotic:2 fully:4 permutation:1 gauged:1 proportional:2 consistent:1 free:4 formal:1 allow:1 perceptron:2 institute:1 curve:3 calculated:1 lett:2 valid:1 transition:14 world:1 author:1 made:1 qualitatively:3 san:3 programme:1 ec:1 lippmann:1 mcgraw:1 global:1 continuous:10 ca:1 symmetry:7 main:1 noise:1 referred:1 fails:1 breaking:1 load:1 symbol:2 decay:1 workshop:1 entropy:3 generalizing:5 smoothly:1 saddle:3 aa:6 goal:1 towards:1 included:2 vote:1 perceptrons:1 support:1 dept:1 |
7,069 | 84 | 154
PRESYNApnC NEURAL INFORMAnON PROCESSING
L. R. Carley
Department of Electrical and Computer Engineering
Carnegie Mellon University, Pittsburgh PA 15213
ABSTRACT
The potential for presynaptic information processing within the arbor
of a single axon will be discussed in this paper. Current knowledge about
the activity dependence of the firing threshold, the conditions required for
conduction failure, and the similarity of nodes along a single axon will be
reviewed. An electronic circuit model for a site of low conduction safety in
an axon will be presented. In response to single frequency stimulation the
electronic circuit acts as a lowpass filter.
I. INTRODUCTION
The axon is often modeled as a wire which imposes a fixed delay on a
propagating signal. Using this model, neural information processing is
performed by synaptically sum m ing weighted contributions of the outputs
from other neurons. However, substantial information processing may be
performed in by the axon itself. Numerous researchers have observed
periodic conruction failures at norma! physiological impulse activity rates
(e.g., in cat, in frog 2 , and in man ). The oscillatory nature of these
conduction failures is a result of the dependence of the firing threshold on
past impulse conduction activity.
The simplest view of axonal (presynaptic) information processing is
as a switch: the axon will either conduct an im pulse or not. The state of
the switch depends on how past impulse activity modulates the firing
threshold, which will result in conduction failure if firing threshold is bigger
than the incoming impulse strength. In this way, the connectivity of a
synaptic neural network could be modulated by past impulse activity at
sites of conduction failure within the network. More sophisticated
presynaptic neural information processing is possible when the axon has
more than one terminus, implying the existence of branch points within the
axon. Section II will present a general description of potential for
presynaptic information processing.
The after-effects of previous activity are able to vary the connectivity
of the axonal arbor at sites of low conduction safety according to the
temporal pattern of the impulse train at each site (Raymond and LeUvin,
1978; Raymond, 1979). In order to understand the inform ation processing
potential of presynaptic networks it is necessary to study the after- effects
of activity on the firing threshold. Each impulse is normally followed by a
brief refractory period (about 10m s in frog sciatic nerve) of increased
? American Institute of Phvl'if:<' 1qR~
155
threshold and a longer superexcitable period (about 1 s in frog sciatic
nerve) during which the threshold is actually below its resting level.
During prolonged periods of activity, there is a gradual increase in firing
threshold which can persist long (> 1 hour in frog nerve) after cessation
of im pulse activity (Raymond and Lettvin, 1978). In section III, the
methods used to measure the firing threshold and the after-effects of
activity will be presented.
In addition to understanding how impulse activity modulates sites of
low conduction safety, it is important to explore possible constraints on
the distribution of sites of low conduction safety within the axon's arbor.
Section IV presents results from a study of the distribution of the aftereffects of activity along an axon.
Section V presents an electronic circuit model for a region of low
conduction safety within an axonal arbor. It has been designed to have a
firing threshold that depends on the past activity in a manner similar to the
activity dependence measured for frog sciatic nerve.
II. PRESYNAPTIC SIGNAL PROCESSING
Conduction failure has been observed in many diffe~e~t organisms,
including man, at normal physiological activity rates. 1 , , The aftereffects of activity can "modulate" conduction failures at a site of low
conduction safety. One common place where the conduction safety is low
is at branch points where an impedance mismatch occurs in the axon.
In order to clarify the meaning of presynaptic information processing,
a simple example is in order. Parnas reported that in crayfish a single
axon separately activates the medial (DEA~~ and lateral (DEAL) branches
of the deep abdominal extensor muscles.' At low stimulus frequencies
(below 40-50 Hz) impulses travel down both branches; however, each
impulse evokes much smaller contractions in DEAL than in DEAM resulting
in contraction of DEAM without significant contraction of DEAL. At higher
stim ulus frequencies conduction in the branch leading to D EAM fails and
DEAL contracts without DEAM contracting. Both DEAL and DEAM can be
stim ulated separately by stim ulus patterns more com plicated than a single
frequency.
The theory of "fallible trees", which has been discussed by Lettvin,
McCulloch and Pitts, Raymond, and Waxman and Grossman among
others, suggests that one axon which branches many times forms an
information processing element with one input and many outputs. Thus,
the after-effects of previous activity are able to vary the connectivity of
the axonal arbor at regions of low conduction safety according to the
temporal pattern of the impulse train in each branch. The transfer function
of the fallible tree is determined by the distribution of sites of low
conduction safety and the distribution of superexcitability and depressibility
at those sites. Thus, a single axon with 1000 terminals can potentially be
in 2 1000 different states as a function of the locations of sites of conduction
failure within the axonal arbor. And, each site of low conduction safety is
156
modulated by the past impulse activity at that site.
Fallible trees have a number of interesting properties. They can be
used to cause different input frequencies to excite different axonal
terminals. Also, fallible trees, starting at rest, will preserve timing
information in the input signal; Le., starting from rest, all branches will
respond to the first impulse.
III. AFTER- EFFECTS OF ACTIVITY
In this section, the firing threshold will be defined and an experimental
method for its measurem ent will be described. In addition, the aftereffects of activity will be characterized and typical results of the
characterization process will be given.
The following method was used to measure the firing threshold.
Whole nerves were placed in the experimental setup (shown in figure 1).
The whole nerve fiber was stim ulated with a gross electrode. The
response from a single axon was recorded using a suction microelectrode.
Firing threshold was measured by applying test stimuli through the gross
stimulating electrode and looking for a response in the suction
m icroelectrode.
F ixed-duration
variable-amplitude
current stimulator
.
??
,
Ag-AgCI
Motordriven
vernier
micrometer
electrode
0?4 mm diameter
f--MOVES~
A
~;
Suction electrOdep'.
Single axon
\
Whole nerve
t'.
t.
lh
Refer~nce
~ suctIon
electrode
Figure 1. Drawing of the experimental recording chamber.
Threshold Hunting, a
was used to characterize
test stimulus which fails
increase the strength of
process forschoosin g the test stimulus strength,
the axons. It uses the following paradigm. A
to elicit a conducting impulse causes a small
subsequent test stimuli. A test stim ulus which
157
elicits an im pulse causes a small decrease in the strength of subsequent
test stimuli. Conditioning Stimuli, ones large enough to guarantee firing an
impulse, can be interspersed between test stimuli in order to achieve a
controlled overall activity rate. Rapid variations in threshold following one
or more conditioning impulses can be measured by slowly increasing the
time delay between the conditioning stimuli and the test stimulus. Several
phases follow each impulse. First, there is a refractory period of short
duration (about 10ms in frog nerve) during which another impulse cannot
be initiated. Following the refractory period the axon actually becomes
more excitable than at rest for a period (ranging from 200ms to 1 s in frog
nerve, see figure 2). The superexcitable period is measured by applying a
conditioning stimulus and then delaying by a gradually increasing time
delay and applying a test stimulus (see figure 3). There is only a slight
increase in the peak of the superexcitable period following multiple
im pulses? The superexcitability of an axon was characterized by the %
decrease of the threshold from its resting level at the peak of the
superexcitable period.
5'(1) fo, P, 0.50
?
5
+
:......-TO~
:_TO'Ald-~
1
1
CONO I TlONING
o~!_ _ _ _~~~I----~~~I~--17~~'---IOc~~Td
INTERVAL 'm.. c)
Figure 2. Typical superexcitable
period in axon from frog sciatic
nerve.
T[ST 5t IMULU5
.
T [5T 5T IMULUS
co NOI T 10NING
:_FRAMC 1 - ; - r R A M C
?-:-
Figure 3. Stim ulus pattern used
for measuring superexcitability.
During a period of repetitive impulse conduction, the firing threshold may
gradually increase. After the period of increased im pulse activity ends, the
threshold gradually recovers from its maximum over the course of several
minutes or more with complete return of the threshold to its resting level
taking as long as an hour or two (in frog nerve) depending on the extent of
the preceding im pulse activity. The depressibility of an axon can be
characterized by the initial upward slope of the depression and the time
158
constant of the recovery phase (see figure 4). The pattern of conditioning
and test stimuli used to generate the curve in figure 4 is shown in figure 5.
Depression may be correlated with microanatomical changes which
occur ira the glial cells in the nodal region during periods of increased
activity. During periods of repetitive stim ulation the size and num ber of
extracellular paranodal intramyelinic vacuoles increases causing changes
in the paranodal geom etry.
Cond.t.on.,,!!
burst
Test
Threshold \percenl.gt of rHling level)
200
120
40
?o+-'--5~-tO--15--2-0--2+-5--:'30
Time (min)
l'
r--
On
Figure 4. Typical depression in an
axon from frog sciatic nerve. The
average activity rate was 4
impulses/sec between the 5 min
mark and the 10 min mark.
5 min
>"
T
Time
~
Off
Figure 5. Stim ulus pattern used
for measuring depression.
IV. CONSTRAINTS ON FALLIBLE TREES
The basic fallible tree theo ry places no constraints on the distribution
of sites of conduction failure among the branches of a single axon. In this
section one possible constraint on the distribution of sites of conduction
failure will be presented. Experiments have been performed in an attempt
to determine if the extremely wide variations in superexcitability anS
depressibility found between nodes from different axons in a single nerve
(particularly for depressibility) also occur between nodes from the same
axon.
A study of the distribution of the after-effects of activity along an
unbranching length of frog sciatic nerve isund only sm all variations in the
after- effects along a single axon.
Both superexcitability and
depressibility were extremely consistent for nodes from along a single
unbranching length of axon (see figures 6 and 7). This suggests that there
may be a cell-wide regulatory system that maintains the depressibility and
159
superexcitability at com parable levels throug hout the extent of the axon.
Thus, portions of a fallible tree which have the same axon diameter would
be expected to have the same superexcitability and depressibility.
3.()
30
95
Superexcitability (%1
Figure 6. PDF of SuperexcitabiliThe upper trace represents
the PDF of the entire population
of nodes studied and the two
lower
traces
represent
the
separate populations of nodes
from two different axons.
ty.
0 -8
8 -0
2-5
25
80
Upward slope ("'/minl
Figure 7. PDF of Depressibility.
The upper trace represents the
PDF of the entire population of
nodes studied and the two lower
traces represent the separate
populations of nodes from two
different axons.
This study did not examine axons which branched, therefore it cannot be
concluded that superexcitability and depressibility must remain constant
throughout a fallible tree. For example, it is quite likely that the cell
actually regulates quantities like pump- site density, not depressibility. In
that case, daughter branches of smaller diameter might be expected to
show consistently higher depressibility. Further research is needed to
determine how the activity dependence of the threshold scales with axon
diameter along a single axon before the consistency of the after-effects
along an unbranching axon can be used as a constraint on presynaptic
information processing networks.
V. ELECTRICAL AXON CIRCUIT
This section presents a simple electronic circuit which has been
designed to have a firing threshold that depends on the past states of the
output in a manner similar to the activity dependence measured for frog
sciatic nerve. In response to constant frequency stimuli, the circuit acts as
160
a low pass filter whose corner frequency depends on the coefficients which
determine the after-effects of activity.
Figure B shows the circuit diagram for a switched capacitor circuit
which approximates the after- effects of activity found in the frog sciatic
nerve. The circuit employs a two phase nonoverlapping clock, e for the
even clock and 0 for the odd clock, typical of switched capacitor circuits.
It incorporates a basic model for superexcitability and depressibility. VTH
represents the resting threshold of the axon. On each clock cycle the V'N
is com pared with VTH+ Vo- Vs.
The two capacitors and three switches at the bottom of figure B model
the change in threshold caused by superexcitability. Note that each
impulse resets the comparator's minus input to (1-cx.)VTH, which decays
back to VTH on subsequent clock cycles with a time constant inversely
proportional to Ps. This is a slight deviation from the actual physiological
situation in which multiple conditioning im pulses will generate slightly more
superexcitability than a single impulse?
The two capacitors and two switches at the upper left of figure B
model the depressibility of the axon. The current source represents a
fixed increment in the firing threshold with every past impulse. The
depression voltage decays back to 0 on subsequent clock cycles with a
time constant inversely proportional to PO.
Figure B. Circuit diagram for electrical circuit analog of nerve threshold.
The electrical circuit exhibits response patterns similar to those of
neurons that are conducting intermittently (see figure 9). During bursts of
conduction, the depression voltage increases linearly until the comparator
161
fails to fire. The electrical axon then fails to fire until the depression
voltage decays back to (1 +aOV)VTH' The connectivity between the input
and output of the axon is defined to be the average fraction of impulses
which are conducted. In terms of connectivity, the electrical axon model
acts as a lowpass filter (see figure 10).
riftiNG
VD '
YES
tll4 Vs
,
??
NO
..
.
rlUINC "'I1ACTI(lN
??
~
\
o :
vS
,00
10
300
T,,.a: ?Sl:CONUS I
Figure 9. Typical waveform s for
intermittent
conduction.
The
upper trace indicates whether
impulses are conducted or not.
VD and Vs are the depression
voltage and the superexcitable
voltage respectively.
o. :.I--~~----;-t-----;2r-------.c.
INruT
rR(: QVE~'
Figure 10. Frequency response of
electrical
axon
model.
The
connectivity is reflected by the
fraction of impulses which are
conducted out of a seq uence of
100.000
stimuli
where
the
frequency is in stim uli/second.
For a fixed stim ulus frequency. the average fraction of im pulses
which are conducted by the electrical model can be predicted analytically.
The expressions can be greatly simplified by making the assumption that
VD increases and decreases in a linear fashion. Under that assumption. in
terms of the variables indicated on the schematic diagram,
where M is the number of clock cycles between input stimuli. which is
inversely proportional to the input frequency. The frequency at which only
half of the impulses are conducted is defined as the corner frequency of
the low pass filter. The corner frequency is
162
f(P == 0.5)
_...!.
M
==
log(1-~D)
aD
log(1--)
aOV
Using the above equations, lowpass filters with any desired cutoff
frequency can be designed.
The analysis indicates that the corner frequency of the lowpass filter
can be varied by changing the degree of conduction safety (aov) without
changing either depressibility or superexcitability. This suggests that the
existence of a cell- wide regulatory system maintaining the depressibility
and superexcitability at comparable levels throughout the extent of the
axon would not prevent the construction of a bank of low pass filters since
their corner frequencies could still be varied by varying the degree of
conduction safety (aov).
VI. CONCLUSIONS
Recent studies report that the primary effect of several common
anesthetics is to abolish the activity dependence of the firing threshold
without interfering with impulse conduction. 11 This suggests that
presynaptic processing may play an important role in human
consciousness. This paper has explored some of the basic ideas of
presynaptic information processing, especially the after- effects of activity
and their modulation of impulse conduction at sites of low conduction
safety. A switched capacitor circuit which sim ulates the activity dependent
conduction block that occurs in axons has been designed and simulated.
Simulation results are very similar to the intermittent conduction patterns
measured experimentally in frog axons. One potential information
processing possibility for the arbor of a single axon, suggested by the
analysis of the electronic circuit, is to act as a filterbank; every terminal
could act as a lowpass filter with a different corner frequency.
BIBLIOGRAPHY
[1] Barron D. H. and B. H. C. Matthews, Intermittent conduction in the
spinal chord. J. Physiol. 85, p. 73-103 (1935).
163
[2]
Fuortes M. G. F., Action of strychnine on the "intermittent
conduction" of impulses along dorsal columns of the spinal chord of
frogs. J. Physiol. 112, p.42 (1950).
[3] Culp W. and J. Ochoa, Nerves and Muscles as Abnormal Impulse
Generators. (Oxford University Press, London, 1980).
[4] Grossman V., I. Parnas, and M. E. Spira, Ionic mechanisms involved
in differential conduction of action potentials at high frequency in a
branching axon. J. Physiol. 295, p.307 - 322 (1978).
[5] Parnas I., Differential block at high frequency of branches of a
single axon innervating two muscles. J. Physiol. 35, p. 903-914,
1972.
[6]
Carley, L.R. and S.A. Raymond, Threshold Measurement:
Applications to Excitable Membranes of Nerve and Muscle. J.
Neurosci. Meth. 9, p. 309 - 333 (1983).
[7] Raymond S. A. and J. V. Lettvin, After-effects of activity in
peripheral axons as a clue to nervous coding. In Physiology and
Pathobiology of Axons, S. G. Waxman (ed.), (Raven Press, New York,
1978), p. 203 - 225.
[8] Wurtz C. C. and M. H. Ellisman, Alternations in the ultrastructure of
peripheral nodes of Ranvier associated with repetitive action
potential propagation. J. Neurosci. 6(11), 3133- 3143 (1986).
[9] Raym ond S. A., Effects of nerve im pulses on threshold of frog
sciatic nerve fibers. J. Physiol. 290,273- 303 (1979).
[10] Carley, L.R. and S.A. Raymond, Com parison of the after- effects of
impulse conduction on threshold at nodes of Ranvier along single
frog Sciatic axons. J. Physiol. 386, p. 503 - 527 (1987).
[11] Raymond S. A. and J. G. Thalhammer, Endogenous activitydependent mechanisms for reducing hyperexcitability ofaxons:
Effects of anesthetics and CO 2 , In Inactivation of Hypersensistive
Neurons, N. Chalazonitis and M. Gola, (eds.), (Alan R. Liss Inc., New
Vork, 1987), p. 331-343.
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7,070 | 840 | Learning in Compositional Hierarchies:
Inducing the Structure of Objects from Data
Joachim Utans
Oregon Graduate Institute
Department of Computer Science and Engineering
P.O. Box 91000
Portland, OR 97291-1000
[email protected]
Abstract
I propose a learning algorithm for learning hierarchical models for object recognition. The model architecture is a compositional hierarchy
that represents part-whole relationships: parts are described in the local context of substructures of the object. The focus of this report is
learning hierarchical models from data, i.e. inducing the structure of
model prototypes from observed exemplars of an object. At each node
in the hierarchy, a probability distribution governing its parameters must
be learned. The connections between nodes reflects the structure of the
object. The formulation of substructures is encouraged such that their
parts become conditionally independent. The resulting model can be
interpreted as a Bayesian Belief Network and also is in many respects
similar to the stochastic visual grammar described by Mjolsness.
1 INTRODUCTION
Model-based object recognition solves the problem of invariant recognition by relying on
stored prototypes at unit scale positioned at the origin of an object-centered coordinate
system. Elastic matching techniques are used to find a correspondence between features of
the stored model and the data and can also compute the parameters of the transformation the
observed instance has undergone relative to the stored model. An example is the TRAFFIC
system (Zemel, Mozer and Hinton, 1990) or the Frameville system (Mjolsness, Gindi and
285
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Utans
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Figure I: Example of a compositional
hierarchy. The simple figure can be
represented as hierarchical composition of parts. The hierarchy can
be represented as a graph (a tree in
this case). Nodes represent parts and
edges represent the structural relationship. Nodes at the bottom represent
individual parts of the object; nodes
at higher levels denote more complex
substructures. The single node at the
top of the tree represents the entire object.
Anandan, 1989; Gindi, Mjolsness and Anandan, 1991; Vtans, 1992). Frameville stores
models as compositional hierarchies and by matching at each level in the hierarchy reduces
the combinatorics of the match.
The attractive feature of feed-forward neural networks for object recognition is the relative
ease with which their parameters can be learned from training data. Multilayer feed-forward
networks are typically trained on input/output pairs (supervised learning) and thus are tuned
to recognize instances of objects as seen during training. Difficulties arise if the observed
object appears at a different position in the input image, is scaled or rotated, or has been
subject to distortions. Some of these problems can be overcome by suitable preprocessing or
judicious choice of features. Other possibilities are weight sharing (LeCun, Boser, Denker,
Henderson, Howard, Hubbard and Jackel, 1989) or invariant distance measures (Simard,
LeCun and Denker, 1993).
Few attempts have been reported in the neural network literature to learn the prototype
models for model based recognition from data. For example, the Frameville system uses
hand-designed models. However, models learned from data and reflecting the statistics of
the data should be superior to the hand-designed models used previously. Segen (1988a;
1988b) reports an approach to learning structural descriptions where features are clustered
to substructures using a Minimum Description Length (MDLJ criterion to obtain a sparse
representation. Saund (1993) has proposed a algorithm for constructing tree presentation
with multiple "causes" where observed data is accounted for by multiple substructures at
higher levels in the hierarchy. Veda and Suzuki (1993) have developed an algorithm for
learning models from shape contours using multiscale convex/concave structure matching
to find a prototype shape typical for exemplars from a given class.
2
LEARNING COMPOSITIONAL HIERARCHIES
The algorithm described here merges parts by means of grouping variables to form substructures. The model architecture is a compositional hierarchy, i.e. a part-whole hierarchy
(an example is shown in Figure 1). The nodes in the graph represent parts and substructures, the arcs describe the structure of the object. At each node a probability density for
part parameters is stored. A prominent advocate of such models has been Marr (1982)
and models of this type are used in the Frameville system (Mjolsness et ai., 1989; Gindi
et al., 1991; Vtans, 1992). The nodes in the graph represent parts and substructures, the
Learning in Compositional Hierarchies: Inducing the Structure of Objects from Data
Figure 2: Examples of different compositional hierarchies for
the same object (the digit 9 for
a seven-segment LED display).
One model emphasizes the parallel lines making up the square in
the top part of the figure while for
another model angles are chosen
as intermediate substructures. The
example on the right shows a hierarchy that "reuses" parts.
arcs describe the structure of the object. The arcs can be regarded as "part-of" or "ina"
relationships (similar to the notion used in semantic networks). At each node a probability
density for part parameters such as position, size and orientation is stored.
The model represents a typical prototype object at unit scale in an object-centered coordinate
system. Parameters of parts are specified relative to parameters of the parent node in the
hierarchy. Substructures thus provide a local context for their parts and decouple their parts
from other parts and substructures in the model. The advantages of this representation are
sparseness, invariance with respect to viewpoint transformations and the ability to model
local deformations. In addition, the model explicitly represents the structure of an object
and emphasizes the importance of structure for recognition (Cooper, 1989).
Learning requires estimating the parameters of the distributions at each node (the mean and
variance in the case of Gaussians) and finding the structure of model. The emphasis in this
report is on learning structure from exemplars. The parameterization of substructures may
be different than for the parts at the lowest level and become more complex and require more
parameters as the substructures themselves become more complex. The representation as
compositional hierarchy can avoid overfitting since at higher levels in the hierarchy more
exemplars are available for parameter estimation due to the grouping of parts (Omohundro,
1991).
2.1
Structure and Conditional Independence: Bayesian Networks
In what way should substructures be allocated? Figure 2 shows examples of different
compositional hierarchies for the same object (the digit 9 for a seven-segment LED display).
One model emphasizes the parallel lines making up the square in the top part of the figure
while for another model angles are chosen as intermediate substructures. It is not clear
which of these models to choose.
The important benefit of a hierarchical representation of structure is that parts belonging to
different substructures become decoupled, i.e. they are assigned to a different local context.
The problem of constructing structured descriptions of data that reflect this independence
relationship has been studied previously in the field of Machine Learning (see (Pearl, 1988)
for a comprehensive introduction). The resulting models are Bayesian Belief Networks.
Central to the idea of Bayesian Networks is the assumption that objects can be regarded
as being composed of components that only sparsely interact and the network captures
the probabilistic dependency of these components. The network can be represented as
an interaction graph augmented with conditional probabilities. The structure of the graph
represents the dependence of variables, i.e. connects them with and arc. The strength of the
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Utans
m,.
0.11
Figure 3: Bayesian Networks and conditional
independence (see text).
Figure 4: The model architecture. Circles denote
the grouping variables ina (here a possible valid
model after leaming is shown).
dependence is expressed as forward conditional probability. The conditional independence
is represented by the absence of an arc between two nodes and leads to the sparseness of
the model.
The notion of conditional independence in the context studied here manifest itself as follows.
By just observing two parts in the image, one must assume that they, i.e. their parameters
such as position, are dependent and must be modeled using their joint distribution. However, if one knows that these two parts are grouped to form a substructure then knowing
the parameters of the substructure, the parts become conditionally independent, namely
conditioned on the parameters of the substructure. Thus, the internal nodes representing the
substructures summarize the interaction of their child nodes. The correlation between the
child nodes is summarized in the parent node and what remains is, for example, independent
noise in observed instances of the child nodes.
The probability of observing an instance can be calculated from the model by starting at
the root node and multiplying with the conditional probabilities of nodes traversed until the
leaf nodes are reached. For example, given the graph in Figure 3, the joint distribution can
be factored as
P(Xl' Yl, Y2, zl, Z2, z3, Z4) =
P(Xd P (Yllxd P (Zllyd P (ZlIYl)P(Z2IYl )P(z3IY2)P(Z4IY2)
(I)
(note that the hidden nodes are treatedjust like the nodes corresponding to observable parts).
Note that the stochastic visual grammar described by Mjolsness (1991) is equivalent to this
model. The model used there is a stochastic forward (generative) model where each level
of the compositional hierarchy corresponds to a stochastic production rule that generates
nodes in the next lower level. The distribution of parameters at the next lower level
are conditioned on the parameters of the parent node. Thus, the model obtained from
constructing a Bayesian network is equivalent to the stochastic grammar if the network is
constrained to a directed acyclic graph (DAG).
If all the nodes of the network correspond to observable events, techniques exist for finding
the structure of the Bayesian Network and estimate its parameters (Pearl, 1988) (see also
(Cooper and Herskovits, 1992)}. However, for the hierarchical models considered here,
only the nodes at the lowest layer (the leaves of the tree) correspond to observable instances
of parts of the object in the training data. The learning algorithm must induce hidden,
unobservable substructures. That is, it is assumed that the observables are "caused" by
internal nodes not directly accessible. These are represented as nodes in the network just
Learning in Compositional Hierarchies: Inducing the Structure of Objects from Data
like the observables and their parameters must be estimated as well. See (Pearl, 1988) for
an extensive discussion and examples of this idea.
Learning Bayesian networks is a hard problem when the network contains hidden nodes
but a construction algorithm exists if it is known that the data is in fact tree-decomposable
(Pearl, 1988). The methods is based on computing the correlations p between child nodes
and constraints on the correlation coefficients dictated by a particular structure. The entire
tree can be constructed recursively using this method. Here, the case of Normal-distributed
real-valued random variables is of interest:
p(XI, ... , Xn)
where x =
1 Vdet'f
1
= v2?r
~
exp (1
--(x detL
2
(XI, X2, ...
p) T :E -I (x
- p) )
(2)
,xn ) with mean p = E{x} and covariance matrix :E = E{(x -
p)(x - p)T} The method is based on a condition under which a set of random variables
is star-decomposable. The question one ask is whether a set of n random variables can
be represented as the marginal distribution of n + 1 variables XI, ... , X n , W such that the
XI, ... , Xn are conditionally independent given w, i.e.
(3)
J
p(XI, ... , Xn , w)dw
(4)
In the graph representation of the Bayesian Network w is the central node relating the
XI, ... ,X n , hence the name star-decomposable. In the general case of n variables this is
hard to verify but a result by Xu and Pearl (1987) is available for 3 variables: A necessary
and sufficient condition for 3 random variables with a joint normal distribution to be stardecomposable is that the pairwise correlation coefficients satisfy the triangle inequality
pjk ~ PjiPik
with
(5)
for all i, j, k E [1,2,3] and i "I j "I k. Equality holds if node w coincides with node i. For
the lowest level of the hierarchy, nodes j and k represent parts and node i = w represents
the common substructure.
2.2
An Objective Function for Grouping Parts
The algorithm proposed here is based on "soft" grouping by means of grouping variables ina
where both the grouping variables and the parameter estimates are updated concurrently.
The learning algorithms described in (Pearl, 1988) incrementally construct a Bayesian
network and decisions made at early stages cannot be reversed. It is hoped that the method
proposed here is more robust with regard to inaccuracies of the estimates. However, if the
true distribution is not a star-decomposable normal distribution it can only be approximated.
Let inaij be a binary variable associated with the arc connecting node i and node j; inaij =
1 if the arc is present in the network (ina is the adjacency matrix of the graph describing the
structure of the model). The model architecture is restricted to a compositional hierarchy (a
departure from the more general structure of a Bayesian Network, i.e. nodes are preassigned
to levels of the hierarchy (see Figure 4)). Based on the condition in equation (5) a cost
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Utans
function term for the grouping variables ina is
Ep =
L
inawjinawk (PwjPwk - Pjk)2
(6)
w,j,kt-j
The term penalizes the grouping of two part nodes to the same parent if the term in
parentheses is large (i and k index part nodes, w nodes at the next higher level in the
hierarchy) The inawj can be regarded as assignment variables the assign child nodes j to
parent nodes w. The parameters at each node and the assignment variables ina are estimated
using an EM algorithm (Dempster, Laird and Rubin, 1977; Utans, 1993; Yuille, Stolorz and
Utans, 1994). For the details of the implementation of grouping with match networks see
(Mjolsness et at., 1989; Mjolsness, 1991; Gindi et at., 1991; Utans, 1992; Utans, 1994).
At each node for each parameter a probability distribution is stored. Nodes at the lowest
level of the hierarchy represent parts in the input data. For the Gaussian distributions used
here for all nodes, the parameters are the mean J-t and the variance (J' and can be estimated
from data. Each part node can potentially be grouped to any substructure at the next
higher level in the hierarchy. The parameters of the distributions at this level are estimated
from data as well but using the current value of the grouping variables inaij to weight the
contribution from each part node. Because each child node j can have only one parent node
i, an additional constraint for a unique assignment is Lw inawj = 1.
3
ANEXAMPLE
Initial simulations of the proposed algorithm were performed using a hierarchial model for
dot clusters. The training data was generated using the three-level model shown in Figure 5.
Each node is parameterized by its position (x, y). The node at the top level represents the
entire dot cluster. At the intermediate level nodes represent subcluster centers. The leaf
nodes at the lowest level represent individual dots that are output by the model and observed
in the image. The top level node represents the position of the entire cluster. At each level
+ 1 stored offsets 1 are added to the parent coordinates x~ to obtain the coordinates
of the child nodes. Then, independent, zero-mean Gaussian distributed noise ( is added:
xj+l = x! + d~jl + ( The training data consists of a vector of positions at the lowest level
{Xj} with Xj (Xj, Yj), j
1 ... 9 for each exemplar.
1
d!t
=
=
The identity of the parts in the training data is assumed known. In addition, the data consists
of parts from a single object. For the simulations, the model architecture is restricted to a
three-level hierarchy. Since at the top level a single node represents the entire object, only
the grouping variables from the lowest to the intermediate level are unknown (the nodes
at the intermediate level are implicitly grouped to the single node at the top level). In the
current implementation the parameters of a parent node are defined as the average over the
parameters of its child nodes: x~ = Jv Lj i~jxj+l
For this problem the algorithm has recovered the structure of the model that generated the
training data. Thus in this case it is possible to use the correlation coefficients to learn
the structure of an object from noisy training exemplars. However, the algorithm does
not recover the same parameter values x used in the generative model at the intermediate
layers. These cannot uniquely specified due to the ambiguity between the parameters Xi
and offsets d ij (a different choice for Xi leads to different values for d ij ).
Learning in Compositional Hierarchies: Inducing the Structure of Objects from Data
0
0
0
0
?
0
0
?
0
DaIs
)(
Global Position
?
)(
?
0
CI ustar Center
0
0
Dot
Figure 5: The model used to generated training data. The structure of the model is a three-level
hierarchy. The model parameters are chosen such that the generated dot cluster spatially overlap. On
the left, an example of an instance of a dot cluster generated from the model is shown (these constitute
the training data).
4 EXTENSIONS
The results of the initial experiments are encouraging but more research needs to be done
before the algorithm can be applied to real data. For the example used here, the training
data was generated by a hierarchical model. Thus the distribution of the training exemplars
could, in principle, be learned exactly using the proposed model architecture. I plan to
study the effect of approximating the distribution of real-world data by applying the method
to the problem of learning models for handwritten digit recognition.
The model should be extended to include provisions to deal with missing data. Instead of
being binary variables, inaij could be the conditional probability that part j is present in a
typical instance of the object given that the parent node i itself is present (similar to the dot
deletion rule described in (Mjolsness, 1991)}. These probabilities must also be estimated
from data. Under this interpretation the inaij are similar to the mixture coefficients in the
mixture of experts model (Jordan and Jacobs, 1993)
The robustness of the algorithm can be improved when the desired locality of the model is
explicitly favored via an additional constraint.
E\ocal
= .A L
inaij inaik
IXj -
Xk
12
ij k
In this sense, the toy problem shown here is unnecessarily difficult. Preliminary experiments
indicate that including this term reduces the sensitivity to spurious correlations between
parts that are far apart.
As described the algorithm performs unsupervised grouping; learning the hierarchical model
does not take in to account the recognition performance obtained when using the model.
While the problem of learning and representing models in a hierarchical form is interesting
in its own right, the final criteria for judging the model in the context of a recognition
problem should be recognition performance. The assumption is that the model should pick
up substructures that are specific to a particular class of objects and maximally discriminate
between objects belonging to other classes. For example, after a initial model is obtained
that roughly captures the structure of the training data, it can be refined on-line during the
recognition stage.
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Utans
Acknowledgements
Initial work on this project was performed while the author was with the International
Computer Science Institute, Berkeley, CA. At OGI supported was provided in part under
grant ONR N00014-92-J-4062. Discussions with S. Knerr, E. Mjolsness and S. Omohundro
were helpful in preparing this work.
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7,071 | 841 | Hoeffding Races: Accelerating Model
Selection Search for Classification and
Function Approximation
Oded Maron
Artificial Intelligence Laboratory
Massachusetts Institute of Technology
Cambridge, MA 02139
Andrew W. Moore
Robotics Institute
School of Computer Science
Carnegie Mellon University
Pittsburgh, PA 15213
Abstract
Selecting a good model of a set of input points by cross validation
is a computationally intensive process, especially if the number of
possible models or the number of training points is high. Techniques such as gradient descent are helpful in searching through
the space of models, but problems such as local minima, and more
importantly, lack of a distance metric between various models reduce the applicability of these search methods. Hoeffding Races is
a technique for finding a good model for the data by quickly discarding bad models, and concentrating the computational effort at
differentiating between the better ones. This paper focuses on the
special case of leave-one-out cross validation applied to memorybased learning algorithms, but we also argue that it is applicable
to any class of model selection problems.
1
Introduction
Model selection addresses "high level" decisions about how best to tune learning
algorithm architectures for particular tasks. Such decisions include which function
approximator to use, how to trade smoothness for goodness of fit and which features are relevant. The problem of automatically selecting a good model has been
variously described as fitting a curve, learning a function, or trying to predict future
59
60
Maron and Moore
0.22
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9
k Nearest Neigh bors Used
Figure 1: A space of models consisting of local-weighted-regression models with
different numbers of nearest neighbors used. The global minimum is at one-nearestneighbor, but a gradient descent algorithm would get stuck in local minima unless
it happened to start in in a model where k < 4.
instances of the problem. One can think of this as a search through the space of
possible models with some criterion of "goodness" such as prediction accuracy, complexity of the model, or smoothness. In this paper, this criterion will be prediction
accuracy. Let us examine two common ways of measuring accuracy: using a test
set and leave-one-out cross validation (Wahba and Wold, 1975) .
? The test set method arbitrarily divides the data into a training set and a
test set. The learner is trained on the training set, and is then queried with
just the input vectors of the test set. The error for a particular point is the
difference between the learner's prediction and the actual output vector .
? Leave-one-out cross validation trains the learner N times (where N is
the number of points), each time omitting a different point. We attempt to
predict each omitted point. The error for a particular point is the difference
between the learner's prediction and the actual output vector.
The total error of either method is computed by averaging all the error instances.
The obvious method of searching through a space of models, the brute force approach, finds the accuracy of every model and picks the best one. The time to find
the accuracy (error rate) of a particular model is proportional to the size of the test
set IT EST!, or the size of the training set in the case of cross validation . Suppose
that the model space is discretized into a finite number of models IMODELSI then the amount of work required is O(IMODELSI x ITEST!), which is expensive.
A popular way of dealing with this problem is gradient descent. This method can
be applied to find the parameters (or weights) of a model. However, it cannot be
used to find the structure (or architecture) of the modeL There are two reasons for
Hoeffding Races: Accelerating Model Selection
this. First, we have empirically noted many occasions on which the search space is
peppered with local minima (Figure 1). Second, at the highest level we are selecting
from a set of entirely distinct models, with no numeric parameters over which to
hill-climb. For example, is a neural net with 100 hidden units closer to a neural net
with 50 hiden units or to a memory-based model which uses 3 nearest neighbors?
There is no viable answer to this question since we cannot impose a viable metric
on this model space.
The algorithm we describe in this paper, Hoeffding Races, combines the robustness
of brute force and the computational feasibility of hill climbing. We instantiated the
algorithm by specifying the set of models to be memory-based algorithms (Stanfill and Waltz, 1986) (Atkeson and Reinkensmeyer, 1989) (Moore, 1992) and the
method of finding the error to be leave-one-out cross validation. We will discuss
how to extend the algorithm to any set of models and to the test set method in the
full paper. We chose memory-based algorithms since they go hand in hand with
cross validation. Training is very cheap - simply keep all the points in memory, and
all the algorithms of the various models can use the same memory. Finding the
leave-one-out cross validation error at a point is cheap as making a prediction: simply "cover up" that point in memory, then predict its value using the current model.
For a discussion of how to generate various memory-based models, see (Moore et
al., 1992).
2
Hoeffding Races
The algorithm was inspired by ideas from (Haussler, 1992) and (Kaelbling, 1990)
and a similar idea appears in (Greiner and Jurisica, 1992). It derives its name from
Hoeffding's formula (Hoeffding, 1963), which concerns our confidence in the sample
mean of n independently drawn points Xl, ??. , X n . The probability of the estimated
mean Ee3t
~ 2::l<i<n Xi being more than epsilon far away from the true mean
E true after n independently drawn points is bounded by:
=
where B bounds the possible spread of point values.
We would like to say that with confidence 1 - 8, our estimate of the mean is within
? of the true mean; or in other words, Pr(IEtrue - Ee3tl > f) < 8. Combining the
two equations and solving for ? gives us a bound on how close the estimated mean
is to the true mean after n points with confidence 1 - 8:
_j
? -
B 2 1og (2/6)
2n
The algorithm starts with a collection of learning boxes. We call each model a
learning box since we are treating the models as if they were black boxes. We
are not looking at how complex or time-consuming each prediction is, just at the
input and output of the box. Associated with each learning box are two pieces of
information: a current estimate of its error rate and the number of points it has
been tested upon so far. The algorithm also starts with a test set of size N. For
leave-one-out cross validation, the test set is simply the training set.
61
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Maron and Moore
I
ERROR
----------
----------;;
Uppez
Bound
o
~------r_----_+------~----~~----_r------+_----~------------learning
box #0
learning
box 411
learning
box 112
learning
box 413
learning
box 114
learning
box lIS
learning
box 116
Figure 2: An example where the best upper bound of learning box #2 eliminates
learning boxes #1 and #5. The size of f varies since each learning box has its own
upper bound on its error range, B.
At each point in the algorithm, we randomly select a point from the test set. We
compute the error at that point for all learning boxes, and update each learning
box's estimate of its own total error rate. In addition, we use Hoeffding's bound
to calculate how close the current estimate is to the true error for each learning
box. We then eliminate those learning boxes whose best possible error (their lower
bound) is still greater than the worst error of the best learning box (its upper
bound); see Figure 2. The intervals get smaller as more points are tested, thereby
"racing" the good learning boxes, and eliminating the bad ones.
We repeat the algorithm until we are left with just one learning box, or until we
run out of points. The algorithm can also be stopped once f has reached a certain
threshhold. The algorithm returns a set of learning boxes whose error rates are
insignificantly (to within f) different after N test points.
3
Proof of Correctness
The careful reader would have noticed that the confidence {; given in the previous
section is incorrect. In order to prove that the algorithm indeed returns a set of
learning boxes which includes the best one, we'll need a more rigorous approach.
We denote by ~ the probability that the algorithm eliminates what would have
been the best learning box. The difference between ~ and {; which was glossed over
in the previous section is that 1 - ~ is the confidence for the success of the entire
algrithm, while 1 - {; is the confidence in Hoeffding's bound for one learning box
Hoeffding Races: Accelerating Model Selection
during one iteration of the algorithm.
We would like to make a formal connection between Ll and {;. In order to do that, let
us make the requirement of a correct algorithm more stringent. We'll say that the
algorithm is correct if every learning box is within f of its true error at every iteration
of the algorithm. This requirement encompasses the weaker requirement that we
don't eliminate the best learning box. An algorithm is correct with confidence Ll if
Pr{ all learning boxes are within f on all iterations} :2: 1 - Ll.
We'll now derive the relationship between {; and Ll by using the disjunctive probability inequality which states that Pr{A V B} ~ Pr{A} + Pr{B}.
Let's assume that we have n iterations (we have n points in our test set), and that
we have m learning boxes (LBl .. ?LBm). By Hoeffding's inequality, we know that
Pr{ a particular LB is within
f
on a particular iteration} :2: 1 - {;
Flipping that around we get:
Pr{ a particular LB is wrong on a particular iteration} < {;
Using the disjunctive inequality we can say
Pr{
a particular LB is
a particular LB is
wrong on iteration 1 V
wrong on iteration 2 V
a particular LB is
wrong on iteration n}
~
{; . n
Let's rewrite this as:
Pr{ a particular LB is wrong on any iteration}
~
{; . n
N ow we do the same thing for all learning boxes:
LBl is wrong on
LB2 is wrong on
any iteration
any iteration
LBm is wrong on
any iteration}
~
{; . n . m
Pr{ some LB is wrong in some iteration}
~
{; . n . m
Pr{
V
V
or in other words:
We flip this to get:
Pr{ all LBs are within
f
on all iterations} :2: 1 - {; . n . m
Which is exactly what we meant by a correct algorithm with some confidence.
Therefore, {; = n~m. When we plug this into our expression for f from the previous
section, we find that we have only increased it by a constant factor. In other words,
by pumping up f, we have managed to ensure the correctness of this algorithm with
confidence Ll. The new f is expressed as:
f
= V~B-~-(l-Og-(-2-nm-n-)--I-O-g(-~-)-)
63
64
Maron and Moore
Table 1: Test problems
Problem
ROBOT
PROTEIN
ENERGY
POWER
POOL
DISCONT
DescrIption
10 input attributes, 5 outputs. Given an initial and a final description
of a robot arm, learn the control needed in order to make the robot
perform devil-sticking (Schaal and Atkeson, 1993).
3 inputs, output is a classification into one of three classes. This is the
famous protein secondary structure database, with some preprocessing
(Zhang et al., 1992).
Given solar radiation sensing, predict the cooling load for a building.
This is taken from the Building Energy Predictor Shootout.
Market data for electricity generation pricing period class for the new
United Kingdom Power Market.
The visually perceived mapping from pool table configurations to shot
outcome for two-ball collisions (Moore, 1992).
An artificially constructed set of points with many discontinuities. Local models should outperform global ones.
Clearly this is an extremely pessimistic bound and tighter proofs are possible (Omohundro, 1993).
4
Results
We ran Hoeffding Races on a wide variety of learning and prediction problems.
Table 1 describes the problems, and Table 2 summarizes the results and compares
them to brute force search.
For Table 2, all ofthe experiments were run using Ll = .01. The initial set of possible
models was constructed from various memory based algorithms: combinations of
different numbers of nearest neighbors, different smoothing kernels, and locally
constant vs. locally weighted regression. We compare the algorithms relative to
the number of queries made, where a query is one learning box finding its error at
one point. The brute force method makes ITESTI x ILEARNING BOXESI queries.
Hoeffding Races eliminates bad learning boxes quickly, so it should make fewer
querIes.
5
Discussion
Hoeffding Races never does worse than brute force. It is least effective when all
models perform equally well. For example, in the POOL problem, where there
were 75 learning boxes left at the end of the race, the number of queries is only
slightly smaller for Hoeffding Races than for brute force . In the ROBOT problem,
where there were only 6 learning boxes left, a significant reduction in the number of
queries can be seen. Therefore, Hoeffding Races is most effective when there exists
a subset of clear winners within the initial set of models. We can then search over
a very broad set of models without much concern about the computational expense
Hoeffding Races: Accelerating Model Selection
Table 2: Results of Brute Force vs. Hoeffding Races.
Problem
points
Initial #
learning
boxes
ROBOT
PROTEIN
ENERGY
POWER
POOL
DISCONT
972
4965
2444
210
259
500
95
95
189
95
95
95
queries
with
Brute
Force
queries
with
Hoeffding
Races
learning
boxes
left
92340
471675
461916
19950
24605
47500
15637
349405
121400
13119
22095
25144
6
60
40
48
75
29
60000
60000
400 00
:;';0000
Figure 3: The x-axis is the size of a set of initial learning boxes (chosen randomly)
and the y-axis is the number of queries to find a good model for the ROBOT
problem. The bottom line shows performance by the Hoeffding Race algorithm)
and the top line by brute force.
65
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Maron and Moore
of a large initial set. Figure 3 demonstrates this. In all the cases we have tested,
the learning box chosen by brute force is also contained by the set returned from
Hoeffding Races. Therefore, there is no loss of performance accuracy.
The results described here show the performance improvement with relatively small
problems. Preliminary results indicate that performance improvements will increase
as the problems scale up. In other words, as the number of test points and the
number of learning boxes increase, the ratio of the number of queries made by
brute force to the number of queries made by Hoeffding Races becomes larger.
However, the cost of each query then becomes the main computational expense.
Acknowledgements
Thanks go to Chris Atkeson, Marina Meila, Greg Galperin, Holly Yanco, and
Stephen Omohundro for helpful and stimulating discussions.
References
[Atkeson and Reinkensmeyer, 1989] C. G. Atkeson and D. J. Reinkensmeyer. Using associative content-addressable memories to control robots. In W. T. Miller, R. S. Sutton,
and P. J. Werbos, editors, Neural Networks for Control. MIT Press, 1989.
[Greiner and Jurisica, 1992] R. Greiner and I. Jurisica. A statistical approach to solving the EBL utility problem. In Proceedings of the Tenth International conference on
Artificial Intelligence (AAAI-92). MIT Press, 1992.
[Haussler, 1992] D. Haussler. Decision theoretic generalizations of the pac model for neural
net and other learning applications. Information and Computation, 100:78-150, 1992.
[Hoeffding, 1963] Wassily Hoeffding. Probability inequalities for sums of bounded random
variables. Journal of the American Statistical Association, 58:13-30, 1963.
[Kaelbling, 1990] 1. P. Kaelbling. Learning in Embedded Systems. PhD. Thesis; Technical
Report No. TR-90-04, Stanford University, Department of Computer Science, June 1990.
[Moore et al., 1992] A. W. Moore, D. J. Hill, and M. P. Johnson. An empirical investigation of brute force to choose features, smoothers and function approximators. In
S. Hanson, S. Judd, and T. Petsche, editors, Computational Learning Theory and Natural Learning Systems, Volume 9. MIT Press, 1992.
[Moore, 1992] A. W. Moore. Fast, robust adaptive control by learning only forward models. In J. E. Moody, S. J. Hanson, and R. P. Lippman, editors, Advances in Neural
Information Processing Systems 4. Morgan Kaufmann, April 1992.
[Omohundro, 1993] Stephen Omohundro. Private communication, 1993.
[Pollard, 1984] David Pollard. Convergence of Stochastic Processes. Springer-Verlag, 1984.
[Schaal and Atkeson, 1993] S. Schaal and C. G. Atkeson. Open loop stable control strategies for robot juggling. In Proceedings of IEEE conference on Robotics and Automation,
May 1993.
[Stanfill and Waltz, 1986] C. Stanfill and D. Waltz. Towards memory-based reasoning.
Communications of the A CM, 29(12):1213-1228, December 1986.
[Wahba and Wold, 1975] G. Wahba and S. Wold. A completely automatic french curve:
Fitting spline functions by cross-validation. Communications in Statistics, 4(1), 1975.
[Zhang et al., 1992] X. Zhang, J.P. Mesirov, and D.L. Waltz. Hybrid system for protein
secondary structure prediction. Journal of Molecular Biology, 225: 1049-1 063, 1992.
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7,072 | 842 | A Network Mechanism for the Determination of
Shape-From-Texture
Ko Sakai and Leif H. Finkel
Department of Bioengineering and
Institute of Neurological Sciences
University of Pennsylvania
220 South 33rd Street, Philadelphia, PA 19104-6392
[email protected], [email protected]
Abstract
We propose a computational model for how the cortex discriminates
shape and depth from texture. The model consists of four stages: (1)
extraction of local spatial frequency, (2) frequency characterization, (3)
detection of texture compression by normalization, and (4) integration
of the normalized frequency over space. The model accounts for a
number of psychophysical observations including experiments based on
novel random textures. These textures are generated from white noise
and manipulated in Fourier domain in order to produce specific
frequency spectra. Simulations with a range of stimuli, including real
images, show qualitative and quantitative agreement with human
perception.
1 INTRODUCTION
There are several physical cues to shape and depth which arise from changes in projection
as a surface curves away from view, or recedes in perspective. One major cue is the
orderly change in the spatial frequency distribution of texture along the surface. In
machine vision approaches, various techniques such as Fourier transformation or wavelet
decomposition are used to determine spatial frequency spectra across a surface. The
determination of the transformation relating these spectra is a difficult problem, and
several techniques have been proposed such as an affine transformation (Super and Bovik
953
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Sakai and Finkel
1992) or a momentum method (Krumm and Shafer 1992). We address the question of
how a biological system which has access only to limited spatial frequency infonnation
and has constrained computational capabilities can nonetheless accurately detennine
shape and depth from texture. For example, the visual system might avoid the direct
comparison of frequency spectra themselves and instead rely on a simpler
characterization of the spectra such as the mean frequency, peak frequency, or the
gradient of a frequency component (Sakai and Finkel 1993; Turner, Gerstein, Bajcsy
1991). In order to study what frequency infonnation is actually utilized by humans, we
created novel random texture patterns and carried out psychophysical experiments with
these stimuli. These patterns are generated by manipulating the frequency components of
white noise stimuli in the Fourier domain so as to produce stimuli with exactly specified
frequency spectra. Based on these experiments, we propose a network mechanism for the
perception of shape-from-texture which takes into account physiological and anatomical
constraints as well as computational considerations.
2 MODEL FOR SHAPE FROM TEXTURE
The model consists of four major processes: extraction of the local spatial frequency at
each orientation, frequency characterization, detennination of texture compression by
frequency nonnalization, and the integration of the nonnalized frequency over space. A
schematic illustration of the model is shown in figure 1. Our psychophysical experiments
suggest that the visual system may use spatially averaged peak frequency for
characterizing the frequency distribution.
The change of surface orientation is
determined from the locally aligned compression of texture which is detected by
frequency normalization followed by lateral inhibition among different orientations.
Depth is then computed from the integration of the normalized frequency over space.
The model is implemented in feed-forward distributed networks and simulated using the
NEXUS neural network simulator (Sajda, Sakai, Yen and Finkel 1993).
3 MOTIVATION FOR EACH STAGE OF THE MODEL
The frequency extraction is carried out by units modeling complex cells in area VI.
These units have subunits with On and Off center difference of Gaussian(DOG) masks
tuned to specific frequencies and orientations. The units take local maximum of the
subunits. As in energy-based models (Bergen and Adelson 1989; Malik and Perona
1990), these units accomplish some major aspects of complex cell functions in the space
domain including invariance to the direction of contrast and spatial phase.
The second stage of the model extracts spatially averaged peak frequency. In order to
examine what frequency infonnation is actually utilized by humans, we created random
texture patterns with specific frequency spectra generated by manipulating the frequency
components of a white noise pattern in Fourier domain. Figure 2 shows a vertical
cylinder and a tilted perspective plane constructed by this technique from white noise.
We are able to see the three dimensional shape of the cylinder in (1). The stimuli were
constructed by making each frequency component undergo a step change at some
A Network Mechanism for the Determination of Shape-from-Texture
Early Vision
Stage
Frequency
CharacterizatIOn
Frequency
Normalization and
Lateral Inhibition
Integration
Figure 1. A schematic illustration of the shape-from-texture model consisting of four
major stages. The early vision stage models major spatial properties of complex cells in
order to decompose local spatial frequency. The second stage characterizes the
frequency by the spatially averaged peak frequency. The third stage detects locally
aligned texture compression by normalizing frequency and taking lateral inhibition
among orientation channels. The last stage determines 3D depth by integrating the
amount of texture compression - which corresponds to the local surface slant. Indices "f'
and "0" denote frequency and orientation channels, respectively. max, min, ave, and LI
stand for taking maximum, minimum, average, and lateral inhibition. The vertical bar
indicates that the function is processed independently within each of denoted channels.
955
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Sakai and Finkel
position along the cylinder; higher frequencies undergo the change at positions closer to
the cylinder's edges. Since the gradient of each frequency component is always either
zero or infinity, this suggests that gradients of individual frequency components over
space do not serve as a dominant cue for three dimensional shape perception. Similar
experiments have been conducted using various stimuli with controlled frequency
spectra. The results of these experiments suggest that averaged peak frequency is a
strong cue for the human perception of three dimensional shape and depth.
The third stage of the model normalizes local frequencies by the global lowest frequency
on the surface. We assume that the region containing the global lowest frequency is the
frontal plane standing vertically with respect to the viewer. One of the justifications for
this assumption can be seen in simple artificial images shown in figure 3. In both (l) and
(2), the bottom region looks vertical to us, and the planes above this region looks slanted,
although the patterns of the center region of (1) and the lower region of (2) are identical.
From a computational point of view, the normalization of frequency corresponds to an
approximation of the relation between local slant and spatial frequency. Depth, Z, as a
function of X (see figure 4) is given by:
Z(x) = JX tan { cos- I ( Fo ) }dx =
xo
F(x)
Jx
eq.(l)
Xo
where Fo is the global lowest frequency. Considering a boundary condition, Z(x) = 0, if
F(x) = Fo, the integrand can be reasonably approximated by (F(x) - Fo) I Fo . The second
stage of the model actually computes this value, and a later stage carries out the
integration.
Figure 2. Random texture patterns generated by manipulating the frequency components
of white noise in Fourier domain. A horizontal cylinder embedded in white noise (1) ,
and a tilted plane (2).
A Network Mechanism for the Determination of Shape-frorn-Texture
The second half of this stage detects the local alignment of texture compression. This
local alignment is detected by taking the lateral inhibition of normalized frequencies
among different orientations. Recent psychophysical experiments (Todd and Akerstrom
1987; Cumming, Johnson, and Parker 1993) show that the compression of texture in a
single orientation is a cue for the perception of shape-from-texture. We can confirm this
result from figure 5. Three images on the top of this figure have compression in a single
orientation, but those on the bottom do not. We clearly see smooth three dimensional
ellipsoids from the top images but not from the bottom images.
The last stage of the model computes the integral of the nonnalized frequency in order to
obtain depth. This integration begins from the region with lowest spatial frequency and
follows the path of the local steepest descent in spatial frequency .
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?? ?????????
?????????
?????????
??????????
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Figure 3. Objects consist of three planes(left), and two planes(right). In both stimuli, the
bottom regions look vertical to us, and the planes above this region look slanted, although
the patterns of the center region of (1) and the lower region of (2) are identical.
Depth: Z(x)
Z(X-V
Xo
x
Figure 4. The coordinate system for the equation (1). Depth, Z, is given as a function of
position, X.
957
958
Sakai and Finkel
4 SIMULATIONS
A quantitative test of the model was carried out by constructing ellipsoids with different
eccentricities and texture patterns shown in figure 5. Results are plotted in figure 6. For
the regular ellipsoids, there is a linear relation between real depth and that determined by
the model. This linear relation agrees with psychophysical experiments (Todd and
Akerstrom 1987; Biilthoff 1991) showing similar human performance for such stimuli.
All of the irregular texture patterns produced little perception of depth, in agreement with
human performance.
Many artificial and real images have been tested with the model and show good
agreement with human perception. For an example, a real image of a part of cantaloupe,
and its computed depth are shown in figure 7. Real images were obtained with a CCD
camera and were input to NEXUS via an Imaging Technology's S151 image processor.
et!~,
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Figure 5. (Top) Regular ellipsoids with eccentricities of 1,2, and 4. (Bottom) Irregular
texture patterns: (left) no compression with regular density change, (middle) randomly
oriented regular compression, (right) pan-orientational regular compression.
A Network Mechanism for the Determination of Shape-from-Texture
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6
Eccentricity
Figure 6. Depth perceived by the model as a function of actual eccentricity. The
simulated depth of regular ellipsoids shows a linear relation to the actual depth. Irregular
patterns produced little depth, in agreement with human perception.
Figure 7. An example of the model's response to a real image. A part of cantaloupe (left),
and its depth computed by the model(right).
5 CONCLUSIONS
(1) We propose a biologically-based network model of shape-from-texture based on the
determination of change in spatial frequency.
(2) Preliminary psychophysical evidence suggests that the spatially averaged peak
frequency is employed to characterize the spatial frequency distribution rather than using
a frequency spectrum or each component of frequency.
959
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Sakai and Finkel
(3) This characterization is validated by psychophysical experiments using novel random
textures with specified frequency spectra. The patterns are generated from white noise
and manipulated in Fourier domain in order to realize specific frequency characteristics.
(4) The model has been tested with a number of artificial stimuli and real images taken
by video camera. Responses show qualitative and quantitative agreements with human
perception.
Acknowledgments
This work is supported by grants from The Office of Naval Research (NOOOI4-90-J-1864,
NOOOI4-93-1-0681), The Whitaker Foundation, and The McDonnell-Pew Program in
Cognitive Neuroscience.
References
Super, B.J. and Bovik, A.C. (1992), Shape-from-texture by wavelet-based measurement
of local spectral moments. Proc. IEEE CVPR 1992, p296-300
Krumm, J. and Shafer, S.A. (1992), Shape from periodic texture using the spectrogram.
Proc. IEEE CVPR 1992, p284-289
Sakai, K. and Finkel, L.H. (1994), A cortical mechanism underlying the perception of
shape-from-texture. In F.Eeckman, et al.(ed.), Computation and Neural Systems 1993 ,
Norwell, MA: Kluwer Academic Publisher [in press]
Sajda, P., Sakai, K., Yen, S-c., and Finkel, L.H. (1993), In Skrzypek, J. (ed.), Neural
Network Simulation Environments, Norwell, MA: Kluwer Academic Publisher[in press]
Bergen, J.R. and Adelson, E.H. (1988), Visual texture segmentation and early vision.
Nature, 333, p363-364
Malik, J. and Perona, P. (1990), Preattentive texture discrimination with early vision
mechanisms. J. Opt. Soc. Am., A Vol.7, No.5, p923-932
Cumming, B.G., Johnston, E.B., and Parker, A.J. (1993), Effects of different texture cues
on curved surfaces viewed stereoscopically. Vision Res. Vol.33, N05, p827-838
Todd, J. T. and Akerstrom, R.A. (1987), Perception of three-dimensional form from
patterns of optical texture. Journal of Experimental Psychology, vol. I 3, No.2, p242-255,
Turner, M.R., Gerstein, G.L., and Bajcsy, R. (1991), Underestimation of visual texture
slant by human observers: a model. Bioi. Cybern. 65, p215-226
Btilthoff, H.H. (1991), Shape from X: Psychophysics and computation. In Landy, M.S.,
et al.(ed.) Computational Models of Visual Processing, Cambridge, MA: MIT press,
p305-330
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7,073 | 843 | Robust Reinforcement Learning
Motion Planning
?
In
Satinder P. Singh'"
Department of Brain and Cognitive Sciences
Massachusetts Institute of Technology
Cambridge, MA 02139
[email protected]
Andrew G. Barto, Roderic Grupen, and Christopher Connolly
Department of Computer Science
University of Massachusetts
Amherst, MA 01003
Abstract
While exploring to find better solutions, an agent performing online reinforcement learning (RL) can perform worse than is acceptable. In some cases, exploration might have unsafe, or even catastrophic, results, often modeled in terms of reaching 'failure' states
of the agent's environment. This paper presents a method that uses
domain knowledge to reduce the number of failures during exploration. This method formulates the set of actions from which the
RL agent composes a control policy to ensure that exploration is
conducted in a policy space that excludes most of the unacceptable
policies. The resulting action set has a more abstract relationship
to the task being solved than is common in many applications of
RL. Although the cost of this added safety is that learning may
result in a suboptimal solution, we argue that this is an appropriate tradeoff in many problems. We illustrate this method in the
domain of motion planning.
"'This work was done while the first author was finishing his Ph.D in computer science
at the University of Massachusetts, Amherst.
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Singh, Barto, Grupen, and Connolly
An agent using reinforcement learning (Sutton et al., 1991; Barto et al., to appear)
(RL) to approximate solutions to optimal control problems has to search, or explore,
to improve its policy for selecting actions. Although exploration does not directly
affect performance (Moore & Atkeson, 1993) in off-line learning with a model of
the environment, exploration in on-line learning can lead the agent to perform
worse than is acceptable. In some cases, exploration might have unsafe, or even
catastrophic, results, often modeled in terms of reaching 'failure' states of the agent's
environment. To make on-line RL more practical, especially if it involves expensive
hardware, task-specific minimal levels of performance should be ensured during
learning, a topic not addressed by prior RL research.
Although the need for exploration cannot be entirely removed, domain knowledge
can sometimes be used to define the set of actions from which the RL agent composes
a control policy so that exploration is conducted in a space that excludes most of
the unacceptable policies. We illustrate this approach using a simulated dynamic
mobile robot in two different environments.
1
Closed-loop policies as actions
RL agents search for optimal policies in a solution space determined in part by
the set of actions available to the agent. With a few exceptions (e.g., Mahadevan
& Connell, 1990; Singh, 1992), researchers have formulated RL tasks with actions
that are primitive in the sense that they are low-level, are available in very state,
are executed open-loop, and last a single time-step. We propose that this is an
arbitrary, and self-imposed, restriction, and that in general the set of actions can
have a much more abstract relationship to the problem being solved. Specifically,
what are considered 'actions' by the RL algorithm can themselves be closed-loop
control policies that meet important subgoals of the task being solved.
In this paper, the following general advantages afforded by using closed-loop policies
as actions are demonstrated in the domain of motion planning:
1. It is possible to design actions to meet certain hard constraints so that RL
maintains acceptable performance while simultaneously improving performance over time.
2. It is possible to design actions so that the action space for the learning
problem has fewer dimensions than the actual dimension of the physical
action space.
The robustness and greatly accelerated learning resulting from the above factors
can more than offset the cost of designing the actions. However, care has to be
taken in defining the action space to ensure that the resulting policy space contains
at least one policy that is close to optimal.
2
RL with Dirichlet and Neumann control policies
The motion planning problem arises from the need to give an autonomous robot
the ability to plan its own motion, i.e., to decide what actions to execute in order
to achieve a task specified by initial and desired spatial arrangements of objects.
Robust Reinforcement Learning in Motion Planning
First consider geometric path planning, i.e., the problem of finding safe paths for a
robot with no dynamical constraints in an environment with stationary obstacles .
A safe path in our context is one that avoids all obstacles and terminates in a
desired configuration. Connolly (1992) has developed a method that generates safe
paths by solving Laplace's equation in configuration space with boundary conditions
determined by obstacle and goal configurations (also see, Connolly & Grupen, 1993).
Laplace's equation is the partial differential equation
n
V2ljJ
{j2ljJ
L {)x~ = 0,
i=l
(1)
I
whose solution is a harmonic function, ljJ, with no interior local minima. In practice,
a finite difference approximation to Equation 1 is solved numerically via Gauss Sidel
relaxation on a mesh over configuration space. Safe paths are generated by gradient
descent on the resulting approximate harmonic function. In the general motion
planning problem, we are interested in finding control policies that not only keep
the robot safe but also extremize some performance criterion, e.g., minimum time,
minimum jerk, etc.
Two types of boundary conditions, called Dirichlet and Neumann boundary conditions, give rise to two different harmonic functions , <I> D and <I> N, that generate different types of safe paths . Robots following paths generated from <I> D tend to be repelled perpendicularly from obstacles. In contrast, robots following paths generated
from <I>N tend to skirt obstacles by moving parallel to their boundaries. Although
the state space in the motion planning problem for a dynamic robot in a planar
environment is {x, x, y, if}, harmonic functions are derived in two-dimensional position space. These functions are inexpensive to compute (relative to the expense of
solving the optimal control problem) because they are independent of the robot dynamics and criterion of optimal control. The closed-loop control policies that follow
the gradient of the Dirichlet or Neumann solutions, respectively denoted 1rD and
1rN, are defined approximately as follows: 1rD(S) V<I>D(?), and 1rN(S) V<I>N(?),
where ? is the projection of the state s onto position space .1
=
=
Instead of formulating the motion planning problem as a RL task in which a control
policy maps states into primitive control actions, consider the formulation in which
a policy maps each state s to a mixing parameter k( s) so that the actual action is
: [1- k(S)]1rD(S) + [k(S)]1rN(S) , where 0 ~ k(s) ~ 1. Figure 1B shows the structure
of this kind of policy. In Appendix B, we present conditions guaranteeing that for
a robot with no dynamical constraints, this policy space contains only acceptable
policies, i.e., policies that cause the robot to reach the goal configuration without
contacting an obstacle. Although this guarantee does not strictly hold when the
robot has dynamical constraints, e.g., when there are bounds on acceleration, this
formulation still reduces the risk of unacceptable behavior.
3
Simulation Results
In this paper we present a brief summary of simulation results for the two environments shown in Figures 2A and 3A. See Singh (1993) for details. The first
1 In
practice, the gradients of the harmonic functions act as reference signals to a PDcontroller.
657
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Singh, Barto, Grupen, and Connolly
environment consists of two rooms connected by a corridor. The second environment is a horseshoe-shaped corridor. The mobile robot is simulated as a unit-mass
that can accelerate in any direction. The only dynamical constraint is a bound on
the maximum acceleration.
Q(state. action)
A
B
(s)
Policy 1
k(s)
State
(s)
Policy
1 - k(s)
X
?
X
Y
state
(s)
?
Y
k
PoliCy 2
Neumann
mixing
coefficient
(s)
Figure 1: Q-Iearning Network and Policy Structure. Panel A: 2-layer Connectionist
Network Used to Store Q-values. Network inversion was used to find the maximum
Q-value (Equation 2) at any state and the associated greedy action. The hidden
layer consists of radial-basis units. Panel B: Policy Structure. The agent has to learn
a mapping from state s to a mixing coefficient 0 < k( s) < 1 that determines the
proportion in which to mix the actions specifies by the pure Dirichlet and Neumann
policies.
The learning task is to approximate minimum time paths from every point inside
the environment to the goal region without contacting the boundary wall. A reinforcement learning algorithm called Q-Iearning (Watkins, 1989) (see Appendix A)
was used to learn the mixing function, k. Figure lA shows the 2-layer neural network architecture used to store the Q-values. The robot was trained in a series
of trials, each trial starting with the robot placed at a randomly chosen state and
ending when the robot enters the goal region. The points marked by stars in Figures 2A and 3A were the starting locations for which statistics were collected to
produce learning curves.
Figures 2B, 2C, 3A and 3B show three robot trajectories from two randomly chosen
start states; the black-filled circles mark the Dirichlet trajectory (labeled D), the
white-filled circles mark the Neumann trajectory (labeled N), and the grey-filled
circles mark the trajectory after learning (labeled Q). Trajectories are shown by
taking snapshots of the robot at every time step; the velocity of the robot can
be judged by the spacing between successive circles on the trajectory. Figure 2D
shows the mixing function for zero-velocity states in the two-room environment,
while Figure 3C shows the mixing function for zero velocity states in the horseshoe
environment. The darker the region, the higher the proportion of the Neumann
Robust Reinforcement Learning in Motion Planning
policy in the mixture. In the two-room environment, t.he agent learns to follow
the Neumann policy in the left-hand room and to follow the Dirichlet policy in the
right-hand room.
Figure 2E shows the average time to reach the goal region as a function of the
number of trials in the two-room environment. The solid-line curve shows the
performance of the Q-Iearning algorithm. The horizontal lines show the average
time to reach the goal region for the designated pure policies. Figure 3D similarly
presents the results for the horseshoe environment. Note that in both cases the
RL agent learns a policy that is better than either the pure Dirichlet or the pure
Neumann policies. The relative advantage of the learned policy is greater in the
two-room environment than in the horseshoe environment .
On the two-room environment we also compared Q-Iearning using harmonic functions, as described above, with Q-Iearning using primitive accelerations of the mobile
robot as actions. The results are summarized along three dimensions: a) speed of
learning: the latter system took more than 20,000 trials to achieve the same level
of performance achieved by the former in 100 trials, b) safety: the simulated robot
using the latter system crashed into the walls more than 200 times, and c) asymptotic performance: the final solution found by the latter system was 6% better than
the one found by the former.
4
Conclusion
Our simulations show how an RL system is capable of maintaining acceptable performance while simultaneously improving performance over time. A secondary motivation for this work was to correct the erroneous impression that the proper, if
not the only, way to formulate RL problems is with low-level actions. Experience
on large problems formulated in this fashion has contributed to the perception that
RL algorithms are hopelessly slow for real-world applications. We suggest that a
more appropriate way to apply RL is as a "component technology" that uses experience to improve on partial solutions that have already been found through either
analytical techniques or the cumulative experience and intuitions of the researchers
themselves. The RL framework is more abstract, and hence more flexible, than
most current applications of RL would lead one to believe. Future applications of
RL should more fully exploit the flexibility of the RL framework.
A
Q-learning
On executing action a in state
function is performed:
St
at time t, the following update on the Q-value
where R( St, a) is the payoff, 0 ::; I ::; 1 is the discount factor, and a is a learning
rate parameter. See Watkins (1989) for further details.
659
660
Singh, Barto, Grupen, and Connolly
A
*
*
*
*
*
*
*
*
*
.
-----------------GOAl
.
*
B
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7000
8000
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Q-Iearning
Neumann
Dirichlet
5000
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9000
18000
27000
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45000
Number of Trials
Figure 2: Results for the Two-Room Environment . Panel A: Two-Room Environment. The stars mark the starting locations for which statistics were computed .
Panel B: Sample Trajectories from one Starting Location. The black-filled circles
labeled D show a pure Dirichlet trajectory, the white-filled circles labeled N show a
pure Neumann trajectory, and the grey-filled circles labeled Q show the trajectory
after learning. The trajectories are shown by taking snapshots at every time step;
the velocity of the robot can be judged by the distance between successive points
on the trajectory. Panel C: Three Sample Trajectories from a Different Starting
Location. Panel D: Mixing Function Learned by the Q-Iearning Network for Zero
Velocity States. The darker the region the higher the proportion of the Neumann
policy in the resulting mixture. Panel E: Learning Curve. The curve plots the time
taken by the robot to reach the goal region, averaged over the locations marked
with stars in Panel A, as a function of the number of Q-Iearning trials. The dashed
line shows the average time using the pure Neumann policy; the dotted line for the
pure Dirichlet policy; and the solid line for Q-Iearning. The mixed policy formed
by Q-Iearning rapidly outperforms both pure harmonic function policies.
Robust Reinforcement Learning in Motion Planning
A
*
*
*
*
*
B
*
??
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? ??? ? M ?? ? ? M ???? _
? ? ? ? M . . . . . . . . . . . . . . . . ._
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Number of_Trials
Figure 3: Results for the Horseshoe Environment . Panel A: Horseshoe-Shaped
Environment. Locations marked with stars are the starting locations for which
statistics were computed. It also shows sample trajectories from one starting location; the black-filled circles marked D show a Dirichlet trajectory, the white-filled
circles marked N show a Neumann trajectory, and the grey-filled circles marked Q
show the trajectory after learning. The trajectories are shown by taking snapshots
at every time step; the velocity of the robot can be judged by the distance between
successive points on the trajectory. Panel B: Three Sample Trajectories from a
Different Starting Location. Panel C: Mixing Function Learned by the Q-Iearning
Network for Zero Velocity States. The darker the region the higher the proportion
of the Neumann policy in the resulting mixture. Panel D: Learning Curve. The
curve plots the time taken by the robot to reach the goal region, averaged over the
locations marked with stars in Panel A, as a function of the number of Q-Iearning
trials. The dashed line shows the average time for the pure Neumann policy; the
dotted line for the pure Dirichlet policy; and the solid line for Q-Iearning. Q-Iearning
rapidly outperforms both pure harmonic function policies.
661
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Singh, Barto, Grupen, and Connolly
B
Safety
Let L denote the surface whose gradients at any point are given by the closed-loop
policy under consideration. Then for there to be no minima in L, the gradient of L
should not vanish in the workspace, i.e., (1- k(S?\7<1>D(S) + k(S)\7<1>N(S) ;/; O. The
only way it can vanish is if 'Vi
k(s)
1- k(s)
(3)
where [?Ji is the ith component of vector [.J. The algorithm can explicitly check for
that possibility and prevent it. Alternatively, note that due to the finite precision
in any practical implementation, it is extremely unlikely that Equation 3 will hold
in any state. Also note that 7r( s) for any point s on the boundary will always point
away from the boundary because it is the convex sum of two vectors, one of which
is normal to the boundary, and the other of which is parallel to the boundary.
Acknowledgements
This work was supported by a grant ECS-9214866 to Prof. A. G. Barto from the
National Science Foundation, and by grants IRI-9116297, IRI-9208920, and CDA8922572 to Prof. R. Grupen from the National Science Foundation.
References
Barto, A.G., Bradtke, S.J., & Singh, S.P. (to appear). Learning to act using realtime dynamic programming. Artificial Intelligence.
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Mahadevan, S. & Connell, J. (1990). Automatic programming of behavior-based
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Moore, A.W. & Atkeson, C.G. (1993). Prioritized sweeping: Reinforcement learning
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Singh, S.P. (1993). Learning to Solve Markovian Decision Processes. PhD thesis,
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Technical Report 93-77.
Sutton, R.S ., Barto, A.G., & Williams, R.J. (1991). Reinforcement learning is direct
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Watkins, C.J.C.H. (1989). Learning from Delayed Rewards. PhD thesis, Cambridge
Univ ., Cambridge, England.
| 843 |@word trial:8 inversion:1 proportion:4 open:1 grey:3 simulation:3 solid:3 initial:1 configuration:5 contains:2 series:1 selecting:1 outperforms:2 current:1 mesh:1 plot:2 update:1 stationary:1 greedy:1 fewer:1 intelligence:1 ith:1 location:10 successive:3 height:1 unacceptable:3 along:1 direct:1 differential:1 corridor:2 symposium:1 grupen:8 consists:2 inside:1 behavior:2 themselves:2 planning:11 brain:1 actual:2 panel:13 mass:1 what:2 kind:1 developed:1 finding:2 guarantee:1 every:4 act:2 iearning:15 ensured:1 control:14 unit:2 grant:2 appear:2 safety:3 local:1 sutton:2 meet:2 path:9 approximately:1 might:2 black:3 averaged:2 practical:2 practice:2 projection:1 radial:1 suggest:1 cannot:1 close:1 interior:1 onto:1 judged:3 context:1 risk:1 restriction:1 imposed:1 demonstrated:1 center:1 map:2 primitive:3 iri:2 starting:8 williams:1 convex:1 formulate:1 pure:12 his:1 autonomous:1 laplace:2 programming:2 us:2 designing:1 velocity:7 expensive:1 labeled:6 solved:4 enters:1 region:9 connected:1 removed:1 intuition:1 environment:22 reward:1 dynamic:4 trained:1 singh:11 solving:2 division:1 basis:1 accelerate:1 univ:1 artificial:1 whose:2 solve:1 ability:1 statistic:3 final:1 online:1 advantage:2 analytical:1 took:1 propose:1 loop:6 rapidly:2 mixing:8 flexibility:1 achieve:2 elemental:1 neumann:17 produce:1 guaranteeing:1 executing:1 object:1 illustrate:2 andrew:1 involves:1 direction:1 safe:6 correct:1 exploration:8 wall:2 exploring:1 strictly:1 hold:2 considered:1 normal:1 mapping:1 mit:1 always:1 reaching:2 mobile:3 barto:9 derived:1 finishing:1 check:1 greatly:1 contrast:1 sense:1 unlikely:1 cmpsci:1 hidden:1 interested:1 flexible:1 denoted:1 plan:1 spatial:1 shaped:2 future:1 connectionist:1 report:2 intelligent:1 few:1 perpendicularly:1 randomly:2 simultaneously:2 national:2 delayed:1 possibility:1 mixture:3 capable:1 partial:2 experience:3 filled:9 desired:2 circle:10 skirt:1 minimal:1 obstacle:6 markovian:1 formulates:1 cost:2 connolly:9 conducted:2 st:2 international:1 amherst:2 workspace:1 off:1 thesis:2 worse:2 cognitive:1 american:1 star:5 summarized:1 coefficient:2 explicitly:1 vi:1 performed:1 closed:5 start:1 maintains:1 parallel:2 formed:1 trajectory:20 researcher:2 composes:2 reach:5 failure:3 inexpensive:1 associated:1 massachusetts:4 knowledge:2 oooo:1 cj:2 higher:3 follow:3 planar:1 ooo:1 formulation:2 done:1 execute:1 hand:2 horizontal:1 christopher:1 believe:1 former:2 hence:1 moore:2 white:3 during:2 self:1 yorktown:1 criterion:2 impression:1 motion:11 bradtke:1 roderic:1 harmonic:10 consideration:1 common:1 rl:22 physical:1 ji:1 subgoals:1 he:1 numerically:1 cambridge:3 rd:3 automatic:1 similarly:1 moving:1 robot:25 surface:1 etc:1 own:1 store:2 certain:1 watson:1 minimum:5 greater:1 care:1 signal:1 dashed:2 mix:1 reduces:1 technical:2 england:1 sometimes:1 robotics:2 achieved:1 spacing:1 addressed:1 tend:2 mahadevan:2 jerk:1 affect:1 architecture:1 suboptimal:1 reduce:1 tradeoff:1 cause:1 action:25 discount:1 ph:1 hardware:1 extremize:1 generate:1 specifies:1 dotted:2 prevent:1 excludes:2 relaxation:1 sum:1 decide:1 realtime:1 decision:1 acceptable:5 appendix:2 entirely:1 bound:2 layer:3 constraint:5 afforded:1 generates:1 speed:1 extremely:1 formulating:1 connell:2 performing:1 department:3 designated:1 terminates:1 psyche:1 taken:3 equation:6 available:2 apply:1 away:1 appropriate:2 robustness:1 dirichlet:13 ensure:2 maintaining:1 exploit:1 especially:1 prof:2 added:1 arrangement:1 contacting:2 already:1 gradient:5 distance:2 simulated:3 topic:1 argue:1 collected:1 modeled:2 relationship:2 executed:1 expense:1 repelled:1 rise:1 design:2 implementation:1 proper:1 policy:44 perform:2 contributed:1 snapshot:3 finite:2 horseshoe:6 descent:1 defining:1 payoff:1 rn:3 arbitrary:1 sweeping:1 crashed:1 specified:1 learned:3 unsafe:2 dynamical:4 perception:1 improve:2 technology:2 brief:1 ljj:1 prior:1 geometric:1 acknowledgement:1 relative:2 asymptotic:1 fully:1 mixed:1 foundation:2 agent:12 ibm:1 summary:1 placed:1 last:1 supported:1 institute:1 taking:3 boundary:9 dimension:3 curve:6 ending:1 avoids:1 world:1 cumulative:1 author:1 reinforcement:10 adaptive:1 atkeson:2 ec:1 approximate:3 keep:1 satinder:1 robotic:1 alternatively:1 search:2 learn:2 transfer:1 robust:4 ca:1 composing:1 improving:2 domain:4 motivation:1 fashion:1 darker:3 slow:1 ny:1 precision:1 position:2 vanish:2 watkins:3 learns:2 erroneous:1 specific:1 offset:1 sequential:1 phd:2 boston:1 explore:1 hopelessly:1 determines:1 ma:3 goal:10 formulated:2 marked:7 acceleration:3 prioritized:1 room:10 hard:1 determined:2 specifically:1 called:2 secondary:1 catastrophic:2 gauss:1 la:1 exception:1 mark:4 latter:3 arises:1 accelerated:1 ex:1 |
7,074 | 844 | Complexity Issues in Neural
Computation and Learning
V. P. Roychowdhnry
School of Electrical Engineering
Purdue University
West Lafayette, IN 47907
Email: [email protected]
K.-Y. Sin
Dept.. of Electrical & Compo Engr.
U ni versit.y of California at Irvine
Irvine, CA 92717
Email: [email protected]
The general goal of this workshop was to bring t.ogether researchers working toward
developing a theoretical framework for the analysis and design of neural networks.
The t.echnical focus of the workshop was to address recent. developments in understanding the capabilities and limitations of variolls modds for neural computation
and learning. The primary topics addressed the following three areas: 1) Computational complexity issues in neural networks, 2) Complexity issues in learning,
and 3) Convergence and numerical properties of learning algorit.hms. Other topics included experiment.al/simulat.ion results on neural llet.works, which seemed to
pose some open problems in the areas of learning and generalizat.ion properties of
feedforward networks.
The presentat.ions and discussions at the workshop highlighted the int.erdisciplinary
nature of research in neural net.works . For example, several of the present.at.ions
discussed recent contributions which have applied complexity-theoretic techniques
to characterize the computing power of neural net.works, t.o design efficient neural
networks, and t.o compare the computational capabilit.ies of neural net.works wit.h
that. of convent.ional models for comput.ation . Such st.udies, in t.urn, have generated
considerable research interest. among computer scient.ists, as evidenced by a significant number of research publications on related topics . A similar development can
be observed in t.he area of learning as well: Techniques primarily developed in the
classical theory of learning are being applied to understand t.he generalization and
learning characteristics of neural networks. In [1, 2] attempts have been made to integrate concept.s from different areas and present a unifie(i treatment of the various
results on the complexity of neural computation ancllearning. In fact, contributions
from several part.icipants in the workshop are included in [2], and interested readers
could find det.ailed discussions of many of the n-~sults IHesented at t.he workshop in
[2] .
Following is a brief descriptioll of the present.ations, along with the Hames and email addresses of the speakers. W. Maass (maa.~.~@igi . tu-gT?(Jz.(!(" . at) and A . Sakurai
([email protected].,ip) made preseutatiol1s Oll tlw VC-dimension and
t.he comput.ational power of feedforwarcl neural net.works . Many neural net.s of depth
3 (or larger) with linear threshold gat.es have a VC-dimf'usion t.hat. is superlinear in
t.he number of weights of the net. The talks presPllted llPW results which establish
1161
1162
Roychowdhury and Siu
effective upper bounds and almost. t.ight lower boun(ls on t.he VC-dimension of
feedforward networks with various activation functions including linear threshold
and sigmoidal functions. Such nonlinear lower bounds on t.he VC-dimension were
also discussed for networks with bot.h integer and rea.l weights . A presentation
by G. Turan (@VM.CC.PURDUE.EDU:Ul1557@UICVM) discussed new result.s on
proving lower bounds on t.he size of circuits for comput.ing specific Boolean functions
where each gate comput.es a real-valued function. In particular the results provide
a lower bound for t.he size of formulas (i.e., circuit.s wit.h fan-out 1) of polynomial
gates, computing Boolean func.t.ions in t.he sensp. of sign-representation.
The presentations on learning addressed both sample allli algorithmic complexity.
The t.alk by V. Cast.elli ([email protected]) and T. Cover st.udip.d the role of
labeled and unlabeled samples in pat.tern recognit.ion. Let. samples be chosen from
two populations whose distribut.ions are known, and ld the proport.ion (mixing parameter) of the two classes be unknown. Assume t.hat a t.raining set composed of
independent observations from the t.wo classes is given, where part. of the samples
are classified and part are not. The talk present.ed new rt~sults which investigate the
relative value of the labeled and unlabeled samples in reducing the probability of
error of the classifier. In particular, it was shown that. uuder the above hypotheses
t.he relative value of labeled and unlabeled samples is proportional t.o the (Fisher)
Informat.ion they carry about, the unknown mixing parameter. B. Dasgupta ([email protected]), on the othE'r hand, addressed tlw issue of the trad.ability of
the t.raining problem of neural net.works. New rp.sults showing tha.t. the training
problem remains NP-complete when the act.iva.t.ion functions are piecewise linear
were presented.
The talk by B. Hassibi ([email protected]/oni.uill.) provided a minimax interpretation of instant.aneous-gradient-based learning algorit.hms such as LMS and backpropagation. When t.he underlying model is linear, it was shown t.hat the LMS
algorithm minimizes the worst C3.<;e ratio of pl'f~ clicted error energy to disturbance
energy. When the model is nonlinear, which arises in t.hE' contp.xt. of neural net.works,
it was shown that t.he backpropagation algorithm performs this minimizat.ion in a
local sense. These results provide theoretical justificat.ioll for the widely observed
excellent robustness properties of the LMS and backpropagatioll algorithms.
The last. t.alk by R. Caruana ([email protected]'.CMU.EDU) presented a set.
of int.eresting empirical results on the learning properties of neural networks of
different sizes. Some of the issues (based on empirical evidence) raised during
the talk are: 1) If cross-validation is used to prevent overt.raining, excess capacity
rarely reduces the generalization performance of fully connected feed-forward backpropagation net.works. 2) Moreover, too little capacity usn ally hurt.s generalization
performance more than too much capacit.y.
References
[1] K.-Y . Siu, V. P. Roychowdhnry, and T. Kailath. Di.H:r'fi(; Nfllml Computation:
A Theordical Foundation. Englewood Cliffs, N.1: Prent.ice-H all , 1994.
[2] V. P. Roychowdhury, K.-Y. Siu, and A. Orlitsky, edit.ors. ThwT'(;tical Advances
in N(;uT'ai Compltiation and LUlT'Tl.ing. Bost.on: Kluwer Academic Publishers,
1994.
| 844 |@word establish:1 c:1 concept:1 boun:1 polynomial:1 classical:1 thwt:1 open:1 maass:1 primary:1 vc:4 rt:1 eng:1 sin:1 during:1 gradient:1 hitachi:1 speaker:1 ld:1 carry:1 capacity:2 generalization:3 topic:3 theoretic:1 complete:1 toward:1 minimizat:1 performs:1 pl:1 bring:1 bost:1 activation:1 ratio:1 fi:1 algorithmic:1 lm:3 numerical:1 ecn:1 discussed:3 he:14 overt:1 interpretation:1 kluwer:1 design:2 unknown:2 significant:1 upper:1 observation:1 edit:1 ai:1 purdue:3 compo:1 pat:1 ional:1 sigmoidal:1 along:1 gt:1 vwani:1 publication:1 evidenced:1 recent:2 focus:1 cast:1 c3:1 california:1 sense:1 address:2 little:1 echnical:1 ight:1 provided:1 interested:1 underlying:1 moreover:1 circuit:2 generalizat:1 issue:5 among:1 distribut:1 reduces:1 ing:2 development:2 minimizes:1 raised:1 academic:1 developed:1 turan:1 iva:1 scient:1 cross:1 disturbance:1 minimax:1 y:1 sults:3 brief:1 stan:1 act:1 orlitsky:1 power:2 hm:2 cmu:1 classifier:1 np:1 piecewise:1 ailed:1 primarily:1 func:1 understanding:1 ion:11 composed:1 rea:1 ice:1 relative:2 engineering:1 local:1 addressed:3 fully:1 io:1 publisher:1 limitation:1 proportional:1 cliff:1 attempt:1 interest:1 validation:1 englewood:1 investigate:1 foundation:1 including:1 integrate:1 umn:1 integer:1 co:1 feedforward:2 uill:1 lafayette:1 tical:1 usion:1 ioll:1 last:1 backpropagation:3 understand:1 det:1 area:4 empirical:2 theoretical:2 alk:2 dimension:3 depth:1 raining:3 wo:1 seemed:1 boolean:2 elli:1 forward:1 cover:1 made:2 superlinear:1 unlabeled:3 sakurai:1 caruana:1 ations:1 excess:1 siu:4 too:2 l:1 characterize:1 wit:2 st:2 roychowdhury:2 bot:1 sign:1 nature:1 proving:1 population:1 vm:1 jz:1 hurt:1 dasgupta:2 ca:1 othe:1 threshold:2 excellent:1 capacit:1 hypothesis:1 prevent:1 sp:1 labeled:3 observed:2 role:1 electrical:2 west:1 worst:1 algorit:2 int:2 almost:1 reader:1 connected:1 tl:1 igi:1 ational:1 hassibi:2 informat:1 comput:4 complexity:6 bound:4 capability:1 fan:1 convent:1 formula:1 engr:1 contribution:2 specific:1 xt:1 ni:1 showing:1 characteristic:1 oll:1 evidence:1 workshop:5 various:2 talk:4 urn:1 gat:1 researcher:1 cc:1 effective:1 classified:1 lwd:1 recognit:1 developing:1 oni:1 ed:1 email:3 whose:1 larger:1 valued:1 stanford:1 widely:1 energy:2 simulat:1 ability:1 tlw:2 di:1 maa:1 highlighted:1 irvine:2 ch:1 ip:1 treatment:1 tha:1 remains:1 net:9 car:1 ut:1 goal:1 presentation:2 kailath:1 tu:1 uci:1 feed:1 fisher:1 considerable:1 included:2 mixing:2 reducing:1 usn:1 e:2 robustness:1 convergence:1 hand:1 working:1 ally:1 gate:2 llet:1 hat:3 nonlinear:2 rp:1 rarely:1 tern:1 arises:1 pose:1 instant:1 dept:1 school:1 ation:1 |
7,075 | 845 | Developing Population Codes By
Minimizing Description Length
Richard S. Zemel
CNL, The Salk Institute
10010 North Torrey Pines Rd.
La J oUa, CA 92037
Geoffrey E. Hinton
Department of Computer Science
University of Toronto
Toronto M5S 1A4 Canada
Abstract
The Minimum Description Length principle (MDL) can be used to
train the hidden units of a neural network to extract a representation that is cheap to describe but nonetheless allows the input to
be reconstructed accurately. We show how MDL can be used to
develop highly redundant population codes. Each hidden unit has
a location in a low-dimensional implicit space. If the hidden unit
activities form a bump of a standard shape in this space, they can
be cheaply encoded by the center ofthis bump. So the weights from
the input units to the hidden units in an autoencoder are trained
to make the activities form a standard bump. The coordinates of
the hidden units in the implicit space are also learned, thus allowing flexibility, as the network develops a discontinuous topography
when presented with different input classes. Population-coding in
a space other than the input enables a network to extract nonlinear
higher-order properties of the inputs.
Most existing unsupervised learning algorithms can be understood using the Minimum Description Length (MDL) principle (Rissanen, 1989). Given an ensemble
of input vectors, the aim of the learning algorithm is to find a method of coding
each input vector that minimizes the total cost, in bits, of communicating the input
vectors to a receiver. There are three terms in the total description length:
? The code-cost is the number of bits required to communicate the code
that the algorithm assigns to each input vector.
11
12
Zemel and Hinton
? The model-cost is the number of bits required to specify how to reconstruct input vectors from codes (e.g., the hidden-to-output weights) .
? The reconstruction-error is the number of bits required to fix up any
errors that occur when the input vector is reconstructed from its code.
Formulating the problem in terms of a communication model allows us to derive an
objective function for a network (note that we are not actually sending the bits).
For example, in competitive learning (vector quantization), the code is the identity
of the winning hidden unit, so by limiting the system to 1i units we limit the
average code-cost to at most log21i bits. The reconstruction-error is proportional
to the squared difference between the input vector and the weight-vector of the
winner, and this is what competitive learning algorithms minimize. The model-cost
is usually ignored.
The representations produced by vector quantization contain very little information
about the in put (at most log21i bits). To get richer representations we must allow
many hidden units to be active at once and to have varying activity levels. Principal
components analysis (PCA) achieves this for linear mappings from inputs to codes.
It can be viewed as a version of MDL in which we limit the code-cost by only
having a few hidden units, and ignoring the model-cost and the accuracy with which
the hidden activities must be coded. An autoencoder (see Figure 2) that tries to
reconstruct the input vector on its output units will perform a version of PCA if the
output units are linear. We can obtain novel and interesting unsupervised learning
algorithms using this MDL approach by considering various alternative methods of
communicating the hidden activities. The algorithms can all be implemented by
backpropagating the derivative of the code-cost for the hidden units in addition to
the derivative of the reconstruction-error backpropagated from the output units.
Any method that communicates each hidden activity separately and independently
will tend to lead to factorial codes because any mutual information between hidden
units will cause redundancy in the communicated message, so the pressure to keep
the message short will squeeze out the redundancy. In (Zemel, 1993) and (Hinton
and Zemel, 1994), we present algorithms derived from this MDL approach aimed
at developing factorial codes. Although factorial codes are interesting, they are not
robust against hardware failure nor do they resemble the population codes found in
some parts of the brain. Our aim in this paper is to show how the MDL approach
can be used to develop population codes in which the activities of hidden units are
highly correlated. For a more complete discussion of the details of this algorithm,
see (Zemel, 1993).
Unsupervised algorithms contain an implicit assumption about the nature of the
structure or constraints underlying the input set. For example, competitive learning
algorithms are suited to datasets in which each input can be attributed to one of
a set of possible causes. In the algorithm we present here, we assume that each
input can be described as a point in a low-dimensional continuous constraint space.
For instance, a complex shape may require a detailed representation, but a set of
images of that shape from multiple viewpoints can be concisely represented by first
describing the shape, and then encoding each instance as a point in the constraint
space spanned by the viewing parameters. Our goal is to find and represent the
constraint space underlying high-dimensional data samples.
Developing Population Codes by Minimizing Description Length
size
?
?
?
?
?
?
?
?
?x.
? ?
??
?
?
?
?
?
?
?
?
?
orientation
Figure 1: The population code for an instance in a two-dimensional implicit space.
The position of each blob corresponds to the position of a unit within the population,
and the blob size corresponds to the unit's activity. Here one dimension describes
the size and the other the orientation of a shape. We can determine the instantiation
parameters of this particular shape by computing the center of gravity of the blob
activities, marked here by an "X".
1
POPULATION CODES
In order to represent inputs as points drawn from a constraint space, we choose
a population code style of representation. In a population code, each code unit is
associated with a position in what we call the implicit space, and the code units'
pattern of activity conveys a single point in this space. This implicit space should
correspond to the constraint space. For example, suppose that each code unit
is assigned a position in a two-dimensional implicit space, where one dimension
corresponds to the size of the shape and the second to its orientation in the image
(see Figure 1). A population of code units broadly-tuned to different positions can
represent any particular instance of the shape by their relative activity levels.
This example illustrates that population codes involve three quite different spaces:
the input-vector space (the pixel intensities in the example); the hidden-vector space
(where each hidden, or code unit entails an additional dimension); and this third,
low-dimensional space which we term the implicit space. In a learning algorithm
for population codes, this implicit space is intended to come to smoothly represent
the underlying dimensions of variability in the inputs, i.e., the constraint space.
For instance, the Kohonen (1982) algorithm defines the implicit space topology
through fixed neighborhood relations, and the algorithm then manipulates hiddenvector space so that neighbors in implicit space respond to similar inputs.
This form of coding has several computational advantages, in addition to its significance due to its prevalence in biological systems. Population codes contain some
redundancy and hence have some degree of fault-tolerance, and they reflect underlying structure of the input, in that similar inputs are mapped to nearby implicit
positions. They also possess a hyperacuity property, as the number of implicit
positions that can be represented far exceeds the number of code units.
13
14
Zemel and Hinton
2
LEARNING POPULATION CODES WITH MDL
Autoencoders are a general way of addressing issues of coding, in which the hidden
unit activities for an input are the codes for that input which are produced by the
input-hidden weights, and in which reconstruction from the code is done by the
hidden-output mapping. In order to allow an autoencoder to develop population
codes for an input set, we need some additional structure in the hidden layer that
will allow a code vector to be interpreted as a point in implicit space. While most
topographic-map formation algorithms (e.g., the Kohonen and elastic net (Durbin
and Willshaw, 1987) algorithms) define the topology of this implicit space by fixed
neighborhood relations, in our algorithm we use a more explicit representation.
Each hidden unit has weights coming from the input units that determine its activity
level. But in addition to these weights, it has another set of adjustable parameters
that represent its coordinates in the implicit space. To determine what implicit
position is represented by a vector of hidden activities, we can average together the
implicit coordinates of the hidden units, weighting each coordinate vector by the
activity level of the unit.
Suppose, for example, that each hidden unit is connected to an 8x8 retina and has
2 implicit coordinates that represent the size and orientation of a particular kind of
shape on the retina, as in our earlier example. If we plot the hidden activity levels
in the implicit space (not the input space), we would like to see a bump of activity
of a standard shape (e.g., a Gaussian) whose center represents the instantiation
parameters of the shape (Figure 2 depicts this for a 1D implicit space). If the
activities form a perfect Gaussian bump of fixed variance we can communicate
them by simply communicating the coordinates of the mean of the Gaussian; this
is very economical if there are many less implicit coordinates than hidden units.
It is important to realize that the activity of a hidden unit is actually caused by the
input-to-hidden weights, but by setting these weights appropriately we can make
the activity match the height under the Gaussian in implicit space. If the activity
bump is not quite perfect, we must also encode the bump-error-the misfit between
the actual activity levels and the levels predicted by the Gaussian bump. The
cost of encoding this misfit is what forces the activity bump in implicit space to
approximate a Gaussian.
The reconstruction-error is then the deviation of the output from the input. This
reconstruction ignores implicit space; the output activities only depend on the vector
of hidden activities and weights.
2.1
The objective function
Currently, we ignore the model-cost, so the description length to be minimized is:
Et
Bt
+ Rt
N
1?
I)bj j=l
bj)2 /2VB
+ L(a~ -
c~)2 /2VR
(1)
k=l
where a, b, c are the activities of units in the input, hidden, and output layers,
respectively, VB and VR are the fixed variances of the Gaussians used for coding the
Developing Population Codes by Minimizing Description Length
NETWOHI<
IMPLICIT SPACE (1/
= 1)
Output
Activity (b)
(1...N)
II}
,;/J
beRt-fit
...
IIidden
(l...H)
Inpllt.
~
Gaussian
- ------~
I;
0 () ... () 0
(l...N)
.l
Xl
JJ
x\
X(j
-
i7
X2
Xi
I:
I:
I:
I:
I'
Ii
X8
X7
Posit ion (x)
.'
X~
Figure 2: Each of the 1t hidden units in the autoencoder has an associated position
of each hidden
in implicit space. Here we show a ID implicit space. The activity
unit j on case t is shown by a solid line. The network fits the best Gaussian to this
pattern of activity in implicit space. The predicted activity, h;, of unit j under this
h;
Gaussian is based on the distance from Xj to the mean j..lt; it serves as a target for
hj.
bump-errors and the reconstruction-errors, and the other symbols are explained in
the caption of Figure 2.
h;,
We compute the actual activity of a hidden unit,
as a normalized exponential
1
of its total input. Note that a unit's actual activity is independent of its position
in implicit space. Its expected activity is its normalized value under the predicted
Gaussian bump:
1{.
hj
= exp( -(Xj - j..lt)2 /2(7'2)/
L exp( -(xi -
j..lt)2/2(7'2)
(2)
i=l
where (7' is the width of the bump, which we assume for now is fixed throughout
training.
We have explored several methods for computing the mean of this bump. Simply
computing the center of gravity of the representation units' positions, weighted
by their activity, produces a bias towards points in the center of implicit space.
Instead, on each case, a separate minimization determines j..lt; it is the position in
implicit space that minimizes Bt given {Xj' hj} . The network has full inter-layer
connectivity, and linear output units. Both the network weights and the implicit
coordinates of the hidden units are adapted to minimize E.
1b~ = exp(net~)/ 2::::1 exp(net~), where net~ is the net input into unit j on case t.
15
16
Zemel and Hinton
Unit 18 - Epoch 0
Unit 18 - Epoch 23
0. 08
0.2
0. 06
0.15
ActivityO.04
Activity 0. 1
0.05
Yposition
x posi tion
10
y position
Xposition
10
Figure 3: This figure shows the receptive field in implicit space for a hidden unit.
The left panel shows that before learning, the unit responds randomly to 100 different test patterns, generated by positioning a shape in the image at each point in a
10xlO grid. Here the 2 dimensions in implicit space correspond to x and y positions.
The right panel shows that after learning, the hidden unit responds to objects in
a particular position, and its activity level falls off smoothly as the object position
moves away from the center of the learned receptive field.
3
EXPERIMENTAL RESULTS
In the first experiment, each 8x8 real-valued input image contained an instance of a
simple shape in a random (x, y)-position. The network began with random weights,
and each of 100 hidden units in a random 2D implicit position; we trained it using
conjugate gradient on 400 examples. The network converged after 25 epochs. Each
hidden unit developed a receptive field so that it responded to inputs in a limited
neighborhood that corresponded to its learned position in implicit space (see Figure
3). The set of hidden units covered the range of possible positions.
In a second experiment, we also varied the orientation of the shape and we gave
each hidden unit three implicit coordinates. The network converged after 60 epochs
of training on 1000 images. The hidden unit activities formed a population code
that allowed the input to be accurately reconstructed.
A third experiment employed a training set where each image contained either a
horizontal or vertical bar, in some random position. The hidden units formed an
interesting 2D implicit space in this case: one set of hidden units moved to one
corner of the space, and represented instances of one shape, while the other group
moved to an opposite corner and represented the other (Figure 4). The network
was thus able to squeeze a third dimension (i.e., which shape) into the 2D implicit
space. This type of representation would be difficult to learn in a Kohonen network;
the fact that the hidden units learn their implicit coordinates allows more flexibility
than a system in which these coordinates are fixed in advance.
Developing Population Codes by Minimizing Description Length
Implicit Spare (Epoch 0)
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Figure 4: This figure shows the positions of the hidden units and the means in the 2D
implicit space before and after training on the horizontal/vertical task. The means
in the top right of the second plot all correspond to images containing vertical bars,
while the other set correspond to horizontal bar images. Note that some hidden
units are far from all the means; these units do not playa role in the coding of the
input, and are free to be recruited for other types of input cases.
4
RELATED WORK
This new algorithm bears some similarities to several earlier algorithms. In the
experiments presented above, each hidden unit learns to act as a Radial Basis
Function (RBF) unit. Unlike standard RBFs, however, here the RBF activity serves
as a target for the activity levels, and is determined by distance in a space other
than the input space.
Our algorithm is more similar to topographic map formation algorithms, such as the
Kohonen and elastic-net algorithms. In these methods, however, the populationcode is in effect formed in input space. Population coding in a space other than
the input enables our networks to extract nonlinear higher-order properties of the
inputs.
In (Saund, 1989), hidden unit patterns of activity in an autoencoder are trained to
form Gaussian bumps, where the center of the bump is intended to correspond to
the position in an underlying dimension of the inputs. In addition to the objective
functions being quite different in the two algorithms, another crucial difference
exists: in his algorithm, as well as the other earlier algorithms, the implicit space
topology is statically determined by the ordering of the hidden units, while units in
our model learn their implicit coordinates.
17
18
Zemel and Hinton
5
CONCLUSIONS AND CURRENT DIRECTIONS
We have shown how MDL can be used to develop non-factorial, redundant representations. The objective function is derived from a communication model where
rather than communicating each hidden unit activity independently, we instead
communicate the location of a Gaussian bump in a low-dimensional implicit space.
If hidden units are appropriately tuned in this space their activities can then be
inferred from the bump location.
Our method can easily be applied to networks with multiple hidden layers , where
the implicit space is constructed at the last hidden layer before the output and
derivatives are then backpropagated; this allows the implicit space to correspond
to arbitrarily high-order input properties. Alternatively, instead of using multiple
hidden layers to extract a single code for the input, one could use a hierarchical
system in which the code-cost is computed at every layer.
A limitation of this approach (as well as the aforementioned approaches) is the
need to predefine the dimensionality of implicit space. We are currently working
on an extension that will allow the learning algorithm to determine for itself the
appropriate number of dimensions in implicit space. We start with many dimensions
but include the cost of specifying j-tt in the description length. This obviously
depends on how many implicit coordinates are used. If all of the hidden units have
the same value for one of the implicit coordinates, it costs nothing to communicate
that value for each bump. In general, the cost of an implicit coordinate depends
on the ratio between its variance (over all the different bumps) and the accuracy
with which it must be communicated. So the network can save bits by reducing
the variance for unneeded coordinates. This creates a smooth search space for
determining how many implicit coordinates are needed.
Acknowledgements
This research was supported by grants from NSERC , the Ontario Information Technology
Research Center, and the Institute for Robotics and Intelligent Systems. Geoffrey Hinton
is the Noranda Fellow of the Canadian Institute for Advanced Research. We thank Peter
Dayan for helpful discussions.
References
Durbin, R. and Willshaw, D. (1987). An analogue approach to the travelling salesman
problem. Nature, 326:689-691.
Hinton, G. and Zemel, R. (1994). Autoencoders, minimum description length, and
Helmholtz free energy. To appear in Cowan, J.D., Tesauro, G., and Alspector,
J. (eds.), Advances in Neural Information Processing Systems 6. San Francisco,
CA: Morgan Kaufmann.
Kohonen, T. (1982). Self-organized formation of topologically correct feature maps.
Biological Cybernetics, 43:59-69.
Rissanen, J. (1989). Stochastic Complexity in StatisticalInquiry. World Scientific Publishing Co., Singapore.
Saund, E. (1989). Dimensionality-reduction using connectionist networks. IEEE
Transactions on Pattern Analysis and Machine Intelligence, 11(3):304-314.
Zemel, R. (1993). A Minimum Description Length Framework for Unsupervised Learning. Ph.D. Thesis, Department of Computer Science, University of Toronto.
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7,076 | 846 | A Unified Gradient-Descent/Clustering
Architecture for
Finite State Machine Induction
Sreerupa Das and Michael C. Mozer
Department of Computer Science
University of Colorado
Boulder, CO 80309-0430
Abstract
Although recurrent neural nets have been moderately successful
in learning to emulate finite-state machines (FSMs), the continuous internal state dynamics of a neural net are not well matched
to the discrete behavior of an FSM. We describe an architecture,
called DOLCE, that allows discrete states to evolve in a net as learning progresses. DOLCE consists of a standard recurrent neural net
trained by gradient descent and an adaptive clustering technique
that quantizes the state space. DOLCE is based on the assumption
that a finite set of discrete internal states is required for the task,
and that the actual network state belongs to this set but has been
corrupted by noise due to inaccuracy in the weights. DOLCE learns
to recover the discrete state with maximum a posteriori probability from the noisy state. Simulations show that DOLCE leads to a
significant improvement in generalization performance over earlier
neural net approaches to FSM induction.
1
INTRODUCTION
Researchers often try to understand-post hoc-representations that emerge in the
hidden layers of a neural net following training. Interpretation is difficult because
these representations are typically highly distributed and continuous. By "continuous," we mean that if one constructed a scatterplot over the hidden unit activity
space of patterns obtained in response to various inputs, examination at any scale
would reveal the patterns to be broadly distributed over the space.
Continuous representations aren't always appropriate. Many task domains seem to
require discrete representations-representations selected from a finite set of alternatives. If a neural net learned a discrete representation, the scatterplot over hidden
activity space would show points to be superimposed at fine scales of analysis. Some
19
20
Das and Mozer
examples of domains in which discrete representations might be desirable include:
finite-state machine emulation, data compression, language and higher cognition
(involving discrete symbol processing), and categorization in the context of decision
making. In such domains, standard neural net learning procedures, which have
a propensity to produce continuous representations, may not be appropriate. The
work we report here involves designing an inductive bias into the learning procedure
in order to encourage the formation of discrete internal representations.
In the recent years, various approaches have been explored for learning discrete
representations using neural networks (McMillan, Mozer, & Smolensky, 1992; Mozer
& Bachrach, 1990; Mozer & Das, 1993; Schiitze, 1993; Towell & Shavlik, 1992).
However, these approaches are domain specific, making strong assumptions about
the nature of the task. In our work, we describe a general methodology that makes
no assumption about the domain to which it is applied, beyond the fact that discrete
representations are desireable.
2
FINITE STATE MACHINE INDUCTION
We illustrate the methodology using the domain of finite-state machine (FSM)
induction. An FSM defines a class of symbol strings. For example, the class (lOt
consists of all strings with one or more repetitions of 10; 101010 is a positive example
of the class, 111 is a negative example. An FSM consists principally of a finite set
of states and a function that maps the current state and the current symbol of the
string into a new state. Certain states of the FSM are designated "accept" states,
meaning that if the FSM ends up in these states, the string is a member of the
class. The induction problem is to infer an FSM that parsimoniously characterizes
the positive and negative exemplars, and hence characterizes the underlying class.
A generic recurrent net architecture that could be used for FSM emulation and
induction is shown on the left side of Figure 1. A string is presented to the input
layer of the net, one symbol at a time. Following the end of the string, the net
should output whether or not the string is a member of the class. The hidden unit
activity pattern at any point during presentation of a string corresponds to the
internal state of an FSM.
Such a net, trained by a gradient descent procedure, is able to learn to perform this
or related tasks (Elman, 1990; Giles et al., 1992; Pollack, 1991; Servan-Schreiber,
Cleeremans, & McClelland, 1991; Watrous & Kuhn, 1992). Although these models
have been relatively successful in learning to emulate FSMs, the continuous internal
state dynamics of a neural net are not well matched to the discrete behavior of FSMs.
Roughly, regions of hidden unit activity space can be identified with states in an
FSM, but because the activities are continuous, one often observes the network
drifting from one state to another. This occurs especially with input strings longer
than those on which the network was trained.
To achieve more robust dynamics, one might consider quantizing the hidden state.
Two approaches to quantization have been explored previously. In the first, a net
is trained in the manner described above. After training, the hidden state space is
partitioned into disjoint regions and each hidden activity pattern is then discretized
by mapping it to the center of its corresponding region (Das & Das, 1991; Giles
A Unified Gradient-Descent/Clustering Architecture for Finite State Machine Induction
Figure 1: On the left is a generic recurrent architecture that could be used for FSM induction. Each box corresponds to a layer of units, and arrows depict complete connectivity
between layers. At each time step, a new symbol is presented on the input and the input
and hidden representations are integrated to form a new hidden representation. On the
right is the general architecture of DOLCE.
et al., 1992). In a second approach, quantization is enforced during training by
mapping the the hidden state at each time step to the nearest corner of a [0,1]"
hypercube (Zeng, Goodman, & Smyth, 1993).
Each of these approaches has its limitations. In the first approach, because learning
does not consider the latter quantization, the hidden activity patterns that result
from learning may not lie in natural clusters. Consequently, the quantization step
may not group together activity patterns that correspond to the same state. In the
second approach, the quantization process causes the error surface to have discontinuities and to be flat in local neighborhoods of the weight space. Hence, gradient
descent learning algorithms cannot be used; instead, even more heuristic approaches
are required. To overcome the limitations of these approaches, we have pursued an
approach in which quantization is an integral part of the learning process.
3
DOLCE
Our approach incorporates a clustering module into the recurrent net architecture,
as shown on the right side of Figure 1. The hidden layer activities are processed by
the clustering module before being passed on to other layers. The clustering module
maps regions in hidden state space to a single point in the same space, effectively
partitioning or clustering the hidden state space. Each cluster corresponds to a
discrete internal state. The clusters are adaptive and dynamic, changing over the
course of learning. We call this architecture DOLCE, for gynamic Qn-!ine ?lustering
and state extraction.
The DOLCE architecture may be explored along two dimensions: (1) the clustering
algorithm used (e.g., a Gaussian mixture model, ISODATA, the Forgy algorithm,
vector quantization schemes), and (2) whether supervised or unsupervised training
is used to identify the clusters. In unsupervised mode, the performance error on
the FSM induction task has no effect on the operation of the clustering algorithm;
instead, an internal criterion characterizes goodness of clusters. In supervised mode,
the primary measure that affects the goodness of a cluster is the performance error.
Regardless of the training mode, all clustering algorithms incorporate a pressure to
21
22
Das and Mozer
o
Figure 2: Two dimensions of a typical state space. The true states needed to perform
the task are Cl, C3, and C3, while the observed hidden states, asswned to be corrupted by
noise, are distributed about the Ci.
produce a small number of clusters. Additionally, as we elaborate more specifically
below, the algorithms must allow for a soft or continuous clustering during training,
in order to be integrated into a gradient-based learning procedure.
We have explored two possibilities for the clustering module. The first involves
the use of Forgy's algorithm in an unsupervised mode. Forgy's (1965) algorithm
determines both the number of clusters and the partitioning of the space. The
second uses a Gaussian mixture model in a supervised mode, where the mixture
model parameters are adjusted so as to minimize the performance error. Both
approaches were successful, but as the latter approach obtained better results, we
describe it in the next section.
4
CLUSTERING USING A MIXTURE MODEL
Here we motivate the incorporation of a Gaussian mixture model into DOLCE, using an argument that gives the approach a solid theoretical foundation. Several
assumptions underly the approach. First, we assume that the task faced by DOLCE
is such that it requires a finite set of internal or true states, C = {Clt C2, ??. , CT}.
This is simply the premise that motivates this line of work. Second, we assume
that any observed hidden state-i.e., a hidden activity pattern that results from
presentation of a symbol sequence-belongs to C but has been corrupted by noise
due to inaccuracy in the network weights. Third, we assume that this noise is Gaussian and decreases as learning progresses (i.e., as the weights are adjusted to better
perform the task). These assumptions are depicted in Figure 2.
Based on these assumptions, we construct a Gaussian mixture distribution that
models the observed hidden states:
T
p( hlC
tT
"
q) = ~
qi
e-lh-c.12 /2q~
L...J (27r0'~)H/2
i=l
?
where h denotes an observed hidden state, 0'; the variance of the noise that corrupts state Ci, qi is the prior probability that the true state is Ci, and H is the
dimensionality of the hidden state space. For pedagogical purposes, a.ssume for the
time being that the parameters of the mixture distribution-T, C, tT, and q-are
all known; in a later section we discuss how these parameters are determined.
A Unified Gradient-Descent/Clustering Architecture for Finite State Machine Induction
h
o
o
0
000
OOOO!,~OO
0
0
~
7
~O
00
0
A
0
before training
after successful training
Figure 3: A schematic depiction of the hidden state space before and after training. The
horizontal plane represents the state space. The bumps indicate the probability density
under the mixture model. Observed hidden states are represented by small open circles.
Given a noisy observed hidden state, h, DOLCE computes the maximum a posteriori
(MAP) estimator of h in C. This estimator then replaces the noisy state and is used
in all subsequent computation. The MAP estimator, h, is computed as follows. The
probability of an observed state h being generated by a given true state i is
p(hltrue state i)
= (27rlTi)-!fe-lh-cill/2u:.
Using Bayes' rule, one can compute the posterior probability of true state i, given
that h has been observed:
.Ih)
p ( true state z
=
p(hltrue state i)qi
L:j p(hltrue state j)qj
=---'---'-------'----
Finally, the MAP estimator is given by it = Cargmax,p(true state ilh). However,
because learning requires that DOLCE's dynamics be differentiable, we use a soft
L:i cip(true state ilh) instead of hand
version of MAP which involves using ii
incorporating a "temperature" parameter into lTi as described below.
=
An important parameter in the mixture model is T, the number of true states
(Gaussians bumps). Because T directly corresponds to the number of states in
the target FSM, if T is chosen too small, DOLCE could not emulate the FSM.
Consequently, we set T to a large value, and the training procedure includes a
technique for eliminating unnecessary true states. (If the initially selected T is not
large enough, the training procedure will not converge to zero error on the training
set, and the procedure can be restarted with a larger value of T.)
At the start of training, each Gaussian center I Ci, is initialized to a random location
in the hidden state space. The standard deviations of each Gaussian, lTi, are initially
set to a large value. The priors, qi, are set to liT. The weights are set to initial
values chosen from the uniform distribution in [-.25,.25]. All connection weights
feeding into the hidden layer are second order.
The network weights and mixture model parameters-C, iT, and q-are adjusted by
gradient descent in a cost measure, C. This cost includes two components: (a) the
performance error, ?, which is a squared difference between the actual and target
network output following presentation of a training string, and (b) a complexity
23
24
Das and Mozer
c:
o
800,------~....,
II
2000,--------,
language
language
0600
i
language S
200
400
E
100
'0
2
NO ROLO OF DG
NO ROLO OF DG
o NO
RO LO OF DG
language
language 6
400
200
OL.......l.:.O=~
NO ROLO OF 00
Figure 4: Each graph depicts generalization performance on one of the Tomita languages
for 5 alternative neural net approaches: no clustering [Ne), rigid quantization [RQ), learn
then quantize [LQ], DOLCE in unsupervised mode using Forgy's algorithm [DF], DOLCE
in supervised mode using mixture model [DG) . The vertical axis shows the number of
misclassification of 3000 test strings. Each bar is the average result across 10 replications
with different initial weights.
cost, which is the entropy of the prior distribution, q:
where ..\ is a regularization parameter. The complexity cost is minimal when only
one Gaussian has a nonzero prior, and maximal when all priors are equal. Hence,
the cost encourages unnecessary Gaussians to drop out of the mixture model.
The particular gradient descent procedure used is a generalization of back propagation through time (Rumelhart, Hinton, & Williams, 1986) that incorporates the
mixture model. To better condition the search space and to avoid a constrained
search, optimization is performed not over iT and q directly but rather over hyperparameters a and h, where
= exp(ai)/,B and qi = exp( -bl)/E j exp( -bj).
u;
The global parameter ,B scales the overall spread of the Gaussians, which corresponds to the level of noise in the model. As performance on the training set
improves, we assume that the network weights are coming to better approximate
the target weights, hence that the level of noise is decreasing. Thus, we tie ,B to
the performance error e. We have used various annealing schedules and DOLCE
appears robust to this variation; we currently use {3 ex 1/ e. Note that as ? --+ 0,
{3 --+ 00 and the probability density under one Gaussian at h will become infinitely
greater than the density under any other; consequently, the soft MAP estimator,
h, becomes equivalent to the MAP estimator h, and the transformed hidden state
becomes discrete. A schematic depiction of the probability landscape both before
and after training is depicted in Figure 3.
A Unified Gradient-Descent/Clustering Architecture for Finite State Machine Induction
5
SIMULATION STUDIES
The network was trained on a set ofregular languages first studied by Tomita (1982).
The languages, which utilize only the symbols 0 and 1, are: (1) 1?; (2) (10)?; (3) no
odd number of consecutive 1 's is directly followed by an odd number of consecutive
O's; (4) any string not containing the substring "000"; (5) , [(01110)(01110)].; (6)
the difference between the number of ones and number of zeros in the string is a
multiple of three; and (7) 0?1? 0?1? .
A fixed training corpus of strings was generated for each language, with an equal
number of positive and negative examples. The maximum string length varied from
5 to 10 symbols and the total number of examples varied from 50 to 150, depending
on the difficulty of the induction task.
Each string was presented one symbol at a time, after which DOLCE was given a
target output that specified whether the string was a positive or negative example
of the language. Training continued until DOLCE converged on a set of weights
and mixture model parameters. Because we assume that the training examples are
correctly classified, the error ? on the training set should go to zero when DOLCE has
learned. If this did not happen on a given training run, we restarted the simulation
with different initial random weights.
For each language, ten replications of DOLCE (with the supervised mixture model)
were trained, each with different random initial weights. The learning rate and
regularization parameter .\ were chosen for each language by quick experimentation
with the aim of maximizing the likelihood of convergence on the training set. We
also trained a version of DOLCE that clustered using the unsupervised Forgy algorithm, as well as several alternative neural net approaches: a generic recurrent net,
as shown on the left side of Figure 1, which used no clustering [NC]; a version with
rigid quantization during training [RQ], comparable to the earlier work of Zeng,
Goodman, and Smyth (1993); and a version in which the unsupervised Forgyalgorithm was used to quantize the hidden state following training [LQ], comparable to
the earlier work of Das and Das (1991). In these alternative approaches, we used
the same architecture as DOLCE except for the clustering procedure. We selected
learning parameters to optimize performance on the training set, ran ten replications for each language, replaced runs which did not converge, and used the same
training sets.
6
RESULTS AND CONCLUSION
In Figure 4, we compare the generalization performance of DOLCE-both the unsupervised Forgy [DF] and supervised mixture model [DG]-to the NC, RQ, and LQ
approaches. Generalization performance was tested using 3000 strings not in the
training set, half positive examples and half negative. The two versions of DOLCE
outperformed the alternative neural net approaches, and the DG version of DOLCE
consistently outperformed the DF version.
To summarize, we have described an approach that incorporates inductive bias into
a learning algorithm in order to encourage the evolution of discrete representations
during training. This approach is a quite general and can be applied to domains
25
26
Das and Mozer
other than grammaticality judgement where discrete representations might be desirable. Also, this approach is not specific to recurrent networks and may be applied
to feedforward networks. We are now in the process of applying DOLCE to a much
larger, real-world problem that involves predicting the next symbol in a string. The
data base comes from a case study in software engineering, where each symbol
represents an operation in the software development process. This data is quite
noisy and it is unlikely that the data can be parsimoniously described by an FSM.
Nonetheless, our initial results are encouraging: DOLCE produces predictions at
least three times more accurate than a standard recurrent net without clustering.
Acknowledgements
This research was supported by NSF Presidential Young Investigator award IRI9058450 and grant 90-21 from the James S. McDonnell Foundation.
References
S. Das & R. Das. (1991) Induction of discrete state-machine by stabilizing a continuous recurrent network using clustering. Computer Science and Informatics 21(2):35-40. Special
Issue on Neural Computing.
J.L. Elman. (1990) Finding structure in time. Cognitive Science 14:179-212.
E. Forgy. (1965) Cluster analysis of multivariate data: efficiency versus interpretability of
classifications. Biometrics 21:768-780.
M.C. Mozer & J.D Bachrach. (1990) Discovering the structure of a reactive environment
by exploration. Neural Computation 2( 4):447-457.
C. McMillan, M.C. Mozer, & P. Smolensky. (1992) Rule induction through integrated
symbolic and subsymbolic processing. In J.E. Moody, S.J. Hanson, & R.P. Lippmann
(eds.), Advances in Neural Information Proceuing Sy6tems 4, 969-976. San Mateo, CA:
Morgan Kaufmann.
C.L. Giles, D. Chen, C.B. Miller, H.H. Chen, G.Z. Sun, & Y.C. Lee. (1992) Learning
and extracting finite state automata with second-order recurrent neural network. Neural
Computation 4(3):393-405.
H. Schiitze. (1993) Word space. In S.J. Hanson, J.D. Cowan, & C.L. Giles (eds.), Advances
in Neural Information Proceuing Systems 5, 895-902. San Mateo, CA: Morgan Kaufmann.
M. Tomita. (1982) Dynamic construction of finite-state automata from examples using hillclimbing. Proceedings of the Fourth Annual Conference of the Cognitive Science Society,
105-108.
G. Towell & J. Shavlik. (1992) Interpretion of artificial neural networks: mapping
knowledge-based neural networks into rules. In J .E. Moody, S.J. Hanson, & R.P. Lippmann (eds.), Advances in Neural Information Proceuing Systems 4, 977-984. San Mateo,
CA: Morgan Kaufmann.
R.L. Watrous & G.M. Kuhn. (1992) Induction of finite state languages using second-order
recurrent networks. In J.E. Moody, S.J. Hanson, & R.P. Lippmann (eds.), Advances in
Neural Information Proceuing Systems 4, 969-976. San Mateo, CA: Morgan Kaufmann.
Z. Zeng, R. Goodman, & P. Smyth. (1993) Learning finite state machines with selfclustering recurrent networks. Neural Computation 5(6):976-990.
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7,077 | 848 | Transition Point Dynamic Programming
Kenneth M. Buckland'"
Dept. of Electrical Engineering
University of British Columbia
Vancouver, B.C, Canada V6T 1Z4
[email protected]
Peter D. Lawrence
Dept. of Electrical Engineering
University of British Columbia
Vancouver, B.C, Canada V6T 1Z4
[email protected]
Abstract
Transition point dynamic programming (TPDP) is a memorybased, reinforcement learning, direct dynamic programming approach to adaptive optimal control that can reduce the learning
time and memory usage required for the control of continuous
stochastic dynamic systems. TPDP does so by determining an
ideal set of transition points (TPs) which specify only the control
action changes necessary for optimal control. TPDP converges to
an ideal TP set by using a variation of Q-Iearning to assess the merits of adding, swapping and removing TPs from states throughout
the state space. When applied to a race track problem, TPDP
learned the optimal control policy much sooner than conventional
Q-Iearning, and was able to do so using less memory.
1
INTRODUCTION
Dynamic programming (DP) approaches can be utilized to determine optimal control policies for continuous stochastic dynamic systems when the state spaces of
those systems have been quantized with a resolution suitable for control (Barto et
al., 1991). DP controllers, in lheir simplest form, are memory-based controllers
that operate by repeatedly updating cost values associated with every state in the
discretized state space (Barto et al., 1991). In a slate space of any size the required
quantization can lead to an excessive memory requirement, and a related increase
in learning time (Moore, 1991). This is the "curse of dimensionality".
?Nowat: PMC-Sierra Inc., 8501 Commerce Court, Burnaby, B.C., Canada V5A 4N3.
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Q-Iearning (Watkins, 1989, Watkins et al., 1992) is a direct form of DP that avoids
explicit system modeling - thereby reducing the memory required for DP control.
Further reductions are possible if Q-Ieal'l1ing is modified so that its DP cost values
(Q-values) are associated only with states where control action changes need to be
specified. Transition point dynamic programming (TPDP), the control approach
described in this paper, is designed to take advantage of this DP memory reduction
possibility by determining the states where control action changes must be specified
for optimal control, and what those optimal changes are.
2
2.1
GENERAL DESCRIPTION OF TPDP
TAKING ADVANTAGE OF INERTIA
TPDP is suited to the control of continuous stochastic dynamic systems that have
inertia. In such systems "uniform regions" are likely to exist in the state space
where all of the (discretized) states have the same optimal control action (or the
same set of optimal actions l ). Considering one such uniform region, if the optimal
action for that region is specified at the "boundary states" of the region and then
maintained throughout the region until it is left and another uniform region is
entered (where another set of boundary states specify the next action), none of the
"dormant states" in the middle of the region need to specify any actions themselves.
Thus dormant states do not have to be represented in memory. This is the basic
premise of TPDP.
The association of optimal actions with boundary states is done by "transition
points" (TPs) at those states. Boundary states include all of the states that can
be reached from outside a uniform region when that region is entered as a result of
stochastic state transitions. The boundary states of anyone uniform region form a
hyper-surface of variable thickness which mayor may not be closed. The TPs at
boundary states must be represented in memory, but if they are small in number
compared to the dormant states the memory savings can be significant.
2.2
ILLUSTRATING THE TPDP CONCEPT
Figure 1 illustrates the TPDP concept when movement control of a "car" on a
one dimensional track is desired. The car, with some initial positive velocity to the
right, must pass Position A and return to the left. The TPs in Figure 1 (represented
by boxes) are located at boundary states. The shaded regions indicate all of the
states that the system can possibly move through given the actions specified at the
boundary states and the stochastic response of the car. Shaded states without TPs
are therefore dormant states. Uniform regiolls consist of adjacent boundary states
where the same action is specified, as well as the shaded region through which that
action is maintained before another boundary is encountered. Boundary states that
do not seem to be on the main sta.te transition routes (the one identified in Figure 1
for example) ensure that any stochastic deviations from those routes are realigned.
Unshaded states are "external states" the system does not reach.
IThe simplifying assumption t.hat t.here is ouly oue optimal action in each uniform
region will be made throughout this paper. TPDP operates the same regardless.
Transition Point Dynamic Programming
+
13
Each
is a
transition point (TP),
~
'0
00
Q)
>
niform
Region
Boundary
State
A
Position
Figure 1: Application of TPDP to a One Dimension Movement Control Task
2.3
MINIMAL TP OPTIMAL CONTROL
The main benefit of the TPDP approach is that, where uniform regions exist, they
can be represented by a relatively small number of DP elements (TPs) - depending
on the shape of the boundaries and the size of the uniform regions they encompass. This reduction in memory usage results in an accompanying reduction in the
learning time required to learn optimal control policies (Chapman et al., 1991).
TPDP operates by learning optimal points of transition in the control action specification, where those points can be accurately located in highly resolved state spaces.
To do this TPDP must determine which states are boundary states that should
have TPs, and what actions those TPs should specify. In other words, TPDP must
find the right TPs for the right states. When it has done so, "minimal TP optimal
control" has been achieved. That is, optimal control with a minimal set of TPs.
3
3.1
ACHIEVING MINIMAL TP OPTIMAL CONTROL
MODIFYING A SET OF TPs
Given an arbitrary initial set of TPs, TPDP must modify that set so that it is
transformed into a minimal TP optimal control set. Modifications can include the
"addition" and "removal" of TPs throughout the state space, and the "swapping"
of one TP for another (each specifying a different action) at the same state. These
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modifications are performed one at a time in arbitl'ary order, and can continue
indefinitely. TPDP operates so that each TP modification results in an incremental
movement towards minimal TP optimal control (Buckland, 1994).
3.2
Q-LEARNING
TPDP makes use of Q-Iearning (Watkins, 1989, Watkins et ai., 1992) to modify the
TP set. Normally Q-Iearning is used to determine the optimal control policy J-t for
a stochastic dynamic system subjected to immediate costs c(i, u) when action u is
applied in each state i (Barto et ai., 1991). Q-learning makes use of "Q-values"
Q( i, u), which indicate the expected total infini te-horizon discounted cost if action
u is applied in state i, and actions defined by the existing policy J-t are applied in
all future states. Q-values are learned by using the following updating equation:
Qt+l(St, Ut) = (1 - Ctt)Qt(St, ud + at [c(St, ud + 'YVt(St+l)]
(1)
Where at is the update rate, l' is the discount factor, and St and Ut are respectively
the state at time step t and the action taken at that time step (all other Q-values
remain the same at time step t). The evaluation function value lit (i) is set to the
lowest Q-value action of all those possible U(i) in each state i:
Vt(i) = min Qt(i, u)
(2)
UEU(i)
If Equations 1 and 2 are employed during exploratory movement of the system, it has
been proven that convergence to optimal Q-values Q* (i, u) and optimal evaluation
function values VI-'. (i) will result (given that the proper constraints are followed,
Watkins, 1989, Watkins et ai., 1992, Jaakkola et ai., 1994). From these values the
optimal action in each state can be determined (the action that fulfills Equation 2).
3.3
ASSESSING TPs WITH Q-LEARNING
TPDP uses Q-Iearning to determiue how an existing set of TPs should be modified
to achieve minimal TP optimal control. Q-values can be associated with TPs, and
the Q-values of two TPs at the same "TP state", each specifying different actions,
can be compared to determine which should be maintained at that state - that is,
which has the lower Q-value. This is how TPs are swapped (Buckland, 1994).
States which do not have TPs, "non-TP states", have no Q-values from which
evaluation function values vt(i) can be determined (using Equation 2). As a result,
to learn TP Q-values, Equation 1 must be modified to facilitate Q-value updating
when the system makes d state transitions from one TP state through a number of
non-TP states to another TP state:
Qt+.( St, Ut) = (1 - a,jQt (5t, Ut)
+ "t [ (~'Yn c( St+n, Ut)) + 'Y.v,( St+.)]
(3)
=
When d 1, Equation 3 takes the form of Equation 1. When d > 1, the intervening
non-TP states are effectively ignored and treated as inherent parts of the stochastic
dynamic behavior of the system (Buckla.nd, 1994).
If Equation 3 is used to determine the costs incurred when no action is specified
at a state (when the action specified at some previous state is maintained), an "Rvalue" R( i) is the result. R-values can be used to expediently add and remove TPs
Transition Point Dynamic Programming
from each state. If the Q-value of a TP is less than the R-value of the state it is
associated with, then it is worthwhile having that TP at that state; otherwise it is
not (Buckland, 1994).
3.4
CONVERGENCE TO MINIMAL TP OPTIMAL CONTROL
It has been proven that a random sequence of TP additions, swaps and removals
attempted at states throughout the state space will result in convergence to minimal TP optimal control (Buckland, 1994). This proof depends mainly on all TP
modifications "locking-in" any potential cost reductions which are discovered as the
result of learning exploration.
The problem with this proof of convergence, and the theoretical form of TPDP
described up to this point, is that each modification to the existing set of TPs (each
addition, swap and removal) requires the determination of Q-values and R-values
which are negligibly close to being exact. This means that a complete session of
Q-Iearning must occur for every TP modification. 2 The result is excessive learning
times - a problem circumvented by the practical form of TPDP described next.
4
4.1
PRACTICAL TPDP
CONCURRENT TP ASSESSMENT
To solve the problem of the protracted learning time required by the theoretical
form of TPDP, many TP modifications can be assessed concurrently. That is,
Q-Iearning can be employed not just to determine the Q-values and R-values for a
single TP modification, but instead to learn these values for a number of concurrent
modifications. Further, the modification attempts, and the learning of the values
required for them, need not be initiated simultaneously. The determination of each
value can be made part of the Q-Iearning process whenever new modifications are
randomly attempted. This approa.ch is called "Pra.ctical TPDP". Practical TPDP
consists of a continually running Q-Ieal'l1ing process (based on Equations 2 and 3),
where the Q-values and R-values of a constantly changing set of TPs are learned.
4.2
USING WEIGHTS FOR CONCURRENT TP ASSESSMENT
The main difficulty that arises when TPs are assessed concurrently is that of determining when an assessment is complete. That is, when the Q-values and R-values
associated with each TP ha.ve been learned well enough for a TP modification to
be made based on them. The technique employed to address this problem is to
associate a "weight" wei, u) with ea.ch TP that indicates the general merit of that
TP. The basic idea of weights is to facilita.te the random addition of trial TPs to
a TP "assessment group" with a low initial weight Winitial. The Q-values and Rvalues of the TPs in the assessment group are learned in an ongoing Q-Iearning
process, and the weights of the TPs are adjusted heuristically using those values.
Of those TPs at any state i whose weights wei, u) have been increased above Wthr
2The TPDP proof allows for more than one TP swap to be assessed simultaneously,
but this does little to reduce the overall problem being described (Buckland, 1994).
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100
Conventional
Q-Iearning
-
..c
C>
C
Q)
.....J
-
..c
CU
a...
50
Q)
C>
~
Q)
~
Practical TPDP
o
o
2500
Epoch Number
Figure 2: Performance of Practical TPDP on a Race Track Problem
(Winitial < Wthr < w max ), the one with the lowest Q-value Q(i, u) is swapped into
the "policy TP" role for that state. The heuristic weight adjustment rules are:
1. New, trial TPs are given an initial weight of Wjnitial (0 < Winitial < Wthr).
2. Each time the Q-value of a TP is updated, the weight w(i, u) of that TP is
incremented if Q(i, u) < R(i) and decremented otherwise.
3. Each TP weight w( i, u) is limited to a maximum value of w max . This
prevents anyone weight from becoming so large that it cannot readily be
reduced again.
4. If a TP weight w(i, u) is decremented to 0 the TP is removed.
An algorithm for Practical TPDP implementation is described in Buckland (1994).
4.3
PERFORMANCE OF PRACTICAL TPDP
Practical TPDP was applied to a continuous version of a control task described by
Barto et al. (1991) - that of controlling the acceleration of a car down a race track
(specifically the track shown in Figures 3 and 4) when that car randomly experiences
control action non-responsiveness. As shown in Figure 2 (each epoch in this Figure
consisted of 20 training trials and 500 testing trials), Practical TPD P learned the
optimal control policy much sooner than conventional Q-Iearning, and it was able
to do so when limited to only 15% of the possible number of TPs (Buckland, 1994).
The possible number of TPs is the full set of Q-values required by conventional
Q-Iearning (one for each possible state and action combination).
The main advantage of Practical TPDP is that it facilitates rapid learning of preliminary control policies. Figure 3 shows typical routes followed by the car early
Transition Point Dynamic Programming
Finishing
Positions
Starting
Positions
Figure 3: Typical Race Track Routes After 300 Epochs
Finishing
Positions
Starting
Positions
Figure 4: Typical Race Track Routes After 1300 Epochs
in the learning process. With the addition of relatively few TPs, the policy of accelerating wildly down the track, smashing into the wall and continuing on to the
finishing positions was learned. Further learning centered around this preliminary
policy led to the optimal policy of sweeping around the left turn. Figure 4 shows
typical routes followed by the car during this shift in the learned policy - a shift
indicated by a slight drop in the learning curve shown in Figure 2 (around 1300
epochs). After this shift, learning progressed rapidly until roughly optimal policies
were consistently followed.
A problem which occurs in Practical TPDP is that of the addition of superfluous
TPs after the optimal policy has bac;ically been learned. The reasons this occurs
are described in Buckland (1994), ac; well as a number of solutions to the problem.
5
CONCLUSION
The practical form of TPDP performs very well when compared to conventional
Q-Iearning. When applied to a race track problem it was able to learn optimal
policies more quickly while using less memory. Like Q-learning, TPDP has all the
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advantages and disadvantages that result from it being a direct control approach
that develops no explicit system model (Watkins, 1989, Buckland, 1994).
In order to take advantage of the sparse memory usage that occurs in TPDP, TPs
are best represented by ACAMs (associative content addressable memories, Atkeson, 1989). A localized neural network design which operates as an ACAM and
which facilitates Practical TPDP control is described in Buckland et al. (1993) and
Buckland (1994).
The main idea of TPDP is to, "try this for a while and see what happens". This
is a potentially powerful approach, and the use of TPs associated with abstracted
control actions could be found to have substantial utility in hierarchical control
systems.
Acknowledgements
Thanks to John Ip for his help on this work. This work was supported by an NSERC
Postgraduate Scholarship, and NSERC Operating Grant A4922.
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7,078 | 849 | Structured Machine Learning For 'Soft'
Classification with Smoothing Spline
ANOVA and Stacked Tuning, Testing
and Evaluation
Yuedong Wang
Dept of Statistics
University of Wisconsin
Madison, WI 53706
Grace Wahba
Dept of Statistics
University of Wisconsin
Madison, WI 53706
Chong Gu
Dept of Statistics
Purdue University
West Lafayette, IN 47907
Ronald Klein, MD
Dept of Ophthalmalogy
University of Wisconsin
Madison, WI 53706
Barbara Klein, MD
Dept of Ophthalmalogy
University of Wisconsin
Madison, WI 53706
Abstract
We describe the use of smoothing spline analysis of variance (SSANOVA) in the penalized log likelihood context, for learning
(estimating) the probability p of a '1' outcome, given a training set with attribute vectors and outcomes. p is of the form
pet) = eJ(t) /(1 + eJ(t)), where, if t is a vector of attributes, f
is learned as a sum of smooth functions of one attribute plus a
sum of smooth functions of two attributes, etc. The smoothing
parameters governing f are obtained by an iterative unbiased risk
or iterative GCV method. Confidence intervals for these estimates
are available.
1. Introduction to 'soft' classification and the bias-variance tradeoff.
In medical risk factor analysis records of attribute vectors and outcomes (0 or 1)
for each example (patient) for n examples are available as training data. Based on
the training data, it is desired to estimate the probability p of the 1 outcome for any
415
416
Wahba, Wang, Gu, Klein, and Klein
new examples in the future, given their attribute vectors. In 'soft' classification, the
estimate p of p is of particular interest, and might be used, say, by a physician to
tell a patient that if he reduces his cholesterol from t to t', then he will reduce his
risk of a heart attack from p(t) to p(t'). We assume here that p varies 'smoothly'
with any continuous attribute (predictor variable).
It is long known that smoothness penalties and Bayes estimates are intimately re-
lated (see e.g. Kimeldorf and Wahba(1970, 1971), Wahba(1990) and references
there). Our philosophy with regard to the use of priors in Bayes estimates is to
use them to generate families of reasonable estimates (or families of penalty functionals) indexed by those smoothing or regularization parameters which are most
relevant to controlling the generalization error. (See Wahba(1990) Chapter 3, also
Wahba(1992)). Then use cross-validation, generalized cross validation (GCV), unbiased risk estimation or some other performance oriented method to choose these
parameter(s) to minimize a computable proxy for the generalization error. A person
who believed the relevant prior might use maximum likelihood (ML) to choose the
parameters, but ML may not be robust against an unrealistic prior (that is, ML
may not do very well from the generalization point of view if the prior is off), see
Wahba(1985). One could assign a hyperprior to these parameters. However, except
in cases where real prior information is available, there is no reason to believe that
the use of hyperpriors will beat out a performance oriented criterion based on a good
proxy for the generalization error, assuming, of course, that low generalization error
is the true goal.
O'Sullivan et al(1986) proposed a penalized log likelihood estimate of I, this work
was extended to the SS-ANOVA context in Wahba, Gu, Wang and Chappell(1993),
where numerous other relevant references are cited. This paper is available by
ftp from ftp. stat. wise. edu, cd pub/wahba in the file soft-class. ps. Z. An
extended bibliography is available in the same directory as ml-bib. ps. The SSANOVA allows a variety of interpretable structures for the possible relationships
between the predictor variables and the outcome, and reduces to simple relations
in some of the attributes, or even, to a two-layer neural net, when the data suggest
that such a representation is adequate.
2. Soft classification and penalized log likelihood risk factor estimation
To describe our 'worldview', let t be a vector of attributes, tEn E T, where n is
some region of interest in attribute space T. Our 'world' consists of an arbitrarily
large population of potential examples, whose attribute vectors are distributed in
some way over n and, considering all members of this 'world' with attribute vectors
in a small neighborhood about t, the fraction of them that are l's is p(t). Our
training set is assumed to be a random sample of n examples from this population,
whose outcomes are known, and our goal is to estimate p(t) for any tEO. In 'soft'
classification, we do not expect one outcome or the other to be a 'sure thing', that
is we do not expect p(t) to be 0 or 1 for large portions of n.
Next, we review penalized log likelihood risk estimates. Let the training data be
{Yi, t(i), i
1, ... n} where Yi has the value 1 or 0 according to the classification of
example i, and t(i) is the attribute vector for example i. If the n examples are a
random sample from our 'world', then the likelihood function of this data, given
=
"Soft" Classification with Smoothing Spline ANOVA
p( .), is
likelihood{y, p} = II~=lP(t(i))Yi (1 - p(t(i) ))l-Yi,
(1)
which is the product of n Bernoulli likelihoods. Define the logit f(t) by f(t) =
10g[P(t)/(I- p(t))], then p(t) = eJ(t) 1(1 + eJ(t)). Substituting in f and taking logs
gIves
=?(y, f) = L
n
-log likelihood{y, f}
log(1 + eJ(t(i))) - Yif(t(i)).
(2)
i=l
We estimate f assuming that it is in some space 1l of smooth functions. (Technically,
1l is a reproducing kernel Hilbert space, see Wahba(1990), but you don't need to
know what this is to read on). The fact that f is assumed 'smooth' makes the
methods here very suitable for medical data analysis. The penalized log likelihood
estimate f>.. of f will be obtained as the minimizer in 1l of
n
(3)
?(y, f) + "2)"J(J)
where J(J) is a suitable 'smoothness' penalty. A simple example is, T = [0,1] and
1 (J(m) (t))2dt, in which case f>.. is a polynomial spline of degree 2m - 1. If
J(J)
= Jo
(4)
then f>.. is a thin plate spline. The thin plate spline is a linear combination of
polynomials of degree m or less in d variables, and certain radial basis functions.
For more details and other penalty functionals which result in rbf's, see Wahba(1980,
1990, 1992).
?
The likelihood function ?(y, f) will be maximized if p(t(i)) is 1 or according as
Yi is 1 or 0. Thus, in the (full-rank) spline case, as ).. -+ 0, 1>.. tends to +00 or -00
at the data points. Therefore, by letting).. be small, we can come close to fitting
the data points exactly, but unless the 1 's and O's are well separated in attribute
space, f>.. will be a very 'wiggly' function and the generalization error (not precisely
defined yet) may be large.
The choice of ).. represents a tradeoff between overfitting and underfitting the data
(bias-variance tradeoff). It is important in practice good value of )... We now define
what we mean by a good value of )... Given the family PA,).. > 0, we want to choose
).. so that PA is close to the 'true' but unknown p so that, if new examples arrive with
attribute vector in a neighborhood of t, PA (t) will be a good estimate of the fraction
of them that are 1 'so 'Closeness' can be defined in various reasonable ways. We use
the Kullbach-Leibler (K L) distance (not a real distance!). The K L distance between
two probability measures (g, g) is defined as K L(g, g)
Eg [log (g 1g)], where Eg
means expectation given g is the true distribution. If v(t) is some probability
measure on T, (say, a proxy for the distribution ofthe attributes in the population),
then define K Lv (p, PA) (for Bernoulli random variables) with respect to v as
=
K Lv(p, PA)
=
J
[P(t)log
(;(~l)) + (1 -
]
p(t)) log (11 ~ :A(~l)) dv(t).
(5)
417
418
Wahba, Wang, Gu, Klein, and Klein
Since K Lv is not computable from the data, it is necessary to develop a computable
proxy for it, By a computable proxy is meant a function of), that can be calculated
from the training set which has the property that its minimizer is a good estimate
of the minimizer of K Lv, By letting p>.(t) = e!>.(t) /(1 + e!>.(t?) it is seen that to
minimize K Lv, it is only necessary to minimize
J
[log(l
+ e!>.(t?) -
(6)
p(t)f>.(t)]dv(t)
over). since (5) and (6) differ by something that does not depend on )., Leavingout-half cross validation (!CV) is one conceptually simple and generally defensible
(albeit possibly wasteful) way of choosing). to minimize a proxy for K Lv(p, P>.),
The n examples are randomly divided in half and the first n/2 examples are used
to compute P>. for a series of trial values of )., Then, the remaining n/2 examples
are used to compute
KLl.~ cv ().) = ~n ~
~
[log(l + e!>.(t(i?) - Yif>.(t(i))]
(7)
i::~+l
for the trial values of )., Since the expected value of Yi is p(t(i)), (7) is, for each), an
unbiased estimate of (6) with dv the sampling distribution of the {tel), ,." t(n/2)},
). would then be chosen by minimizing (7) over the trial values. It is inappropriate to
just evaluate (7) using the same data that was used to obtain f>., as that would lead
to overfitting the data, Variations on (7) are obtained by successively leaving out
groups of data. Leaving-out-one versions of (7) may be defined, but the computation
may be prohibitive.
3. Newton-Raphson Iteration and the Unbiased Risk estimate of A.
We use the unbiased risk estimate given in Craven and Wahba(1979) for smoothing
spline estimation with Gaussian errors, which has been adapted by Gu(1992a) for
the Bernoulli case, To describe the estimate we need to describe the NewtonRaphson iteration for minimizing (3). Let b(J) = log(l + e f ), then Ley, f) =
E?::db(J(t(i)) - Yif(t(i))], It is easy to show that Ey;
f(t(i))
b'(f(t(i)) and
var Yi = p(t(i))(l - p(t(i)) = b"(f(t(i)). Represent f either exactly by using a
basis for the (known) n-space of functions containing the solution, or approximately
by suitable approximating basis functions, to get
=
=
N
f ~ L CkBk?
(8)
k=l
Then we need to find
C
n
= (C1' . ' . , C N)' to minimize
N
N
1>.(c) = L beL CkBk(t(i))) - Yi(L CkBk(t(i)))
;=1
k=l
+ ~ ).c'~c,
(9)
k=l
where E is the necessarily non-negative definite matrix determined by
J (Ek Ck Bk) = c'Ec. The gradient \l 1>. and the Hessian \l2l.x of l.x are given by
=
X' (Pc - y)
+ n).~c,
(10)
"Soft" Classification with Smoothing Spline ANOVA
=
X' WcX
+ nXE,
(11)
where X is the matrix with ijth entry Bj(t(i)), Pc is the vector with ith entry Pc (t(i))
given by Pc (t(i)) = (1~:c/~g~:?) where fcO = 2::=1 ekBk(?), and Wc is the diagonal
matrix with iith entry Pc(t(i))(I-Pc(t(i))). Given the ith Newton-Raphson iterate
eCl ), e(l+1) is given by
e(l+1)
and
e( l+ 1 )
= eel) -
(X'WC<l) X + nA~)-l(X'(pc(l) - y)
+ nA~e(l))
(12)
is the minimizer of
Iil\e)
= IIz(l) -
Wcl(~~ Xell 2 + nAe'~e.
(13)
where z(l), the so-called pseudo-data, is given by
z(l)
= Wc(l~/2(y -
Pdl?)
+ W:(~~XeCl).
(14)
The 'predicted' value z(l) = W:(~~ X e, where e is the minimizer of (13), is related to
the pseudo-data z(l) by
Z(l) = A(l)(A)Z(l),
(15)
where A(l)(A) is the smoother matrix given by
A(l)(A) = W:(~~ X(X'Wc(l)X + nA~)-l X'W:(~~.
(16)
In Wahba(1990), Section 9.2 1, it was proposed to obtain a GCV score for A
in (9) as follows: For fixed A, iterate (12) to convergence. Define VCl)(A) =
~II(I - A(l) (A))z(l) 112 /(~tr(I - A(l) (A)))2 . Letting L be the converged value of i,
compute
VCL)(A)
= ~II(I -
A(L) (A))z(L) 112 ,. . , ~IIW:clr(Y - pC<L?)1I 2
(~tr(I - A(L)(A)))2
(~tr(I - A(L)(A)))2
(17)
and minimize VeL) with respect to A. Gu(1992a) showed that (since the variance
is known once the mean is known here) that the unbiased risk estimate U (A) in
Craven and Wahba can also be adapted to this problem as
U(l)(A) =
.!.IIW(l~/2(y
- Pc(l?)11 2 + ~tr A(l)(A).
n
c
n
(18)
He also proposed an alternating iteration, different than that described in
Wahba(1990), namely, given eCl)
e(l)(A(l?), find A
ACl+l) to minimize (18).
Given A(l+!) , do a Newton step to get eCl +1 ), get A(l+2) by minimizing (18), continue
until convergence. He showed that the alternating iteration gave better estimates of
A using V than the iteration in Wahba(1990), as measured by the [( L-distance. His
results (with the alternating iteration) suggested U had somewhat of an advantage
over V, and that is what we are using in the present work. Zhao et aI, this volume,
have used V successfully with the alternating iteration.
=
=
lThe definition of A there differs from the definition here by a factor of n/2 . Please
note the typographical error in (9.2.18) there where A should be 2A.
419
420
Wahba, Wang, Gu, Klein, and Klein
4. Smoothing spline analysis of variance (SS-ANOVA)
In SS-ANOVA, /(t)
= l(t1, ... , td) is decomposed as
I(t)
= I-' + L /a(ta) + L
a
/a/3(ta , t/3) + ...
(19)
a</3
where the terms in the expansion are uniquely determined by side conditions which
generalize the side conditions ofthe usual ANOVA decompositions. Let the logit/(t)
be of the form (19) where the terms are summed over Ct EM, Ct, f3 E M, etc. where
M indexes terms which are chosen to be retained in the model after a model selection procedure. Then 1>..,8, an estimate of I, is obtained as the minimizer of
?(y, 1>.,8)
where
J8(1) =
L
(J~lJa(fa)
aEM
+ )"J8 (I)
+
L
(J;JJa/3(fa/3)
(20)
+...
(21)
a,{3EM
The Ja , J a/3, ... are quadratic 'smoothness' penalty functionals, and the (J's satisfy
a single constraint. For certain spline-like smoothness penalties, the minimizer of
(20) is known to be in the span of a certain set of n functions, and the vector
c of coefficients of these functions can (for fixed ().., (J)) be chosen by the Newton
Raphson iteration. Both)" and the (J's are estimated by the unbiased risk estimate
of Gu using RKPACK( available from netlibClresearch. att. com) as a subroutine
at each Newton iteration. Details of smoothing spline ANOVA decompositions may
be found in Wahba(1990) and in Gu and Wahba(1993) (also available by ftp to
ftp.stat.wisc.edu, cd to pub/wahba , in the file ssanova.ps.Z). In Wahba et
al(1993) op cit, we estimate the risk of diabetes given some of the attributes in the
Pima-Indian data base. There M was chosen partly by a screening process using
paramteric GLIM models and partly by a leaving out approximately 1/3 procedure.
Continuing work involves development of confidence intervals based on Gu(1992b),
development of numerical methods suitable for very large data sets based on Girard's(1991) randomized trace estimation, and further model selection issues.
In the Figures we provide some preliminary analyses of data from the Wisconsin Epidemiological Study of Diabetic Retinopathy (WESDR, Klein et al 1988).
The data used here is from people with early onset diabetes participating in the
WESDR study.
Figure 1(left) gives a plot of body mass index (bmi) (a measure of obesity) vs age (age) for 669 instances (subjects) in the WESDR study
that had no diabetic retinopathy or non proliferative retinopathy at the start of
the study. Those subjects who had (progressed) retinopathy four years later, are
marked as * and those with no progression are marked as '. The contours are
lines of estimated posterior standard deviation of the estimate p of the probability of progression. These contours are used to delineate a region in which p
is deemed to be reliable. Glycosylated hemoglobin (gly), a measure of blood
sugar control. was also used in the estimation of p. A model of the form
p eJ /(1 + eJ ), I(age, gly, bmi) I-' + h(age) + b? gly + h(bmi) + ha(age, bmi)
was selected using some of the screening procedures described in Wahba et al(1993),
along with an examination of the estimated multiple smoothing parameters, which
indicated that the linear term in gly was sufficient to describe the (quite strong)
dependence on gly. Figure l(right) shows the estimated probability of progression
=
=
"Soft" Classification with Smoothing Spline ANOVA
given by this model.
1(right), and Figure
interval. Interesting
20's with higher gly
Figure 2(left) gives cross sections of the fitted model of Figure
2(right) gives another cross section, along with its confidence
observations can be made, for example, persons in their late
and bmi are at greatest risk for progression of the disease .
...?.
........... - .: -..-...
............
:
..............
'
.
.... ....."::
:?
????
???
10
20
30
40
50
60
age (yr)
Figure 1: Left: Data and contours of constant posterior standard deviation at
the median gly, as a function of age and bmi. Right: Estimated probability of
progression at the median gly, as a function of age and bmi.
q
CD
o
q1 bmi
q2bmi
q3bmi
q4bmi
gy.q2
gy-q3
l:jI,,-??? <:AJian
bmi-median
o
o
10 20 30 40 50 60
age (yr)
10 20 30 40 50 60
age (yr)
10 20 30 40 50 60
age (yr)
Figure 2: Left: Eight cross sections of the right panel of Figure 1, Estimated probability of progression as a function of age, at four levels of bmi by two of gly.
q1, ... q4 are the quartiles at .125, .375, .625 and .875. Right: Cross section of the
right panel of Figure 1 for bmi and gly at their medians, as a function of age,
with Bayesian 'condifidence interval' (shaded) which generalizes Gu(1992b) to the
multivariate case.
421
422
Wahba, Wang, Gu, Klein, and Klein
Acknowledgements
Supported by NSF DMS-9121003 and DMS-9301511, and NEI-NIH EY09946 and
EY03083
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Wolpert and A. Lapedes, eds, and Proc. CLNL92, T. Petsche, ed, with permission
of all eds.
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7,079 | 85 | 850
Strategies for Teaching Layered Networks
Classification Tasks
Ben S. Wittner 1 and John S. Denker
AT&T Bell Laboratories
Holmdel, New Jersey 07733
Abstract
There is a widespread misconception that the delta-rule is in some sense guaranteed to
work on networks without hidden units. As previous authors have mentioned, there is
no such guarantee for classification tasks. We will begin by presenting explicit counterexamples illustrating two different interesting ways in which the delta rule can fail. We
go on to provide conditions which do guarantee that gradient descent will successfully
train networks without hidden units to perform two-category classification tasks. We
discuss the generalization of our ideas to networks with hidden units and to multicategory classification tasks.
The Classification Task
Consider networks of the form indicated in figure 1. We discuss various methods for
training such a network, that is for adjusting its weight vector, w. If we call the input
v, the output is g(w? v), where 9 is some function.
The classification task we wish to train the network to perform is the following. Given
two finite sets of vectors, Fl and F2, output a number greater than zero when a vector in
Fl is input, and output a number less than zero when a vector in F2 is input. Without
significant loss of generality, we assume that 9 is odd (Le. g( -s) == -g( s?. In that case,
the task can be reformulated as follows. Define 2
F :== Fl U {-v such that v E F2}
(1)
and output a number greater than zero when a vector in F is input. The former
formulation is more natural in some sense, but the later formulation is somewhat more
convenient for analysis and is the one we use. We call vectors in F, training vectors.
A Class of Gradient Descent Algorithms
We denote the solution set by
W :== {w such that g(w? v) > 0 for all v E F},
lCurrently at NYNEX Science and Technology, 500 Westchester Ave., White Plains, NY 10604
2 We use both A := Band B =: A to denote "A is by definition B".
@
American Institute of Physics 1988
(2)
851
output
inputs
Figure 1: a simple network
and we are interested in rules for finding some weight vector in W. We restrict our
attention to rules based upon gradient'descent down error functions E(w) of the form
E(w) =
L h(w . v).
(3)
VEF
The delta-rule is of this form with
1
h(w . v) = h6(W . v) := -(b - g(w . v))2
2
(4)
for some positive number b called the target (Rumelhart, McClelland, et al.). We call
the delta rule error function E 6 .
Failure of Delta-rule Using Obtainable Targets
Let 9 be any function that is odd and differentiable with g'(s) > 0 for all s. In this
section we assume that the target b is in the range of g. We construct a set F of
training vectors such that even though M' is not empty, there is a local minimum of E6
not located in W. In order to facilitate visualization, we begin by assuming that 9 is
linear. We will then indicate why the construction works for the nonlinear case as well.
We guess that this is the type of counter-example alluded to by Duda and Hart (p. 151)
and by Minsky and Papert (p. 15).
The input vectors are two dimensional. The arrows in figure 2 represent the training
vectors in F and the shaded region is W. There is one training vector, vI, in the second
quadrant, and all the rest are in the first quadrant. The training vectors in the first
quadrant are arranged in pairs symmetric about the ray R and ending on the line L.
The line L is perpendicular to R, and intersects R at unit distance from the origin.
Figure 2 only shows three of those symmetric pairs, but to make this construction work
we might need many. The point p lies on R at a distance of g-l(b) from the origin .
We first consider the contribution to E6 due to any single training vector, v. The
contribution is
(5)
(1/2)(b - g(w? v))2,
and is represented in figure 3 in the z-direction. Since 9 is linear and since b is in the
\
p,
,"
.....
\
\
,
,'c"
\
, ,.
, R
,
, L
\
X-axis
853
x-axis
Figure 3: Error surface
We now remove the assumption that 9 is linear. The key observation is that
dh6/ds == h/(s) = (b - g(s?( -g'(s?
(6)
still only has a single zero at g-l(b) and so h(s) still has a single minimum at g-l(b).
The contribution to E6 due to the training vectors in the first quadrant therefore still
has a global minimum on the xy-plane at the point p. So, as in the linear case, if there
are enough symmetric pairs of training vectors in the first quadrant, the value of Eo
at p can be made arbitrarily lower than the value along some circle in the xy-plane
centered around p, and E5 = Eo + El will have a local minimum arbitrarily near p.
Q.E.D.
Failure of Delta-rule Using Unobtainable Targets
We now consider the case where the target b is greater than any number in the range
of g. The kind of counter-example presented in the previous section no longer exists,
but we will show that for some choices of g, including the traditional choices, the delta
rule can still fail. Specifically, we construct a set F of training vectors such that even
though W is not empty, for some choices of initial weights, the path traced out by going
down the gradient of E5 never enters W.
854
y-axis
,
,
",.,-P
4
J'=----~----~--~
L
q , __ ....
:,
,
,
,
,
,:~
x-axis
Figure 4: Counter-example for unobtainable targets
We suppose that 9 has the following property. There exists a number r > 0 such that
. hs'( -rs)
hm h 5'()
=
S
_-00
(7)
o.
An example of such a 9 is
9(S)
2
= tanh(s) = 1 + e- 2., -
1,
(8)
for which any r greater than 1 will do.
The solid arrows in figure 4 represent the training vectors in F and the more darkly
shaded region is W. The set F has two elements,
and
v2
The dotted ray, R lies on the diagonal {y
= x}.
=mm[n
(9)
Since
(10)
855
the gradient descent algorithm follows the vector field
-v E(w) =
-h/(w?
h/(w. V 2 )V 2 .
V1)V 1 -
(11)
The reader can easily verify that for all won R,
(12)
So by equation (7), if we constrain w to move along R,
.
-h/(w. vI)
hm
,
2 = O.
w ...... oo -h o (w . v )
(13)
Combining equations (11) and (13) we see that there is a point q somewhere on R such
that beyond q, - V E( w) points into the region to the right of R, as indicated by the
dotted arrows in figure 4.
Let L be the horizontal ray extending to the right from q. Since for all s,
g'(s) > 0
and
b> g(s),
(14)
o.
(15)
we get that
- h/(s) = (b - g(s?g'(s) >
So since both vI and v 2 have a positive y-component, -V E(w) also has a positive
y-component for all w. So once the algorithm following -V E enters the region above
L and to the right of R (indicated by light shading in figure 4), it never leaves. Q.E.D.
Properties to Guarantee Gradient Descent Learning
In this section we present three properties of an error function which guarantee that
gradient descent will not fail to enter a non-empty W.
We call an error function of the form presented in equation (3) well formed if h is
differentiable and has the following three properties.
1. For all s, -h'( s) ~ 0 (i.e. h does not push in the wrong direction).
2. There exists some f > 0 such that -h'(s)
if there is a misclassification).
~ f
for all s
~
0 (i.e. h keeps pushing
3. h is bounded below.
Proposition 1 If the error junction is well formed, then gradient descent is guaranteed
to enter W, provided W is not empty.
856
The proof proceeds by contradiction. Suppose for some starting weight vector the path
traced out by gradient descent never enters W. Since W is not empty, there is some
non-zero w* in W. Since F is finite,
A := min{w*. v such that v E F} -:> O.
(16)
Let wet) be the path traced out by the gradient descent algorithm. So
w'(t) = -VE(w(t? =
I:: -h'(w(t) ?v)v
for all t.
(17)
vEF
Since we are assuming that at least one training vector is misclassified at all times, by
properties 1 and 2 and equation (17),
w* . w'(t) 2: fA
So
Iw'(t)1 2: fA/lw*1 =:
for all t.
e> 0
(18)
for all t.
(19)
By equations (17) and (19),
dE(w(t?/dt = V E? w'(t)
= -w'(t) . w'(t) ~ -e < 0
for all t.
(20)
This means that
E(w(t?
--+ -00
as
t
--+ 00.
(21)
But property 3 and the fact that F is finite guarantee that E is bounded below. This
contradicts equation (21) and finishes the proof.
Consensus and Compromise
So far we have been concerned with the case in which F is separable (i.e. W is not
empty). What kind of behavior do we desire in the non-separable case? One might
hope that the algorithm will choose weights which produce correct results for as many
of the training vectors as possible. We suggest that this is what gradient descent using
a well formed error function does.
From investigations of many well formed error functions, we suspect the following well
formed error function is representative. Let g( s) = s, and for some b > 0, let
h(S)={ (b-s)2
o
ifs~~;
otherwIse.
(22)
In all four frames of figure 5 there are three training vectors. Training vectors 1 and 2
are held fixed while 3 is rotated to become increasingly inconsistent with the others. In
frames (i) and (ii) F is separable. The training set in frame (iii) lies just on the border
between separability and non-separability, and the one in frame (iv) is in the interior of
857
i)
3
ii )
3
2
1
iv)
iii)
2
3
...
L.1
2
1
3
Figure 5: The transition between seperability and non-seperability
the non-separable regime. Regardless of the position of vector 3, the global minimum
of the error function is the only minimum.
In frames (i) and (ii), the error function is zero on the shaded region and the shaded
region is contained in W. As we move training vector number 3 towards its position in
frame (iii), the situation remains the same except the shaded region moves arbitrarily
far from the origin. At frame (iii) there is a discontinuity; the region on which the
error function is at its global minimum is now the one-dimensional ray indicated by
the shading. Once training vector 3 has moved into the interior of the non-separable
regime, the region on which the error function has its global minimum is a point closer
to training vectors 1 and 2 than to 3 (as indicated by the "x" in frame (iv?.
If all the training vectors can be satisfied, the algorithm does so; otherwise, it tries to
satisfy as many as possible, and there is a discontinuity between the two regimes. We
summarize this by saying that it finds a consensus if possible, otherwise it devises a
compromise.
Hidden Layers
For networks with hidden units, it is probably impossible to prove anything like proposition 1. The reason is that even though property 2 assures that the top layer of weights
858
gets a non-vanishing error signal for misclassified inputs, the lower layers might still get
a vanishingly weak signal if the units above them are operating in the saturated regime.
We believe it is nevertheless a good idea to use a well formed error function when
training such networks. Based upon a probabilistic interpretation of the output of the
network, Baum and Wilczek have suggested using an entropy error function (we thank
J.J. Hopfield and D.W. Tank for bringing this to our attention). Their error function
is well formed. Levin, Solla, and Fleisher report simulations in which switching to the
entropy error function from the delta-rule introduced an order of magnitude speed-up
of learning for a network with hidden units.
Multiple Categories
Often one wants to classify a given input vector into one of many categories. One popular
way of implementing multiple categories in a feed-forward network is the following. Let
the network have one output unit for each category. Denote by oj(w) the output of
the j-th output unit when input v is presented to the network having weights w. The
network is considered to have classified v as being in the k-th category if
or(w) > oj(w) for all j ~ k.
(23)
The way such a network is usually trained is the generalized delta-rule (Rumelhart,
McClelland, et al.). Specifically, denote by c(v) the desired classification of v and let
b"! .= {b
1
?
if j = c(v);
-b otherwise,
(24)
for some target b > O. One then uses the error function
E(w):=
EE
(bj - oj (w?)
v
.
2
?
(25)
3
This formulation has several bothersome aspects. For one, the error function is not will
formed. Secondly, the error function is trying to adjust the outputs, but what we really
care about is the differences between the outputs. A symptom of this is the fact that
the change made to the weights of the connections to any output unit does not depend
on any of the weights of the connections to any of the other output units.
To remedy this and also the other defects of the delta rule we have been discussing, we
suggest the following. For each v and j, define the relative coordinate
(26)
859
What we really want is all the
13 to be positive, so use the error function
E(w):=
E E
h (f3j(w))
(27)
v #c(v)
for some well formed h. In the simulations we have run, this does not always help, but
sometimes it helps quite a bit.
We have one further suggestion. Property 2 of a well formed error function (and the
fact that derivatives are continuous) means that the algorithm will not be completely
satisfied with positive 13; it will try to make them greater than zero by some non-zero
margin. That is a good thing, because the training vectors are only representatives of
the vectors one wants the network to correctly classify. Margins are critically important
for obtaining robust performance on input vectors not in the training set. The problem
is that the margin is expressed in meaningless units; it makes no sense to use the same
numerical margin for an output unit which varies a lot as is used for an output unit
which varies only a little. We suggest, therefore, that for each j and v, keep a running
estimate of uj(w), the variance of f3J(w), and replace f3J(w) in equation (27) by
f3J (w)/uj (w).
(28)
Of course, when beginning the gradient descent, it is difficult to have a meaningful
estimate of uj(w) because w is changing so much, but as the algorithm begins to
converge, your estimate can become increasingly meaningful.
References
1. David Rumelhart, James McClelland, and the PDP Research Group, Parallel Dis-
tributed Processing, MIT Press, 1986
2. Richard Duda and Peter Hart, Pattern Classification and Scene Analysis, John
Wiley & Sons, 1973.
3. Marvin Minsky and Seymour Papert, "On Perceptrons", Draft, 1987.
4. Eric Baum and Frank Wilczek, these proceedings.
5. Esther Levin, Sara A. Solla, and Michael Fleisher, private communications.
| 85 |@word h:1 illustrating:1 private:1 duda:2 r:1 simulation:2 solid:1 shading:2 initial:1 john:2 numerical:1 remove:1 nynex:1 leaf:1 guess:1 plane:2 beginning:1 vanishing:1 draft:1 along:2 become:2 prove:1 ray:4 behavior:1 little:1 begin:3 provided:1 bounded:2 what:4 kind:2 finding:1 guarantee:5 ifs:1 wrong:1 unit:14 positive:5 local:2 seymour:1 switching:1 tributed:1 path:3 might:3 shaded:5 sara:1 range:2 perpendicular:1 bell:1 convenient:1 quadrant:5 suggest:3 get:3 interior:2 layered:1 impossible:1 baum:2 go:1 attention:2 starting:1 regardless:1 contradiction:1 rule:12 coordinate:1 target:7 construction:2 suppose:2 us:1 origin:3 element:1 rumelhart:3 located:1 enters:3 fleisher:2 region:9 solla:2 counter:3 mentioned:1 trained:1 depend:1 compromise:2 upon:2 f2:3 eric:1 completely:1 easily:1 hopfield:1 jersey:1 various:1 represented:1 intersects:1 train:2 quite:1 otherwise:4 differentiable:2 vanishingly:1 combining:1 moved:1 empty:6 extending:1 produce:1 ben:1 rotated:1 help:2 oo:1 odd:2 indicate:1 direction:2 correct:1 centered:1 implementing:1 generalization:1 really:2 investigation:1 proposition:2 secondly:1 mm:1 around:1 considered:1 bj:1 wet:1 tanh:1 iw:1 successfully:1 hope:1 mit:1 always:1 seperability:2 ave:1 sense:3 esther:1 el:1 hidden:6 going:1 misclassified:2 interested:1 tank:1 classification:8 field:1 construct:2 never:3 once:2 having:1 others:1 report:1 richard:1 ve:1 minsky:2 adjust:1 saturated:1 light:1 held:1 closer:1 xy:2 iv:3 circle:1 desired:1 classify:2 levin:2 varies:2 probabilistic:1 physic:1 michael:1 satisfied:2 choose:1 american:1 derivative:1 de:1 satisfy:1 vi:3 later:1 try:2 lot:1 parallel:1 contribution:3 formed:10 variance:1 weak:1 critically:1 classified:1 definition:1 failure:2 james:1 proof:2 adjusting:1 popular:1 f3j:4 obtainable:1 feed:1 dt:1 formulation:3 arranged:1 though:3 symptom:1 generality:1 just:1 d:1 horizontal:1 wilczek:2 nonlinear:1 widespread:1 indicated:5 believe:1 facilitate:1 verify:1 remedy:1 former:1 symmetric:3 laboratory:1 white:1 anything:1 won:1 generalized:1 trying:1 presenting:1 interpretation:1 significant:1 counterexample:1 enter:2 teaching:1 longer:1 surface:1 operating:1 arbitrarily:3 discussing:1 devise:1 minimum:8 greater:5 somewhat:1 care:1 eo:2 converge:1 signal:2 ii:3 multiple:2 wittner:1 hart:2 represent:2 sometimes:1 want:3 rest:1 meaningless:1 bringing:1 probably:1 suspect:1 thing:1 inconsistent:1 call:4 ee:1 near:1 iii:4 enough:1 concerned:1 finish:1 restrict:1 idea:2 peter:1 reformulated:1 band:1 category:6 mcclelland:3 dotted:2 delta:10 correctly:1 group:1 key:1 four:1 nevertheless:1 traced:3 changing:1 v1:1 defect:1 run:1 saying:1 reader:1 holmdel:1 bit:1 fl:3 layer:3 guaranteed:2 marvin:1 constrain:1 your:1 scene:1 aspect:1 speed:1 min:1 separable:5 increasingly:2 contradicts:1 separability:2 son:1 alluded:1 visualization:1 equation:7 remains:1 discus:2 assures:1 fail:3 junction:1 h6:1 denker:1 v2:1 top:1 running:1 pushing:1 somewhere:1 multicategory:1 uj:3 move:3 strategy:1 fa:2 traditional:1 diagonal:1 gradient:12 distance:2 thank:1 consensus:2 reason:1 assuming:2 difficult:1 frank:1 perform:2 observation:1 finite:3 descent:11 situation:1 communication:1 frame:8 pdp:1 introduced:1 david:1 pair:3 connection:2 darkly:1 discontinuity:2 beyond:1 suggested:1 proceeds:1 below:2 usually:1 pattern:1 regime:4 summarize:1 including:1 oj:3 misclassification:1 natural:1 technology:1 axis:4 hm:2 relative:1 loss:1 interesting:1 suggestion:1 course:1 dis:1 institute:1 plain:1 ending:1 transition:1 author:1 made:2 forward:1 far:2 keep:2 global:4 continuous:1 why:1 robust:1 obtaining:1 e5:2 arrow:3 border:1 representative:2 ny:1 wiley:1 papert:2 position:2 explicit:1 wish:1 lie:3 lw:1 down:2 misconception:1 exists:3 magnitude:1 push:1 margin:4 entropy:2 desire:1 contained:1 expressed:1 towards:1 replace:1 change:1 specifically:2 except:1 called:1 meaningful:2 perceptrons:1 e6:3 |
7,080 | 850 | How to Describe Neuronal Activity:
Spikes, Rates, or Assemblies?
Wulfram Gerstner and J. Leo van Hemmen
Physik-Department der TU Miinchen
D-85748 Garching bei Miinchen, Germany
Abstract
What is the 'correct' theoretical description of neuronal activity?
The analysis of the dynamics of a globally connected network of
spiking neurons (the Spike Response Model) shows that a description by mean firing rates is possible only if active neurons fire incoherently. If firing occurs coherently or with spatio-temporal correlations, the spike structure of the neural code becomes relevant.
Alternatively, neurons can be gathered into local or distributed ensembles or 'assemblies'. A description based on the mean ensemble
activity is, in principle, possible but the interaction between different assemblies becomes highly nonlinear. A description with spikes
should therefore be preferred.
1
INTRODUCTION
Neurons communicate by sequences of short pulses, the so-called action potentials
or spikes. One of the most important problems in theoretical neuroscience concerns
the question of how information on the environment is encoded in such spike trains:
Is the exact timing of spikes with relation to earlier spikes relevant (spike or interval
code (MacKay and McCulloch 1952) or does the mean firing rate averaged over several spikes contain all important information (rate code; see, e.g., Stein 1967)? Are
spikes of single neurons important or do we have to consider ensembles of equivalent
neurons (ensemble code)? If so, can we find local ensembles (e.g., columns; Hubel
and Wiesel 1962) or do neurons form 'assemblies' (Hebb 1949) distributed all over
the network?
463
464
Gerstner and van Hemmen
2
SPIKE RESPONSE MODEL
We consider a globally connected network of N neurons with 1 ~ i ~ N. A neuron i
fires, if its membrane potential passes a threshold (). A spike at time t{ is described
by a 6-pulse; thus Sf (t) = L:~=1 6(t - t{) is the spike train of neuron i. Spikes are
labelled such that tt is the most recent spike and tf is the Fth spike going back in
time.
In the Spike Response Model, short SRM, (Gerstner 1990, Gerstner and van Hemmen 1992) a neuron is characterized by two different response junctions, f and "1re f .
Spikes which neuron i receives from other neurons evoke a synaptic potential
(1)
where the response kernel
0
f(S) = { -::-r,,_a exp (,,_a
- -T,T.
tr
tr )
for s < Ll tr
lor s > u r
CAt
(2)
describes a typical excitatory or inhibitory postsynaptic potential; see Fig. 1. The
weight Jij is the synaptic efficacy of a connection from j to i, Ll tr is the axonal (and
synaptic) transmission time, and T" is a time constant of the postsynaptic neuron.
The origin S 0 in (2) denotes the firing time of a presynaptic spike. In simulations
we usually assume T" = 2 ms and for Ll tr a value between 1 and 4 ms
=
Similarly, spike emission induces refractoriness immediately after spiking. This is
modelled by a refractory potential
(3)
with a refractory function
"1
re f ()
s
={
-00
"1o/(s _ ,ref)
for
for
S
~ ,ref
S
> ,ref.
(4)
For 0 ~ s ~ ,ref the neuron is in the absolute refractory period and cannot spike at
all whereas for s > ,ref spiking is possible but difficult (relative refractory period).
To put it differently, () - "1 ref (s) describes an increased threshold immediately after
spiking; cf. Fig. 1. In simulations, ,ref is taken to be 4 ms. Note that, for the sake
of simplicity, we assume that only the most recent spike Sf induces refractoriness
whereas all past spikes Sf contribute to the synaptic potential; cf., Eqs. (1) and (3).
How to Describe Neuronal Activity: Spikes, Rates, or Assemblies?
Fig 1 Response functions.
9-n(s)
w
f 0.5
CD
0.0
'-........o...-_Ll------'-_L-..--'---.----l---=::::t=~
0.0
Immediately after firing at 8 =
the effective threshold is increased to (J - TIre! (8) (dashed).
The form of an excitatory postsynaptic potential (EPSP) is
described by the response function f( 8) (solid). It is delayed by
a time ~ tr. The arrow denotes
the period Tosc of coherent oscillations; d. Section 5.
o
5.0
10.0
5[m5]
15.0
20.0
The total membrane potential is the sum of both parts, i.e.
hi(t) = h~ef (t)
+ h:yn(t).
(5)
Noise is included by introduction of a firing probability
PF(h; 6t) =
r- 1 (h)
6t.
(6)
where 6t is an infinitesimal time interval and r(h) is a time constant which depends
on the momentary value of the membrane potential in relation to the threshold ().
In analogy to the chemical reaction constant we assume
r(h) = ro exp[-,B(h - (})],
(7)
where ro is the response time at threshold. The parameter ,B determines the amount
of noise in the system. For,B --+ 00 we recover the noise-free behavior, i.e., a neuron
fires immediately, if h > () (r --+ 0), but it cannot fire, if h < () (r --+ (0). Eqs. (1),
(3), (5), and (6) define the spiking dynamics in a network of SRM-neurons.
3
FIRING STATISTICS
We start our considerations with a large ensemble of identical neurons driven by the
same arbitrary synaptic potential h 3yn (t) . We assume that all neurons have fired a
first spike at t = t{ . Thus the total membrane potential is h(t) = hsyn(t) + 7]re f (tto. If h(t) slowly approaches (), some of the neurons will fire again. We now ask
for the probability that a neuron which has fired at time t{ will fire again at a later
time t. The conditional probability p~2\tlt{) that the next spike of a given neuron
occurs at time t > t{ is
p~2)(tlt{) = r-l[h(t)] exp {
-1;
r- 1 [h(S')]dS'} .
(8)
The exponential factor is the portion of neurons that have survived from time t{ to
time t without firing again and the prefactor r- 1 [h(t)] is the instantaneous firing
probability (6) at time t. Since the refractory potential is reset after each spike,
the spiking statistics does not depend on earlier spikes, in other words, it is fully
described by p~2)(tlt{). This will be used below; cf. Eq. (14) .
465
466
Gerstner and van Hemmen
=
As a special case, we may consider constant synaptic input h 3yn h O? In this case,
(8) yields the distribution of inter-spike intervals in a spike train of a neuron driven
by constant input h O? The mean firing rate at an input level h O is defined as the
inverse of the mean inter-spike interval. Integration by parts yields
I[h o] =
{J.;dt(t-t{lP~2)(tlt{l}
-I =
{J.oodsexp{-lT-I[hO+~"f (s'l]ds'} }
-I
(9)
Thus both firing rate and interval distribution can be calculated for arbitrary inputs.
4
ASSEMBLY FORMATION AND NETWORK
DYNAMICS
We now turn to a large, but structured network. Structure is induced by the
formation of different assemblies in the system. Each neuronal assembly aP. (Hebb
1949) consists of neurons which have the tendency to be active at the same time.
Following the traditional interpretation, active means an elevated mean firing rate
during some reasonable period of time. Later, in Section 5.3, we will deal with a
different interpretation where active means a spike within a time window of a few
ms. In any case, the notion of simultaneous activity allows to define an activity
pattern {~r, 1 :::; i :::; N} with ~r
1 if i E aP. and ~r
0 otherwise. Each neuron
may belong to different assemblies 1 :::; I-l :::; q. The vector
,~n is the
'identity card' of neuron i, e.g.,
= (1,0,0,1,0) says that neuron i belongs to
assembly 1 and 4 but not to assembly 2,3, and 5.
=
ei
=
ei = (a, ...
Note that, in general, there are many neurons with the same identity card. This
can be used to define ensembles (or sublattices) L(x) of equivalent neurons, i.e.,
L(x) = {ilei = x} (van Hemmen and Kiihn 1991). In general, the number of
neurons IL(x)1 in an ensemble L(x) goes to infinity if N --;. 00, and we write
IL(x)1 = p(x)N. The mean activity of an ensemble L(x) can be defined by
A(x, t)
= at--+o
lim lim
N--+oo
IL(x)I- 1
I
L
iEL(X)
t
t +at
S[ (t)dt.
(10)
In the following we assume that the synaptic efficacies have been adjusted according
to some Hebbian learning rule in a way that allows to stabilize the different activity
patterns or assemblies ap.. To be specific, we assume
J q q
Jij = ~ L L Qp.vpost(~r)pre(~j)
(11)
p.=lv=l
where post(x) and pre(x) are some arbitrary functions characterizing the pre- and
postsynaptic part of synaptic learning. Note that for Qp.v
fJp.v and post(x) and
pre(x) linear, Eq. (11) can be reduced to the usual Hebb rule.
=
With the above definitions we can write the synaptic potential of a neuron i E L(x)
in the following form
q
h 3yn (x , t)
= Jo L
q
(>0
L Qp.vpost(xp.) Lpre(zV) 10
p.=lv=l
z
0
f(s')p(z)A(z, t - s')ds'. (12)
How to Describe Neuronal Activity: Spikes, Rates, or Assemblies?
We note that the index i and j has disappeared and there remains a dependence
upon x and z only. The activity of a typical ensemble is given by (Gerstner and
van Hemmen 1993, 1994)
A(x, t)
=
1
00
p?)(tlt - s)A(x, t - s)ds
where
{-1
p~2)(tlt-s) = r- 1 [h',yn(x, t)+7]re f (s)] exp
(13)
3r - 1 [h3 yn(x, t - s+s' )+7]re f (s')]ds' }
(14)
is the conditional probability (8) that a neuron i E L(x) which has fired at time
t-s fires again at time t. Equations (12) - (14) define the ensemble dynamics of the
network.
5
DISCUSSION
5.1
ENSEMBLE CODE
Equations. (12) - (14) show that in a large network a description by mean ensemble
activities is, in principle, possible. A couple of things, however, should be noted.
First, the interaction between the activity of different ensembles is highly nonlinear.
It involves three integrations over the past and one exponentiation; cf. (12) - (14).
If we had started theoretical modeling with an approach based on mean activities,
it would have been hard to find the correct interaction term.
Second, L(x) defines an ensemble of equivalent neurons which is a subset of a given
assembly al-'. A reduction of (12) to pure assembly activities is, in general, not
possible. Finally, equivalent neurons that form an ensemble L(x) are not necessarily
situated next to each other. In fact, they may be distributed all over the network;
cf. Fig. 2. In this case a local ensemble average yields meaningless results. A
theoretical model based on local ensemble averaging is useful only if we know that
neighboring neurons have the same 'identity card'.
activity
a)
~': t
100
l
150
200
time [ms]
b)
30
_ 20
~
???
..
..
..
..
..
..
..
..
.. ..
.....?:..:.. :': '....:.. :.: .. :..:': ': -:
....
\
. .. .... . . .. .. . .. .. . .. ..
.. -. -...
-. -. .. I
~ 10 ? ? ?- .- .- ..... - .... -.
,
30
r::::::
20
:J
?? :. - . . . . . . . . . . . ..
. .. . . . .. .... .. .. .. .. .. .. .. ..
. " ...... ...- .- .................... " .. " .. " .ill.
I
.- .- .- .-..
o???~????-???????
100
150
200
time [ms]
5.2
rate [Hz]
0)
..
::>
10
:::I
0 0
100
200
Fig. 2
Stationary activity (incoherent
firing). In this case a description by firing rates is possible.
(a) Ensemble averaged activity
A(x, t). (b) Spike raster of 30
neurons out of a network of
4000. (c) Time-averaged mean
firing rate f. We have two different assemblies, one of them
active (d tr
2 ms, f3 5).
=
=
rate [Hz]
RATE CODE
Can the system of Eqs. (12) -(14) be transformed into a rate description? In general,
this is not the case but if we assume that the ensemble activities are constant in
467
468
Gerstner and van Hemmen
1.0
.---~--~----~--~--------~--~---.----~--~----~--,
O.B
O.B
0.4
~.x-2
~.)(-3.5
0.2
0.0
o
100
200
400
300
500
BOO
Zeit [rn5]
Fig. 3 Stability of stationary states.The postsynaptic potential h~yn is plotted as a function
of time. Every 100 ms the delay Ll tr has been increased by 0.5 ms. In the stationary state
(Lltr = 1.5 ms and Ll tr = 3.5 ms), active neurons fire regularly with rate T;l = 1/5.5 ms.
For a delay Ll tr > 3.5 ms, oscillations with period Wl = 27r /Tp build up rapidly. For
intermediate delays 2 ~ Ll tr ~ 2.5 small-amplitude oscillations with twice the frequency
occur. Higher harmonics are suppressed by noise (/3 = 20).
=
time, i.e., A(x, t)
A(x), then an exact reduction is possible. The result
fixed-point equation (Gerstner and van Hemmen 1992)
q
A(x)
= f[Jo L
f[h,yn]
a
q
L Q~lIpost(X~) L pre(zll)p(z)A(z)]
= {J.oo dsexp{-
(15)
z
~=lll=l
where
IS
1.'
r- 1 [h,yn
+ ~"J(8')]ds'}} -1
(16)
is the mean firing rate (9) of a typical neuron stimulated by a synaptic input h3yn.
Constant activities correspond to incoherent, stationary firing and in this case a
rate code is sufficient; cf. Fig. 2.
Two points should, however, be kept in mind. First, a stationary state of incoherent
firing is not necessarily stable. In fact, in a noise-free system the stationary state
is always unstable and oscillations build up (Gerstner and van Hemmen 1993). In
a system with noise, the stability depends on the noise level f3 and the delay Ll tr of
axonal and synaptic transmission (Gerstner and van Hemmen 1994). This is shown
in Fig. 3 where the delay Ll tr has been increased every 100 ms. The frequency of
the small-amplitude oscillation around the stationary state is approximately equal
to the mean firing rate (16) in the stationary state or higher harmonics thereof.
A small-amplitude oscillation corresponds to partially synchronized activity. Note
that for Ll tr = 4 ms a large-amplitude oscillation builds up. Here all neurons fire in
nearly perfect synchrony; cf. Fig. 4. In the noiseless case f3 - 00, the oscillations
period of such a collective or 'locked' oscillation can be found from the threshold
condition
T",
= inf {s I0 = ~"J (8) + Jo~ f(nS)} .
(17)
In most cases the contribution with n = 1 is dominant which allows a simple graphical solution. The first intersection of the effective threshold () - TJ ref (s) with the
How to Describe Neuronal Activity: Spikes. Rates. or Assemblies?
weighted EPSP JOf( s) yields the oscillation period; cf. Fig 1. An analytical argument shows that locking is stable only if ;" dTooc > 0 (Gerstner and van Hemmen
1993).
activity
a)
~:lliHHHlUHHHj
100
1~
200
time [ms]
b)
~
,., 20
~
!
10
................. .
I f S S S S SIS ) S S S S \ 'a \ \
... ...... ... .... ..
. .. . . . . . . . .. . . . . ..
?.\'\\'111".?.?'1111
\ \ \ \ \ , \ I 1 \ ???? , \ \ \ \ \
rate [Hz]
0)
]
10
t==:::;;=
" 1 ' 1 1 1 \ \ 1 \ \ 1 " ' .. 1
o ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ??
100
150
time [ms]
200
0 0
100
200
rate [Hz]
Fig. 4
Oscillatory activity (coherent
firing). In this case a description by firing rates must be combined with a description by ensemble activities. (a) Ensemble
averaged activity A(x, t). (b)
Spike raster of 30 neurons out
of a network of 4000. (c) Timeaveraged mean firing rate f. In
this simulation, we have used
Ll tr = 4 ms and f3 = 8.
Second, even if the incoherent state is stable and attractive, there is always a transition time before the stationary state is assumed. During this time, a rate description
is insufficient and we have to go back to the full dynamic equations (12) - (14). Similarly, if neurons are subject to a fast time-dependent external stimulus, a rate code
fails .
5.3
SPIKE CODE
A superficial inspection of Eqs. (12) - (14) gives the impression that all information
about neuronal spiking has disappeared. This is, however, false. The term A(x, t-s)
in (13) denotes all neurons with 'identity card' x that have fired at time t-s . The
integration kernel in (13) is the conditional probability that one of these neurons
fires again at time t. Keeping t - s fixed and varying t we get the distribution
of inter-spike intervals for neurons in L(x). Thus information on both spikes and
intervals is contained in (13) and (14).
We can make use of this fact, if we consider network states where in every time step a
different assembly is active. This leads to a spatia-temporal spike pattern as shown
in Fig. 5. To transform a specific spike pattern into a stable state of the network
we can use a Hebbian learning rule. However, in contrast to the standard rule, a
synapse is strenthened only if pre- and postsynaptic activity occurs simultaneously
within a time window of a few ms (Gerstner et al. 1993). Note that in this case,
averaging over time or space spoils the information contained in the spike pattern.
5.4
CONCLUSIONS
Equations . (12) - (14) show that in our large and fully connected network an
ensemble code with an appropriately chosen ensemble is sufficient. If, however, the
efficacies (11) and the connection scheme become more involved, the construction
of appropriate ensembles becomes more and more difficult. Also, in a finite network
we cannot make use of the law of large number in defining the activities (10). Thus,
in general, we should always start with a network model of spiking neurons.
469
470
Gerstner and van Hemmen
=: ]
activity
a)
~~C
100
150
200
time [ms]
b)
..
30
o
.. 20
0
g
!5
!
.. .
?
10
0
..
?0
rata [Hz]
n-"~----'
20
?
.....
?
10
???
e.
o '--_ _o-".'___---:-~---"'----~
1()0
0)
30
200
0 ,-,-.
(---''---'.
0-
100
200
Fig. 5
Spatio-temporal spike pattern.
In this case, neither firing rates
nor locally averaged activities
contain enough information to
describe the state of the network. (a) Ensemble averaged
activity A(t). (b) Spike raster of
30 neurons out of a network of
4000. (c) Time-averaged mean
firing rate f.
rata [Hz]
Acknowledgements: This work has been supported by the Deutsche Forschungsgemeinschaft (DFG) under grant No. He 1729/2-1.
References
Gerstner W (1990) Associative memory in a network of 'biological' neurons. In:
Advances in Neural Information Processing Systems 3, edited by R.P. Lippmann,
J .E. Moody, and D.S. Touretzky (Morgan Kaufmann, San Mateo, CA) pp 84-90
Gerstner Wand van Hemmen JL (1992a) Associative memory in a network of
'spiking' neurons. Network 3:139-164
Gerstner W, van Hemmen JL (1993) Coherence and incoherence in a globally coupled ensemble of pulse-emitting units. Phys. Rev. Lett. 71:312-315
Gerstner W, Ritz R, van Hemmen JL (1993b) Why spikes? Hebbian learning and
retrieval of time-resolved excitation patterns. BioI. Cybern. 69:503-515
Gerstner Wand van Hemmen JL (1994) Coding and Information processing in
neural systems. In: Models of neural networks, Vol. 2, edited by E. Domany, J .L.
van Hemmen and K. Schulten (Springer-Verlag, Berlin, Heidelberg, New York) pp
Iff
Hebb DO (1949) The Organization of Behavior. Wiley, New York
van Hemmen JL and Kiihn R(1991) Collective phenomena in neural networks. In:
Models of neural networks, edited by E. Domany, J .L. van Hemmen and K. Schulten
(Springer-Verlag, Berlin, Heidelberg, New York) pp Iff
Hubel DH, Wiesel TN (1962) Receptive fields, binocular interaction and functional
architecture in the cat's visual cortex. J. Neurophysiol. 28:215-243
MacKay DM, McCulloch WS (1952) The limiting information capacity of a neuronal
link. Bull. of Mathm. Biophysics 14:127-135
Stein RB (1967) The information capacity of nerve cells using a frequency code.
Biophys. J. 7:797-826
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7,081 | 851 | Learning Temporal Dependencies in
Connectionist Speech Recognition
Steve Renals
Mike Hocbberg
Tony Robinson
Cambridge University Engineering Department
Cambridge CB2 IPZ, UK
{sjr,mmh,ajr}@eng.cam.ac.uk
Abstract
Hybrid connectionistfHMM systems model time both using a Markov
chain and through properties of a connectionist network. In this paper,
we discuss the nature of the time dependence currently employed in our
systems using recurrent networks (RNs) and feed-forward multi-layer
perceptrons (MLPs). In particular, we introduce local recurrences into a
MLP to produce an enhanced input representation. This is in the form
of an adaptive gamma filter and incorporates an automatic approach for
learning temporal dependencies. We have experimented on a speakerindependent phone recognition task using the TIMIT database. Results
using the gamma filtered input representation have shown improvement
over the baseline MLP system. Improvements have also been obtained
through merging the baseline and gamma filter models.
1 INTRODUCTION
The most common approach to large-vocabulary, talker-independent speech recognition
has been statistical modelling with hidden Markov models (HMMs). The HMM has an
explicit model for time specified by the Markov chain parameters. This temporal model
is governed by the grammar and phonology of the language being modelled. The acoustic
signal is modelled as a random process of the Markov chain and adjoining local temporal
information is assumed to be independent. This assumption is certainly not the case and a
great deal of research has addressed the problem of modelling acoustic context.
Standard HMM techniques for handling the context dependencies of the signal have ex-
1051
1052
Renals, Hochberg, and Robinson
plicitly modelled all the n-tuples of acoustic segments (e.g., context-dependent triphone
models). Typically, these systems employ a great number of parameters and, subsequently,
require massive amounts of training data and/or care in smoothing of the parameters. Where
the context of the model is greater than two segments, an additional problem is that it is
very likely that contexts found in testing data are never observed in the training data.
Recently, we have developed state-of-the-art continuous speech recognition systems using
hybrid connectionistlHMM methods (Robinson, 1994; Renals et aI., 1994). These hybrid
connectionistlHMM systems model context at two levels (although these levels are not
necessarily at distinct scales). As in the traditional HMM, a Markov process is used to
specify the duration and lexical constraints on the model. The connectionist framework
provides a conditional likelihood estimate of the local (in time) acoustic waveform given
the Markov process. Acoustic context is handled by either expanding the network input to
include multiple, adjacent input frames, or using recurrent connections in the network to
provide some memory of the previous acoustic inputs.
2 DEPTH AND RESOLUTION
Following Principe et al. (1993), we may characterise the time dependence displayed by a
particular model in terms of depth and resolution. Loosely speaking, the depth tells us how
far back in time a model is able to look l , and the resolution tells us how accurately the past
to a given depth may be reconstructed. The baseline models that we currently use are very
different in terms of these characteristics.
Multi-layer Perceptron
The feed-forward multi-layer perceptron (MLP) does not naturally model time, but simply
maps an input to an output. Crude temporal dependence may be imparted into the system by
using a delay-lined input (figure 1a); an extension of this approach is the time-delay neural
network (TDNN). The MLP may be interpreted as acting as a FIR filter. A delay-lined
input representation may be characterised as having low depth (limited by the delay line
length) and high resolution (no smoothing).
Recurrent Network
The recurrent network (RN) models time dependencies of the acoustic signal via a fullyconnected, recurrent hidden layer (figure 1b). The RN has a potentially infinite depth
(although in practice this is limited by available training algorithms) and low resolution,
and may be regarded as analogous to an IIR filter. A small amount of future context is
available to the RN, through a four frame target delay.
Experiments
Experiments on the DARPA Resource Management (RM) database have indicated that
the tradeoff between depth and resolution is important. In Robinson et al. (1993), we
compared different acoustic front ends using a MLP and a RN. Both networks used 68
1In the language of section 3, the depth may be expressed as the mean duration, relative to the
target, of the last kernel in a filter that is convolved with the input.
Learning Temporal Dependencies in Connectionist Speech Recognition
p(q" I X~~), Vk = I , .... K
u(t)
y(t-4)
x(t)
Hidden Layer
512 - 1,024 hidden units
Xn_c
.,.
xn_1
xn+1
...
xn+c
(a) Multi-layer Perceptron
(b) Recurrent Network
Figure 1: Connectionist architectures used for speech recognition.
outputs (corresponding to phones); the MLP used 1000 hidden units and the RN used
256 hidden units. Both architectures were trained using a training set containing 3990
sentences spoken by 109 speakers. Two different resolutions were used in the front-end
computation of mel-frequency cepstral coefficients (MFCCs): one with a 20ms Hamming
window and a lOms frame step (referred to as 20110), the other with a 32ms Hamming
window and a 16ms frame step (referred to as 32116). A priori, we expected the higher
resolution frame rate (20/10) to produce a higher performance recogniser because rapid
speech events would be more accurately modelled. While this was the case for the MLP,
the RN showed better results using the lower resolution front end (32/16) (see table 1). For
the higher resolution front-end, both models require a greater depth (in frames) for the same
context (in milliseconds). In these experiments the network architectures were constant so
increasing the resolution of the front end results in a loss of depth.
Net
RN
RN
MLP
MLP
Front End
20/10
32116
20/10
32/16
feb89
6.1
5.9
5.7
6.6
Word Error Rate %
oct89 feb91 sep92
12.1
7.6
7.4
6.3
6.1
11.5
7.1
7.6
12.0
15.0
7.8
8.5
Table 1: Comparison of acoustic front ends using a RN and a MLP for continuous speech
recognition on the RM task, using a wordpair grammar of perplexity 60. The four test sets
(feb89, oct89, feb91 and sep92, labelled according to their date of release by DARPA) each
contain 300 sentences spoken by 10 new speakers.
In the case of the MLP we were able to explicitly set the memory depth. Previous experiments had determined that a memory depth of 6 frames (together with a target delayed by 3
frames) was adequate for problems relating to this database. In the case of the RN, memory
1053
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Renals, Hochberg, and Robinson
P(qlx)
P(qlx)
Output Layer
Hidden Layer (1000 hidden units)
Hidden Layer (1000 hidden units)
x(I+2)
x(l)
(a) Gamma Filtered Input
(b) Gamma Filter + Future Context
Figure 2: Gamma memory applied to the network input. The simple gamma memory in (a)
does not incorporate any information about the future, unless the target is delayed. In (b)
there is an explicit delay line to incorporate some future context.
depth is not determined directly, but results from the interaction between the network architecture (i.e., number of state units) and the training process (in this case, back-propagation
through time). We hypothesise that the RN failed to make use of the higher resolution front
end because it did not adapt to the required depth.
3
GAMMA MEMORY STRUCTURE
The tradeoff between depth and resolution has led us to investigate other network architectures. The gamma filter, introduced by de Vries and Principe (1992) and Principe et al.
(1993), is a memory structure designed to automatically determine the appropriate depth
and resolution (figure 2). This locally recurrent architecture enables lowpass and bandpass
filters to be learned from data (using back-propagation through time or real-time recurrent
learning) with only a few additional parameters.
We may regard the gamma memory as a generalisation of a delay line (Mozer, 1993) in
which the kth tap at time t is obtained by convolving the input time series with a kernel
function, g~(t), and where 11 parametrises the Kth order gamma filter,
gg(t)
=
8(t)
l<k<K.
This family of kernels is attractive, since it may be computed incrementally by
dXk(t)
---;tt = -l1 xk(t) + I1Xk-l (t) .
This is in contrast to some other kernels that have been proposed (e.g., Gaussian kernels
proposed by Bodenhausen and Waibel (1991) in which the convolutions must be performed
Learning Temporal Dependencies in Connectionist Speech Recognition
explicitly). In the discrete time case the filter becomes:
Xk(t)
= (l - Il)Xk(t -
1) + IlXk-l(t - 1)
This recursive filter is guaranteed to be stable when 0 < J1 < 2.
In the experiments reported below we have replaced the input delay line of a MLP with a
gamma memory structure, using one gamma filter for each input feature. This structure is
referred to as a "focused gamma net" by de Vries and Principe (1992).
Owing to the effects of anticipatory coarticulation, information about the future is as
important as past context in speech recognition. A simple gamma filtered input (figure 2a)
does not include any future context. There are various ways in which this may be remedied;
? Use the same architecture, but delay the target (similar to figure Ib);
? Explicitly specify future context by adding a delay line from the future (figure 2b);
? Use two gamma filters per feature: one forward, one backward in time.
A drawback of the first approach is that the central frame corresponding to the delayed target
will have been smoothed by the action of the gamma filter. The third approach necessitates
two passes when either training or running the network.
4 SPEECH RECOGNITION EXPERIMENTS
We have performed experiments using the standard TIMIT speech database. This database
is divided into 462 training speakers and 168 test speakers. Each speaker utters eight
sentences that are used in these experiments, giving a training set of 3696 sentences and a
test set of 1344 sentences. We have used this database for a continuous phone recognition
task: labelling each sentence using a sequence of symbols, drawn from the standard 61
element phone set.
The acoustic data was preprocessed using a 12th order perceptual linear prediction (PLP)
analysis to produce an energy coefficient plus 12 PLP cepstral coefficients for each frame
of data. A 20ms Hamming window was used with a lOms frame step. The temporal
derivatives of each of these features was also estimated (using a linear regression over ? 3
adjacent frames) giving a total of 26 features per frame.
The networks we employed (table 2) were MLPs, with 1000 hidden units, 61 output
units (one per phone) and a variety of input representations. The Markov process used
single state phone models, a bigram phone grammar, and a Viterbi decoder was used for
recognition. The feed-forward weights in each network were initialised with identical sets
of small random values. The gamma filter coefficients were initialised to 1.0 (equivalent
to a delay line). The feed-forward weights were trained using back-propagation and the
gamma filter coefficients were trained in a forward in time back-propagation procedure
equivalent to real-time recurrent learning. An important detail is that the gradient step size
was substantially lower (by a factor of 10) for the gamma filter parameters compared with
the feed-forward weights. This was necessary to prevent the gamma filter parameters from
becoming unstable.
The baseline system using a delay line (Base) corresponds to figure 1a, with ? 3 frames of
context. The basic four-tap gamma filter G4 is illustrated in figure 2a (but using 1 fewer
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System ID
Base
G4
G7
G7i
G4F3
G4F3i
Description
Baseline delay line, ? 3 frames of context
Gamma filter, 4 taps
Gamma filter, 7 taps, delayed target
G7 initialised using weights from Base
Gamma filter, 4 taps, 3 frames future context
G4F3 initialised using weights from Base
Table 2: Input representations used in the experiments. Note that G7i and G4F3i were
initialised using a partially trained weight matrix (after six epochs) from Base.
tap than the picture) and G7 is a 7 frame gamma filter with the target delayed for 3 frames,
thus providing some future context (but at the expense of smoothing the "centre" frame).
Future context is explicitly incorporated in G4F3, in which the three adjacent future frames
are included (similar to figure 2b). Systems G7i and G4F3i were both initialised using
a partially trained weight matrix for the delay line system, Base. This was equivalent to
fixing the value of the gamma filter coefficients to a constant (1.0) during the first six epochs
of training and only adapting the feed-forward weights, before allowing the gamma filter
coefficients to adapt.
The results of using these systems on the TIM IT phone recognition task are given in table
3. Table 4 contains the results of some model merging experiments, in which the output
probability estimates of 2 or more networks were averaged to produce a merged estimate.
System ID
Base
G4
G7
G7i
G4F3
G4F3i
Depth
4.0
8.5
11.7
5.8
9.6
4.9
Correct%
67.6
65.8
65.5
67.3
67.8
68.0
Insert.%
4.1
4.1
4.1
3.8
3.8
3.9
Subst.%
24.7
25.9
26.0
24.5
24.2
24.2
Delet.%
7.7
8.3
8.5
8.2
8.0
7.8
Error %
36.5
38.2
38.6
36.5
36.0
35.9
Table 3: TIMIT phone recognition results for the systems defined in table 2. The Depth
value is estimated as the ratio of filter order to average filter parameter KIJ.!. Future context
is ignored in the estimate of depth, and the estimates for G7 and G7i are adjusted to account
for the delayed target.
System ID
G4F3+Base
G4F3 +G4F3i
G7 + Base
G7+G7i
Correct%
68.1
68.2
67.0
67.4
Insert.%
3.2
3.2
3.2
3.6
Subst.%
23.7
23.5
24.4
24.4
Delet.%
8.2
8.3
8.6
8.2
Error%
35.1
35.0
36.2
36.2
Table 4: Model merging on the TIMIT phone recognition task.
Learning Temporal Dependencies in Connectionist Speech Recognition
O.B
--
0.6
PLP Coefficients
Derivatives
0.4
0.2
E
Cl
C2
I
C3
C4
C5
I
C6
C7
CB
C9
Cl0
Cll
C12
Feature
Figure 3: Gamma filter coefficients for G4F3. The coefficients correspond to energy (E)
and 12 PLP cepstral coefficients (C1-C12) and their temporal derivatives.
5
DISCUSSION
Several comments may be made about the results in section 4. As can be seen in table 3,
replacing a delay line with an adaptive gamma filter can lead to an improvement in performance. Knowledge of future context is important. This is shown by G4, which had no
future context or delayed target information, and had poorer performance than the baseline.
However, incorporating future context using a delay line (G4F3) gives better performance
than a pure gamma filter representation with a delayed target (G7). Training the locally
recurrent gamma filter coefficients is not trivial. Fixing the gamma filter coefficients to
1.0 (delay line) whilst adapting the feed-forward weights during the first part of training
is beneficial. This is demonstrated by comparing the performance of G7 with G7i and
G4F3 with G4F3i. Finally, table 4 shows that model merging generally leads to improved
recognition performance relative to the component models. This also indicates that the
delay line and gamma filter input representations are somewhat complementary.
Figure 3 displays the trained gamma filter coefficients for G4F3. There are several points
to make about the learned temporal dependencies.
? The derivative parameters are smaller compared with the static PLP parameters.
This indicates the derivative filters have greater depth and lower resolution compared with the static PLP filters.
? If a gamma filter is regarded as a lowpass IIR filter, then lower filter coefficients
indicate a greater degree of smoothing. Better estimated coefficients (e.g., static
PLP coefficients Cl and C2) give rise to gamma filters with less smoothing.
? The training schedule has a significant effect on filter coefficients. The depth
estimates of G4F3 and G4F3i in table 3 demonstrate that very different sets of
filters were arrived at for the same architecture with identical initial parameters,
but with different training schedules.
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We are investigating the possibility of using gamma filters to model speaker characteristics.
Preliminary experiments in which the gamma filters of speaker independent networks were
adapted to a new speaker have indicated that the gamma filter coefficients are speaker
dependent. This is an attractive approach to speaker adaptation, since very few parameters
(26 in our case) need be adapted to a new speaker.
Gamma filtering is a simple, well-motivated approach to modelling temporal dependencies
for speech recognition and other problems. It adds minimal complexity to the system
(in our case a parameter increase of 0.01 %), and these initial experiments have shown an
improvement in phone recognition performance on the TIM IT database. A further increase
in performance resulted from a model merging process. We note that gamma filtering and
model merging may be regarded as two sides of the same coin: gamma filtering smooths
the input acoustic features, while model merging smooths the output probability estimates.
Acknowledgement
This work was supported by ESPRIT BRA 6487, WERNICKE. SR was supported by a SERC
postdoctoral fellowship and a travel grant from the NIPS foundation. TR was supported by
a SERC advanced fellowship.
References
Bodenhausen, D ., & Waibel, A. (1991). The Tempo 2 algorithm: Adjusting time delays
by supervised learning. In Lippmann, R. P., Moody, J. E., & Touretzky, D. S. (Eds.),
Advances in Neural Information Processing Systems, Vol. 3, pp. 155-161. Morgan
Kaufmann, San Mateo CA.
de Vries, B., & Principe, J. C. (1992). The gamma model-a new neural model for temporal
processing. Neural Networks, 5,565-576.
Mozer, M. C. (1993). Neural net architectures for temporal sequence processing. In
Weigend, A. S., & Gershenfeld, N. (Eds.), Predicting the future and understanding
the past. Addison-Wesley, Redwood City CA.
Principe, J. C., de Vries, B., & de Oliveira, P. G. (1993). The gamma filter-a new class of
adaptive IIR filters with restricted feedback. IEEE Transactions on Signal Processing,
41, 649-656.
Renals, S., Morgan, N., Bourlard, H., Cohen, M., & Franco, H. (1994). Connectionist
probability estimators in HMM speech recognition. IEEE Transactions on Speech
and Audio Processing. In press.
Robinson, A. J., Almeida, L., Boite, J.-M., Bourlard, H., Fallside, F., Hochberg, M., Kershaw, D., Kohn, P., Konig, Y., Morgan, N., Neto, J. P., Renals, S., Saerens, M., &
Wooters, C. (1993). A neural network based, speaker independent, large vocabulary, continuous speech recognition system: the WERNICKE project. In Proceedings
European Conference on Speech Communication and Technology, pp. 1941-1944
Berlin.
Robinson, T. (1994). The application of recurrent nets to phone probability estimation.
IEEE Transactions on Neural Networks. In press.
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incorporate:2 c9:1 audio:1 handling:1 |
7,082 | 852 | Emergence of Global Structure from
Local Associations
Thea B. Ghiselli-Crippa
Paul W. Munro
Department of Infonnation Science
University of Pittsburgh
Pittsburgh PA 15260
Department of Infonnation Science
University of Pittsburgh
Pittsburgh PA 15260
ABSTRACT
A variant of the encoder architecture, where units at the input and output layers represent nodes on a graph. is applied to the task of mapping
locations to sets of neighboring locations. The degree to which the resuIting internal (i.e. hidden unit) representations reflect global properties of the environment depends upon several parameters of the learning
procedure. Architectural bottlenecks. noise. and incremental learning of
landmarks are shown to be important factors in maintaining topographic relationships at a global scale.
1 INTRODUCTION
The acquisition of spatial knowledge by exploration of an environment has been the subject of several recent experimental studies. investigating such phenomena as the relationship between distance estimation and priming (e.g. McNamara et al .? 1989) and the influence of route infonnation (McNamara et al., 1984). Clayton and Habibi (1991) have gathered data suggesting that temporal contiguity during exploration is an important factor in
detennining associations between spatially distinct sites. This data supports the notion
that spatial associations are built by a temporal process that is active during exploration
and by extension supports Hebb's (1949) neurophysiological postulate that temporal associations underlie mechanisms of synaptic learning. Local spatial infonnation acquired
during the exploration process is continuously integrated into a global representation of
the environment (cognitive map). which is typically arrived at by also considering global
constraints. such as low dimensionality. not explicitly represented in the local relationships.
1101
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Ghiselli-Crippa and Munro
2 NETWORK ARCHITECTURE AND TRAINING
The goal of this network design is to reveal structure among the internal representations
that emerges solely from integration of local spatial associations; in other words. to show
how a network trained to learn only local spatial associations characteristic of an environment can develop internal representations which capture global spatial properties. A variant of the encoder architecture (Ackley et al .? 1985) is used to associate each node on a 2D graph with the set of its neighboring nodes. as defined by the arcs in the graph. This 2D neighborhood mapping task is similar to the I-D task explored by Wiles (1993) using
an N-2-N architecture. which can be characterized in terms of a graph environment as a
circular chain with broad neighborhoods.
In the neighborhood mapping experiments described in the following, the graph nodes are
visited at random: at each iteration, a training pair (node-neighborhood) is selected at random from the training set. As in the standard encoder task, the input patterns are all or-'
thogonal. so that there is no structure in the input domain that the network could exploit
in constructing the internal representations; the only information about the structure of
the environment comes from the local associations that the network is shown during
training.
2.1
N?H?N NETWORKS
The neighborhood mapping task was first studied using a strictly layered feed-forward NH-N architecture, where N is the number of input and output units. corresponding to the
number of nodes in the environment, and H is the number of units in the single hidden
layer. Experiments were done using square grid environments with wrap-around (toroidal)
and without wrap-around (bounded) at the edges. The resulting hidden unit representations
reflect global properties of the environment to the extent that distances between them correlate with distances between corresponding points on the grid. These two distance measures are plotted against one another in Figure 1 for toroidal and bounded environments.
5x5 Grid
5x5 Grid
4 Hidden Units
U5?, . - - - - - - - - - - - : " ' 1
R"2 = 0.499
1.4
e::
_o
e::
1.2
e:: ?
:Ie::
~ lIS
06
i?1.0
.,0
!i
0.6
i"~
a:c
g.~
a:c
1.5
o
iii
1.0
.!!
?0
4 Hidden Units
2.0-,.----------.......,
0.4
0.2
O.ol---~----"T"""---i
3
o
1
2
Grid Distance
With wrap-around
0.5
:
----r--__-...-..-4
O.O ......_,......._ _
234
o
5
Grid Distance
No wrap-around
Figure 1: Scatterplots of Distances between Hidden Unit Representations vs. Distances
between Corresponding Locations in the Grid Environment.
Emergence of Global Structure from Local Associations
2.2
N?2?H?N Networks
A hidden layer with just two units forces representations into a 2-D space. which matches
the dimensionality of the environment. Under this constraint. the image of the environment in the 2-D space may reflect the topological structure of the environment. This conjecture leads to a further conjecture that the 2-D representations will also reveal global relationships of the environment. Since the neighborhoods in a 2-D representation are not
linearly separable regions. another layer (H-Iayer) is introduced between the two-unit layer
and the output (see Figure 2). Thus. the network has a strictly layered feed-forward
N-2-H-N architecture. where the N units at the input and output layers correspond to the
N nodes in the environment. two units make up the topographic layer. and H is the number of units chosen for the new layer (H is estimated according to the complexity of the
graph). Responses for the hidden units (in both the T- and H-layers) are computed using
the hyperbolic tangent (which ranges from -1 to +1). while the standard sigmoid (0 to +1)
is used for the output units. to promote orthogonality between representations (Munro.
1989). Instead of the squared error. the cross entropy function (Hinton. 1987) is used to
avoid problems with low derivatives observed in early versions of the network.
~ooe@o~oo
.~
1.::; .::'.,. : ...............
<: .? ??.? ????.:.:.??.
:' <.
o
3
:>/<::>. : .???.1.??.?.?? ??.??.?.:?.?? ?. ?.:.???..?:.??. :?.?. i: ?.i.\.: . j)
?!..??.
' .. :.'/> .'> < '
2
f
5
00
6
7
8
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Figure 2: A 3x3 Environment and the Corresponding Network. When input unit 3 is
activated, the network responds by activating the same unit and all its neighbors.
3 RESULTS
3.1
T?UNIT RESPONSES
Neighborhood mapping experiments were done using bounded square grid environments
and N-2-H-N networks. After training, the topographic unit activities corresponding to
each of the N possible inputs are plotted, with connecting lines representing the arcs from
1103
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Ghiselli-Crippa and Munro
the environment. Each axis in Figure 3 represents the activity of one of the T-units.
These maps can be readily examined to study the relationship between their global structure and the structure of the environment. The receptive fields of the T-units give an alternative representation of the same data: the response of each T-unit to all N inputs is represented by N circles arranged in the same configuration as the nodes in the grid environment. Circle size is proportional to the absolute value of the unit activity; filled circles
indicate negative values, open circles indicate positive values. The receptive field represents the T-unit's sensitivity with respect to the environment.
???
?
?
0
000
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1:8
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~c8
...
26~
oCXX)
.
0
00
????
????o
leo
? 00
. ?8
.?0
Figure 3: Representations at the Topographic Layer. Activity plots and receptive fields
for two 3x3 grids (left and middle) and a 4x4 grid(right).
The two 3x3 cases shown in Figure 3 illustrate alternative solutions that are each locally
consistent, but have different global structure. In the first case, it is evident how the first
unit is sensitive to changes in the vertical location of the grid nodes, while the second
unit is sensitive to their horizontal location. The axes are essentially rotated 45 degrees in
the second case. Except for this rotation of the reference axes, both representations captured the global structure of the 3x3 environment.
3.2
NOISE IN THE HIDDEN UNITS
While networks tended to fonn maps in the T -layer that reflect the global structure of the
environment, in some cases the maps showed correspondences that were less obvious:
i.e., the grid lines crossed, even though the network converged. A few techniques have
proven valuable for promoting global correspondence between the topographic representations and the environment, including Judd and Munro's (1993) introduction of noise as
pressure to separate representations. The noise is implemented as a small probability for
Emergence of Global Structure from Local Associations
reversing the sign of individual H-unit outputs. As reported in a previous study
(Ghiselli-Crippa and Munro, 1994), the presence of noise causes the network to develop
topographic representations which are more separated, and therefore more robust, so that
the correct output units can be activated even if one or more of the H-units provides an
incorrect output. From another point of view, the noise can be seen as causing the network to behave as if it had an effective number of hidden units which is smaller than the
given number H. The introduction of noise as a means to promote robust topographic
representations can be appreciated by examining Figure 4, which illustrates the representations of a 5x5 grid developed by a 25-2-20-25 network trained without noise (left) and
with noise (middle) (the network was initialized with the same set of small random
weights in all cases). Note that the representations developed by the network subject to
noise are more separated and exhibit the same global structure as the environment. To
avoid convergence problems observed with the use of noise throughout the whole training
process, the noise can be introduced at the beginning of training and then gradually reduced over time.
A similar technique involves the use of low-level noise injected in the T-Iayer to directly
promote the formation of well-separated representations. Either Gaussian or uniform
noise directly added to the T-unit outputs gives comparable results. The use of noise in
either hidden layer has a beneficial influence on the formation of globally consistent representations. However. since the noise in the H-units exerts only an indirect influence on
the T -unit representations, the choice of its actual value seems to be less crucial than in
the case where the noise is directly applied at the T-Iayer.
The drawback for the use of noise is an increase in the number of iterations required by
the network to converge, that scales up with the magnitude and duration of the noise.
Figure 4: Representations at the Topographic Layer. Training with no noise (left) and
with noise in the hidden units (middle); training using landmarks (right).
3.3
LANDMARK LEARNING
Another effective method involves the organization of training in 2 separate phases, to
model the acquisition of landmark information followed by the development of route
and/or survey knowledge (Hart and Moore, 1973; Siegel and White, 1975). This method
is implemented by manipulating the training set during learning, using coarse spatial resolution at the outset and introducing interstitial features as learning progresses to the second phase. The first phase involves training the network only on a subset of the possible
1105
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Ghiselli-Crippa and Munro
N patterns (landmarks). Once the landmarks have been learned. the remaining patterns are
added to the training set. In the second phase. training proceeds as usual with the full set
of training patterns; the only restriction is applied to the landmark points. whose topographical representations are not allowed to change (the corresponding weights between
input units and T-units are frozen). thus modeling the use of landmarks as stable reference
points when learning the details of a new environment. The right pane of Figure 4 illustrates the representations developed for a 5x5 grid using landmark training; the same 25-220-25 network mentioned above was trained in 2 phases. first on a subset of 9 patterns
(landmarks) and then on the full set of 25 patterns (the landmarks are indicated as white
circles in the activity plot).
3.4
NOISE IN LANDMARK LEARNING
The techniques described above (noise and landmark learning) can be combined together to
better promote the emergence of well-structured representation spaces. In particular, noise
can be used during the first phase of landmark learning to encourage a robust representation of the landmarks: Figure 5 illustrates the representations obtained for a 5x5 grid
using landmark training with two different levels of noise in the H-units during the first
phase. The effect of noise is evident when comparing the 4 comer landmarks in the right
pane of Figure 4 (landmark learning with no noise) with those in Figure 5. With increasing levels of noise. the T-unit activities corresponding to the 4 comer landmarks approach
the asymptotic values of +1 and -1; the activity plots illustrate this effect by showing
how the comer landmark representations move toward the comers of T-space, reaching a
configuration which provides more resistance to noise. During the second phase of training, the landmarks function as reference points for the additional features of the environment and their positioning in the representational space therefore becomes very important. A well-fonned, robust representation of the landmarks at the end of the first phase
is crucial for the fonnation of a map in T-space that reflects global structure, and the use
of noise can help promote this.
Figure 5: Representations at the Topographic Layer. Landmark training using noise in
phase 1: low noise level (left). high noise level (right).
4
DISCUSSION
Large scale constraints intrinsic to natural environments. such as low dimensionality, are
not necessarily reflected in local neighborhood relations, but they constitute infonnation
which is essential to the successful development of useful representations of the environ-
Emergence of Global Structure from Local Associations
ment. In our model, some of the constraints imposed on the network architecture effectively reduce the dimensionality of the representational space. Constraints have been introduced several ways: bottlenecks, noise, and landmark learning; in all cases, these constraints have had constructive influences on the emergence of globally consistent representation spaces. The approach described presents an alternative to Kohonen's (1982)
scheme for capturing topography; here, topographic relations emerge in the representational space, rather than in the weights between directly connected units.
The experiments described thus far have focused on how global spatial structure can
emerge from the integration of local associations and how it is affected by the introduction of global constraints. As mentioned in the introduction, one additional factor influencing the process of acquisition of spatial knowledge needs to be considered: temporal
contiguity during exploration. that is. how temporal associations of spatially adjacent locations can influence the representation of the environment. For example, a random type
of exploration ("wandering") can be considered. where the next node to be visited is selected at random from the neighbors of the current node. Preliminary studies indicate that
such temporal contiguity during training reSUlts in the fonnation of hidden unit representations with global properties qualitatively similar to those reported here. Alternatively,
more directed exploration methods can be studied. with a systematic pattern guiding the
choice of the next node to be visited. The main purpose of these studies will be to show
how different exploration strategies can affect the formation and the characteristics of cognitive maps of the environment.
Higher order effects of temporal and spatial contiguity can also be considered. However,
in order to capture regularities in the training process that span several exploration steps.
simple feed-forward networks may no longer be sufficient; partially recurrent networks
(Elman, 1990) are a likely candidate for the study of such processes.
Acknowledgements
We wish to thank Stephen Hirtle, whose expertise in the area of spatial cognition greatly
benefited our research. We are also grateful for the insightful comments of Janet Wiles.
References
D. H. Ackley. G. E. Hinton, and T. J. Sejnowski (1985) "A learning algorithm for
Boltzmann machines," Cognitive Science, vol. 9. pp. 147-169.
K. Clayton and A. Habibi (1991) "The contribution of temporal contiguity to the spatial
priming effect," Journal of Experimental Psychology: Learning, Memory, and
Cognition. vol. 17, pp. 263-27l.
J. L. Elman (1990) "Finding structure in time," Cognitive Science, vol. 14, pp. 179211.
T. B. Ghiselli-Crippa and P. W. Munro (1994) "Learning global spatial structures from
local associations," in M. C Mozer, P. Smolensky, D. S. Touretzky, J. L. Elman, and A.
S. Weigend (Eds.), Proceedings of the 1993 Connectionist Models Summer School,
Hillsdale, NJ: Erlbaum.
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R. A. Hart and G. T. Moore (1973) "The development of spatial cognition: A review," in
R. M. Downs and Stea (Eds.), Image and Environment, Chicago, IL: Aldine.
D. O. Hebb (1949) The Organization of Behavior, New York, NY: Wiley.
G. E. Hinton (1987) "Connectionist learning procedures," Technical Report CMU-CS87-115, version 2, Pittsburgh, PA: Carnegie-Mellon University, Computer Science
Department.
S. Judd and P. W. Munro (1993) "Nets with unreliable hidden nodes learn error-correcting
codes," in C. L. Giles, S. J. Hanson, and J. D. Cowan, Advances in Neural Information
Processing Systems 5, San Mateo, CA: Morgan Kaufmann.
T. Kohonen (1982) "Self-organized fonnation of topological correct feature maps,"
Biological Cybernetics, vol. 43, pp. 59-69.
T. P. McNamara, J. K. Hardy, and S. C. Hirtle (1989) "Subjective hierarchies in spatial
memory," Journal of Experimental Psychology: Learning, Memory, and Cognition, vol.
15, pp. 211-227.
T. P. McNamara, R. Ratcliff, and G. McKoon (1984) "The mental representation of
knowledge acquired from maps," Journal of Experimental Psychology: Learning,
Memory, and Cognition, vol. 10, pp. 723-732.
P. W. Munro (1989) "Conjectures on representations in backpropagation networks,"
Technical Report TR-89-035, Berkeley, CA: International Computer Science Institute.
A. W. Siegel and S. H. White (1975) "The development of spatial representations of
large-scale environments," in H. W. Reese (Ed.), Advances in Child Development and
Behavior, New York, NY: Academic Press.
J. Wiles (1993) "Representation of variables and their values in neural networks," in
Proceedings of the Fifteenth Annual Conference of the Cognitive Science Society,
Hillsdale, NJ: Erlbaum.
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topographical:1 phenomenon:1 |
7,083 | 853 | Classification of Multi-Spectral Pixels
by the
Binary Diamond Neural Network
Yehuda Salu
Department of Physics and CSTEA, Howard University, Washington, DC 20059
Abstract
A new neural network, the Binary Diamond, is presented and its use
as a classifier is demonstrated and evaluated. The network is of the
feed-forward type. It learns from examples in the 'one shot' mode,
and recruits new neurons as needed. It was tested on the problem of
pixel classification, and performed well. Possible applications of the
network in associative memories are outlined.
1
INTRODUCTION: CLASSIFICATION BY CLUES
Classification is a process by which an item is assigned to a class. Classification is
widely used in the animal kingdom. Identifying an item as food is classification.
Assigning words to objects, actions, feelings, and situations is classification. The
purpose of this work is to introduce a new neural network, the Binary Diamond,
which can be used as a general purpose classification tool. The design and
operational mode of the Binary Diamond are influenced by observations of the
underlying mechanisms that take place in human classification processes.
An item to be classified consists of basic features. Any arbitrary combination of basic
features will be called a clue. Generally, an item will consist of many clues. Clues are
related not only to the items which contain them, but also to the classes. Each class,
that resides in the memory, has a list of clues which are associated with it. These clues
1143
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Salu
are the basic building blocks of the classification rules. A classification rule for a class
X would have the following general form:
Classification rule: If an item contains clue Xl, or clue X2, ... , or clue X n ? and if it
does not contain clue Xl. nor clue X2, ...? nor clue X m ? it is classified as belonging
to class X.
Clues Xl ?...,X n are the excitatory clues of class X, and clues Xl, ... ,xmare the
inhibitory clues of class X.
When classifying an item, we frrst identify the clues that it contains. We then match
these clues with the classification rules, and fmd the class of the item. It may happen
that a certain item satisfies classification rules of different classes. Some of the clues
match one class, while others match another. In such cases, a second set of rules,
disambiguation rules, are employed. These rules select one class out of those tagged
by the classification rules. The disambiguation rules rely on a hierarchy that exists
among the clues. a hierarchy that may vary from one classification scheme to another.
For example, in a certain hierarchy clue A is considered more reliable than clue B, if
it contains more features. In a different hierarchy scheme, the most frequent clue is
considered the most reliable. In the disambiguation process, the most reliable clue,
out of those that has actively contributed to the classification. is identified and serves
as the pointer to the selected class. This classification approach will be called
classification by clues (CRC).
The classification rules may be 'loaded' into our memory in two ways. FIrst, the
precise rules may be spelled out and recorded (e.g. 'A red light means stop'). Second,
we may learn the classification rules from examples presented to us, utilizing innate
common sense learning mechanism. These mechanisms enable us to deduce from the
examples presented to us, what clues should serve in the classification rules of the
adequate classes, and what clues have no specificity, and should be ignored. For
example, by pointing to a red balloon and saying red, an infant may associate each of
the stimuli red and balloon as pointers to the word red. After presenting a red car,
and saying red, and presenting a green balloon and saying green. the infant has
enough information to deduce that the stimulus red is associated with the word red,
and the stimulus balloon should not be classified as red.
2
THE BINARY DIAMOND
2.1
STRUCTURE
In order to perform a CRe in a systematic way, all the clues that are present in the
item to be classified have to be identified frrst, and then compared against the
classification rules. The Binary Diamond enables carrying these tasks fast and
Classification of Multi-Spectral Pixels by the Binary Diamond Neural Network
effectively. Assume that there are N different basic features in the environment. Each
feature can be assigned to a certain bit in an N dimensional binary vector. An item
will be represented by turning-on (from the default value of to the value of 1) all the
bits that correspond to basic features, that are present in the item. The total number
?
of possible clues in this environment is at most 2N. One way to represent these
possible clues is by a lattice, in which each possible clue is represented by one node.
The Binary Diamond is a lattice whose nodes represent clues. It is arranged in layers.
The frrst (bottom) layer has N nodes that represent the basic features in the
environment. The second layer has N'(N-l)/2 nodes that represent clues consisting of
2 basic features. The K'th layer has nodes that represent clues, which consist of K
basic features. Nodes from neighboring layers which represent clues that differ by
exactly one basic feature are connected by a line. Figure 1 is a diagram of the Binary
Diamond for N =4.
Figure 1: The Binary Diamond of order 4. The numbers inside the nodes are the
binary codes for the feature combination that the node represents, e.g 1 < = >
(0,0,0,1),5< = >(0,1,0,1),14 < = > (1,1,1,0), 15 < => (1,1,1,1).
2.2
THE BINARY DIAMOND NEURAL NElWORK
The Binary Diamond can be turned into a feed-forward neural network by treating
each node as a neuron, and each line as a synapse leading from a neuron in a lower
layer (k) to a neuron in the higher layer (k + 1). All synaptic weights are set to 0.6, and
1145
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Salu
all thresholds are set to 1, in a standard Pitts McCulloch neuron. The output of a
flring neuron is 1. An item is entered into the network by turning-on the neurons in
the flrst layer, that represent the basic features constituting this item. Signals
propagate forward one layer at a time tick, and neurons stay active for one time tick.
It is easy to verify that all the clues that are part of the input item, and only such clues,
will be turned on as the signals propagate in the network. In other words, the network
identifies all the clues in the item to be classified. An item consisting of M basic
features will activate neurons in the fIrst M layers. The activated neuron in the M'th
layer is the representation of the entire item. As an example, consider the input item
with feature vector (0,1,1,1), using the notations of figure 1. It is entered by activating
neurons 1, 2, and 4 in the first layer. The signals will propagate to neurons 3, 5, 6, and
7, which represent all the clues that the input item contains.
2.3
INCORPORATING CLASS INFORMATION
Each neuron in the Binary Diamond represent a possible clue in the environment
spun by N basic features. When an item is entered in the frrst layer, all the clues that it
contains activate their representing neurons in the upper layers. This is the first step
in the classification process. Next, these clues have to point to the appropriate class,
based upon the classification rule. The possible classes are represented by neurons
outside of the Binary Diamond. Let x denote the neuron, outside the Binary
Diamond, that represents class X. An excitatory clue Xi (from the Binary Diamond)
will synapse onto x with a synaptic weight of 1. An inhibitory clue
Xl
(in the Binary
Diamond) will synapse onto x with an inhibitory weight of -z, where z is a very large
number (larger than the maximum number of clues that may point to a class). This
arrangement ensures that the classification rule formulated above is carried out. In
cases of ambiguity, where a number of classes have been activated in the process, the
class that was activated by the clue in the highest layer will prevail. This clue has the
largest number of features, as compared with the other clues that actively participated
in the classification.
2.4
GROWING A BINARY DIAMOND
A possible limitation on the processes described in the two previous sections is that, if
there are many basic features in the environment, the 2N nodes of the Binary
Diamond may be too much to handle. However, in practical situations, not all the
clues really occur, and there is no need to actually represent all of them by nodes.
One way of taking advantage of this simplifying situation is to grow the network one
event (a training item and its classification) at a time. At the beginning, there is just
the frrst layer with N neurons, that represent the N basic features. Each event adds its
neurons to the network, in the exact positions that they would occupy in the regular
Classification of Multi-Spectral Pixels by the Binary Diamond Neural Network
Binary Diamond. A clue that has already been represented in previous events, is not
duplicated. After the new clues of the event have been added to the network, the
information about the relationships between clues and classes is updated. This is done
for all the clues that are contained in the new event. The new neurons send synapses
to the neuron that represent the class of the current event. Neurons of the current
event, that took part in previous events, are checked for consistency. If they point to
other classes, their synapses are cut-off. They have just lost their specificity. It should
be noted that there is no need to present an event more than one time for it to be
correctly recorded (' one shot learning'). A new event will never adversely interfere
with previously recorded information. Neither the order of presenting the events, nor
repetitions in presenting them will affect the final structure of the network. Figure 2
illustrates how a Binary Diamond is grown. It encodes the information contained in
two events, each having three basic features, in an environment that has four basic
features. The first event belongs to class A, and the second to class B.
(0,1,1,1) -> A
@
Figure 2. Growing a Binary Diamond. Left: All the feature combinations of the threefeature item (0,1,1,1) are represented by a 3'rd order Binary Diamond, which is grown
from the basic features represented by neurons 1, 2, and 4. All these combinations,
marked by a wavy background, are, for the time being, specific clues to class A.
Right: The three-feature item, (1,1,1,0) is added, as another 3'rd order Binary
Diamond. At this point, only neurons l,3~,and 7 represent specific clues to class A.
Neurons 8,10,12, and 14 represent specific clues to class B, and neurons 2,4, and 6
represent non-specific clues.
3
CLASSIFICATION OF MULTI-SPECfRAL PIXELS
3.1
THE PROBLEM
Spectral information of land pixels, which is collected by satellites, is used in
preparation of land cover maps and similar applications. Depending on the satellite
and its instrumentation, the spectral information consists of the intensities of several
1147
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Salu
light bands, usually in the visible and infra-red ranges, which have been reflected from
the land pixels. One method of classification of such pixels relies on independent
knowledge of the land cover of some pixels in the scene. These classified pixels serve
as the training set for a classification algorithm. Once the algorithm is trained, it
classifies the rest of the pixels.
The actual problem described here involves testing the Binary Diamond in a pixel
classification problem. The tests were done on four scenes from the vicinity of
Washington DC, each consisting of approximately 22,000 pixels. The spectral
information of each pixel consisted of intensities of four spectral bands, as collected
by the Thematic Mapper of the Landsat 4 satellite. Ground covers of these scenes
were determined independently by ground and aerial surveys. There were 17 classes
of ground covers. The following list gives the number of pixels per class in one of the
scenes. The distributions in the other scenes were similar.
1) water (28). 2) miscellaneous crops (299). 3) corn-standing (0). 4) com-stubble
(349). 5) shrub-land (515). 6) grass/ pasture (3,184). 7) soybeans (125). 8) baresoil, clear land (535). 9) hardwood, canopy> 50% (10,169). 10) hardwood, canopy
< 50% (945). 11) conifer forest (2,051). 12) mixed wood forest (616). 13)
asphalt (390). 14) single family housing (2,220). 15) multiple family housing (26).
16) industrial/ commercial (118). 17) bare soil-plowed field (382). Total 21,952.
3.2
METHODS
Approximately 10% of the pixels in each of the four scenes were randomly selected to
become a training set. Four Binary Diamond networks were grown, based on these
four training sets. In the evaluation phase, each network classified each scene.
The intensity of the light in each band was discretized into 64 intervals. Each interval
was considered as a basic feature. So, each pixel was characterized by four basic
features (one for each band), out of 4x64=256 possible basic features. The fust layer
of the Binary Diamond consisted of 256 neurons, representing these basic features.
Pixels of the training set were treated like events. They were presented sequentially,
one at a time, for one time, and the neurons that represent their clues were added to
the network, as explained in section 2.4. After the training phase, the rest of the pixels
were presented, and the network classified them. The results of this classification
were kept for comparisons with the observed ground cover values.
The same training sets were used to train two other classification algorithms; a
backpropagation neural network, and a nearest neighbor classifier. The backpropagation network had four neurons in the input layer, each representing a spectral
band. It had seventeen neurons in the output layer, each representing a class, and a
hidden layer of ten neurons. The nearest neighbor classifier used the pixels of the
training set as models. The Euclidean distance between the feature vector of a pixel to
Classification of Multi-Spectral Pixels by the Binary Diamond Neural Network
be classified and each model pixel was computed. The pixel was classified according
to the class of its closest model.
3.3
RESULTS
In auto-classification, the pixels of a scene are classified by an algorithm that was
trained using pixels from the same scene. In cross-classification, the classification of a
scene is done by an algorithm that was trained by pixels of another scene. It was found
that in both auto-classification and cross-classification, the results depend on the
consistency of the training set. Boundary pixels, which form the boundary (on the
ground) between two classes, may contain a combination of two ground cover classes.
If boundary pixels were excluded from the scene, the results of all the classification
methods improved significantly. Table 1 compares the overall performance of the
three classification methods in auto-classification and cross-classification, when only
boundary pixels were considered. Similar ordering of the classification methods was
obtained when all the pixels were considered.
1
1
2
3
4
2 3 4
83 58 71 74
41 78 50 44
49 48 75 52
54 44 57 76
1
2
3
4
1
83
27
43
52
2 3 4
41 46 61
73 28 17
38 62 35
36 39 70
1
1
2
3
4
73
25
33
48
2
3 4
60 33 64
55 38 26
38 52 37
43 42 60
Binary Diamond
Nearest Neighbor
Back-Propagation
Table 1: The percent of correctly classified pixels for the implementations of the
three methods, for non-boundary pixels only, as tested on the four maps. Column's
index is the training map, rows index is the testing map.
Table 2 compares the performances of the three methods class by class, as obtained in
the classification of the flfst scene. Similar results were obtained for the other scenes.
1=
BD
BDp
bNN
BP
1
2
48 10
57 8
63 54
68 5
3
0
0
0
0
4
33
5 6 7 8 9
7 44 10 53 88
48 10 14 10 58 87
72 47 19 52 77 60
68 0 1 11 66 80
10
37
37
63
74
11 U
34
35
5
6
48 60
11
1
14 15 16
32 69 33 34
30 69 42 25
70 43 64 62
26 45 54 52
13
17
41
27
72
27
Table 2: The percent of pixels from category I that have been classified as category I.
Auto-classification of scene 1. All the pixels are included. BD; results of Binary
Diamond where the feature vectors are in the standard Cartesian representation.
BDp =results of Binary Diamond where the feature vectors are in four dimensional
polar coordinates. bNN results of nearest neighbor, and BP of back-propagation.
1149
1150
Salu
The overall performance of the Binary Diamond was better than those of the nearest
neighbor and the back-propagation classifiers. This was the case in auto-classification
and in cross-classification, in scenes that included all the pixels, and in scenes that
consisted only of non-boundary pixels. However, when comparing individual classes, it
was found that different classes may have different best classifiers. In practical
applications, the prices of correct or the wrong classifications of each class, as well as
the frequency of the classes in the environment will determine the optimal classifier.
All the networks recruited their neurons as needed, during the training phase. They
all started with 256 neurons in the first layer, and with seventeen neuron in the class
layer, outside the Binary Diamond. At the end of the training phase of the first scene,
The Binary Diamond consisted of 5,622 neurons, in four layers. This is a manageable
number, and it is much smaller than the maximum number of possible clues,
644 =224.
4
OrnER APPLICATIONS OF mE BINARY DIAMOND
The Binary Diamond, as presented here, was the core of a network that was used as a
classifier. Because of its special structure, the Binary Diamond can be used in other
related problems, such as in associative memories. In associative memory, a presented
clue has to retrieve all the basic features of an associated item. If we start from any
node in the Binary Diamond, and cascade down in the existing lines, we reach all the
basic features of this clue in the frrst layer. So, to retrieve an associated item, the
signals of the input clue have frrst to climb up the binary diamond till they reach a
node, which is the best generalization of this clue, and then to cascade down and to
activate the basic features of this generalization. The synaptic weights in the upward
direction can encode information about causality relationships and the frequency of
co-activations of the pre and post-synaptic neurons. This information can be used in
the retrieval of the most appropriate generalization to the given clue. An associative
memory of this kind retrieves information in ways similar to human associative
retrieval (paper submitted).
REFERENCES
A reference list, as well as more details about pixel classification can be found in:
Classification of Multi-Spectral Image Data by the Binary Diamond Neural Network
and by Non-Parametric Pixel-by-Pixel Methods, by Yehuda Salu and James Tilton.
IEEE Transactions On Geoscience And Remote Sensing, 1993 (in press).
| 853 |@word manageable:1 propagate:3 simplifying:1 shot:2 contains:5 existing:1 current:2 com:1 comparing:1 activation:1 assigning:1 bd:2 visible:1 happen:1 enables:1 treating:1 grass:1 infant:2 selected:2 item:26 beginning:1 core:1 pointer:2 node:13 become:1 consists:2 inside:1 introduce:1 nor:3 growing:2 multi:6 discretized:1 food:1 actual:1 classifies:1 notation:1 underlying:1 pasture:1 mcculloch:1 what:2 kind:1 recruit:1 exactly:1 classifier:7 wrong:1 approximately:2 co:1 range:1 practical:2 testing:2 yehuda:2 block:1 lost:1 backpropagation:2 significantly:1 cascade:2 word:4 pre:1 regular:1 specificity:2 onto:2 map:4 demonstrated:1 send:1 independently:1 survey:1 identifying:1 rule:17 utilizing:1 retrieve:2 handle:1 x64:1 coordinate:1 updated:1 hierarchy:4 commercial:1 exact:1 associate:1 cut:1 bottom:1 observed:1 ensures:1 connected:1 balloon:4 ordering:1 highest:1 remote:1 environment:7 trained:3 carrying:1 depend:1 serve:2 upon:1 represented:6 retrieves:1 grown:3 train:1 fast:1 activate:3 outside:3 whose:1 widely:1 larger:1 final:1 associative:5 housing:2 advantage:1 took:1 frequent:1 neighboring:1 turned:2 entered:3 till:1 frrst:7 satellite:3 spelled:1 wavy:1 object:1 depending:1 nearest:5 involves:1 differ:1 direction:1 correct:1 human:2 enable:1 crc:1 activating:1 generalization:3 really:1 considered:5 ground:6 pitt:1 pointing:1 vary:1 purpose:2 polar:1 largest:1 repetition:1 tool:1 encode:1 industrial:1 sense:1 landsat:1 entire:1 fust:1 hidden:1 pixel:40 overall:2 classification:54 among:1 upward:1 animal:1 special:1 field:1 once:1 never:1 having:1 washington:2 represents:2 others:1 stimulus:3 infra:1 randomly:1 individual:1 phase:4 consisting:3 evaluation:1 light:3 activated:3 euclidean:1 column:1 cover:6 lattice:2 too:1 canopy:2 cre:1 stay:1 standing:1 systematic:1 physic:1 off:1 ambiguity:1 recorded:3 spun:1 soybean:1 adversely:1 leading:1 actively:2 performed:1 red:11 start:1 loaded:1 correspond:1 identify:1 classified:13 submitted:1 flrst:1 influenced:1 synapsis:2 reach:2 synaptic:4 checked:1 orner:1 against:1 frequency:2 james:1 associated:4 stop:1 seventeen:2 duplicated:1 knowledge:1 car:1 actually:1 back:3 feed:2 higher:1 reflected:1 improved:1 synapse:3 arranged:1 evaluated:1 done:3 just:2 propagation:3 interfere:1 mode:2 innate:1 building:1 contain:3 verify:1 consisted:4 tagged:1 assigned:2 vicinity:1 excluded:1 bnn:2 during:1 noted:1 presenting:4 percent:2 image:1 common:1 rd:2 outlined:1 consistency:2 mapper:1 had:2 deduce:2 add:1 closest:1 belongs:1 instrumentation:1 certain:3 binary:43 flring:1 employed:1 determine:1 signal:4 nelwork:1 multiple:1 match:3 characterized:1 cross:4 retrieval:2 post:1 basic:24 crop:1 represent:16 background:1 participated:1 interval:2 diagram:1 grow:1 rest:2 recruited:1 climb:1 enough:1 easy:1 affect:1 identified:2 flfst:1 action:1 adequate:1 ignored:1 generally:1 clear:1 band:5 ten:1 category:2 occupy:1 inhibitory:3 per:1 correctly:2 four:11 threshold:1 neither:1 kept:1 wood:1 place:1 saying:3 family:2 disambiguation:3 bit:2 layer:24 occur:1 bp:2 x2:2 scene:18 encodes:1 corn:1 department:1 according:1 combination:5 aerial:1 belonging:1 smaller:1 explained:1 previously:1 mechanism:3 needed:2 serf:1 end:1 spectral:10 appropriate:2 arrangement:1 already:1 added:3 parametric:1 distance:1 me:1 collected:2 water:1 code:1 index:2 relationship:2 kingdom:1 design:1 implementation:1 diamond:41 contributed:1 perform:1 upper:1 neuron:35 observation:1 howard:1 situation:3 precise:1 dc:2 arbitrary:1 intensity:3 usually:1 reliable:3 memory:6 green:2 event:14 treated:1 rely:1 turning:2 representing:4 scheme:2 identifies:1 started:1 carried:1 auto:5 bare:1 mixed:1 limitation:1 bdp:2 classifying:1 land:6 row:1 excitatory:2 soil:1 tick:2 neighbor:5 taking:1 boundary:6 default:1 resides:1 forward:3 clue:64 feeling:1 constituting:1 transaction:1 active:1 sequentially:1 xi:1 table:4 learn:1 operational:1 forest:2 fmd:1 causality:1 position:1 thematic:1 xl:5 learns:1 down:2 specific:4 sensing:1 list:3 consist:2 exists:1 incorporating:1 effectively:1 prevail:1 illustrates:1 cartesian:1 contained:2 geoscience:1 satisfies:1 relies:1 marked:1 formulated:1 miscellaneous:1 price:1 included:2 determined:1 called:2 total:2 select:1 preparation:1 tested:2 |
7,084 | 854 | Correlation Functions in a Large
Stochastic Neural Network
Iris Ginzburg
School of Physics and Astronomy
Raymond and Beverly Sackler Faculty of Exact Sciences
Tel-Aviv University
Tel-Aviv 69978, Israel
Haim Sompolinsky
Racah Institute of Physics and Center for Neural Computation
Hebrew University
Jerusalem 91904, Israel
Abstract
Most theoretical investigations of large recurrent networks focus on
the properties of the macroscopic order parameters such as population averaged activities or average overlaps with memories. However, the statistics of the fluctuations in the local activities may
be an important testing ground for comparison between models
and observed cortical dynamics. We evaluated the neuronal correlation functions in a stochastic network comprising of excitatory
and inhibitory populations. We show that when the network is in
a stationary state, the cross-correlations are relatively weak, i.e.,
their amplitude relative to that of the auto-correlations are of order of 1/N, N being the size of the interacting population. This
holds except in the neighborhoods of bifurcations to nonstationary
states. As a bifurcation point is approached the amplitude of the
cross-correlations grows and becomes of order 1 and the decay timeconstant diverges. This behavior is analogous to the phenomenon
of critical slowing down in systems at thermal equilibrium near a
critical point. Near a Hopf bifurcation the cross-correlations exhibit damped oscillations.
471
472
Ginzburg and Sompolinsky
1
INTRODUCTION
In recent years there has been a growing interest in the study of cross-correlations
between the activities of pairs of neurons in the cortex. In many cases the crosscorrelations between the activities of cortical neurons are approximately symmetric
about zero time delay. These have been taken as an indication of the presence of
"functional connectivity" between the correlated neurons (Fetz, Toyama and Smith
1991, Abeles 1991). However, a quantitative comparison between the observed
cross-correlations and those expected to exist between neurons that are part of a
large assembly of interacting population has been lacking.
Most of the theoretical studies of recurrent neural network models consider only time
averaged firing rates, which are usually given as solutions of mean-field equations.
They do not account for the fluctuations about these averages, the study of which
requires going beyond the mean-field approximations. In this work we perform a
theoretical study of the fluctuations in the neuronal activities and their correlations,
in a large stochastic network of excitatory and inhibitory neurons. Depending on the
model parameters, this system can exhibit coherent undamped oscillations. Here we
focus on parameter regimes where the system is in a statistically stationary state,
which is more appropriate for modeling non oscillatory neuronal activity in cortex.
Our results for the magnitudes and the time-dependence of the correlation functions
can provide a basis for comparison with physiological data on neuronal correlation
functions.
2
THE NEURAL NETWORK MODEL
We study the correlations in the activities of neurons in a fully connected recurrent
network consisting of excitatory and inhibitory populations. The excitatory connections between all pairs of excitatory neurons are assumed to be equal to J / N
where N denotes the number of excitatory neurons in the network. The excitatory
connections from each of the excitatory neurons to each of the inhibitory neurons
are J' / N. The inhibitory coupling of each of the inhibitory neurons onto each of
the excitatory neurons is K / M where M denotes the number of inhibitory neurons.
Finally, the inhibitory connections between pairs of inhibitory neurons are ](' / M.
The values of these parameters are in units of the amplitude of the local noise (see
below). Each neuron has two possible states, denoted by Si
?1 and Ui
?1
for the i-th excitatory and inhibitory neurons, respectively. The value -1 denotes
a quiet state. The value +1 denotes an active state that corresponds to a state
with high firing rate. The neurons are assumed to be exposed to local noise resulting in stochastic dynamics of their states. This dynamics is specified by transition
probabilities between the -1 and +1 states that are sigmoidal functions of their
local fields. The local fields of the i-th excitatory neuron, Ei and the i-th inhibitory
neuron, Ii, at time t, are
=
Ei(t)
=
=
J s(t) - K u(t) - ()
(1)
J's(t) - K' u(t) - ()
(2)
Correlation Functions in a Large Stochastic Neural Network
where () represents the local threshold and sand 0' are the population-averaged
activities s(t) = l/N"'?j Sj(t), and O'(t) = l/M"'?j O'j(t) of the excitatory and
inhibitory neurons, respectively.
3
AVERAGE FIRING RATES
The macroscopic state of the network is characterized by the dynamics of s(t)
and O'(t). To leading order in l/N and l/M, they obey the following well known
equations
TO
ds
dt
dO'
dt
TO-
=
=
-s + tanh(Js - J{O' - 0)
(3)
-0' + tanh (I
J s - K-,I 0' - 0)
(4)
where TO is the microscopic time constant of the system. Equations of this form for
the two population dynamics have been studied extensively by Wilson and Cowan
(Wilson and Cowan 1972) and others (Schuster and Wagner 1990, Grannan, Kleinfeld and Sompolinsky 1992)
Depending on the various parameters the stable solutions of these equations are
either fixed-points or limit cycles. The fixed-point solutions represent a stationary
state of the network in which the popUlation-averaged activities are almost constant
in time. The limit-cycle solutions represent nonstationary states in which there
is a coherent oscillatory activity. Obviously in the latter case there are strong
oscillatory correlations among the neurons. Here we focus on the fixed-point case.
It is described by the following equations
So = tanh ( J So - K 0'0 - 0)
(5)
0'0 = tanh (J ' So - K'O'o - 0)
(6)
where So and 0'0 are the fixed-point values of sand O'. Our aim is to estimate the
magnitude of the correlations between the temporal fluctuations in the activities of
neurons in this statistically stationary state.
4
CORRELATION FUNCTIONS
There are two types of auto-correlation functions, for the two different populations.
For the excitatory neurons we define the auto-correlations as:
(7)
where 6s i (t) = Sj(t)-so and < ... >t means average over time t. A similar definition
holds for the auto-correlations of the inhibitory neurons. In our network there are
three different cross-correlations: excitatory-excitatory, inhibitory- inhibitory, and
inhibitory-excitatory. The excitatory-excitatory correlations are
Cij(T)
= {8s i (t)8sj(t + T)}t
Similar definitions hold for the other functions.
(8)
473
474
Ginzburg and Sompolinsky
We have evaluated these correlation functions by solving the equations for the correlations of 6Si(t) in the limit of large Nand M. We find the following forms for
the correlations:
Gii(T) ~ (1- s~)exp(-A1T)
+
1 3
N La,exp(-AI T)
(9)
'=1
1
3
Gij(T)~ NLb,exP(-A,T) .
(10)
1=1
The coefficients a, and b, are in general of order 1. The three A, represent three
inverse time-constants in our system, where Re(AI) ~ Re(A2) ~ Re(A3)' The first
inverse time constant equals simply to Al = liTo, and corresponds to a purely
local mode of fluctuations. The values of A2 and A3 depend on the parameters of
the system. They represent two collective modes of fluctuations that are coherent
across the populations. An important outcome of our analysis is that A2 and A3
are exactly the eigenvalues of the stability matrix obtained by linearizing Eqs. (3)
and (4) about the fixed-point Eqs. (5) and (6) .
The above equations imply two differences between the auto-correlations and the
cross-correlations. First, Gi i are of order 1 whereas in general Gij is of 0(1/ N).
Secondly, the time-dependence of Gii is dominated by the local, fast time constant
TO, whereas Gij may be dominated by the slower, collective time-constants.
The conclusion that the cross-correlations are small relative to the auto-correlations
might break down if the coefficients b, take anomalously large values. To check these
possibility we have studied in detail the behavior of the correlations near bifurcation
points, at which the fixed point solutions become unstable. For concreteness we will
discuss here the case of Hopfbifurcations. (Similar results hold for other bifurcations
as well). Near a Hopf bifurcation A2 and A3 can be written as A? ~ ? ? iw,
where ? > 0 and vanishes at the bifurcation point. In this parameter regime, the
amplitudes b1 ? b2, b3 and b2 ~ b3 ~ ~. Similar results hold for a2 and a3. Thus,
near the bifurcation, we have
Gii (T) ~ (1 - s~) exp( -T /ro)cos(wr)
(11)
B
Gij(r) ~ N? exp(-?r)cos(wr) .
(12)
Note that near a bifurcation point ? is linear in the difference between any of the
parameters and their value at the bifurcation. The above expressions hold for
??
1 but large compared to l/N.When ? ~ liN the cross-correlation becomes of
order 1, and remains so throughout the bifurcation.
Figures 1 and 2 summarize the results of Eqs. (9) and (10) near the Hopf
bifurcation point at J,J',K,K',O 225,65, 161,422,2.4. The population sizes
are N
10000, M
1000. We have chosen a parameter range so that the fixed
point values of So and lTo will represent a state with low firing rate resembling
the spontaneous activity levels in the cortex. For the above parameters the rates
relative to the saturation rates are 0.01 and 0.03 for the excitatory and inhibitory
populations respectively.
=
=
=
Correlation Functions in a Large Stochastic Neural Network
0.45
04
035
0.3
025
02
015
01
005
O~~==C==C==~~--~~--~~
180
185
190
195
200
205
210
215
220
225
J
FIG URE 1. The equal-time cross-correlations between a pair of excitatory neurons, and the real part of its inverse time-constant,f, vs. the excitatory coupling
parameter J.
The values of Cij (0) and of the real-part of the inverse-time constants of Cij are
plotted (Fig. 1) as a function of the parameter J holding the rest of the parameters
fixed at their values at the bifurcation point. Thus in this case f a(225 - J). The
Figure shows the growth of Cij and the vanishing of the inverse time constant as
the bifurcation point is approached.
0.15 ..-----r--..-----,---.,....---,---r----,----,--.,.----,
0.1
0.05
o
-0 .05
-0 .1
-0 .15
L-_....l.-_--L_---l._ _.L..-_-'--_-'-_~_-.-I_ _~_-'
o
5
10
15
20
25
delay
30
35
40
45
50
The time-dependence of the cross-correlations near the bifurcation (J = 215) is
shown in Fig. 2. Time is plotted in units of TO. The pronounced damped oscillations
are, according to our theory, characteristic of the behavior of the correlations near
but below a Hopf bifurcation.
475
476
Ginzburg and Sompolinsky
5
CONCLUSION
Most theoretical investigations of large recurrent networks focus on the properties of
the macroscopic order parameters such as population averaged activity or average
overlap with memories. However, the statistics of the fluctuations in the activities
may be an important testing ground for comparison between models and observed
cortical dynamics. We have studied the properties of the correlation functions in a
stochastic network comprising of excitatory and inhibitory populations. We have
shown that the cross-correlations are relatively weak in stationary states, except in
the neighborhoods of bifurcations to nonstationary states. The growth of the amplitude of these correlations is coupled to a growth in the correlation time-constant.
This divergence of the correlation time is analogous to the phenomenon of critical
slowing down in systems at thermal equilibrium near a critical point. Our analysis
can be extended to stochastic networks consisting of a small number of interacting
homogeneous populations.
Detailed comparison between the model's results and experimental values of autoand cross- correlograms of extracellularly measured spike trains in the neocortex
have been carried out (Abeles, Ginzburg and Sompolinsky). The tentative conclusion of this study is that the magnitude of the observed correlations and their
time-dependence are inconsistent with the expected ones for a system in a stationary state. They therefore indicate that cortical neuronal assemblies are in a
nonstationary (but aperiodic) dynamic state.
Acknowledgements: We thank M. Abeles for most helpful discussions. This work
is partially supported by the USA-Israel Binational Science Foundation.
REFERENCES
Abeles M., 1991. Corticonics: Neural Circuits of the Cerebral Cortex. Cambridge
University Press.
Abeles M., Ginzburg I. & Sompolinsky H. Neuronal Cross-Correlations and Organized Dynamics in the Neocortex. to appear
Fetz E., Toyama K. & Smith W., 1991. Synaptic Interactions Between Cortical
Neurons. Cerebral Cortex, edited by A. Peters & G. Jones Plenum Press,NY. Vol
9. 1-43.
Grannan E., Kleinfeld D. & Sompolinsky H., 1992. Stimulus Dependent Synchronization of Neuronal Assemblies. Neural Computation 4,550-559.
Schuster H. G. & Wagner P., 1990. BioI. Cybern. 64, 77.
Wilson H. R. & Cowan J. D., 1972. Excitatory and Inhibitory Interactions m
Localized Populations of Model Neurons. Biophy. J. 12, 1-23.
| 854 |@word faculty:1 indicate:1 aperiodic:1 symmetric:1 spike:1 stochastic:8 dependence:4 exhibit:2 microscopic:1 quiet:1 sand:2 thank:1 iris:1 linearizing:1 investigation:2 unstable:1 secondly:1 hold:6 ground:2 si:2 exp:5 hebrew:1 written:1 equilibrium:2 cij:4 functional:1 holding:1 a2:5 binational:1 cerebral:2 v:1 stationary:6 collective:2 perform:1 tanh:4 iw:1 slowing:2 i_:1 cambridge:1 neuron:27 ai:2 smith:2 vanishing:1 thermal:2 b3:2 extended:1 aim:1 sigmoidal:1 interacting:3 stable:1 anomalously:1 cortex:5 correlograms:1 become:1 hopf:4 wilson:3 j:1 recent:1 focus:4 pair:4 specified:1 connection:3 tentative:1 check:1 coherent:3 expected:2 behavior:3 helpful:1 growing:1 dependent:1 beyond:1 usually:1 below:2 nand:1 regime:2 summarize:1 going:1 becomes:2 comprising:2 ii:1 saturation:1 memory:2 circuit:1 among:1 overlap:2 israel:3 denoted:1 critical:4 characterized:1 cross:14 bifurcation:17 lin:1 field:4 astronomy:1 equal:3 corticonics:1 imply:1 temporal:1 quantitative:1 represents:1 toyama:2 jones:1 carried:1 growth:3 exactly:1 ro:1 others:1 stimulus:1 represent:5 auto:6 unit:2 coupled:1 raymond:1 appear:1 acknowledgement:1 whereas:2 divergence:1 relative:3 local:8 lacking:1 fully:1 limit:3 synchronization:1 consisting:2 macroscopic:3 rest:1 ure:1 fluctuation:7 firing:4 approximately:1 interest:1 might:1 possibility:1 gii:3 cowan:3 studied:3 usa:1 inconsistent:1 undamped:1 foundation:1 co:2 nonstationary:4 near:10 presence:1 range:1 statistically:2 averaged:5 damped:2 excitatory:23 supported:1 testing:2 l_:1 institute:1 fetz:2 wagner:2 re:3 plotted:2 expression:1 biophy:1 theoretical:4 cortical:5 transition:1 modeling:1 peter:1 aviv:2 localized:1 onto:1 cybern:1 sj:3 detailed:1 center:1 delay:2 resembling:1 jerusalem:1 neocortex:2 extensively:1 active:1 b1:1 crosscorrelations:1 assumed:2 exist:1 abele:5 nlb:1 timeconstant:1 inhibitory:19 wr:2 schuster:2 racah:1 population:16 stability:1 physic:2 correlated:1 analogous:2 plenum:1 vol:1 spontaneous:1 tel:2 connectivity:1 exact:1 threshold:1 homogeneous:1 a1t:1 noise:2 leading:1 concreteness:1 account:1 year:1 observed:4 inverse:5 neuronal:7 fig:3 b2:2 coefficient:2 almost:1 throughout:1 connected:1 cycle:2 sompolinsky:8 ny:1 oscillation:3 break:1 edited:1 extracellularly:1 vanishes:1 ui:1 haim:1 sackler:1 dynamic:8 down:3 depend:1 solving:1 activity:14 exposed:1 purely:1 beverly:1 characteristic:1 lto:1 decay:1 basis:1 physiological:1 a3:5 dominated:2 weak:2 various:1 magnitude:3 train:1 relatively:2 fast:1 according:1 approached:2 oscillatory:3 neighborhood:2 outcome:1 synaptic:1 across:1 definition:2 simply:1 lito:1 statistic:2 gi:1 partially:1 ginzburg:6 corresponds:2 taken:1 obviously:1 equation:7 indication:1 eigenvalue:1 remains:1 discus:1 bioi:1 organized:1 interaction:2 amplitude:5 dt:2 except:2 obey:1 evaluated:2 appropriate:1 gij:4 pronounced:1 experimental:1 la:1 correlation:41 d:1 slower:1 diverges:1 ei:2 denotes:4 assembly:3 latter:1 kleinfeld:2 depending:2 recurrent:4 coupling:2 mode:2 phenomenon:2 measured:1 school:1 grows:1 eq:3 strong:1 |
7,085 | 855 | Classification of Electroencephalogram using
Artificial Neural Networks
A C Tsoi*, D S C So*, A Sergejew**
*Department of Electrical Engineering
**Department of Psychiatry
University of Queensland
St Lucia, Queensland 4072
Australia
Abstract
In this paper, we will consider the problem of classifying electroencephalogram (EEG) signals of normal subjects, and subjects suffering from psychiatric disorder, e.g., obsessive compulsive disorder, schizophrenia, using a
class of artificial neural networks, viz., multi-layer perceptron . It is shown
that the multilayer perceptron is capable of classifying unseen test EEG
signals to a high degree of accuracy.
1
Introduction
The spontaneous electrical activity of the brain was first observed by Caton in 1875.
Although considerable investigations on the electrical activity of the non-human
brain have been undertaken, it was not until 1929 that a German neurologist Hans
Berger first published studies on the electroencephalogram (EEG) recorded on the
scalp of human. He lay the foundation of clinical and experimental applications of
EEG between 1929 and 1938.
Since then EEG signals have been used in both clinical and experimental work
to discover the state which the brain is in (see e.g., Herrmann, 1982, Kolb and
Whishaw, 1990, Lindsay and Holmes, 1984). It has served as a direct indication of
any brain activities. It is routinely being used in clinical diagnosis of epilepsy (see
e.g., Basar, 1980; Cooper, 1980).
Despite advances in technology, the classification of EEG signals at present requires
a trained personnel who either "eyeballs" the direct EEG recordings over time,
1151
1152
Tsoi, So, and Sergejew
or studies the contour maps representing the potentials generated from the "raw"
electrical signal (see e.g., Cooper, 1980). This is both a highly skillful job, as well as
a laborious task for a neurologist. With the current advances in computers, a logical
question to ask: can we use the computer to perform an automa'(.ic classification of
EEG signals into different classes denoting the psychiatric states of the subjects?
This type of classification studies is not new. In fact, in the late 1960's there were
a number of attempts in performing the automatic classification using discriminant
analysis techniques. However, this work was largely abandoned as most researchers
concluded that classification based on discriminant techniques does not generalise
well, i.e., while it has very good classification accuracies in classifying the data which
is used to train the automatic classification system, it may not have high accuracy
in classifying the unseen data which are not used to train the system in the first
instance.
Recently, a class of classification techniques, called artificial neural network (ANN),
based on nonlinear models, has become very popular (see e.g., Touretzky, 1989,
1990, Lippmann et aI, 1991). This type of networks claims to be inspired by biological neurons, and their many inter-connections. This type of artificial neural
networks has limited pattern recognition capabilities. Among the many applications
which have been applied so far are sonar signal classification (see e.g., Touretzky,
1989), handwritten character recognition (see .e.g., Touretzky, 1990), facial expression recognition (see e.g., Lippmann et a1. 1991).
In this paper, we will investigate the possibility of using an ANN for EEG classifications. While it is possible to extract features from the time series using either time
domain or frequency domain techniques, from some preliminary work, it is found
that the time domain techniques give much better results.
The structure of this paper is as follows: In section 2, we will give a brief discussion
on a popular class of ANNs, viz., multi-layer perceptrons (MLP). In section 3, we
will discuss various feature extractions using time domain techniques. In section 4,
we will present results in classifying a set of unseen EEG signals.
2
Multi-layer Perceptrons
Artificial neural network (ANN) consists of a number of artificial neurons interconnected together by synaptic weights to form a network (see e.g, Lippmann,
1987). Each neuron is modeled by the following mechanical model:
n
y
= f(L WiXi + 0)
(1)
i=l
=
where y is the output of the neuron, Wi, i
1,2, ... , n are the synaptic weights,
Xi, i
1,2 ... , n are the inputs, and 0 is a threshold function. The nonlinear
function f(.) can be a sigmoid function, or a hyperbolic tangent function. An ANN
is a network of inter-connected neurons by synapses (Hertz, Krogh and Palmer,
1991).
=
There are many possible ANN architectures (Hertz, Krogh, Palmer, 1991). A pop-
Classification of Electroencephalogram Using Artificial Neural Networks
ular architecture is the multi-layer perceptron (MLP) (see e.g., Lippmann, 1987).
In this class of ANN, signal travels only in a forward direction. Hence it is also
known as a feedforward network. Mathematically, it can be described as follows:
Y = !(Az
+ 0ll)
z=!(Bu+O z )
(2)
(3)
where y is a m x 1 vector, representing the output of the output layer neurons; z
is a p x 1 vector, representing the outputs of the hidden layer neurons; u is a n x 1
vector, representing the input feature vector; OJ! is a m x 1 vector, known as the
threshold vector for the output layer neurons; Oz is a p x 1 vector, representing
the threshold vector for the hidden layer neurons; A and B are matrices of m x p
and p x n respectively. The matrices A, and B are the synaptic weights connecting
the hidden layer neuron to the output layer neuron; and the input layer neurons,
and the hidden layer neurons respectively. For simplicity sake, we will assume the
nonlinearity function to be a sigmoid function, i.e.,
1
f(a)=I+e- a
(4)
The unknown parameters A, B, OJ!, Oz can be obtained by minimizing an error criterion:
p
J
= L(di .=1
Yi)2
(5)
where P is the total number of examplars, di , i = 1,2, ... , P are the desired outputs
which we wish the MLP to learn.
By differentiating the error criterion J with respect to the unknown parameters,
learning algorithms can be obtained.
The learning rules are as follows:
(6)
where Anew is the next estimate of the matrix A, T denotes the transpose of a
vector or a matrix. TJ is a learning constant. A(y) is a m x m diagonal matrix,
whose dia~onal elements are / (Y')' i 1,2, ... , m. The vector e is m x 1, and it is
given by e
[(d 1 - yd, (d 2 - Y2), ... , (d m - Ym)]T.
=
=
The updating equation for the B matrix is given by the following
(7)
where 6 is a p x 1 vector, given by
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Tsoi, So, and Sergejew
fJ = AT A(y)e
(8)
and the other parameters are as defined above.
The threshold vectors can be obtained as follows:
(9)
and
(10)
Thus it is observed that once a set of initial conditions for the unknown parameters
are given, this algorithm will find a set of parameters which will converge to a value,
representing possibly a local minimum of the error criterion.
3
Pre-processing of the EEG signal
A cursory glance at a typical EEG signal of a normal subject, or a psychiatrically ill
subject would convince anyone that one cannot hope to distinguish the signal just
from the raw data alone. Consequently, one would need to perform considerable
feature extraction (data pre-processing) before classification can be made. There
are two types of simple feature extraction techniques, viz., frequency domain and
time domain (see e.g., Kay, 1988, Marple, 1987). In the frequency domain, one
performs a fast Fourier transform (FFT) on the data. Often it is advantageous
to modify the signal by a window function. This will reduce the sidelobe leakage
(Kay, and Marple, 1981, Harris, 1978). it is possible to use the average spectrum,
obtained by averaging the spectrum over a number of frames, as the input feature
vector to the MLP.
In the time domain, one way to pre-process the data is to fit a parametric model to
the underlying data. There are a number of parametric models, e.g., autoregressive
(AR) model, an autoregressive moving average (ARMA) model (see e.g., Kay, 1988,
Marple, 1987).
The autoregressive model can be described as follows:
N
Se
= L OjSe_j + fe
(11)
j=1
where Se is the signal at time t; ft is assumed to be a zero mean Gaussian variable with variance (T2. The unknown parameters OJ, j = 1,2, ... , N describe the
spectrum of the signal. They can be obtained by using standard methods, e.g.,
Yule-Walker equations, or Levinson algorithm (Kay, 1988, Marple, 1987).
The autoregressive moving average (ARMA) model can be seen as a parsimonious
model for an AR model with a large N. Hence, as long as we are not concerned
Classification of Electroencephalogram Using Artificial Neural Networks
about the interpretation of the AR model obtained, there is little advantage to
use the more complicated ARMA model. Subsequently, in this paper, we will only
consider the AR models.
Once the AR parameters are determined, then they can be used as the input features to the MLP. It is known that the AR parametric model basically produces
a smoothed spectral envelope (Kay, 1988, Marple, 1987). Thus, the model parameters of AR is another way to convey the spectral information to the MLP. This
information is different in quality to that given by the FFT technique in that the
FFT transforms both signal and noise alike, while the parametric models tend to
favor the signal more and is more effective in suppressing the noise effect.
In some preliminary work, we find that the frequency domain extracted features do
not give rise to good classification results using MLP. Henceforth we will consider
only the AR parameters as input feature vectors.
4
Classification Results
In this section, we will summarise the results of the experiments in using the AR
parametric method of feature extraction as input parameters to the MLP.
We obtained EEG data pertaining to normal subjects, subjects who have been diagnosed as suffering from severe obsessive compulsive disorder (OCD), and subjects
who have been diagnosed as suffering from severe schizophrenia. Both the OCD and
the schizophrenic subjects are under medication. The subjects are chosen so that
their medication as well as their medical conditions are at a steady state, i.e., they
have not changed over a long period of time. The diagnosis is made by a number of
trained neurologists. The data files are chosen only if the diagnosis from the experts
concur.
We use the standard 10-20 recording system (Cooper, 1980), i.e., there are 19
channels of EEG recording, each sampled at 128 Hz. The recording were obtained
while the subject is at rest. Some data screening has been performed to screen out
the segment of data which contains any artifact. In addition, the data is anti-aliased
first by a low pass filter before being sampled. The sampled data is then low pass
filtered at 30 Hz to get rid of any higher frequency components.
We have chosen one channel, viz., the C z channel (the channel which is the recording
of the signal at the azimuth of the scalp). This channel can be assumed to be
representative of the brain state from the overall EEG recording of the scalp. 1
This time series is employed for feature extraction purposes.
For time domain feature extraction, we first convert the time series into a zero
mean one. Then a data frame of one second duration is chosen 2 as the basic time
segmentation of the series. An AR model is fitted to this one second time frame to
1 From some preliminary work, it can be shown that this channel can be considered as
a linear combination of the other channels, in the sense that the prediction error variance
is small.
2It has been found that the EEG signal is approximately stationary for signal length of
one second. Hence employing a data frame width of one second ensures that the underlying
assumptions in the AR modelling technique are valid (Marple, 1988)
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Tsoi, So, and Sergejew
extract a feature vector formed by the resulting AR coefficients.
An average feature vector is acquired from the first 250 seconds, as in practice, the
first 250 seconds usually represent a state of calm in the patient, and therefore the
EEG is less noisy. After the first 250 seconds, the patient may enter an unstable
condition, such as breathing faster and muscle contraction which can introduce
artifacts. We use an AR model of length between 8 to 15.
We have chosen 15 such data file to form our training data set. This consists of 5
data files from normal subjects, 5 from OeD subjects, and 5 from subjects suffering
from schizophrenia.
In the time domain extracted feature vectors, we use a MLP with 8 input neurons,
15 hidden layer neurons, and 3 output neurons. The MLP's are trained accordingly.
We use a learning gain of 0.01. Once trained, the network is used to classify unseen
data files. These unseen data files were pre-classified by human experts. Thus the
desired classification of the unseen data files are known. This can then be used to
check the usefulness of the MLP in generalising to unseen data files.
The results
3
are shown in table 1.
The unseen data set consists of 6 normal subjects, 8 schizophrenic subjects, and
10 obsessive compulsive disorder subjects. It can be observed that the network
correctly classifies all the normal cases, makes one mistake in classifying the
schizophrena cases, and one mistake in classifying the OeD cases.
Also we have experimented on varying the number of hidden neurons. It is found
that the classification accuracy does not vary much with the variation of hidden
layer neurons from 15 to 50.
We have also applied the MLP on the frame by frame data, i.e., before they are being
averaged over the 250 second interval. However, it is found that the classification
results are not as good as the ones presented. We were puzzled by this result as
intuitively, we would expect the frame by frame results to be better than the ones
presented.
A plausible explanation for this puzzle is given as follows: the EEG data is in
general quite noisy. In the frame by frame analysis, the features extracted may
vary considerably over a short time interval, while in the approach taken here, the
noise effect is smoothed out by the averaging process.
One may ask: why would the methods presented work at all? In traditional EEG
analysis (Lindsay & Holmes, 1984), FFT technique is used to extract the frame
by frame frequency responses. The averaged frequency response is then obtained
over this interval. Traditionally only four dominant frequencies are observed, viz.,
the "alpha", "beta", "delta", and "theta" frequencies. It is a basic result in EEG
research that these frequencies describe the underlying state of the subject. For
example, it is known that the "alpha" wave indicates that the subject is at rest. An
EEG technologist uses data in this form to assist in the diagnosis of the subject.
On the other hand, it is relatively well known in signal processing literature (Kay,
3The results shown are typical results. We have used different data files for training
and testing. In most cases, the classification errors on the unseen data files are small,
similar to those presented here.
Classification of Electroencephalogram Using Artificial Neural Networks
original
classes
normall
normal2
normal3
normal4
normal5
norma16
schiz1
schiz2
schiz3
schiz4
schiz5
schiz6
schiz7
schiz8
ocdl
ocd2
ocd3
ocd4
ocd5
ocd6
ocd7
ocd8
ocd9
ocdlO
activation of
normal
0.905
0.963
0.896
0.870
0.760
0.752
0.000
0.000
0.002
0.015
0.000
0.377
0.062
0.006
0.017
0.027
0.000
0.000
0.015
0.000
0.002
0.006
0.045
0.085
activation of
schiz
0.008
0.006
0.021
0.057
0.237
0.177
0.981
0.941
0.845
0.989
0.932
0.695
0.898
0.086
0.134
0.007
0.033
0.014
0.138
0.150
0.034
0.960
0.005
0.046
activation of
ocd
0.201
0.103
0.086
0.020
0.000
0.065
0.042
0.163
0.050
0.004
0.061
0.014
0.000
0.921
0.922
0.940
0.993
0.997
0.889
0.946
0.985
0.003
0.940
0.585
predicted
classes
normal
normal
normal
normal
normal
normal
schiz
schiz
schiz
schiz
schiz
schiz
schiz
ocd
ocd
ocd
ocd
ocd
ocd
ocd
ocd
schiz
ocd
ocd
Table 1: Classification of unseen EEG data files
1988, Marple, 1987) to view the AR model as indicative of the underlying frequency
content of the signal. In fact, an 8th order AR model indicates that the signal
can be considered to consist of 4 underlying frequencies. Thus, intuitively, the
8th order AR model averaged over the first 250 seconds represents the underlying
dominant frequencies in the signal. Given this interpretation, it is not surprising
that the results are so good. The features extracted are similar to those used in the
diagnosis of the subjects. The classification technique, which in this case, the MLP,
is known to have good generalisation capabilities (Hertz, Krogh, Palmer, 1991).
This contrasts the techniques used in previous attempts in the 1960's, e.g., the
discriminant analysis, which is known to have poor generalisation capabilities. Thus,
one of the reasons why this approach works may be attributed to the generalisation
capabilities of the MLP.
5
Conclusions
In this paper, a method for classifying EEG data obtained from subjects who are
normal, OCD or schizophrenia has been obtained by using the AR parameters as
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Tsoi, So, and Sergejew
input feature vectors. It is found that such a network has good generalisation
capabili ties .
6
Acknowledgments
The first and third author wish to acknowledge partial financial support from the
Australian National Health and Medical Research Council. In addition, the first
author wishes to acknowledge partial financial support from the Australian Research
Council.
7
References
Basar, E. (1980). EEG-Brain Dynamics - Relation between EEG and Brain Evoked
Potentials. Elsevier/North Holland Biomedical Press.
Cooper, R. (1980). EEG Technology. Butterworths. Third Editions.
Harris, F.J. (1978). "On the Use of windows for Harmonic Analysis with the
Discrete Fourier Transform". Proceedings IEEE. Vol. 66, pp 51-83.
Herrmann, W.M. (1982). Electroencephalography in Drug Research. Butterworths.
Hertz, J. Krogh, A, Palmer, R. (1991) Introduction to The Theory of Neural Computation. Addison Wesley, Redwood City, Calif.
Kay, S.M., Marple, S.L., Jr. (1981). "Spectrum Analysis - A Modern Perspective".
Proceeding IEEE. Vol. 69, No. 11, Nov. pp 1380 - 1417.
Kay, S.M. (1988) Modern Spectral Estimation - Theory and Applications Prentice
hall.
Kolb, B., Whishaw, I.Q. (1990). Fundamentals of Human Neuropsychology. Freeman, New York.
Lindsay, D.F., Holmes, J.E. (1984). Basic Human Neurophysiology. Elsevier.
Lippmann, R.P. (1987) " An introduction to computing with neural nets" IEEE
Acoustics Speech and Signal Processing Magazine. Vol. 4, No.2, pp 4-22.
Lippmann, R.P., Moody, J., Touretzky, D.S. (Ed.) (1991). Advances in Neural
Information Processing Systems 9. Morgan Kaufmann, San Mateo, Calif.
Marple, S.L., Jr. (1987). Digital Spectral Analysis with Applications. Prentice Hall.
Touretzky, D.S. (Ed.) (1989). Advances in Neural Information Processing Systems
1. Morgan Kaufmann, San Mateo, Calif.
Touretzky, D.S. (Ed.) (1990). Advances in Neural Information Processing Systems
2. Morgan Kaufmann, San Mateo, Calif.
PART XII
WORKSHOPS
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7,086 | 856 | Postal Address Block Location Using A
Convolutional Locator Network
Ralph Wolf and John C. Platt
Synaptics, Inc.
2698 Orchard Parkway
San Jose, CA 95134
Abstract
This paper describes the use of a convolutional neural network
to perform address block location on machine-printed mail pieces.
Locating the address block is a difficult object recognition problem
because there is often a large amount of extraneous printing on a
mail piece and because address blocks vary dramatically in size and
shape.
We used a convolutional locator network with four outputs, each
trained to find a different corner of the address block. A simple
set of rules was used to generate ABL candidates from the network
output. The system performs very well: when allowed five guesses,
the network will tightly bound the address delivery information in
98.2% of the cases.
1
INTRODUCTION
The U.S. Postal Service delivers about 350 million mail pieces a day. On this scale,
even highly sophisticated and custom-built sorting equipment quickly pays for itself.
Ideally, such equipment would be able to perform optical character recognition
(OCR) over an image of the entire mail piece. However, such large-scale OCR is
impractical given that the sorting equipment must recognize addresses on 18 mail
pieces a second. Also, the large amount of advertising and other irrelevant text that
can be found on some mail pieces could easily confuse or overwhelm the address
recognition system. For both of these reasons, character recognition must occur
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Wolf and Platt
Figure 1: Typical address blocks from our data set. Notice the wide variety in
the shape, size, justification and number of lines of text. Also notice the detached
ZIP code in the upper right example. Note: The USPS requires us to preserve the
confidentiality of the mail stream. Therefore, the name fields of all address block
figures in this paper have been scrambled for publication. However, the network
was trained and tested using unmodified images.
only on the relevant portion of the envelope: the destination address block. The
system thus requires an address block location (ABL) module, which draws a tight
bounding box around the destination address block.
The ABL problem is a challenging object recognition task because address blocks
vary considerably in their size and shape (see figure 1). In addition, figures 2 and 3
show that there is often a great deal of advertising or other information on the mail
piece which the network must learn to ignore.
Conventional systems perform ABL in two steps (Caviglione, 1990) (Palumbo,
1990). First, low-level features, such as blobs of ink, are extracted from the image. Then, address block candidates are generated using complex rules. Typically,
there are hundreds of rules and tens of thousands of lines of code.
The architecture of our ABL system is very different from conventional systems.
Instead of using low-level features, we train a neural network to find high-level
abstract features of an address block. In particular, our neural network detects
the corners of the bounding box of the address block. By finding abstract features
instead of trying to detect the whole address block in one step, we build a large
degree of scale and shape invariance into the system. By using a neural network,
we do not need to develop explicit rules or models of address blocks, which yields a
more accurate system.
Because the features are high-level, it becomes easy to combine these features into
object hypotheses. We use simple address block statistics to convert the corner
features into object hypotheses, using only 200 lines of code.
Postal Address Block Location Using a Convolutional Locator Network
2
SYSTEM ARCHITECTURE
Our ABL system takes 300 dpi grey scale images as input and produces a list of the
5 most likely ABL candidates as output. The system consists of three parts: the
preprocessor, a convolutional locator network, and a candidate generator.
2.1
PREPROCESSOR
The preprocessor serves two purposes. First, it substantially reduces the resolution
of the input image, therefore decreasing the computational requirements of the
neural network. Second, the preprocessor enhances spatial frequencies in the image
which are associated with address text. The recipe used for the preprocessing is as
follows:
1:
2:
3:
4:
5:
Clip the top 20% of the image.
Spatially filter with a passband of 0.3 to 1.4mm.
Take the absolute value of each pixel.
Low-pass filter and subsample by a factor of 16 in X and Y.
Perform a linear contrast stretch, mapping the darkest
pixel to 1.0 and the lightest pixel to 0.0.
The effect of this preprocessing can be seen in figures 2 and 3.
2.2
CONVOLUTIONAL LOCATOR NETWORK
We use a convolutional locator network (CLN) to find the corners of the bounding
box. Each layer of a CLN convolves its weight pattern in two dimensions over the
outputs of the previous layer (LeCun, 1989) (Fukushima, 1980). Unlike standard
convolutional networks, the output of a CLN is a set of images, in which regions
of activity correspond to recognition of a particular object. We train an output
neuron of a CLN to be on when the receptive field of that neuron is over an object
or feature, and off everywhere else.
CLNs have been previously used to assist in the segmentation step for optical character recognition, where a neuron is trained to turn on in the center of every character,
regardless of the identity of the character (Martin, 1992) (Platt, 1992). The recognition of an address block is a significantly more difficult image segmentation problem
because address blocks vary over a much wider range than printed characters (see
figure 1).
The output of the CLN is a set of four feature maps, each corresponding to one
corner of the address block. The intensity of a pixel in a given feature map represents
the likelihood that the corresponding corner of the address block is located at that
pixel.
Figure 4 shows the architecture of our convolutional locator network (CLN). It has
three layers of trainable weights, with a total of 22,800 free parameters. The network
was trained via weight-shared backpropagation. The network was trained for 23
epochs on 800 mail piece images. This required 125 hours of cpu-time on an i860
based computer. Cross validation and final testing was done with two additional
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Wolf and Platt
Figure 2: The network operating on an example from the test set. The top image
is the original image. The middle image is the image that is fed to the CLN after
preprocessing. The preprocessing enhances the text and suppresses the background
color. The bottom image is the first candidate of the ABL system. The output of the
system is shown with a white and black rectangle. In this case, the first candidate
is correct. Notice that our ABL system does not get confused by the horizontal
lines in the image, which would confound a line-finding-based ABL system.
Postal Address Block Location Using a Convolutional Locator Network
Figure 3: Another example from the test set. The preprocessed image still has a
large amount of background noise. In this example, the first candidate of the ABL
system (shown in the lower left) was almost correct, but the ZIP code got truncated.
The second candidate of the system (shown in the lower right) gives the complete
address.
749
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Wolf and Platt
Output maps
Third layer of weights
4 36x16 windows
Second layer feature maps
Second layer of weights
8 9x9 windows
2x2 subsampled first layer
feature maps
First layer feature maps
First layer of weights
6 9x9 windows
Input image
Figure 4: The architecture of the convolutional locator network used in our ABL
system.
data sets of 500 mail pieces each. All together, these 1800 images represent 6 Gbytes
of raw data, or 25 Mbytes of preprocessed images.
2.3
CANDIDATE GENERATOR
The candidate generator uses the following recipe to convert the output maps of
the CLN into a list of ABL candidates:
1: Find the top 10 local maxima in each feature map.
2: Construct all possible tBL candidates by combining pairs of
local maxima from opposing corners.
3: Discard candidates which have negative length or width.
4: Compute confidence of each candidate.
6: Sort the candidates according to confidence.
6: Remove duplicate and near duplicate candidates.
7: Pad the candidates by a fixed amount on all sides.
The confidence of an address block candidate is:
2
Caddress block
= PsizePIocation II Ci
i=l
where Caddress block is the confidence of the address block candidate, Psize is the
prior probability of finding an address block of the hypothesized size, I\ocation is
the prior probability of finding an address block in the hypothesized location, and
Postal Address Block Location Using a Convolutional Locator Network
Ci are the value of each of the output maxima. The prior probabilities Psize and
.A.ocation were based on smoothed histograms generated from the training set and
validation set truths.
Steps 6 and 7 each contain 4 tuning parameters which we optimized using the
validation set and then froze before evaluating the final test set.
3
SYSTEM PERFORMANCE
Figures 2 and 3 show the performance of the system on two challenging mail pieces
from the final test set. We examined and classified the response of the system to all
500 test images. When allowed to produce five candidates, the ABL system found
98.2% of the address blocks in the test images.
More specifically, 96% of the images have a compact bounding box for the complete
address block. Another 2.2% have bounding boxes which contain all of the delivery
information, but omit part of the name field. The remaining 1.8% fail, either
because none of the candidates contain all the delivery information, or because
they contain too much non-address information. The average number of candidates
required to find a compact bounding box is only 1.4.
4
DISCUSSION
This paper demonstrates that using a CLN to find abstract features of an object,
rather than locating the entire object, provides a reasonable amount of insensitivity
to the shape and scale of the obj~ct. In particular, the completely identified address
blocks in the final test set had aspect ratios which ranged from 1.3 to 6.1 and their
absolute X and Y dimensions both varied over a 3:1 range. They contained anywhere
from 2 to 6 lines of text.
In the past, rule-based systems for object recognition 'were designed from scratch
and required a great deal of domain-specific knowledge. CLNs can be trained to
recognize different classes of objects without a lot of domain-specific knowledge.
Therefore, CLNs are a general purpose object segmentation and recognition architecture.
The basic computation of a CLN is a high-speed convolution, which can be costeffectively implemented by using parallel hardware (Sickinger, 1992). Therefore,
CLNs can be used to reduce the complexity and cost of hardware recognition systems.
5
CONCLUSIONS
In this paper, we have described a software implementation for an address block
location system which uses a convolutional locator network to detect the corners of
the destination address on machine printed mail pieces.
The success of this system suggests a general approach to object recognition tasks
where the objects vary considerably in size and shape. We suggest the following
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Wolf and Platt
three-step approach: use a simple preprocessing algorithm to enhance stimuli which
are correlated to the object, use a CLN to detect abstract features of the objects in
the preprocessed image, and construct object hypotheses by a simple analysis of the
network output. The use of CLNs to detect abstract features enables versatile object
recognition architectures with a reasonable amount of scale and shape invariance.
Acknowledgements
This work was funded by USPS Contract No. 104230-90-C-344l. The authors would
like to thank Dr. Binh Phan of the USPS for his generous advice and encouragement. The images used in this work were provided by the USPS.
References
Caviglione, M., Scaiola, (1990), "A Modular Real-time Vision System for Address
Block Location," Proc. 4th Advanced Technology Conference, USPS, 42-56.
Fukushima, K., (1980), "Neocognitron: A Self-Organizing Neural Network Model
for a Mechanism of Pattern Recognition Unaffected by Shift in Position." Biological
Cybernetics, 36, 193-202.
LeCun, Y., Boser, B., Denker, J.S., Henderson, D., Howard, R. E., Hubbard, W.,
Jackel, L. D., (1989), "Backpropagation Applied to Handwritten Zip Code Recognition" Neural Computation, 1, 541-55l.
Martin, G., Rashid, M., (1992), "Recognizing Overlapping Hand-Printed Characters
by Centered-Object Integrated Segmentation and Recognition," Advances in Neural
Information Processing Systems, 4, 504-51l.
Palumbo, P. W., Soh, J., Srihari, S. N., Demjanenjo, V., Sridhar, R., (1990), "RealTime Address Block Location using Pipelining and Multiprocessing," Proc. 4th
Advanced Technology Conference, USPS, 73-87.
Platt, J., Decker, J. E, LeMoncheck, J. E., (1992), "Convolutional Neural Networks
for the Combined Segmentation and Recognition of Machine Printed Characters,"
Proc. 5th Advanced Technology Conference, USPS, 701-713.
Sackinger, E., Boser, B., Bromley, J., LeCun, Y., Jackel, L., (1992) "Application of the ANNA neural network chip to high-speed character recognition," IEEE
Trans. Neural Networks, 3, (3), 498-505.
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7,087 | 857 | Learning Complex Boolean Functions:
Algorithms and Applications
Arlindo L. Oliveira and Alberto Sangiovanni- Vincentelli
Dept. of EECS
UC Berkeley
Berkeley CA 94720
Abstract
The most commonly used neural network models are not well suited
to direct digital implementations because each node needs to perform a large number of operations between floating point values.
Fortunately, the ability to learn from examples and to generalize is
not restricted to networks ofthis type. Indeed, networks where each
node implements a simple Boolean function (Boolean networks) can
be designed in such a way as to exhibit similar properties. Two
algorithms that generate Boolean networks from examples are presented. The results show that these algorithms generalize very
well in a class of problems that accept compact Boolean network
descriptions. The techniques described are general and can be applied to tasks that are not known to have that characteristic. Two
examples of applications are presented: image reconstruction and
hand-written character recognition.
1
Introduction
The main objective of this research is the design of algorithms for empirical learning
that generate networks suitable for digital implementations. Although threshold
gate networks can be implemented using standard digital technologies, for many
applications this approach is expensive and inefficient. Pulse stream modulation
[Murray and Smith, 1988] is one possible approach, but is limited to a relatively
small number of neurons and becomes slow if high precision is required. Dedicated
911
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Oliveira and Sangiovanni-Vincentelli
boards based on DSP processors can achieve very high performance and are very
flexible but may be too expensive for some applications.
The algorithms described in this paper accept as input a training set and generate
networks where each node implements a relatively simple Boolean function. Such
networks will be called Boolean networks. Many applications can benefit from
such an approach because the speed and compactness of digital implementations
is still unmatched by its analog counterparts. Additionally, many alternatives are
available to designers that want to implement Boolean networks, from full-custom
design to field programmable gate arrays. This makes the digital alternative more
cost effective than solutions based on analog designs.
Occam's razor [Blumer et ai., 1987; Rissanen, 1986] provides the theoretical foundation for the development of algorithms that can be used to obtain Boolean networks
that generalize well. According to this paradigm, simpler explanations for the available data have higher predictive power. The induction problem can therefore be
posed as an optimization problem: given a labeled training set, derive the
less complex Boolean network that is consistent I with the training set.
Occam's razor, however, doesn't help in the choice of the particular way of measuring complexity that should be used. In general, different types of problems may
require different complexity measures. The algorithms described in section 3.1 and
3.2 are greedy algorithms that aim at minimizing one specific complexity measure:
the size of the overall network. Although this particular way of measuring complexity may prove inappropriate in some cases, we believe the approach proposed
can be generalized and used with minor modifications in many other tasks. The
problem of finding the smallest Boolean network consistent with the training set is
NP-hard [Garey and Johnson, 1979] and cannot be solved exactly in most cases.
Heuristic approaches like the ones described are therefore required.
2
Definitions
We consider the problem of supervised learning in an attribute based description
language. The attributes (input variables) are assumed to be Boolean and every
exemplar in the training set is labeled with a value that describes its class. Both
algorithms try to maximize the mutual information between the network output
and these labels.
Let variable X take the values {Xl, X2, ... x n } with probabilities p(Xd,P(X2) ... P(x n ).
The entropy of X is given by H(X) = - Lj p(Xj) logp(xj) and is a measure
of the uncertainty about the value of X. The uncertainty about the value
of X when the value of another variable Y is known is given by H(XIY) =
- Li p(Yi) Lj p(Xj Iyd logp(xj Iyd?
The amount by which the uncertainty of X is reduced when the value of variable Y
is known, I(Y, X) = H(X) - H(XIY) is called the mutual information between Y
and X. In this context, Y will be a variable defined by the output of one or more
nottes in the network and X will be the target value specified in the training set.
1 Up
to some specified level.
Learning Complex Boolean Functions: Algorithms and Applications
3
3.1
Algorithms
Muesli - An algorithm for the design of multi-level logic networks
This algorithm derives the Boolean network by performing gradient descent in the
mutual information between a set of nodes and the target values specified by the
labels in the training set.
In the pseudo code description of the algorithm given in figure 1, the function 'L (S)
computes the mutual information between the nodes in S (viewed as a multi-valued
variable) and the target output.
muesli( nlist) {
nlist ;- sorLnlisLby1(nlist,1);
sup;- 2;
while (noLdone(nlist) /\ sup < max_sup) {
act ;- 0;
do {
act + +;
success;- improvLmi(act, nlist, sup);
} while (success = FALSE /\ act < max_act);
if (success = TRUE) {
sup;- 2;
while (success = TRUE)
success;- improve_mi(act, nlist, sup);
}
else sup
+ +;
}
}
improVLmi(act, nlist, sup) {
nlist;- sorLnlisLby1(nlist, act);
1;- besLlunction(nlist, act, sup);
if (I(nlist[l:act-l] U f) > I(nlist[l:act])) {
nlist ;- nlist U I;
return(TRUE);
}
else return(FALSE) j
}
Figure 1: Pseudo-code for the Muesli algorithm.
The algorithm works by keeping a list of candidate nodes, nlist, that initially contains only the primary inputs. The act variable selects which node in nl ist is active.
Initially, act is set to 1 and the node that provides more information about the output is selected as the active node. Function imp1'ove_miO tries to combine the
active node with other nodes as to increase the mutual information.
Except for very simple functions, a point will be reached where no further improve-
913
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Oliveira and Sangiovanni-Vincentelli
ments can be made for the single most informative node. The value of act is then
increased (up to a pre-specified maximum) and improve_mi is again called to select
auxiliary features using other nodes in ntist as the active node. If this fails, the
value of sup (size of the support of each selected function) is increased until no
further improvements are possible or the target is reached.
The function sorLnlisLbyJ(nlist, act) sorts the first act nodes in the list by decreasing value of the information they provide about the labels. More explicitly, the
first node in the sorted list is the one that provides maximal information about the
labels. The second node is the one that will provide more additional information
after the first has been selected and so on.
Function improve_miO calls besLfunction(nlist, act, sup) to select the Boolean
function f that takes as inputs node nlist[act] plus s'up-1 other nodes and maximizes
I(nlist[l : act -1] U f). When sup is larger than 2 it is unfeasible to search all 22 s UP
possible functions to select the desired one. However, given sup input variables,
finding such a function is equivalent to selecting a partition 2 of the 28UP points in
the input space that maximizes a specific cost function. This partition is found using
the Kernighan-Lin algorithm [Kernighan and Lin, 1970] for graph-partitioning.
Figure 2 exemplifies how the algorithm works when learning the simple Boolean
function f = ab + cde from a complete training set. In this example, the value of
sup is always at 2. Therefore, only 2 input Boolean functions are generated.
Select x = ab
mi([]) = 0.0
Fails to fmd f(x,?) with mi([f]) > 0.52
Set act = 2;
a
nlist = [a,b,c,d,e]
act = 1
mi([a]) = 0.16
Selecty = cd
nlist = [x,y,e,a,b,c,d]
act = 2
mi([x,y]) = 0.74
nlist = [x,c,d,e,a,b]
act = 1
mi([xD =0.52
Select w =ye
nlist = [x,y,e,a,b,c,d]
act = 2
mi([x,w]) = 0.93
nlist = [x,c,d,e,a,b]
act = 2
mi([x,c]) = 0.63
Fails to find f(w,?) with mi([x,f]) > 0.93
Set act =0; Select Z =x+w
nlist = [z,x,y,a,b,c,d,e]
act = 1
mi([z]) =0.93
Figure 2: The muesli algorithm, illustrated
2 A single output Boolean function is equivalent to a partition of the input space in two
sets.
Learning Complex Boolean Functions: Algorithms and Applications
3.2
Fulfringe - a network generation algorithm based on decision trees
This algorithm uses binary decision trees [Quinlan, 1986] as the basic underlying
representation. A binary decision tree is a rooted, directed, acyclic graph, where
each terminal node (a node with no outgoing edges) is labeled with one of the
possible output labels and each non-terminal node has exactly two outgoing edges
labeled 0 and 1. Each non-terminal node is also labeled with the name of the
attribute that is tested at that node. A decision tree can be used to classify a
particular example by starting at the root node and taking, until a terminal is
reached, the edge labeled with the value of the attribute tested at the current node.
Decision trees are usually built in a greedy way. At each step, the algorithm greedily
selects the attribute to be tested as the one that provides maximal information about
the label of the examples that reached that node in the decision tree. It then recurs
after splitting these examples according to the value of the tested attribute.
Fulfringe works by identifying patterns near the fringes of the decision tree and
using them to build new features. The idea was first proposed in [Pagallo and
Haussler, 1990].
N
A
1\0
+
~+
+
A
1\
!A "A
o
+
+~
+
p+g
-p&g
A
+
A
+1\
+
+
-p+-g
p+-g
-p+g
0
+
p&g
-p&-g
p&-g
A
0
MMMM
+
+
+
+
+
+
+
+
p(t)g
Figure 3: Fringe patterns identified by fuifringe
Figure 3 shows the patterns that fulfringe identifies . Dcfringe, proposed in [Yang
et al., 1991], identifies the patterns shown in the first two rows. These patterns
correspond to 8 Boolean functions of 2 variables . Since there are only 10 distinct
Boolean functions that depend on two variables 3 , it is natural to add the patterns
in the third row and identify all possible functions of 2 variables. As in dcftinge
and fringe, these new composite features are added (if they have not yet been
generated) to the list of available features and a new decision tree is built. The
3The remaining 6 functions of 2 variables depend on only one or none of the variables.
915
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Oliveira and Sangiovanni-Vincentelli
process is iterated until a decision tree with only one decision node is built. The
attribute tested at this node is a complex feature and can be viewed as the output
of a Boolean network that matches the training set data.
3.3
Encoding multivalued outputs
Both muesli and Julfringe generate Boolean networks with a single binary valued
output. When the target label can have more than 2 values, some encoding must be
used. The prefered solution is to encode the outputs using an error correcting code
[Dietterich and Bakiri, 1991] . This approach preserves most of the compactness of
a digital encoding while beeing much less sensitive to errors in one of the output
variables. Additionally, the Hamming distance between an observed output and the
closest valid codeword gives a measure of the certainty of the classification. This
can be used to our advantage in problems where a failure to classify is less serious
than the output of a wrong classification.
4
Performance evaluation
To evaluate the algorithms, we selected a set of 11 functions of variable complexity.
A complete description of these functions can be found in [Oliveira, 1994]. The first
6 functions were proposed as test cases in [Pagallo and Haussler, 1990] and accept
compact disjoint normal form descriptions. The remaining ones accept compact
multi-level representations but have large two level descriptions. The algorithms
described in sections 3.1 and 3.2 were compared with the cascade-correlation algorithm [Fahlman and Lebiere, 1990] and a standard decision t.ree algorithm analog
to ID3 [Quinlan, 1986]. As in [Pagallo and Haussler, 1990], the number of examples
in the training set was selected to be equal to ~ times the description length of the
function under a fixed encoding scheme, where f was set equal to 0.1. For each
function, 5 training sets were randomly selected. The average accuracy for the 5
runs in an independent set of 4000 examples is listed in table 1.
Table 1: Accuracy of the four algorithms.
Function
dnfl
dnf2
dnf3
dnf4
xor4_16
xor5_32
sm12
sm18
str18
str27
carry8
Average
#
inputs
80
40
32
64
16
32
12
18
18
27
16
#
examples
3292
2185
1650
2640
1200
4000
1540
2720
2720
4160
2017
muesli
99.91
99.28
99.94
100.00
98.35
60.16
99.90
100.00
100.00
98.64
99.50
95 .97
Accuracy
fulfringe
ID3
99.98 82.09
98.89 88.84
100.00 89.98
100.00 72.61
100.00 75.20
100.00 51.41
lUO.OO
99.81
99.92
91.48
100.00 94.55
99.35 94 .24
98.71
96.70
99.71
85.35
CasCor
75.38
73.11
79 .19
58.41
99.91
99.97
98.98
91.30
92.57
93 .90
99 .22
87.45
The results show that the performance of muesli and fulfringe is consistently su-
Learning Complex Boolean Functions: Algorithms and Applications
perior to the other two algorithms. Muesli performs poorly in examples that have
many xor functions, due the greedy nature of the algorithm . In particular, muesli
failed to find a solution in the alloted time for 4 of the 5 runs of xor5_32 and found
the exact solution in only one of the runs.
ID3 was the fastest of the algorithms and Cascade-Correlation the slowest. Fulfringe
and muesli exhibited similar running times for these tasks. 'rVe observed, however,
that for larger problems the runtime for fulfringe becomes prohibitively high and
muesli is comparatively much faster.
5
Applications
To evaluate the techniques described in real problems, experiments were performed
in two domains: noisy image reconstruction and handwritten character recognition.
The main objective was to investigate whether the approach is applicable to problems that are not known to accept a compact Boolean network representation. The
outputs were encoded using a 15 bit Hadamard error correcting code.
5.1
Image reconstruction
The speed required by applications in image processing makes it a very interesting
field for this type of approach. In this experiment, 16 level gray scale images were
corrupted by random noise by switching each bit with 5% probability. Samples of
this image were used to train a network in the reconstruction of the original image.
The training set consisted of .5x5 pixel regions of corrupted images (100 binary
variables per sample) labeled with the value of the center pixel. Figure 4 shows a
detail of the reconstruction performed in an independent test image by the network
obtained using fulfringe.
Original image
corrupted image
Reconstructed image
Figure 4: Image reconstruction experiment
5.2
Handwritten character recognition
The NIST database of handwritten characters was used for this task. Individually
segmented digits were normalized to a 16 by 16 binary grid. A set of 53629 digits
was used for training and the resulting network was tested in a different set of 52467
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Oliveira and Sangiovanni-Vincentelli
digits. Training was performed using muesli. The algorithm was stopped after a prespecified time (48 hours on a DECstation 5000/260) ellapsed. The resulting network
was placed and routed using the TimberWolf [Sechen and Sangiovanni-Vincentelli,
1986] package and occupies an area of 78.8 sq. mm. using 0.8fl technology.
The accuracy on the test set was 93.9%. This value compares well with the performance obtained by alternative approaches that use a similarly sized training set
and little domain knowledge, but falls short of the best results published so far.
Ongoing research on this problem is concentrated on the use of domain knowledge
to restrict the search for compact networks and speed up the training.
Acknowledgements
This work was supported by Joint Services Electronics Program grant F49620-93-C-0014.
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| 857 |@word pulse:2 solid:1 electronics:1 contains:1 xiy:2 selecting:1 current:1 luo:1 yet:1 written:1 must:1 partition:3 informative:1 designed:1 greedy:3 selected:6 intelligence:1 warmuth:1 smith:3 short:1 prespecified:1 provides:4 completeness:1 node:29 simpler:1 direct:1 prove:1 combine:1 indeed:1 multi:3 terminal:4 freeman:1 decreasing:1 little:1 inappropriate:1 becomes:2 underlying:1 maximizes:2 circuit:1 finding:2 pseudo:2 berkeley:3 every:1 certainty:1 act:26 xd:2 runtime:1 exactly:2 prohibitively:1 wrong:1 partitioning:2 grant:1 service:1 switching:1 encoding:4 ree:1 modulation:1 plus:1 mateo:2 fastest:1 limited:1 directed:1 implement:3 sq:1 digit:3 procedure:1 decstation:1 area:1 empirical:2 bell:1 cascade:3 composite:1 pre:1 cannot:1 unfeasible:1 selection:1 context:1 equivalent:2 center:1 starting:1 splitting:1 identifying:1 correcting:3 haussler:6 array:1 annals:1 target:5 construction:1 exact:1 carl:1 us:1 recognition:3 expensive:2 labeled:7 database:1 observed:2 solved:1 region:1 sangiovanni:8 prefered:1 complexity:6 depend:2 predictive:1 joint:1 train:1 distinct:1 effective:1 artificial:1 heuristic:2 posed:1 valued:2 larger:2 encoded:1 ability:1 statistic:1 id3:3 noisy:1 advantage:1 reconstruction:6 maximal:2 hadamard:1 poorly:1 achieve:1 description:7 comparative:1 help:1 derive:1 oo:1 exemplar:1 minor:1 implemented:1 auxiliary:1 attribute:7 stochastic:1 occupies:1 routing:1 require:1 mm:1 normal:1 smallest:1 applicable:1 label:7 sensitive:1 individually:1 always:1 aim:1 encode:1 exemplifies:1 dsp:1 improvement:1 consistently:1 recurs:1 slowest:1 greedily:1 lj:2 accept:5 compactness:2 initially:2 vlsi:1 selects:2 pixel:2 overall:1 classification:2 flexible:1 development:1 uc:2 mutual:5 field:2 equal:2 np:2 serious:1 randomly:1 preserve:1 national:1 floating:1 ab:2 investigate:1 custom:1 evaluation:1 nl:1 edge:3 tree:10 desired:1 theoretical:1 minimal:1 stopped:1 increased:2 classify:2 modeling:1 boolean:26 measuring:2 logp:2 cost:2 johnson:3 too:1 eec:1 corrupted:3 international:1 arlindo:2 again:1 aaai:2 thesis:1 unmatched:1 inefficient:1 return:2 li:1 automation:1 explicitly:1 stream:2 performed:3 try:2 root:1 sup:13 reached:4 sort:1 accuracy:4 xor:1 kaufmann:2 characteristic:1 correspond:1 identify:1 generalize:3 handwritten:3 iterated:1 none:1 published:1 processor:1 touretzky:1 definition:1 failure:1 garey:3 lebiere:3 mi:9 hamming:1 multivalued:1 knowledge:2 higher:1 supervised:1 until:3 correlation:3 hand:1 su:1 kernighan:4 gray:1 believe:1 name:1 dietterich:3 ye:1 consisted:1 true:3 normalized:1 counterpart:1 inductive:2 ehrenfeucht:1 illustrated:1 x5:1 razor:3 rooted:1 generalized:1 complete:2 performs:1 dedicated:1 image:13 rendell:1 volume:1 analog:3 ai:1 rd:1 grid:1 similarly:1 language:1 add:1 closest:1 codeword:1 binary:5 success:5 yi:1 morgan:2 fortunately:1 additional:1 paradigm:1 maximize:1 arithmetic:1 full:1 alan:1 segmented:1 technical:1 match:1 faster:1 dept:1 lin:4 alberto:2 vincentelli:8 basic:1 cell:1 want:1 else:2 exhibited:1 call:1 near:1 yang:3 xj:4 architecture:1 identified:1 restrict:1 idea:1 multiclass:1 whether:1 routed:1 york:1 programmable:1 listed:1 amount:1 oliveira:8 concentrated:1 reduced:1 generate:4 designer:1 disjoint:1 per:1 ist:1 four:1 threshold:1 rissanen:3 graph:3 run:3 package:2 letter:1 uncertainty:3 decision:12 bit:2 fl:1 placement:1 x2:2 speed:3 performing:1 relatively:2 according:2 describes:1 character:4 modification:1 restricted:1 available:3 operation:1 eight:1 alternative:3 gate:2 original:2 remaining:2 running:1 quinlan:4 unifying:1 iyd:2 murray:3 build:1 bakiri:3 february:1 comparatively:1 objective:2 added:1 primary:1 exhibit:1 gradient:1 distance:1 induction:2 code:5 length:1 minimizing:1 implementation:3 design:5 perform:1 neuron:1 nist:1 descent:1 ninth:1 required:3 specified:4 hour:1 usually:1 pattern:6 program:2 built:3 explanation:1 power:1 suitable:1 natural:1 scheme:2 improve:1 technology:2 identifies:2 acknowledgement:1 discovery:1 generation:1 interesting:1 acyclic:1 digital:6 foundation:1 consistent:2 editor:1 intractability:1 occam:3 cd:1 row:2 placed:1 fahlman:3 keeping:1 supported:1 asynchronous:1 guide:1 fall:1 taking:1 benefit:1 f49620:1 valid:1 doesn:1 computes:1 commonly:1 made:1 san:2 far:1 reconstructed:1 compact:5 logic:1 global:1 active:4 assumed:1 search:2 table:2 additionally:2 learn:1 nature:1 ca:1 improving:1 alloted:1 complex:6 anthony:1 domain:3 main:2 noise:1 fmd:1 board:1 slow:1 precision:1 fails:3 xl:1 candidate:1 third:1 rve:1 specific:2 list:4 ments:1 derives:1 ofthis:1 false:2 phd:1 suited:1 entropy:1 failed:1 fringe:4 viewed:2 sorted:1 blumer:3 sized:1 hard:1 except:1 cde:1 called:3 select:6 support:1 preparation:1 ongoing:1 evaluate:2 outgoing:2 tested:6 |
7,088 | 858 | Lipreading by neural networks:
Visual preprocessing, learning
and sensory integration
Gregory J. Wolff
Ricoh California Research Center
2882 Sand Hill Road Suite 115
Menlo Park, CA 94025-7022
[email protected]
David G. Stork
Ricoh California Research Center
2882 Sand Hill Road Suite 115
Menlo Park, CA 94025-7022
stor [email protected]
K. Venkatesh Prasad
Ricoh California Research Center
2882 Sand Hill Road Suite 115
Menlo Park, CA 94025-7022
[email protected]
Marcus Hennecke
Department of Electrical Engineering
Mail Code 4055
Stanford University
Stanford, CA 94305
Abstract
We have developed visual preprocessing algorithms for extracting
phonologically relevant features from the grayscale video image of
a speaker, to provide speaker-independent inputs for an automatic lipreading ("speechreading") system. Visual features such as
mouth open/closed, tongue visible/not-visible, teeth visible/notvisible, and several shape descriptors of the mouth and its motion
are all rapidly computable in a manner quite insensitive to lighting
conditions. We formed a hybrid speechreading system consisting
of two time delay neural networks (video and acoustic) and integrated their responses by means of independent opinion pooling
- the Bayesian optimal method given conditional independence,
which seems to hold for our data. This hybrid system had an error rate 25% lower than that of the acoustic subsystem alone on a
five-utterance speaker-independent task, indicating that video can
be used to improve speech recognition.
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Wolff, Prasad, Stork, and Hennecke
1
INTRODUCTION
Automated speech recognition is notoriously hard, and thus any predictive source
of information and constraints that could be incorporated into a computer speech
recognition system would be desirable. Humans, especially the hearing impaired,
can utilize visual information - "speech reading" - for improved accuracy (Dodd
& Campbell, 1987, Sanders & Goodrich, 1971). Speech reading can provide direct
information about segments, phonemes, rate, speaker gender and identity, and subtle information for segmenting speech from background noise or multiple speakers
(De Filippo & Sims, 1988, Green & Miller, 1985).
Fundamental support for the use of visual information comes from the complementary nature of the visual and acoustic speech signals. Utterances that are difficult to
distinguish acoustically are the easiest to distinguish. visually, and vice versa. Thus,
for example /mi/ H /ni/ are highly confusable acoustically but are easily distinguished based on the visual information of lip closure. Conversely, /bi/ H /pi/ are
highly confusable visually ("visemes"), but are easily distinguished acoustically by
the voice-onset time (the delay between the burst sound and the onset of vocal fold
vibration). Thus automatic lipreading promises to help acoustic speech recognition
systems for those utterances where they need it most; visual information cannot
contribute much information to those utterances that are already well recognized
acoustically.
1.1
PREVIOUS SYSTEMS
The system described below differs from recent speech reading systems. Whereas
Yuhas et al. (1989) recognized static images and acoustic spectra for vowel recognition, ours recognizes dynamic consonant-vowel (CV) utterances. Whereas Petajan,
Bischoff & Bodoff (1988) used thresholded pixel based representations of speakers,
our system uses more sophisticated visual preprocessing to obtain phonologically
relevant features. Whereas Pentland and Mase (1989) used optical flow methods
for estimating the motion of four lip regions (and used no acoustic subsystem), we
obtain several other features from intensity profiles. Whereas Bregler et al. (1993)
used direct pixel images, our recognition engine used a far more compressed visual
representation; our method of integration, too, was based on statistical properties
of our data. We build upon the basic recognizer architecture of Stork, Wolff and
Levine (1992), but extend it to grayscale video input.
2
VISUAL PREPROCESSING
The sheer quantity of image data presents a hurdle to utilizing video information for
speech recognition. Our approach to video preprocessing makes use of several simple
computations to reduce the large amount of data to a manageable set of low-level
image statistics describing the region of interest around the mouth. These statistics
capture such features as the positions of the upper and lower lip, the mouth shape,
and their time derivatives. The rest of this section describes the computation of
these features.
Grayscale video images are captured at 30 frames/second with a standard NTSC
Lipreading by Neural Networks: Visual Preprocessing, Learning, and Sensory Integration
pixel posiCion
pixel position
Figure 1: (Left) The central bands of the automatically determined ROI from two frames
of the video sequence of the utterance /ba/ and their associated luminance profiles along
the central marked line. Notice that the lowest valley in this profile changes drastically
in intensity as the mouth changes from closed to open. In addition, the linear separation
between the peaks adjacent to the lowest valley also increases as the mouth opens. These
features are identified on the ROI from a single frame (right). The position, intensity, and
temporal variation of these features provide input to our recognizer.
camera, and subsampled to give 150 x 150 pixel image sequence. A 64 x 64 pixel
region of interest (ROI) is detected and tracked by means of the following operations
on the full video images:
?
?
?
?
?
Convolve with 3 x 3 pixel low-pass filter
Convolve with 3 x 3 pixel edge detector
Convolve with 3 x 3 pixel low-pass filter
Threshold at (Imax - I min)/2
Triangulate eyes with mouth
(to
(to
(to
(to
(to
remove spatial noise)
detect edges)
smooth edges)
isolate eyes and mouth)
obtain ROI)
We also use temporal coherence in frame-to-frame correlations to reduce the effects
of noise in the profile or missing data (such as "closed" eyes). Within the ROI the
phonological features are found by the following steps (see Figure 1):
?
?
?
?
?
Convolve with 16 x 16 pixel low-pass filter
Extract a vertical intensity profile
Extract a horizontal intensity profile
Locate and label intensity peaks and valleys
Calculate interframe peak motion
(to remove noise)
(mouth height)
(mouth width)
(candidates for teeth, tongue)
(speed estimates)
Video preprocessing tasks, including temporal averaging, are usually complicated
because they require identifying corresponding pixels across frames. We circumvent
this pixel correspondence problem by matching labeled features (such as intensity
extrema - peaks and valleys) on successive frames.
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Wolff, Prasad, Stork, and Hennecke
2.1
FEATURES
The seventeen video features which serve as input to our recognizer are:
? Horizontal separation between the left and right mouth corners
? Vertical separation between the top and bottom lips
For each of the three vertically aligned positions:
? Vertical position:
? Intensity value:
? Change in intensity versus time:
Pv
I
!:::.I /!:::.t
For both of the mouth corner positions:
? Horizontal position:
Ph
? Intensity value:
? Change in intensity versus time:
I
!:::.I /!:::.t
For each speaker, each feature was scaled have a zero mean and unit standard
deviation.
3
DATA COLLECTION AND NETWORK TRAINING
We trained the modified time delay neural network (Waibel, 1989) shown in Figure 2
on both the video and acoustic data. (See Stork, Wolff and Levine (1992) for
a complete description of the architecture.) For the video only (VO) network, the
input layer consists of 24 samples of each of the 17 features, corresponding to roughly
0.8 seconds. Each (sigmoidal) hidden unit received signals from a receptive field of
17 features for five consecutive frames. Each of the different hidden units (there
were 3 for the results reported below) is replicated to cover the entire input space
with overlapping receptive fields. The next layer consisted of 5 rows of x-units (one
row for each possible utterance), with exponential transfer functions. They received
inputs from the hidden units for 11 consecutive frames, thus they indirectly received
input from a total of 18 input frames corresponding to roughly 0.6 seconds. The
activities of the x-units encode the likelihood that a given letter occurs in that
interval. The final layer consists of five p-units (probability units), which encode
the relative probabilities of the presence of each of the possible utterances across
the entire input window. Each p-unit sums the entire row of corresponding x-units,
normalized by the sum over all x-units. (Note that "weights" from the x-units to
the p-units are fixed.)
The acoustic only (AO) network shared the same architecture, except that the input
consisted of 100 frames (1 second) of 14 mel scale coefficients each, and the x-units
received fan in from 25 consecutive hidden units.
In the TDNN architecture, weights are shared, i.e., the pattern of input-to-hidden
weights is forced to be the same at each interval. Thus the total number of independent weights in this VO network is 428, and 593 for the AO network.
These networks were trained using Backpropagation to minimize the KullbackLeibler distance (cross-entropy) between the targets and outputs,
E
=D(t II p) = Ltdn(--.!..).
t?
l
.
Pi
(1)
Here the target probability is 1 for the target category, and 0 for all other categories.
In this case Equation 1 simplifies to E
-In(pc) where c is the correct category.
=
Lipreading by Neural Networks: Visual Preprocessing. Learning. and Sensory Integration
Outputs for utterance ma3
bada!a lama
c>
ci
en
'"Q)c>.
-0
-0
c>
I
Q
'"
~--------~~----------~
5
10
15
20
Time
Figure 2: Modified time delay neural network architecture (left) and unit activities
for a particular pattern (right). The output probabilities are calculated by integrating over the entire input window and normalizing across categories. Note the
temporal segmentation which naturally occurs in the layer of X-units.
3.1
SENSORY INTEGRATION
Given the output probability distributions of the two networks, we combine them
assuming conditional independence and using Bayes rule to obtain:
(2)
That is, the joint probability of the utterance belonging to category
normalized product of the outputs for category Ci of each network.
Ci
is just the
This "independent opinion pooling" (Berger, 1985) offers several ad vantages over
other methods for combining the modalities. First, it is optimal if the two signals
really are conditionally independent, which appears to be the case for our data.
(Proving that two signals are not conditionally independent is difficult.) Moreover,
Massaro and Cohen (1983) have shown that human recognition performance is
consistent with the independence assumption. A second advantage is simplicity.
The combination adds no extra parameters beyond those used to model each signal,
thus generalization performance should be good. Furthermore, the independent
recognizers can be developed and trained separately, the only requirement is that
they both output probability estimations.
A third advantage is that this system automatically compensates for noise and
assigns more importance to the network which is most sure of its classification. For
example, if the video data were very noisy (or missing), the video network would
1031
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Wolff, Prasad, Stork, and Hennecke
judge all utterances equally likely. In this case the video contribution would cancel
out, and the final output probabilities would be determined solely by the audio
network. Bregler et al. (1993) attempt to compensate for the variance between
channels by using the entropy of the output of the individual networks as a weighting
on their contribution to the final outputs. Their ad hoc method suffers several
drawbacks. For example, it does not distinguish the case where a one category is
highly likely and the rest equiprobable, from the case where several categories are
moderately likely.
A final advantage of Eq. 2 is that it does not require synchrony of the acoustic and
visual features. The registration between the two signals could be off substantially
(as long as the same utterance is present in the input to both networks). On the
contrary, methods which attempt to detect cross-modal features would be very
sensitive to the relative timing of the two signals.
4
RESULTS
Audio Test
Video Test
ma
o
0
o
0
0
la
o
0
0
0
0
-.e-
fa
o
o
da
0000
::J
::J
ba
0
00
0
0
ba da fa
0
0
0
0
la ma
Input
54% correct
maooooO
maooooO
_
la
::J
0
AV Test
.e::J
o
fa
da
:OQ~ o-- 00:c9~
00
::J
o
Q.
::J
0
baOO
0
0
ooQ
ba da fa
la ma
Input
64% correct
::
0
da
0
baOo
ba da fa
0
0
0
0
la ma
Input
72% correct
Figure 3: Confusion matrices for the video only (VO), acoustic only (AO), and the AV
networks. Each vertical column is labeled by the spoken CV pair presented as input; each
horizontal row represents the output by the network. The radius of each disk in the array
is proportional to the output probability given an input letter. The recognition accuracy
(measured as a percentage of novel test patterns properly classified by maximum network
output) is shown.
The video and audio networks were trained separately on several different consonants in the same vowel context (/ba/, Ida/, Ifa/, Ila/, Ima/) recorded from
several different speakers. (For the results reported below, there were 10 speakers,
repeating each of 5 CV pairs 5 times. Four of these were used for training, and one
for testing generalization.)
For the video only networks, the correct classification (using the Max decision rule)
on unseen data is typically 40-60%. As expected, the audio networks perform better
with classification rates in the 50-70% range on these small sets of similar utterances.
Lipreading by Neural Networks: Visual Preprocessing, Learning, and Sensory Integration
Figure 3 shows the confusion matrices for the network outputs. We see that for the
video only network the confusion matrix is fairly diagonal, indicating generally good
performance. However the video network does tend to confuse utterances such as
/ba/ H /maj. The audio network generally makes fewer errors, but confuses other
utterances, such as /ba/ H / da/.
The performance for the combined outputs (the AV network) is much better than
either of the individual networks, achieving classification rates above 70%. (In
previous work with only 4 speakers, classifications rates of up to 95% have been
achieved.) We also see a strongly diagonal confusion matrix for the AV network,
indicating that complementary nature of the the confusions made by the individual
networks.
5
RELATIONSHIP TO HUMAN PERCEPTION
Visual
Acoustic
o
.,#
0
. ?0
:;
Q.
:;
o
"t?
o .
'6 . .
?0
o? ?
o
o
:~6 ? ?:
o?
?? 0 ?
o
??? ? . .
. 0
.
? 0? 0.?
????? ?
?
.
.
.
o?
??0
O~~9hj kI~ r~'~~h
Input
-
-
0
::s
::s
Q.
.
:;
0
AV
Q.
:;
~?~~;k;~;S~h
Input
0
Input
Figure 4: Confusion matrices from human recognition performance for video only,
acoustic only, and combined speech for CV pairs (Massaro et aI., 1993).
Interestingly, our results are qualitatively similar to findings in human perception.
Massaro et aI. (1993) presented Visual only, Acoustic only, and combined speech to
subjects and collected response probabilities. As can be seen in the confusion matrices of Figure 4, subjects are not so bad at lipreading. The Visual only confusion
matrix shows a strong diagonal component, though confusions such as /ma/ H /ba/
are common . Performance on acoustic speech is better, of course, but there are still
confusions such as /ba/ H / daj. Combined speech yields even better recognition
performance, eliminating most confusions. In fact, Massaro et aI. found that the
response probabilities of combined speech are accurately predicted by the product
of the two single mode response probabilities. Massaro uses this and other evidence
to argue quite convincingly that humans treat acoustic and visual speech channels
independently, combining them only at a rather late stage of processing.
1033
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Wolff, Prasad, Stork, and Hennecke
6
CONCLUSIONS AND FUTURE WORK
The video pre-processing presented here represents a first pass at reducing the
amount of visual data to a manageable level in order to enable on-line processing. Our results indicate that even these straightforward, computationally tractable
methods can significantly enhance speech recognition. Future efforts will concentrate on refining the pre-processing to capture more information, such as rounding
and f-tuck, and testing the efficacy of our recognition system on larger datasets. The
complementary nature of the acoustic and visual signals lead us to believe that a
further refined speech reading system will significantly improve the state-of-the-art
acoustic recognizers, especially in noisy environments.
References
J. O . Berger. (1985) Statistical decision theory and Bayesian analysis (2nd ed'). 272-275,
New York: Springer-Verlag.
C. Bregler, S. Manke, H. Hild & A. Waibel. (1993) Bimodal Sensor Integration on the
example of "Speech-Reading". Proc. ICNN-93, Vol. II 667-677.
C. L. De Filippo & D. G. Sims (eds.), (1988) New Reflections on Speechreading (Special
issue of The Volta Review). 90(5).
B. Dodd & R. Campbell (eds.). (1987) Hearing by Eye: The Psychology of Lip-reading.
Hillsdale, N J: Lawrence Erlbaum Press.
K. P. Green & J. L. Miller. (1985) On the role of visual rate information in phonetic
perception. Perception and Psychophysics 38, 269-276.
D. W. Massaro & M. M. Cohen (1983) Evaluation and integration of visual and auditory
information in speech perception J. Exp. Psych: Human Perception and Performance 9,
753-771.
D. W. Massaro, M. M. Cohen & A. T. Gesi (1993). Long-term training, transfer, and
retention in learning to lipread. Perception ?3 Psychophysics, 53, 549-562.
A. Pentland & K. Mase (1989) Lip reading: Automatic visual recognition of spoken words.
Proc. Image Understanding and Machine Vision, Optical Society of America, June 12-14.
E. D. Petajan, B. Bischoff & D. Bodoff. (1988) An improved automatic lipreading system
to enhance speech recognition. ACM SIGCHI-88, 19-25.
D. Sanders & S. Goodrich. (1971) The relative contribution of visual and auditory components of speech to speech intelligibility as a function. of three conditions of frequency
distortion. J. Speech and Hearing Research 14, 154-159.
D. G . Stork, G. Wolff & E. Levine. (1992) Neural network lipreading system for improved
speech recognition. Proc. IJCNN-92, Vol. II 285-295.
A. Waibel. (1989) Modular construction of time-delay neural networks for speech recognition. Neural Computation 1, 39-46.
B. P. Yuhas, M. H. Goldstein, Jr., T. J. Sejnowski & R. E. Jenkins. (1988) Neural network
models of sensory integration for improved vowel recognition. Proc. IEEE 78(10), 16581668.
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7,089 | 859 | Connectionism for Music and Audition
Andreas S. Weigend
Department of Computer Science
and Institute of Cognitive Science
University of Colorado
Boulder, CO 80309-0430
Abstract
This workshop explored machine learning approaches to 3 topics:
(1) finding structure in music (analysis, continuation, and completion of an unfinished piece), (2) modeling perception of time (extraction of musical meter, explanation of human data on timing),
and (3) interpolation in timbre space.
In recent years, NIPS has heard neural networks generate tunes and harmonize
chorales. With a large amount of music becoming available in computer readable
form, real data can be used to train connectionist models. At the beginning of
this workshop, Andreas Weigend focused on architectures to capture structure
on multiple time scales. J. S. Bach's last (unfinished) fugue from Die Kunst der
Fuge served as an example (Dirst & Weigend, 1994).1 The prediction approach to
continuation and completion, as well as to modeling expectations, can be characterized by the question "What's next?". Moving to time as the primary medium
of musical communication, the inquiry in music perception and cognition shifted to
the question "When next?" .
In other words, so far we have considered patterns in time. They assume prior identification and subsequent processing of events. Bob Port, coming from the speech
community, considered patterns of time, discussing timing in linguistic polyrhythms
(e.g., hot cup of tea). He also drew parallels between timing in Japanese language
and timing in music, supporting the hypothesis that perceptional rhythms entrain
attentional rhythms. As a mechanism for entrainment, Devin McAuley presented
adaptive oscillators: the oscillators adapt their frequencies such that their "firing"
coincides with the beat of the music (McAuley, 1994).
As the beat can be viewed as entrainment of an individual oscillator, the meter
can be viewed as entrainment of multiple oscillators. Ed Large described human
perception of metrical structure in analogy to two pendulum clocks that synchronize
their motions by hanging on the same wall. An advantage of these entrainment
1 This fugue is available via anonymous ftp from ftp. santafe. edu as data set F. dat of
the Santa Fe Time Series Analysis and Prediction Competition.
1163
1164
Weigend
approaches (which focus on time as time) over traditional approaches (which focus
on music notation and treat time symbolically) is their ability to model phenomena
in music performance, such as expressive timing.
Taking a Gibsonian perspective, Fred Cummins emphasized the relevance of ecological constraints on audition: perceptually relevant features are not easily spotted
in the wave form or the spectrum. Among the questions he posed were: what
"higher-order" features might be useful for audition, and whether recurrent networks could be useful to extract such features.
The last contribution also addressed the issue of representation, but with sound
synthesis in mind: wouldn't a musician like to control sound in a perceptually relevant space, rather than fiddling with non-intuitive coefficients of an FM-algorithm?
Such a space was constructed with human input: subjects were asked to similarityjudge sounds from different instruments (normalized in pitch, duration and volume).
Multidimensional scaling was used to define a low-dimensional sub-space keeping
the distance relations. Michael Lee first trained a network to find a map from
timbre space to the space of the first 33 harmonics (Lee, 1994). He then used the
network to generate rich new sounds by interpolating in this perceptually relevant
space, through physical gestures, such as from a data glove, or through an interface
musicians might be comfortable with, such as a cello.
The discussion turned to the importance of working with perceptually adequate,
"ecologically sound" representations (e.g., by using a cochlea model as pre-processor,
or a speech model as post-processor for sonification applications). Finally, to probe
human cognition, we discussed synthetic sounds, designed to reveal fundamental
characteristics of the auditory system, independent of our daily experience. Returning to the title, the workshop turned out to be problem driven: people presented a
problem or a finding and searched for a solution-connectionist or otherwise-rather
than applying canned connectionist ideas to music and cognition.
I thank the speakers, Fred Cummins ([email protected]), Ed Large
([email protected]), Michael Lee ([email protected]), Devin McAuley
([email protected]), Robert Port ([email protected]), as well as all participants. I also thank Tom Ngo ([email protected]) for sending me the notes he took
at the workshop, and Eckhard Kahle ([email protected]) for discussing this summary.
References
Dirst, M., and A. S. Weigend (1994) "Baroque Forecasting: On Completing
J. S. Bach's Last Fugue." In Time Series Prediction: Forecasting the Future and
Understanding the Past, edited by A. S. Weigend and N. A. Gershenfeld, pp. 151172. Addison-Wesley.
Lee, M., and D. Wessel (1992) "Connectionist Models for Real-Time Control of Synthesis and Compositional Algorithms." In Proceedings of the International Computer Music Conference, pp. 277-280. San Francisco, CA: International Computer
Music Association.
McAuley, J. D. (1994) "Finding metrical structure in time." In Proceedings of the
1993 Connectionist Models Summer School, edited by M. C. Mozer, P. Smolensky,
D. S. Touretzky, J. L. Elman and A. S. Weigend, pp. 219-227. Lawrence Erlbaum .
| 859 |@word c:1 normalized:1 dat:1 question:3 primary:1 human:4 traditional:1 distance:1 speaker:1 die:1 rhythm:2 mcauley:5 coincides:1 attentional:1 thank:2 series:2 wall:1 anonymous:1 baroque:1 me:1 topic:1 connectionism:1 past:1 motion:1 interface:1 com:1 considered:2 harmonic:1 ohio:1 lawrence:1 cognition:3 devin:2 subsequent:1 fe:1 robert:1 physical:1 designed:1 volume:1 discussed:1 he:4 association:1 title:1 cup:1 beginning:1 supporting:1 beat:2 communication:1 language:1 rather:2 harmonize:1 moving:1 constructed:1 community:1 linguistic:1 recent:1 focus:2 perspective:1 driven:1 ecological:1 discussing:2 elman:1 nip:1 der:1 perception:3 pattern:2 smolensky:1 relation:1 cummins:2 notation:1 multiple:2 issue:1 medium:1 among:1 sound:6 what:2 explanation:1 hot:1 event:1 characterized:1 gesture:1 bach:2 adapt:1 synchronize:1 finding:3 indiana:3 extraction:1 post:1 spotted:1 chorale:1 berkeley:1 prediction:3 multidimensional:1 pitch:1 expectation:1 future:1 returning:1 connectionist:5 cochlea:1 extract:1 control:2 prior:1 understanding:1 meter:2 comfortable:1 individual:1 timing:5 treat:1 addressed:1 interval:1 analogy:1 firing:1 interpolation:1 becoming:1 subject:1 might:2 wessel:1 ngo:2 co:1 port:3 unfinished:2 summary:1 metrical:2 ircam:1 last:3 architecture:1 fm:1 keeping:1 daily:1 experience:1 andreas:2 idea:1 institute:1 taking:1 whether:1 fred:2 word:1 pre:1 eckhard:1 modeling:2 forecasting:2 rich:1 adaptive:1 wouldn:1 speech:2 san:1 compositional:1 far:1 adequate:1 applying:1 useful:2 heard:1 santa:1 tune:1 map:1 amount:1 musician:2 erlbaum:1 duration:1 focused:1 continuation:2 generate:2 francisco:1 synthetic:1 spectrum:1 shifted:1 cello:1 fundamental:1 international:2 lee:5 ca:1 michael:2 synthesis:2 tea:1 colorado:1 japanese:1 interpolating:1 hypothesis:1 gershenfeld:1 cognitive:1 audition:3 kahle:2 symbolically:1 year:1 weigend:7 capture:1 coefficient:1 sub:1 piece:1 edited:2 scaling:1 mozer:1 pendulum:1 entrain:1 wave:1 participant:1 asked:1 parallel:1 completing:1 summer:1 trained:1 contribution:1 santafe:1 emphasized:1 constraint:1 musical:2 characteristic:1 explored:1 timbre:2 easily:1 workshop:4 identification:1 drew:1 importance:1 ci:1 train:1 ecologically:1 served:1 bob:1 perceptually:4 processor:2 department:1 hanging:1 inquiry:1 touretzky:1 ed:2 posed:1 frequency:1 otherwise:1 pp:3 ability:1 fugue:3 boulder:1 auditory:1 advantage:1 took:1 mechanism:1 viewed:2 coming:1 mind:1 fr:1 addison:1 instrument:1 relevant:3 turned:2 wesley:1 higher:1 sending:1 available:2 oscillator:4 tom:1 glove:1 probe:1 entrainment:4 intuitive:1 competition:1 clock:1 working:1 expressive:1 searched:1 people:1 canned:1 ftp:2 recurrent:1 completion:2 readable:1 relevance:1 reveal:1 music:11 school:1 phenomenon:1 |
7,090 | 86 | 467
SPONTANEOUS AND INFORMATION-TRIGGERED SEGMENTS OF SERIES
OF HUMAN BRAIN ELECTRIC FIELD MAPS
D. lehmann, D. Brandeis*, A. Horst, H. Ozaki* and I. Pal*
Neurol09Y Department, University Hospital, 8091 Zurich, Switzerland
ABSTRACT
The brain works in a state-dependent manner: processin9
strate9ies and access to stored information depends on the momentary
functional state which is continuously re-adjusted. The state is
manifest as spatial confi9uration of the brain electric field.
Spontaneous and information-tri9gered brain electric activity is a
series of momentary field maps. Adaptive segmentation of spontaneous
series into spatially stable epochs (states) exhibited 210 msec mean
segments, discontinuous changes. Different maps imply different
active neural populations, hence expectedly different effects on
information processing: Reaction time differred between map classes
at stimulus arrival. Segments might be units of brain information
processin9 (content/mode/step), possibly operationalizin9
consciousness time. Related units (e.9. tri9gered by stimuli durin9
fi9ure perception and voluntary attention) mi9ht specify brain submechanisms of information treatment.
BRAIN FUNCTIONAL STATES AND THEIR CHANGES
The momentary functional state of the brain is reflected by the
confi9uration of the brain's electro-ma9netic field. The state
manifests the strate9Y, mode, step and content of brain information
processing, and the state constrains the choice of strate9ies and
modes and the access to memory material available for processin9 of
incoming information (1). The constraints include the available
range of changes of state in PAVLOV's classical ?orienting reaction"
as response to new or important informations. Different states mi9ht
be viewed as different functional connectivities between the neural
elements.
The orienting reaction (see 1,2) is the result of the first
(Mpre-attentiveM) stage of information processing. This stage
operates automatically (no involvement of consciousness) and in a
parallel mode, and quickly determines whether (a) the information is
important or unknown and hence requires increased attention and
alertness, i.e. an orienting reaction which means a re-adjustment of
functional state in order to deal adequately with the information
invokin9 consciousness for further processing, or whether (b) the
information is known or unimportant and hence requires no readjustment of state, i.e. that it can be treated further with well* Present addresses: D.B. at Psychiat. Dept., V.A. Med. Center, San
Francisco CA 94121; H.O. at lab. Physiol. for the Developmentally
Handicapped, Ibaraki Univ., Mito, Japan 310; I.P. at Biol09ic
Systems Corp., Mundelein Il 60060.
? American Institute of Physics 1988
468
established (?automatic?) strategies. Conscious strategies are slow
but flexible (offer wide choice), automatic strategies are fast but
rigid.
Examples for functional states on a gross scale are wakefulness,
drowsin.ss and sleep in adults, or developmental stages as infancy,
childhood and adolesc.nce, or drug states induced by alcohol or
other psychoactive agent ?? The different states are associated with
distinctly different ways of information processing. For example, in
normal adults, reality-close, abstracting strategies based on causal
relationships predominate during wakefulness, whereas in drowsiness
and sleep (dreams), reality-remote, visualizing, associative
concatenations of contents are used. Other well-known examples are
drug states.
HUMAN BRAIN ELECTRIC FIELD DATA AND STATES
While alive, the brain produces an ever-changing el.ctromagnetic
fi.ld, which very sensitively reflects global and local states as
effected by spontaneous activity, incoming information, metabolism,
drugs, and diseases. The .lectric component of the brain~s electromagnetic field as non-invasively measured from the intact human
scalp shows voltages between 0.1 and 250 microVolts, temporal
fr.quencies between 0.1 and 30, 100 or 3000 Hz depending on the
examined function, and spatial frequencies up to 0.2 cycles/em.
Brain electric field data are traditionally viewed as time series
of potential differences betwe.n two scalp locations (the
electroencephalogram or EE6). Time series analysis has offered an
effective way to class different gross brain functional states,
typically using EE6 power spectral values. Differences between power
spectra during different gross states typically are greater than
between different locations. States of lesser functional complexity
such as childhood vs adult states, sleep vs wakefulness, and many
drug-state. vs non-drug states tend to increased power in slower
frequencies (e.g. 1,4).
Time series analyses of epochs of intermediate durations between
30 and 10 seconds have demonstrated (e.g. 1,5,6) that there are
significant and reliable relations between spectral power or
coh.rency values of EE6 and characteristics of human mentation
(reality-close thoughts vs free associations, visual vs non-visual
thoughts, po.itive vs negative ~otions).
Viewing brain electric field data as series of momentary field
maps (7,8) opens the possibility to investigate the temporal
microstructure of brain functional states in the sub-second range.
The rationale is that the momentary configuration of activated
neural elements represents a given brain functional state, and that
the spatial pattern of activation is reflected by the momentary
brain electric field which is recordable on the scalp as a momentary
field map. Different configurations of activation (different field
maps) are expected to be associated with different modes,
strategies, steps and contents of information processing.
469
SE(J1ENTATI~
OF BRAIN ELECTRIC HAP SERIES INTO STABLE SE(J1ENTS
When Viewing brain electric activity as series of maps of
momentary potential distributions, changes of functional state are
recognizable as changes of the ?electric landscapes? of these maps.
Typically, several successive maps show similar landscapes, then
quickly change to a new configuration which again tends to persist
for a number of successive maps, suggestive of stable states
concatenated by non-linear transitions (9,10). Stable map landscapes
might be hypothesized to indicate the basic building blocks of
information processing in the brain, the -atoms of thoughts?. Thus,
the task at hand is the recognition of the landscape configurations;
this leads to the adaptive segmentation of time series of momentary
maps into segments of stable landscapes during varying durations.
We have proposed and used a method which describes the
configuration of a momentary map by the locations of its maximal and
minimal potential values, thus invoking a dipole model. The goal
here is the phenomenological recognition of different momentary
functional states using a very limited number of major map features
as classifiers, and we suggest conservative interpretion of the data
as to real brain locations of the generating processes which always
involve millions of neural elements.
We have studied (11) map series recorded from 16 scalp locations
over posterior skull areas from normal subjects during relaxation
with closed eyes. For adaptive segmentation, the maps at the times
of maximal map relief were selected for optimal signal/nOise
conditions. The locations of the maximal and minimal (extrema)
potentials were extracted in each map as descriptors of the
landscape; taking into account the basically periodic nature of
spontaneous brain electric activity (Fig. 1), extrema locations were
treated disregarding polarity information. If over time an extreme
left its pre-set spatial window (say, one electrode distance), the
segment was terminated. The map series showed stable map
configurations for varying durations (Fig. 2), and discontinuous,
step-wise changes. Over 6 subjects, resting alpha-type EEG showed
210 msec mean segment duration; segments longer than 323 msec
covered 50% of total time; the most prominent segment class (1.5% of
all classes) covered 20% of total time (prominence varied strongly
over classes; not all possible classes occurred). Spectral power and
phase of averages of adaptive and pre-determined segments
demonstrated the adequacy of the strategy and the homogeneity of
adaptive segment classes by their reduced within-class variance.
Segmentation using global map dissimilarity (sum of Euklidian
difference vs average reference at all measured points) emulates the
results of the extracted-characteristics-strategy.
FUNCTIONAL SIGNIFICANCE OF MOMENTARY MICRO STATES
Since different maps of momentary EEG fields imply activity of
different neural populations, different segment classes must
manifest different brain functional states with expectedly different
470
189 to 189
117 to 117
125 to 125
132 to 132
WItS
148 to 148
148 to 148
156 to 156
164 to 164
WItS
171 to 171
179 to 179
RECORD=1 FILE=A:VP3EC2A
187 to 187
195 to 195 WItS
NORMAL SUBJECT, EYES CLOSED
Fig. 1. Series of momentary potential distribution maps of the brain
field recorded from the scalp of a normal human during relaxation
with closed eyes. Recording with 21 electrodes (one 5-electrode row
added to the 16-electrode array in Fig. 2) using 128 samples/sec/
channel. Head seen from above, left ear left; white positive, dark
negative, 8 levels from +32 to -32 microVolts. Note the periodic
reversal of field polarity within the about 100 msec (one cycle of
the 8-12Hz so-called ?EEG alpha- activity) while the field configuration remains largely constant. - This recording and display was
done with a BRAIN ATLAS system (Biologic Systems, Mundelein, Il).
effects on ongoing information processing. This was supported by
measurements of selective reaction time to acoustic stimuli which
were randomly presented to eight subjects during different classes
of EEG segments (323 responses for each subject). We found
significant reaction time differences over segment classes (ANOVA p
smaller than .02), but similar characteristics over subjects. This
indicates that the momentary sub-second state as manifest in the
potential distribution map significantly influences the behavioral
consequence of information reaching the brain.
Presentation of information is followed by a sequence of
potential distribution maps (Nevent-related potentials? or ERP's,
averaged over say, 100 presentations of the same stimulus, see 12).
The different spatial configurations of these maps (12) are thought
to reflect the sequential stages of information processing
associated with Mcomponents? of event-related brain activity (see
e.g. 13) which are traditionally defined as times of maximal
voltages after information input (maximal response strength).
471
45
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:
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Fig. 2. Sequence of spatially stable segments durin9 a spontaneous
series of momentary EEG maps of 3.1 sec duration in a normal
volunteer. Each map shows the occurrence of the extreme potential
values durin9 one adaptively determined segment: the momentary maps
were searched for the locations of the two extreme potentials; these
locations were accumulated, and linearly interpolated between
electrodes to construct the present maps. (The number of isofrequency-of-occurrence lines therefore is related to the number of
searched maps). - Head seen from above, left ear left, electrode
locations indicated by crosses, most forward electrode at vertex.
Data FIR filtered to 8-12Hz (alpha EEG). The fi9ure to the left
below each map is a running segment number. The figure to the ri9ht
above each map multiplied by 50 indicates the segment duration in
msec.
Application of the adaptive segmentation procedure described above
for identification of functional components of event-related brain
electric map sequences requires the inclusion of polarity
information (14); such adaptive segmentation permits to separate
different brain functional states without resortin9 to the strength
concept of processing stages.
An example (12) might illustrate the type of results obtained
with this analysis: Given segments of brain activity which were
triggered by visual information showed different map configurations
when subjects paid attention vs when they paid no attention to the
stimulus, and when they viewed figures vs meanin9less shapes as
472
LVF
RVF
Fig. 3. Four difference maps, computed as differences between maps
obtained during (upper row) perception of a visual -illusionarytriangle figure (left picture) minus a visual non-figure (right)
shown to the left and right visual hemi-fields (LVF, RVF) , and
obtained during (lower row) attending minus during ignoring the
presented display. The analysed segment covered the time from 168 to
200 msec after stimulus presentations. - Mean of 12 subjects. Head
seen from above, left ear left, 16 electrodes as in Fig. 2,
isopotential contour lines at 0.1 microVolt steps, dotted negative
referred to mean of all values. The -illusionary- figure stimulus
wa. studied by Kanisza (16); see also (12). - Note that the mirror
symmetric configuration of the difference maps for LVF and RVF is
found for the -figure- effect only, not for the -attention- effect,
but that the anterior-posterior difference is similar for both cases.
stimuli. Fig. 3 illustrates such differences in map configuration.
The -attention--induced and -figureR-induced changes in map
configuration showed certain similarities e.g. in the illustrated
segment 168-200 msec after information arrival, supporting the
hypothesis that brain mechanisms for figure perception draw on brain
resources which in other circumstances are utilized in volontary
attention.
The spatially homogeneous temporal segments might be basic
building blocks of brain information processing, possibly
operationalizing consciousness time (15), and offering a common
concept for analysis of brain spontaneous activity and event related
brain potentials. The functional significance of the segments might
be types/ modes/ steps of brain information processing or
performance. Identification of related building blocks during
different brain functions accordingly could specify brain submechanisms of information treatment.
473
Acknowledgement: Financial support by the Swiss National Science
Foundation (including Fellowships to H.O. and I.P.) and by the 8HDO,
the Hartmann Muller and the SANDOZ Foundation is gratefully
acknowledged.
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G. Kanisza, Organization of Vision (Praeger, New York, 1979).
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isopotential:1 intact:1 searched:2 support:1 ongoing:1 dept:1 |
7,091 | 860 | Encoding Labeled Graphs by Labeling
RAAM
Alessandro Sperduti*
Department of Computer Science
Pisa University
Corso Italia 40, 56125 Pisa, Italy
Abstract
In this paper we propose an extension to the RAAM by Pollack.
This extension, the Labeling RAAM (LRAAM), can encode labeled graphs with cycles by representing pointers explicitly. Data
encoded in an LRAAM can be accessed by pointer as well as by
content. Direct access by content can be achieved by transforming the encoder network of the LRAAM into an analog Hopfield
network with hidden units. Different access procedures can be
defined depending on the access key. Sufficient conditions on the
asymptotical stability of the associated Hopfield network are briefly
introduced.
1
INTRODUCTION
In the last few years, several researchers have tried to demonstrate how symbolic
structures such as lists, trees, and stacks can be represented and manipulated in a
connectionist system, while still preserving all the computational characteristics of
connectionism (and extending them to the symbolic representations) (Hinton, 1990;
Plate, 1991; Pollack, 1990; Smolensky, 1990; Touretzky, 1990). The goal is to highlight the potential of the connectionist approach in handling domains of structured
tasks. The common background of their ideas is an attempt to achieve distal access
and consequently compositionality. The RAAM model, proposed by Pollack (Pollack, 1990), is one example of how a neural network can discover compact recursive
"Work partially done while at the International Computer Science Institute, Berkeley.
1125
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Output Layer
Hidden Layer
Input Layer
Label
Figure 1: The network for a general LRAAM. The first layer of the network implements an encoder; the second layer, the corresponding decoder.
distributed representations of trees with a fixed branching factor.
This paper presents an extension of the RAAM, the Labeling RAAM (LRAAM).
An LRAAM allows one to store a label for each component of the structure to be
represented, so as to generate reduced representations of labeled graphs. Moreover,
data encoded in an LRAAM can be accessed not only by pointer but also by content.
In Section 2 we present the network and we discuss some technical aspects of the
model. The possibility to access data by content is presented in Section 3. Some
stability results are introduced in Section 4, and the paper is closed by discussion
and conclusions in Section 5.
2
THE NETWORK
The general structure of the network for an LRAAM is shown in Figure 1. The
network is trained by backpropagation to learn the identity function. The idea is to
obtain a compressed representation (hidden layer activation) of a node of a labeled
graph by allocating a part of the input (output) of the network to represent the
label (Nl units) and the remaining part to represent one or more pointers. This
representation is then used as pointer to the node. To allow the recursive use of these
compressed representations, the part of the input (output) layer which represents
a pointer must be of the same dimension as the hidden layer (N H units) . Thus, a
general LRAAM is implemented by a NJ - N H - NJ feed-forward network, where
NJ = Nl + nNH, and n is the number of pointer fields.
Labeled graphs can be easily encoded using an LRAAM. Each node of the graph
only needs to be represented as a record, with one field for the label and one
field for each pointer to a connected node. The pointers only need to be logical
pointers, since their actual values will be the patterns of hidden activation of the
network. At the beginning of learning, their values are set at random. A graph is
represented by a list of these records, and this list constitutes the initial training set
for the LRAAM. During training the representations of the pointers are consistently
updated according to the hidden activations. Consequently, the training set is
dynamic. For example, the network for the graph shown in Figure 2 can be trained
as follows:
Encoding Labeled Graphs by Labeling RAAM
input
(Ll
(L2
(L3
(L4
(L5
(L6
hidden
d n2 d n4 d n5 )
d n3 d n4 nil)
d n6 nil nil)
d n6 d n3 nil)
d n4 d n6 nil)
nil nil nil)
----
d~1
d~2
d~3
d~4
d~5
d~6
----
output
(L"1
(L"2
(L"3
(L"4
(L"5
(L~
d"n2 d"n4 d"n5 )
d"n3 d"n4 nil")
d"n6 nil" nil")
d"n6 d"n3 nil")
d"n4 d"n6 nil")
nil" nil" nil")
where Li and d ni are respectively the label and the pointer (reduced descriptor to
the i-th node. For the sake of simplicity, the void pointer is represented by a single
symbol, nil, but each instance of it must be considered as being different. This
statement will be made clear in the next section.
Once the training is complete, the patterns of activation representing pointers can be
used to retrieve information. Thus, for example, if the activity of the hidden units of
the network is clamped to d n1 , the output of the network becomes (Ll ,dn2 ,dn4 ,dn5 ),
enabling further retrieval of information by decoding d n2 , or d n4 , or d n5 , and so on.
Note that more labeled graphs can be encoded in the same LRAAM.
2.1
THE VOID POINTER PROBLEM
In the RAAM model there is a termination problem in the decoding of a compressed
representation: due to approximation errors introduced during decoding, it is not
clear when a decoded pattern is a terminal or a nonterminal. One solution is to test
for "binary-ness", which consists in checking whether all the values of a pattern are
above 1 - T or below T, T > 0, T ? 1. However, a nonterminal may also pass the
test for "binary-ness".
One advantage of LRAAM over RAAM is the possibility to solve the problem by
allocating one bit of the label for each pointer to represent if the pointer is void or
not. This works better than fixing a particular pattern for the void pointer, such
as a pattern with all the bits to 1 or 0 or -1 (if symmetrical sigmoids are used).
Simulations performed with symmetrical sigmoids showed that the configurations
with all bits equal to 1 or -1 were also used by non void pointers, whereas the
configuration with all bits set to zero considerably reduced the rate of convergence.
U sing a part of the label to solve the problem is particularly efficient, since the
pointer fields are free to take on any configuration when they are void, and this
increases the freedom of the system. To facilitate learning, the output activation
of the void pointers in one epoch is used as an input activation in the next epoch.
Experimentation showed fast convergence to different fixed points for different void
Figure 2: An example of a labeled graph.
1127
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Sperduti
pointers. For this reason, we claimed that each occurrence of the void pointer is
different, and that the nil symbol can be considered as a "don't care" symbol.
2.2
REPRESENTATION OF THE TRAINING SET
An important question about the way a graph is represented in the training set
is which aspects of the representation itself can make the encoding task harder
or easier. In (Sperduti, 1993a) we made a theoretical analysis on the constraints
imposed by the representation on the set of weights of the LRAAM, under the
hypotheses of perfect learning (zero total error after learning) and linear output
units. Our findings were:
i) pointers to nodes belonging to the same cycle of length k and represented in
the same pointer field p, must be eigenvectors of the matrix (W(p))k, where
W(p) is the connection matrix between the hidden layer and the output
units representing the pointer field p;
ii) confluent pointers, i.e., pointers to the same node represented in the same
pointer field p (of different nodes), contribute to reducing the rank of the
matrix W(p), the actual rank is however dependent on the constraints imposed by the label field and the other pointer fields.
We have observed that different representations of the same structure can lead to
very different learning performances. However, representations with roughly the
same number of non void pointers for each pointer field, with cycles represented in
different pointer fields and with confluent pointers seem to be more effective.
3
ACCESS BY CONTENT
Retrieval of coded information is performed in RAAM through the pointers. All the
terminals and nonterminals can be retrieved recursively by the pointers to the whole
tree encoded in a RAAM. If direct access to a component of the tree is required,
the pointer to the component must be stored and used on demand.
Data encoded in an LRAAM can also be accessed directly by content. In fact, an
LRAAM network can be transformed into an analog Hopfield network with one
hidden layer and asymmetric connection matrix by feeding back its output into its
input units. 1 Because each pattern is structured in different fields, different access
1 Experimental
results have shown that there is a high correlation between elements of
(the set of weights from the input to the hidden layer) and the corresponding elements
in W(o)T (the set of weights from the hidden to the output layer). This is particularly true
for weights corresponding to units of the label field. Such result is not a total surprise,
since in the case of a static training set, the error function of a linear encoder network has
been proven to have a unique global minimum corresponding to the projection onto the
subspace generated by the first principal vectors of a covariance matrix associated with the
training set (Baldi & Hornik, 1989). This implies that the weights matrices are transposes
of each other unless there is an invertible transformation between them (see also (Bourlard
& Kamp, 1988)) .
W(h)
Encoding Labeled Graphs by Labeling RAAM
n2=.-r..~=
n5
=100=00=-==.=1.1
n9
nlO
~IQl R",.~.=.
nl4
/n15 \
101??101.1. 1 O~.=lctJ~.=
I.1
Figure 3: The labeled graph encoded in a 16-3-16 LRAAM (5450 epochs), and the
labeled tree encoded in a 18-6-18 LRAAM (1719 epochs).
procedures can be defined on the Hopfield network according to the type of access
key. An access procedure is defined by:
1. choosing one or more fields in the input layer according to the access key(s);
2. clamping the output of such units to the the access key(s);
3. setting randomly the output of the remaining units in the network;
4. letting the remaining units of the network to relax into a stable state.
A validation test of the reached stable state can be performed by:
1. unfreezing the clamped units in the input layer;
2. if the stable state is no longer stable the result of the procedure is considered
wrong and another run is performed;
3. otherwise the stable state is considered a success.
This validation test, however, sometimes can fail to detect an erroneous retrieval
(error) because of the existence of spurious stable states that share the same known
information with the desired one.
The results obtained by the access procedures on an LRAAM codifying the graph
and on an LRAAM codifying the tree shown in Figure 3 are reported in Table
1. For each procedure 100 trials were performed.
The "mean" column in the
table reports the mean number of iterations employed by the Bopfield network to
converge. The access procedure by outgoing pointers was applied only for the tree.
It can be seen from Table 1 that the performances of the access procedures were
high for the graph (no errors and no wrong retrievals), but not so good for the
tree, in particular for the access by label procedure, due to spurious memories. It is
interesting to note that the access by label procedure is very efficient for the leaves
of the tree. This feature can be used to build a system with two identical networks,
one accessed by pointer and the other by content. The search for a label proceeds
simultaneously into the two networks. The network accessed by pointer will be very
fast to respond when the label is located on a node at lower levels of the tree, and
the network accessed by content will be able to respond correctly and very fast "2
when the label is located on a node at higher levels of the tree.
2 Assuming
an analog implementation of the Hopfield network.
1129
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Sperduti
GRAPH: Access by Label
key(s)
success wrong error mean
100%
7.35
0%
0%
io
100%
0%
0%
36.05
i1
6.04
100%
0%
0%
i2
100%
0%
0%
3.99
i3
23.12
100%
0%
0%
i4
18.12
100%
0%
0%
15
29.26
100%
0%
0%
i6
TREE: Access by Children Pointers
(d1 , d 2)
49%
51%
0%
6.29
10%
90%
0%
8.55
(d3,d4)
40%
12.48
60%
0%
(d5, d6)
78%
22%
0%
6.57
(d7 , ds)
(d 9 , d lO )
6.22
91%
9%
0%
d~~)
14.01
14%
86%
0%
(d 12 ,d 13 )
14%
86%
0%
7.87
6.07
28%
72%
0%
ld 14, d 15 )
(*) one pointer
key
io
it
i2
i3
i4
15
16
i7
is
i9
11O
III
lt2
it3
114
115
TREE: Access by Label
success wrong error mean
16.48
100%
0%
0%
14.57
94%
6%
0%
16.92
47%
53%
0%
100%
0%
18.07
0%
32.64
97%
3%
0%
16.03
100%
0%
0%
27.50
49%
51%
0%
27.10
42%
0%
58%
57%
43%
0%
62.45
20%
80%
14.75
0%
19.11
100%
0%
0%
10.83
0%
100%
0%
19.12
100%
0%
0%
23.87
29%
0%
71%
0%
12.09
100%
0%
13.11
0%
100%
0%
Table 1: Results obtained by the access procedures.
4
STABILITY RESULTS
In the LRAAM model two stability problems are encountered. The first one arises
when considering the decoding of a pointer along a cycle of the encoded structures.
Since the decoding process suffers, in general, of approximation errors, it may happen that the decoding diverges from the correct representations of the pointers
belonging to the cycle. Thus, it is fundamental to discover under which conditions
the representations obtained for the pointers are asymptotically stable with respect
to the pointer transformation. In fact, if the representations are asymptotically
stable, the errors introduced by the decoding function are automatically corrected.
The following theorem can be proven (Sperduti, 1993b):
Theorem 1 A decoding sequence
l(i;+I)
= F(p';)(l(iJ?),
j
= 0, .. . ,L
(1)
with l(iL+d = l(t o ) , satisfying
m
L Ibikl
< 1,
i = 1, ... ,m
(2)
k=l
for some index Pi'l' q = 0, ... , L, is asymptotically stable, where btk is the (i, k) th
element of a matrix B, given by
B
= J(P"I) (l( i'l) )J(P"I-l ) (l( i'l_ J)) ... J(p'{J) (l( io) )J(p, L \
l(iL?) ... J(P"I+l ) (d (i'l+d).
In the statement of the theorem, F(p;) (l) = F(D(p; )l+~;?) is the transformation
of the reduced descriptor (pointer) d by the pointer field Pj, and J(pJ)(l) is its
Encoding Labeled Graphs by Labeling RAAM
Jacobian matrix. As a corollary of this theorem we have that if at least one pointer
belonging to the cycle has saturated components, then the cycle is asymptotically
stable with respect to the decoding process. Moreover, the theorem can be applied
with a few modifications to the stability analysis of the fixed points of the associated
Hopfield network.
The second stability problem consists into the discovering of sufficient conditions
under which the property of asymptotical stability of a fixed point in one particular
constrained version of the associated Hopfield network, i.e., an access procedure,
can be extended to related fixed points of different constrained versions of it, i.e.,
access procedures with more information or different information. The result of
Theorem 1 was used to derive three theorems regarding this issue (see (Sperduti,
1993b) ).
5
DISCUSSION AND CONCLUSIONS
The LRAAM model can be seen from various perspectives. It can be considered as
an extension of the RAAM model, which allows one to encode not only trees with
information on the leaves, but also labeled graphs with cycles. On the other hand,
it can be seen as an approximate method to build analog Hopfield networks with
a hidden layer. An LRAAM is probably somewhere in between. In fact, although
it extends the representational capabilities of the RAAM model, it doesn't possess
the same synthetic capabilities as the RAAM, since it explicitly uses the concept
of pointer. Different subsets of units are thus used to codify labels and pointers.
In the RAAM model, using the same set of units to codify labels and reduced
representations is a more natural way of integrating a previously developed reduced
descriptor as a component of a new structure. In fact, this ability was Pollack's
original rationale behind the RAAM model, since with this ability it is possible to fill
a linguistic role with the reduced descriptor of a complex sentence. In the LRAAM
model the same target can be reached, but less naturally. There are two possible
solutions. One is to store the pointer of some complex sentence (or structure, in
general), which was previously developed, in the label of a new structure. The
other solution would be to have a particular label value which tells us that the
information we are looking for can be retrieved using one conventional or particular
pointer among the current ones.
An issue strictly correlated with this is that, even if in an LRAAM it is possible
to encode a cycle, what we get from the LRAAM is not an explicit reduced representation of the cycle, but several pointers to the components of the cycle forged
in such a way that the information on the cycle is only represented implicitly in
each of them. However, the ability to synthesize reduced descriptors for structures
with cycles is what makes the difference between the LRAAM and the RAAM. The
only system that we know of which is able to represent labeled graphs is the DUAL
system proposed by Dyer (Dyer, 1991). It is able to encode small labeled graphs
representing relationships among entities. However, the DUAL system cannot be
considered as being on the same level as the LRAAM, since it devises a reduced
representation of a set of functions relating the components of the graph rather
than a reduced representation for the graph. Potentially also Holographic Reduced
Representations (Plate, 1991) are able to encode cyclic graphs.
1131
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Sperduti
The LRAAM model can also be seen as an extension of the Hopfield networks
philosophy. A relevant aspect of the use of the Hopfield network associated with an
LRAAM, is that the access procedures defined on it can efficiently exploit subsets
of the weights. In fact, their use corresponds to generating several smaller networks
from a large network, one for each kind of access procedure, each specialized on a
particular feature of the stored data. Thus, by training a single network, we get
several useful smaller networks.
In conclusion an LRAAM has several advantages over a standard RAAM. Firstly,
it is more powerful, since it allows to encode directed graphs where each node has
a bounded number of outgoing arcs. Secondly, an LRAAM allows direct access to
the components of the encoded structure not only by pointer, but also by content.
Concerning the applications where LRAAMs can be exploited, we believe there are
at least three possibilities: in knowledge representation, by encoding Conceptual
Graphs (Sowa, 1984); in unification, by representing terms in restricted domains
(Knight, 1989); in image coding, by storing Quadtrees (Samet, 1984);
References
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from examples without local minima. Neural Networks, 2:53-58.
H. Bourlard & Y. Kamp. (1988) Auto-association by multilayer perceptrons and singular
value decomposition. Biological Cybernetics, 59:291-294.
M. G. Dyer. (1991) Symbolic NeuroEngineering for Natural Language Processing: A Multilevel Research Approach., volume 1 of Advances in Connectionist and Neural Computation
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(1989) Unification: A multidisciplinary survey. A CM Computing Surveys,
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H. Samet. (1984) The quadtree and related hierarchical data structures. A CM Computing
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A rtificial Intelligence, 46:5-46.
PART XI
ADDENDA TO
NIPS 5
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7,092 | 861 | SPEAKER RECOGNITION USING
NEURAL TREE NETWORKS
Kevin R. Farrell and Richard J. Marnrnone
CAIP Center, Rutgers University
Core Building, Frelinghuysen Road
Piscataway, New Jersey 08855
Abstract
A new classifier is presented for text-independent speaker recognition. The
new classifier is called the modified neural tree network (MNTN). The
NTN is a hierarchical classifier that combines the properties of decision
trees and feed-forward neural networks. The MNTN differs from the standard NTN in that a new learning rule based on discriminant learning
is used, which minimizes the classification error as opposed to a norm
of the approximation error. The MNTN also uses leaf probability measures in addition to the class labels. The MNTN is evaluated for several
speaker identification experiments and is compared to multilayer perceptrons (MLPs) , decision trees, and vector quantization (VQ) classifiers. The
VQ classifier and MNTN demonstrate comparable performance and perform significantly better than the other classifiers for this task. Additionally, the MNTN provides a logarithmic saving in retrieval time over that
of the VQ classifier. The MNTN and VQ classifiers are also compared
for several speaker verification experiments where the MNTN is found to
outperform the VQ classifier.
1
INTRODUCTION
Automatic speaker recognition consists of having a machine recognize a person based
on his or her voice. Automatic speaker recognition is comprised of two categories:
speaker identification and speaker verification. The objective of speaker identification is to identify a person within a fixed population based on a test utterance
from that person. This is contrasted to speaker verification where the objective is
to verify a person's claimed identity based on the test utterance.
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Farrell and Mammone
Speaker recognition systems can be either text dependent or text independent.
Text-dependent speaker recognition systems require that the speaker utter a specific
phrase or a given password. Text-independent speaker identification systems identify the speaker regardless of the utterance. This paper focuses on text-independent
speaker identification and speaker verification tasks.
A new classifier is introduced and evaluated for speaker recognition. The new classifier is the modified neural tree network (MNTN). The MNTN incorporates modifications to the learning rule of the original NTN [1] and also uses leaf probability
measures in addition to the class labels. Also, vector quantization (VQ) classifiers,
multilayer perceptrons (MLPs), and decision trees are evaluated for comparison.
This paper is organized as follows. Section 2 reviews the neural tree network and
discusses the modifications. Section 3 discusses the feature extraction and classification phases used here for text-independent speaker recognition. Section 4 describes
the database used and provides the experimental results. The summary and conclusions of the paper are given in Section 5.
2
THE MODIFIED NEURAL TREE NETWORK
The NTN [1] is a hierarchical classifier that uses a tree architecture to implement
a sequential linear decision strategy. Each node at every level of the NTN divides
the input training vectors into a number of exclusive subsets of this data. The
leaf nodes of the NTN partition the feature space into homogeneous subsets, i.e.,
a single class at each leaf node. The NTN is recursively trained as follows. Given
a set of training data at a particular node, if all data within that node belongs to
the same class, the node becomes a leaf. Otherwise, the data is split into several
subsets, which become the children of this node. This procedure is repeated until
all the data is completely uniform at the leaf nodes.
For each node the NTN computes the inner product of a weight vector wand an
input feature vector x, which should be approximately equal to the the output
label y E {O,1}. Traditional learning algorithms minimize a norm of the error
? = (y- < w, x ?, such as the L2 or L1 norm. The splitting algorithm of the
modified NTN is based on discriminant learning [2]. Discriminant learning uses a
cost function that minimizes the classification error.
For an M class NTN, the discriminant learning approach first defines a misclassification measure d( x) as [2]:
d( x)
= - < Wi, X > + {
M
~ 1 I) <
1
Wj,
x
>t
}n ,
(1)
jf;i
where n is a predetermined smoothing constant. If x belongs to class i, then d(x)
will be negative, and if x does not belong to class i, d( x) will be positive. The
misclassification measure d( x) is then applied to a sigmoid to yield:
g[d(x)]
= 1 + e-1 d( x ).
(2)
The cost function in equation (2) is approximately zero for correct classifications
and one for misclassifications. Hence, minimizing this cost function will tend to
Speaker Recognition Using Neural Tree Networks
,
\??,~,o
\
0
0
0 1 0./
, 1 ~/
.', .....
o~/
.............
.--;r~
1
o
LABEL=O
CONFIDENCE c 1.0
\1
...
1 \
o ...
1
....
1 0\
'. 1
...
0
1
"... 1
'.
LABEL= 1
CONFIDENCE = 0.6
'.
\\.
LABEL = 0
CONFIDENCE c 0.8
LABEL = 1
CONFIDENCE = 0.7
Figure 1: Forward Pruning and Confidence Measures
mmlmize the classification error. It is noted that for binary NTNs, the weight
updates obtained by the discriminant learning approach and the Ll norm of the
error are equivalent [3].
The NTN training algorithm described above constructs a tree having 100% performance on the training set. However, an NTN trained to this level may not have
optimal generalization due to overtraining. The generalization can be improved
by reducing the number of nodes in a tree, which is known as pruning. A technique known as backward pruning was recently proposed [1] for the NTN. Given a
fully grown NTN, i.e., 100% performance on the training set, the backward pruning
method uses a Lagrangian cost function to minimize the classification error and
the number of leaves in the tree. The method used here prunes the tree during its
growth, hence it is called forward pruning.
The forward pruning algorithm consists of simply truncating the growth of the tree
beyond a certain level. For the leaves at the truncated level, a vote is taken and
the leaf is assigned the label of the majority. In addition to a label, the leaf is
also assigned a confidence. The confidence is computed as the ratio of the number
of elements for the vote winner to the total number of elements. The confidence
provides a measure of confusion for the different regions of feature space. The
concept of forward pruning is illustrated in Figure 1.
3
FEATURE EXTRACTION AND CLASSIFICATION
The process of feature extraction consists of obtaining characteristic parameters of
a signal to be used to classify the signal. The extraction of salient features is a
key step in solving any pattern recognition problem. For speaker recognition, the
features extracted from a speech signal should be invariant with regard to the desired
1037
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Farrell and Mammone
Speaker 1
yl i
,----.. (NTN, VQ r--=--. Codebook)
Speaker 2
y2 i
(NTN, VQ ..:.-=-. Decision
Codebook)
Feature Xi
Vector
?
Speaker
Identity
or
Authenticity
?
?
~
Speaker N
(NTN, VQ
Codebook)
yN i
r:----=--
Figure 2: Classifier Structure for Speaker Recognition
speaker while exhibiting a large distance to any imposter. Cepstral coefficients are
commonly used for speaker recognition [4] and shall be considered here to evaluate
the classifiers.
The classification stage of text-independent speaker recognition is typically implemented by modeling each speaker with an individual classifier. The classifier structure for speaker recognition is illustrated in Figure 2. Given a specific feature vector,
each speaker model associates a number corresponding to the degree of match with
that speaker. The stream of numbers obtained for a set of feature vectors can be
used to obtain a likelihood score for each speaker model. For speaker identification,
the feature vectors for the test utterance are applied to all speaker models and the
corresponding likelihood scores are computed. The speaker is selected as having
the largest score. For speaker verification, the feature vectors are applied only to
the speaker model for the speaker to be verified. If the likelihood score exceeds a
threshold, the speaker is verified or else is rejected.
The classifiers for the individual speaker models are trained using either supervised
or unsupervised training methods. For supervised training methods the classifier for
each speaker model is presented with the data for all speakers. Here, the extracted
feature vectors for that speaker are labeled as "one" and the extracted feature vectors for everyone else are labeled as "zero" . The supervised classifiers considered
here are the multilayer perceptron (MLP), decision trees, and modified neural tree
network (MNTN). For unsupervised training methods each speaker model is presented with only the extracted feature vectors for that speaker. This data can
then be clustered to determine a set of centroids that are representative of that
speaker. The unsupervised classifiers evaluated here are the full-search and treestructure vector quantization classifiers, henceforth denoted as FSVQ and TSVQ .
Speaker models based on supervised training capture the differences of that speaker
to other speakers, whereas models based on unsupervised training use a similarity
measure.
Specifically, a trained NTN can be applied to speaker recognition as follows. Given
a sequence of feature vectors x from the test utterance and a trained NTN for
Speaker Recognition Using Neural Tree Networks
speaker Si, the corresponding speaker score is found as the "hit" ratio:
(3)
Here. M is the number of vectors classified as "one" and N is the number of vectors classified as "zero" . The modified NTN computes a hit ratio weighed by the
confidence scores:
",M
PMNTN(xISi) =
1
L..Jj=l Cj
",N
0
",M
l'
L..JJ=l Cj
L..JJ=l cj
(4)
+
where c 1 and cO are the confidence scores for the speaker and antispeaker, respectively. These scores can be used for decisions regarding identification or verification.
4
4.1
EXPERIMENTAL RESULTS
Database
The database considered for the speaker identification and verification experiments
is a subset of the DARPA TIMIT database. This set represents 38 speakers of the
same (New England) dialect. The preprocessing of the TIMIT speech data consists
of several steps. First, the speech is downsampled from 16KHz to 8 KHz sampling
frequency. The downsampling is performed to obtain a toll quality signal. The
speech data is processed by a silence removing algorithm followed by the application
of a pre-emphasis filter H(z) = 1-0.95z- 1 . A 30 ms Hamming window is applied to
the speech every 10 ms. A twelfth order linear predictive (LP) analysis is performed
for each speech frame. The features consist of the twelve cepstral coefficients derived
from this LP polynomial.
There are 10 utterances for each speaker in the selected set. Five of the utterances
are concatenated and used for training. The remaining five sentences are used
individually for testing. The duration of the training data ranges from 7 to 13
seconds per speaker and the duration of each test utterance ranges from 0.7 to 3.2
seconds.
4.2
Speaker Identification
The first experiment is for closed set speaker identification using 10 and 20 speakers
from the TIMIT New England dialect. The identification is closed set in that the
speaker is assumed to be one of the 10 or 20 speakers, i.e., no "none of the above"
option. The NTN, MLP [5], and VQ [4] classifiers are each evaluated on this data in
addition to the ID3 [6], C4 [7], CART [8], and Bayesian [9] decision trees. The VQ
classifier is trained using a K-means algorithm and tested for codebook sizes varying
from 16 to 128 centroids. The MNTN used here is pruned at levels ranging from the
fourth through seventh. The MLP is trained using the backpropagation algorithm
[10] for architectures having 16, 32, and 64 hidden nodes (within one hidden layer).
The results are summarized in Table 1. The * denotes that the CART tree could
not be grown for the 20 speaker experiment due to memory limitations.
1039
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Farrell and Mammone
Classifier
ID3
CART
C4
Table 1: Speaker Identification Experiments
4.3
Speaker Verification
The FSVQ classifier and MNTN are evaluated next for speaker verification. The
first speaker verification experiment consists of 10 speakers and 10 imposters (i.e.,
people not used in the training set). The second speaker verification experiment
uses 20 enrolled speakers and 18 imposters. The MNTN is pruned at the seventh
level (128 leaves) and the FSVQ classifier has a codebook size of 128 entries.
Speaker verification performance can be enhanced by using a technique known as
cohort normalization [11]. Traditional verification systems accept a speaker if:
p(XII) > T(I),
(5)
where p( X II) is the likelihood that the sequence of feature vectors X was generated
by speaker I and T( I) is the corresponding likelihood threshold. Instead of using
the fixed threshold criteria in equation (5), an adaptive threshold can be used via
the likelihood measure:
P(XII)
T(I).
P(XII) >
(6)
Here, the speaker score is first normalized by the probability that the feature vectors
X were generated by a speaker other than I. The likelihood p(XII) can be estimated
with the scores of the speaker models that are closest to I, denoted as 1's cohorts
[11]. This estimate can consist of a maximum, minimum, average, etc., depending
on the classifier used.
The threshold for the VQ and MNTN likelihood scores are varied from the point
of 0% false acceptance to 0% false rejection to yield the operating curves shown
in Figures 3 and 4 for the 10 and 20 speaker populations, respectively. Note that
all operating curves presented in this section for speaker verification represent the
posterior performance of the classifiers, given the speaker and imposter scores. Here
it can be seen that the MNTN and VQ classifiers are both improved by the cohort
normalized scores. The equal error rates for the MNTN and VQ classifier are
summarized in Table 2.
For both experiments (10 and 20 speakers), the MNTN provides better performance
than the VQ classifier, both with and without cohort normalization, for most of the
operating curve.
Speaker Recognition Using Neural Tree Networks
MNTN
10 Speakers
20 Speakers
Table 2: Equal error rates for speaker verification
Speaker Verification (10 speakers)
0.35r----~-----r---_r_---..___--___,---_,
- -_ .. ..... ...:............ .. .. :.......... ... ...:...... ....... ...;......... ....... ;....... ...... .
?
?
?
.
I
~
?
?
-+
?
va .~
-. va. ...with
cohort
.
... . ... ..... ??....... - .. . ... ' :.:' .... ...... .......::..........
..:..... ....... ... . .. ..... ..... .
.
...
..
..
.
.
.
.
..
,
~
_
~
"
(])
Vl
~ 0.15
CL
0.1
0.05
0.01
0.02
0.03
0.04
P(Falsa Accept}
0.05
0.06
Figure 3: Speaker Verification (10 Speakers)
Speaker Verification (20 speakers)
0.45,------r----.---_r_---..------,-----,
0 .4 ... ..... . .. ... : .. ..... .. . __... . ; ... ....... ..... ;.... .. ..... .. . -. ~ - .- .... .. ...... : .... ...... .... .
...
...
..
....
."
,
.,
.
.
.
.
....... .... .. ..;. .. .. .. .. ... .... : ... ..... .... ... ,.. .... ?? ???? ????:? ????? ??? ???? ??? i?????? ?? ?? ?? ??
:
~
: -+ va
~
:
??,
.
0.35
: -.. va
with cohort
0.3 i' .... ??.. .. .. ::.......... ?.. ?.. :: ...... .. ? .. ? .. ?:??
............
f.............. ?~:.. ?.. ?? ....... .
MNTN ,
.
?f.. ?.... ?? .. ??.. ~? ::?MNI~?1i!~..~~.9n? ? ?i ?? ? ??? ...... ???
0.25 ~: .. ..
n.
1
.
.
. -.
?:
.:
.:
????????j? .. ?????? ......
.
0.1
... .... - . .. ..... . -_ ... , .;. . ..... . . ... . ..
0.05
-.~
??
.... .......... .
..
~
oL-----~~~~~--~--~~====~~--~
o
0.02
0.04
0.06
0.08
P(Faisa Accept)
0.1
Figure 4: Speaker Verification (20 Speakers)
0.12
1041
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Farrell and Mammone
5
CONCLUSION
A new classifier called the modified NTN is examined for text-independent speaker
recognition. The performance of the MNTN is evaluated for several speaker recognition experiments using various sized speaker sets from a 38 speaker corpus. The
features used to evaluate the classifiers are the LP-derived cepstrum. The MNTN is
compared to full-search and tree-structured VQ classifiers, multi-layer perceptrons,
and decision trees. The FSVQ and MNTN classifiers both demonstrate equivalent
performance for the speaker identification experiments and outperform the other
classifiers. For speaker verification, the MNTN consistently outperforms the FSVQ
classifier. In addition to performance advantages for speaker verification, the MNTN
also demonstrates a logarithmic saving in retrieval time over that of the FSVQ classifier . This computational advantage can be obtained by using TSVQ, although
TSVQ will reduce the performance with respect to FSVQ.
6
ACKNOWLEDGEMENTS
The authors gratefully acknowledge the support of Rome Laboratories, Contract
No. F30602-91-C-OI20. The decision tree simulations utilized the IND package
developed by W. Buntine of NASA.
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[11] A.E. Rosenberg, J. Delong, C.H. Lee, B.H. Juang, and F.K. Soong. The use of
cohort normalized scores for speaker recognition. In Proc. ICSLP, Oct. 1992.
| 861 |@word polynomial:1 norm:4 twelfth:1 simulation:1 simplifying:1 recursively:1 score:14 imposter:4 outperforms:1 si:1 belmont:1 partition:1 predetermined:1 update:1 leaf:11 selected:2 core:1 provides:4 codebook:5 node:11 five:2 become:1 consists:5 combine:1 growing:1 multi:1 ol:1 window:1 becomes:1 minimizes:2 developed:1 every:2 growth:2 classifier:39 hit:2 demonstrates:1 yn:1 positive:1 approximately:2 emphasis:1 examined:1 co:1 range:2 testing:1 implement:1 differs:1 backpropagation:1 procedure:1 significantly:1 confidence:10 road:1 pre:1 downsampled:1 weighed:1 equivalent:2 lagrangian:1 center:1 regardless:1 truncating:1 duration:2 splitting:1 rule:2 his:1 population:2 enhanced:1 homogeneous:1 us:6 associate:1 element:2 rumelhart:1 recognition:23 utilized:1 database:4 labeled:2 capture:1 wj:1 region:1 edited:1 trained:7 solving:1 predictive:1 completely:1 icassp:2 darpa:1 jersey:1 various:1 grown:2 dialect:2 london:1 artificial:1 treestructure:1 kevin:1 mammone:6 otherwise:1 statistic:1 id3:2 sequence:2 toll:1 advantage:2 product:1 juang:2 depending:1 implemented:1 exhibiting:1 correct:1 filter:1 require:1 icslp:1 generalization:2 clustered:1 considered:3 hall:1 tsvq:3 proc:1 label:9 individually:1 largest:1 mit:1 modified:8 r_:2 breiman:1 varying:1 password:1 rosenberg:2 derived:2 focus:1 consistently:1 likelihood:8 centroid:2 dependent:2 vl:1 typically:1 accept:3 her:1 hidden:2 classification:10 denoted:2 smoothing:1 delong:1 wadsworth:1 equal:3 construct:1 saving:2 having:4 extraction:5 sampling:1 chapman:1 represents:1 unsupervised:4 richard:1 recognize:1 individual:2 phase:1 friedman:1 mlp:3 acceptance:1 tree:30 divide:1 gaines:1 desired:1 classify:1 modeling:1 phrase:1 cost:4 subset:4 entry:1 uniform:1 comprised:1 seventh:2 buntine:2 person:4 twelve:1 international:1 contract:1 yl:1 lee:1 thesis:1 opposed:1 henceforth:1 summarized:2 coefficient:2 farrell:6 stream:1 performed:2 closed:2 option:1 parallel:1 timit:3 mlps:2 minimize:2 characteristic:1 yield:2 identify:2 rabiner:1 identification:14 bayesian:1 none:1 classified:2 overtraining:1 acquisition:1 frequency:1 hamming:1 knowledge:1 organized:1 cj:3 nasa:1 feed:1 supervised:4 improved:2 cepstrum:1 evaluated:7 rejected:1 stage:1 until:1 defines:1 quality:1 building:1 verify:1 concept:1 y2:1 normalized:3 hence:2 assigned:2 laboratory:1 illustrated:2 ind:1 ll:1 during:1 speaker:106 noted:1 m:2 criterion:1 stone:1 demonstrate:2 confusion:1 l1:1 ranging:1 recently:1 sigmoid:1 winner:1 khz:2 belong:1 cambridge:2 automatic:2 gratefully:1 katagiri:1 similarity:1 operating:3 etc:1 closest:1 posterior:1 belongs:2 claimed:1 certain:1 binary:1 seen:1 minimum:1 prune:1 determine:1 mntn:25 signal:4 ii:1 full:2 exceeds:1 match:1 academic:1 england:2 retrieval:2 va:4 regression:1 multilayer:3 vision:1 rutgers:2 normalization:2 represent:1 addition:5 whereas:1 else:2 cart:3 tend:1 incorporates:1 cohort:7 split:1 misclassifications:1 architecture:2 inner:1 regarding:1 reduce:1 speech:7 jj:3 enrolled:1 processed:1 category:1 mcclelland:1 outperform:2 sankar:1 estimated:1 per:1 xii:4 shall:1 group:1 key:1 salient:1 threshold:5 verified:2 backward:2 wand:1 package:1 fourth:1 decision:12 comparable:1 layer:2 followed:1 ntn:22 mni:1 pruned:2 structured:1 piscataway:1 march:1 describes:1 wi:1 lp:3 modification:2 invariant:1 soong:2 taken:1 equation:2 vq:16 discus:2 hierarchical:2 caip:1 voice:1 original:1 denotes:1 remaining:1 quinlan:2 f30602:1 concatenated:1 objective:2 strategy:1 exclusive:1 traditional:2 distance:1 majority:1 discriminant:5 induction:1 ratio:3 minimizing:1 downsampling:1 olshen:1 negative:1 perform:1 xisi:1 acknowledge:1 truncated:1 frame:1 rome:1 varied:1 introduced:1 sentence:1 c4:2 uang:1 beyond:1 pattern:1 memory:1 everyone:1 misclassification:2 utterance:8 text:9 review:1 l2:1 acknowledgement:1 fully:1 limitation:1 utter:1 degree:1 verification:21 summary:1 silence:1 perceptron:1 cepstral:2 distributed:1 regard:1 curve:3 computes:2 forward:5 commonly:1 adaptive:1 preprocessing:1 author:1 transaction:1 pruning:8 corpus:1 assumed:1 xi:1 discriminative:1 search:2 table:4 additionally:1 ca:1 obtaining:1 cl:1 child:1 repeated:1 representative:1 removing:1 specific:2 mason:1 consist:2 quantization:4 false:2 sequential:1 phd:1 rejection:1 logarithmic:2 simply:1 extracted:4 ma:1 oct:2 identity:2 sized:1 jf:1 specifically:1 contrasted:1 reducing:1 total:1 called:3 experimental:2 vote:2 perceptrons:3 people:1 support:1 evaluate:2 authenticity:1 tested:1 |
7,093 | 862 | Non-linear Statistical Analysis and
Self-Organizing Hebbian Networks
Jonathan L. Shapiro and Adam Priigel-Bennett
Department of Computer Science
The University, Manchester
Manchester, UK
M139PL
Abstract
Neurons learning under an unsupervised Hebbian learning rule can
perform a nonlinear generalization of principal component analysis.
This relationship between nonlinear PCA and nonlinear neurons is
reviewed. The stable fixed points of the neuron learning dynamics
correspond to the maxima of the statist,ic optimized under nonlinear PCA. However, in order to predict. what the neuron learns,
knowledge of the basins of attractions of the neuron dynamics is
required. Here the correspondence between nonlinear PCA and
neural networks breaks down. This is shown for a simple model.
Methods of statistical mechanics can be used to find the optima
of the objective function of non-linear PCA. This determines what
the neurons can learn. In order to find how the solutions are partitioned amoung the neurons, however, one must solve the dynamics.
1
INTRODUCTION
Linear neurons learning under an unsupervised Hebbian rule can learn to perform a
linear statistical analysis ofthe input data. This was first shown by Oja (1982), who
proposed a learning rule which finds the first principal component of the variance
matrix of the input data. Based on this model, Oja (1989), Sanger (1989), and
many others have devised numerous neural networks which find many components
of this matrix. These networks perform principal component analysis (PCA), a
well-known method of statistical analysis.
407
408
Shapiro and Priigel-Bennett
Since PCA is a form of linear analysis, and the neurons used in the PCA networks
are linear - the output of these neurons is equal to the weighted sum of inputs;
there is no squashing function of sigmoid - it is obvious to ask whether non-linear
Hebbian neurons compute some form of non-linear PCA? Is this a useful way to
understand the performance of the networks? Do these networks learn to extract
features of the input data which are different from those learned by linear neurons?
Currently in the literature, the phrase "non-linear PCA" is used to describe what
is learned by any non-linear generalization of Oja neurons or other PCA networks
(see for example, Oja, 1993 and Taylor, 1993).
In this paper, we discuss the relationship between a particular form of non-linear
Hebbian neurons (Priigel-Bennett and Shapiro, 1992) and a particular generalization of non-linear PCA (Softky and Kammen 1991). It is clear that non-linear neurons can perform very differently from linear ones. This has been shown through
analysis (Priigel-Bennett and Shapiro, 1993) and in application (Karhuenen and
Joutsensalo, 1992). It can also be very useful way of understanding what the neurons learn. This is because non-linear PCA is equivalent to maximizing some objective function. The features that this extracts from a data set can be studied using
techniques of statistical mechanics. However, non-linear PCA is ambiguous because
there are multiple solutions. What the neuron can learn is given by non-linear PCA.
The likelihood of learning the different solutions is governed by the dyanamics chosen to implement non-linear PCA, and may differ in different implementations of
the dynamics.
2
NON-LINEAR HEBBIAN NEURONS
Neurons with non-linear activation functions can learn to perform very different
tasks from those learned by linear neurons. Nonlinear Hebbian neurons have been
analyzed for general non-linearities by Oja (1991), and was applied to sinusoidal
signal detection by Karhuenen and Joutsensalo (1992).
Previously, we analysed a simple non-linear generalization of Oja's rule (PriigelBennett and Shapiro, 1993). We showed how the shape of the neuron activation
function can control what a neuron learns. Whereas linear neurons learn to a
statistic mixture of all of the input patterns, non-linear neurons can learn to become
tuned to individual patterns, or to small clusters of closely correlated patterns.
In this model, each neuron has weights, Wi is the weight from the ith input, and
responds to the usual sum of input times weights through an activation function
A(y). This is assumed a simple power-law above a threshold and zero below it. I.e.
(1)
Here ? is the threshold, b controls the power of the power-law, xf is the ith component of the pth pattern, and VP = Li xf Wi. Curves of these functions are shown
in figure laj if b = 1 the neurons are threshold-linear. For b > 1 the curves can be
thought of as low activation approximations to a sigmoid which is shown in figure
1b. The generalization of Oja's learning rule is that the change in the weights 8Wi
Non-Linear Statistical Analysis and Self-Organizing Hebbian Networks
Neuron Activation Function
A Sigmoid Activation Function
b>1
b<1
?
psp
Figure 1: a) The form of the neuron activation function. Control by two parameters
band <p. When b > 1, this activation function approximates a sigmoid, which is
shown in b) .
is given by
6Wi
= LA(VP) [xf -
VP Wi ] .
(2)
P
If b < 1, the neuron learns to average a set of patterns. If b = 1, the neuron finds
the principal component of the pattern set. When b > 1, the neuron learns to
distinguish one of the patterns in the presence of the others, if those others are not
too correlated with the pattern. There is a critical correlation which is determined
by b; the neuron learns to individual patterns which are less correlated than the
critical value, but learns to something like the center of the cluster if the patterns
are more correlated. The threshold controls the size of the subset of patterns which
the neuron can respond to.
For these neurons, the relationship between non-PCA and the activation function
was not previously discussed. That is done in the next section.
3
NON-LINEAR peA
A non-linear generalization of PCA was proposed by Softky and Kammen (1991).
In this section, the relationship between non-linear PCA and unsupervised Hebbian
learning is reviewed.
409
410
Shapiro and Priigel-Bennett
3.1
WHAT IS NON-LINEAR PCA
The principal component of a set of data is the direction which maximises the
variance. I.e. to find the principal component of the data set, find the vector tV of
unit length which maximises
(3)
Here, Xi denotes the ith component of an input pattern and < .. . > denotes
the average over the patterns. Sofky and Kammen suggested that an appropriate
generalization is to find the vector tV which maximizes the d-dimensional correlation,
(4)
They argued this would give interesting results if higher order correlations are important, or ifthe shape ofthe data cloud is not second order. This can be generalized
further, of course, maximizing the average of any non-linear function of the input
U(y),
(5)
The equations for the principal components are easily found using Lagrange multipliers. The extremal points are given by
< U'
(1:
>=
AWi.
These points will be (local) maxima if the Hessian
1lij,
WkXk )Xi
(6)
k
1lij
=< U"(I: WkXk)XiXj >
-ADij,
(7)
k
Here, A is a Lagrange multiplier chosen to make
3.2
Iwl 2 =
1.
NEURONS WHICH LEARN PCA
A neuron learning via unsupervised Hebbian learning rule can perform this optimization. This is done by associating Wi with the weight from the ith input to
the neuron, and the data average < . > as the sum over input patterns xf. The
nonlinear function which is optimized is determined by the integral of the activation
function of the neuron
A(y) = U'(y).
In their paper, Softky and Kammen propose a learning rule which does not perform
this optimization in general. The correct learning rule is a generalization of Oja's
rule (equation (2) above), in this notation,
(8)
Non-Linear Statistical Analysis and Self-Organizing Hebbian Networks
This fixed points of this dynamical equation will be solutions to the extremal equation of nonlinear peA, equation (6), when the a.'3sociations
A = (A(V)V) ,
and
A(y)
= U'(y)
are made.
Here (.) is interpreted as sum over patterns; this is batch learning. The rule can also
be used incrementally, but then the dynamics are stochastic and the optimization
might be performed only on average, and then maybe only for small enough learning
rates. These fixed points will be stable when the Hessian ll i j is negative definite at
the fixed point. This is now,
which is the same as the previous, equation (7),in directions perpendicular to the
fixed point, but contains additional terms in direction of the fixed point which
normalize it.
The neurons described in section 2 would perform precisely what Softky and Kammen proposed if the activation function was pure power-law and not thresholded;
as it is they maximize a more complicated objective function.
Since there is a one to one correspondence between the stable fixed points of the
dynamics and the local maxima of the non-linear correlation measure, one says that
these non-linear neurons compute non-linear peA.
3.3
THEORETICAL STUDIES OF NONLINEAR PCA
In order to understand what these neurons learn, we have studied the networks
learning on model data drawn from statistical distributions. For very dense clusters
p ~ 00, N fixed, the stable fixed point equations are algebraic. In a few simple
cases they can be solved. For example, if the data is Gaussian or if the data cloud is
a quadratic cloud (a function of a quadratic form), the neuron learns the principal
component, like the linear neuron. Likewise, if the patterns are not random, the
fixed point equations can be solved in some cases.
For large number of patterns in high dimensions fluctuations in the data are important (N and P goes to 00 together in some way). In this case, methods of
statistical mechanics can be used to average over the data. The objective function
of the non-linear peA acts as (minus) the energy in statistical mechanics. The free
energy is formally,
F =<
IOg(D. JOf, 6 (t wl- I) exp (3U(V) > .
(10)
In the limit that f3 is large, this calculation finds the local maxima of U. In this
form of analysis, the fact that the neuron optimizes an objective function is very
important. This technique was used to produce the results outlined in section 2.
411
412
Shapiro and Priigel-Bennett
3.4
WHAT NON-LINEAR peA FAILS TO REVEAL
In the linear peA, there is one unique solution, or if there are many solutions
it is because the solutions are degenerate. However, for the non-linear situation,
there are many stable fixed points of the dynamics and many local maxima of the
non-linear correlation measure.
This has two effects. First, it means that you cannot predict what the neuron will
learn simply by studying fixed point equations. This tells you what the neuron
might learn, but the probability that this solution will be can only be ascertained if
the dynamics are understood. This also breaks the relationship between non-linear
peA and the neurons, because, in principle, there could be other dynamics which
have the same fixed point structure, but do not have the same basins of attraction.
Simple fixed point analysis would be incapable of predicting what these neurons
would learn.
4
PARTITIONING
An important question which the fixed-point analysis, or corresponding statistical
mechanics cannot address is: what is the likelihood of learning the different solutions? This is the essential ambiguity of non-linear peA - there are many solutions
and the size of the basin of attractions of each is determined by the dynamics, not
by local maxima of the nonlinear correlation measure.
As an example, we consider the partitioning of the neurons described in section 2.
These neurons act much like neurons in competitive networks, they become tuned to
individual patterns or highly correlated clusters. Given that the density of patterns
in the input set is p(i), what is the probability p(i) that a neuron will become
tuned to this pattern. It is often said that the desired result should be p(i) = p(i),
although for Kohonen I-d feature maps ha.~ been shown to be p(i) = p(i)2/3 (see
for example, Hertz, Krogh, and Palmer 1991).
We have found that he partitioning cannot be calculated by finding the optima
of the objective function . For example, in the case of weakly correlated patterns,
the global maxima is the most likely pattern, whereas all of the patterns are local
maxima. To determine the partitioning, the basin of attraction of each pattern
must be computed. This could be different for different dynamics with the same
fixed point structure.
In order to determine the partitioning, the dynamics must be understood. The
details will be described elsewhere (Priigel-Bennett and Shapiro, 1994). For the
case of weakly correlated patterns, a neuron will learn a pattern for which
p(xp)(Vcr/- 1 > p(xq)(Voq)b-l
Vq
f- p.
Here Vcr is the initial overlap (before learning) of the neuron's weights with the pth
pattern. This defines the basin of attraction for each pattern.
In the large P limit and for random patterns
p(i)
~
p(iYx
(11)
where a ~ 210g(P)/(b -1), P is the number of patterns, and where b is a parameter
that controls the non-linearity of the neuron's response. If b is chosen so that a = 1,
Non-Linear Statistical Analysis and Self-Organizing Hebbian Networks
then the probability of a neuron learning a pattern will be proportional to the
frequency with which the pattern is presented.
5
CONCLUSIONS
The relationship between a non-linear generalization of Oja's rule and a non-linear
generalization of PCA was reviewed. Non-linear PCA is equivalent to maximizing a
objective function which is a statistical measure of the data set. The objective function optimized is determined by the form of the activation function of the neuron.
Viewing the neuron in this way is useful, because rather than solving the dynamics,
one can use methods of statistical mechanics or other methods to find the maxima
of the objective function. Since this function has many local maxima, however,
these techniques cannot determine how the solutions are partitioned amoung the
neurons. To determine this, the dynamics must be solved.
Acknowledgements
This work was supported by SERC grant GRG20912.
References
J. Hertz, A. Krogh, and R.G. Palmer. (1991). Introduction to the Theory of Neural
Computation. Addison-Wesley.
J. Karhunen and J. J outsensalo. (1992) Nonlinear Heb bian algorithms for sinusoidal
frequency estimation, in Artificial Neural Networks, 2, I. Akeksander and J . Taylor,
editors, North-Holland.
Erkki Oja. (1982) A simplified neuron model as a principal component analyzer.
em J. Math. Bio., 15:267-273.
Erkki Oja. (1989) Neural networks, principal components, and subspaces. Int. J.
of Neural Systems, 1(1):61-68.
E. Oja, H. Ogawa, and J. Wangviwattan. (1992) Principal Component Analysis
by homogeneous neural networks: Part II: analysis and extension of the learning
algorithms IEICE Trans. on Information and Systems, E75-D, 3, pp 376-382.
E. Oja. (1993) Nonlinear PCA: algorithms and applications, in Proceedings of
World Congress on Neural Networks, Portland, Or. 1993.
A. Prugel-Bennett and Jonathan 1. Shapiro. (1993) Statistical Mechanics of Unsupervised Hebbian Learning. J. Phys. A: 26, 2343.
A. Prugel-Bennett and Jonathan L. Shapiro. (1994) The Partitioning Problem for
Unsupervised Learning for Non-linear Neurons. J. Phys. A to appear.
T. D. Sanger. (1989) Optimal Unsupervised Learning in a Single-Layer Linear
Feedforward Neural Network. Neural Networks 2,459-473.
Jonathan L. Shapiro and A. Prugel-Bennett (1992), Unsupervised Hebbian Learning
and the Shape of the Neuron Activation Function, in Artificial Neural Networks, 2,
I. Akeksander and J. Taylor, editors, North-Holland.
413
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Shapiro and Prugel-Bennett
W . Softky and D. Kammen (1991). Correlations in High Dimensional or Asymmetric Data Sets: Hebbian Neuronal Processing. Neural Networks 4, pp 337-347.
J . Taylor, (1993) Forms of Memory, in Proceedings of World Congress on Neural
Networks, Portland, Or. 1993.
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7,094 | 863 | Non-Intrusive Gaze Tracking Using Artificial
Neural Networks
Shumeet Baluja
Dean Pomerleau
[email protected]
School of Computer Science
Carnegie Mellon University
Pittsburgh, PA 15213
pomerleau @cs.cmu.edu
School of Computer Science
Carnegie Mellon University
Pittsburgh, PA 15213
Abstract
We have developed an artificial neural network based gaze tracking system
which can be customized to individual users. Unlike other gaze trackers,
which normally require the user to wear cumbersome headgear, or to use a
chin rest to ensure head immobility, our system is entirely non-intrusive.
Currently, the best intrusive gaze tracking systems are accurate to approximately 0.75 degrees. In our experiments, we have been able to achieve an
accuracy of 1.5 degrees, while allowing head mobility. In this paper we
present an empirical analysis of the performance of a large number of artificial neural network architectures for this task.
1 INTRODUCTION
The goal of gaze tracking is to determine where a subject is looking from the appearance
of the subject's eye. The interest in gaze tracking exists because of the large number of
potential applications. Three of the most common uses of a gaze tracker are as an alternative to the mouse as an input modality [Ware & Mikaelian, 1987], as an analysis tool for
human-computer interaction (HCI) studies [Nodine et. aI, 1992], and as an aid for the
handicapped [Ware & Mikaelian, 1987].
Viewed in the context of machine vision, successful gaze tracking requires techniques to
handle imprecise data, noisy images, and a potentially infinitely large image set. The most
accurate gaze tracking has come from intrusive systems. These systems either use devices
such as chin rests to restrict head motion, or require the user to wear cumbersome equipment, ranging from special contact lenses to a camera placed on the user's head. The system described here attempts to perform non-intrusive gaze tracking, in which the user is
neither required to wear any special equipment, nor required to keep hislher head still.
753
754
Baluja and Pomerleau
2 GAZE TRACKING
2.1 TRADITIONAL GAZE TRACKING
In standard gaze trackers, an image of the eye is processed in three basic steps. First, the
specular reflection of a stationary light source is found in the eye's image. Second, the
pupil's center is found. Finally, the relative position of the light's reflection to the pupil's
center is calculated. The gaze direction is determined from information about the relative
positions, as shown in Figure 1. In many of the current gaze tracker systems, the user is
required to remain motionless, or wear special headgear to maintain a constant offset
between the position of the camera and the eye.
Specular
Reflection
~~~
Looking at
Light
Looking Above
Light
Looking Below
Light
Looking Left of
Light
Figure 1: Relative position of specular reflection and pupil. This diagram assumes that
the light is placed in the same location as the observer (or camera).
2.2 ARTIFICIAL NEURAL NETWORK BASED GAZE TRACKING
One of the primary benefits of an artificial neural network based gaze tracker is that it is
non-intrusive; the user is allowed to move his head freely. In order to account for the shifts
in the relative positions of the camera and the eye, the eye must be located in each image
frame. In the current system, the right eye is located by searching for the specular reflection of a stationary light in the image of the user's face. This can usually be distinguished
by a small bright region surrounded by a very dark region. The reflection's location is used
to limit the search for the eye in the next frame. A window surrounding the reflection is
extracted; the image of the eye is located within this window.
To determine the coordinates of the point the user is looking at, the pixels of the extracted
window are used as the inputs to the artificial neural network. The forward pass is simulated in the ANN, and the coordinates of the gaze are determined by reading the output
units. The output units are organized with 50 output units for specifying the X coordinate,
and 50 units for the Y coordinate. A gaussian output representation, similar to that used in
the ALVINN autonomous road following system [Pomerleau, 1993], is used for the X and
Y axis output units. Gaussian encoding represents the network's response by a Gaussian
shaped activation peak in a vector of output units. The position of the peak within the vector represents the gaze location along either the X or Y axis. The number of hidden units
and the structure of the hidden layer necessary for this task are explored in section 3.
The training data is collected by instructing the user to visually track a moving cursor. The
cursor moves in a predefined path. The image of the eye is digitized, and paired with the
(X,Y) coordinates of the cursor. A total of 2000 image/position pairs are gathered. All of
the networks described in this paper are trained with the same parameters for 260 epochs,
using standard error back propagation. The training procedure is described in greater
Non-Intrusive Gaze Tracking Using Artificial Neural Networks
detail in the next section.
3 THE ARTIFICIAL NEURAL NETWORK IMPLEMENTATION
In designing a gaze tracker, the most important attributes are accuracy and speed. The
need for balancing these attributes arises in deciding the number of connections in the
ANN, the number of hidden units needed, and the resolution of the input image. This section describes several architectures tested, and their respective performances.
3.1 EXAMINING ONLY THE PUPIL AND CORNEA
Many of the traditional gaze trackers look only at a high resolution picture of the subject's
pupil and cornea. Although we use low resolution images, our first attempt also only used
an image of the pupil and cornea as the input to the ANN. Some typical input images are
shown below, in Figure 2(a). The size of the images is 15x15 pixels. The ANN architecture used is shown in Figure 2(b). This architecture was used with varying numbers of hidden units in the single, divided, hidden layer; experiments with 10, 16 and 20 hidden units
were performed.
As mentioned before, 2000 image/position pairs were gathered for training. The cursor
automatically moved in a zig-zag motion horizontally across the screen, while the user
visually tracked the cursor. In addition, 2000 image/position pairs were also gathered for
testing. These pairs were gathered while the user tracked the cursor as it followed a vertical zig-zag path across the screen. The results reported in this paper, unless noted otherwise, were all measured on the 2000 testing points. The results for training the ANN on
the three architectures mentioned above as a function of epochs is shown in Figure 3. Each
line in Figure 3 represents the average of three ANN training trials (with random initial
weights) for each of the two users tested.
Using this system, we were able to reduce the average error to approximately 2.1 degrees,
which corresponds to 0.6 inches at a comfortable sitting distance of approximately 17
inches. In addition to these initial attempts, we have also attempted to use the position of
the cornea within the eye socket to aid in making finer discriminations. These experiments
are described in the next section.
50 X Output Units
50 Y Output Units
15 x 15
Input
Retina
Figure 2: (a-left) 15 x 15 Input to the ANN. Target outputs also shown. (b-right)
the ANN architecture used. A single divided hidden layer is used.
755
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Baluja and Pomerleau
JSdSlmages
to Hidden
iii Hidden
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Figure 3: Error vs. Epochs for the 15x15
images. Errors shown for the 2000
image test set. Each line represents
three ANN trainings per user; two
users are tested.
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210
3.2 USING THE EYE SOCKET FOR ADDITIONAL INFORMATION
In addition to using the information present from the pupil and cornea, it is possible to
gain information about the subject's gaze by analyzing the position of the pupil and cornea
within the eye socket. Two sets of experiments were performed using the expanded eye
image. The first set used the network described in the next section. The second set of
experiments used the same architecture shown in Figure 2(b), with a larger input image
size. A sample image used for training is shown below, in Figure 4.
Figure 4: Image of the pupil and the
eye socket, and the corresponding
target outputs. 15 x 40 input image
shown.
3.2.1. Using a Single Continuous Hidden Layer
One of the remaining issues in creating the ANN to be used for analyzing the position of
the gaze is the structure of the hidden unit layer. In this study, we have limited our exploration of ANN architectures to simple 3 layer feed-forward networks. In the previous
architecture (using 15 x 15 images) the hidden layer was divided into 2 separate parts, one
for predicting the x-axis, and the other for the y-axis. Selecting this architecture over a
fully connected hidden layer makes the assumption that the features needed for accurate
prediction of the x-axis are not related to the features needed for predicting the y-axis. In
this section, this assumption is tested. This section explores a network architecture in
which the hidden layer is fully connected to the inputs and the outputs.
In addition to deciding the architecture of the ANN, it is necessary to decide on the size of
the input images. Several input sizes were attempted, 15x30, 15x40 and 20x40. Surprisingly, the 20x40 input image did not provide the most accuracy. Rather, it was the 15x40
image which gave the best results. Figure 5 provides two charts showing the performance
of the 15x40 and 20x40 image sizes as a function of the number of hidden units and
epochs. The 15x30 graph is not shown due to space restrictions, it can be found in [Baluja
& Pomerleau, 1994]. The accuracy achieved by using the eye socket information, for the
15x40 input images, is better than using only the pupil and cornea; in particular, the 15x40
input retina worked better than both the 15x30 and 20x40.
Non-Intrusive Gaze Tracking Using Artificial Neural Networks
IS x 40 Image
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Figure 5: Performance of 15x40, and 20x40 input image sizes as a function of
epochs and number of hidden units. Each line is the average of 3 runs. Data
points taken every 20 epochs, between 20 and 260 epochs.
3.2.2. Using a Divided Hidden Layer
The final set of experiments which were performed were with 15x40 input images and 3
different hidden unit architectures: 5x2, 8x2 and 10x2. The hidden unit layer was divided
in the manner described in the first network, shown in Figure 2(b). Two experiments were
performed, with the only difference between experiments being the selection of training
and testing images. The first experiment was similar to the experiments described previously. The training and testing images were collected in two different sessions, one in
which the user visually tracked the cursor as it moved horizontally across the screen and
the other in which the cursor moved vertically across the screen. The training of the ANN
was on the "horizontally" collected images, and the testing of the network was on the
"vertically" collected images. In the second experiment, a random sample of 1000 images
from the horizontally collected images and a random sample of 1000 vertically collected
images were used as the training set. The remaining 2000 images from both sets were used
as the testing set. The second method yielded reduced tracking errors. If the images from
only one session were used, the network was not trained to accurately predict gaze position independently of head position. As the two sets of data were collected in two separate
sessions, the head positions from one session to the other would have changed slightly.
Therefore, using both sets should have helped the network in two ways. First, the presentation of different head positions and different head movements should have improved the
ability of the network to generalize. Secondly, the network was tested on images which
were gathered from the same sessions as it was trained. The use of mixed training and testing sets will be explored in more detail in section 3.2.3.
The results of the first and second experiments are presented here, see Figure 6. In order to
compare this architecture with the previous architectures mentioned, it should be noted
that the performance of this architecture, with 10 hidden units, more accurately predicted
gaze location than the architecture mentioned in section 3.2.1, in which a single continuous hidden layer was used. In comparing the performance of the architectures with 16 and
20 hidden units, the performances were very similar. Another valuable feature of using the
757
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Baluja and Pomerleau
divided hidden layer is the reduced number of connections decreases the training and simulation times. This architecture operates at approximately 15hz. with 10 and 16 hidden
units, and slightly slower with 20 hidden units.
Eno,-o.gr... Separate Hidden Layer & 15x40 Image - Test Set 1
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Figure 6: (Left) The average of 2 users with the 15x40 images, and a divided hidden
layer architecture, using test setup #1. (Right) The average performance tested on
5 users, with test setup #2. Each line represents the average of three ANN
trainings per user per hidden unit architecture.
3.2.3. Mixed Training and Testing Sets
It was hypothesized, above, that there are two reasons for the improved performance of a
mixed training and testing set. First, the network ability to generalize is improved, as it is
trained with more than a single head position. Second, the network is tested on images
which are similar, with respect to head position, as those on which it was trained. In this
section, the first hypothesized benefit is examined in greater detail using the experiments
described below.
Four sets of 2000 images were collected. In each set, the user had a different head position
with respect to the camera. The first two sets were collected as previously described. The
first set of 2000 images (horizontal train set 1) was collected by visually tracking the cursor as it made a horizontal path across the screen. The second set (vertical test set 1) was
collected by visually tracking the cursor as it moved in a vertical path across the screen.
For the third and fourth image sets, the camera was moved, and the user was seated in a
different location with respect to the screen than during the collection of the first training
and testing sets. The third set (horizontal train set 2) was again gathered from tracking the
cursor's horizontal path, while the fourth (vertical test set 2) was from the vertical path of
the cursor.
Three tests were performed. In the first test, the ANN was trained using only the 2000
images in horizontal training set 1. In the second test, the network was trained using the
2000 images in horizontal training set 2. In the third test, the network was trained with a
random selection of 1000 images from horizontal training set 1, and a random selection of
1000 images of horizontal training set 2. The performance of these networks was tested on
both of the vertical test sets. The results are reported below, in Figure 7. The last experiment, in which samples were taken from both training sets, provides more accurate results
Non-Intrusive Gaze Tracking Using Artificial Neural Networks
when testing on vertical test set I, than the network trained alone on horizontal training set
1. When testing on vertical test set 2, the combined network performs almost as well as the
network trained only on horizontal training set 2.
These three experiments provide evidence for the network's increased ability to generalize
if sets of images which contain multiple head positions are used for training. These experiments also show the sensitivity of the gaze tracker to movements in the camera; if the
camera is moved between training and testing, the errors in simulation will be large.
Vertil:al Test Set I
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Figure 7: Comparing the performance between networks trained with only one head
position (horizontal train set 1 & 2), and a network trained with both.
4 USING THE GAZE TRACKER
The experiments described to this point have used static test sets which are gathered over
a period of several minutes, and then stored for repeated use. Using the same test set has
been valuable in gauging the performance of different ANN architectures. However, a
useful gaze tracker must produce accurate on-line estimates of gaze location. The use of
an "offset table" can increase the accuracy of on-line gaze prediction. The offset table is a
table of corrections to the output made by a gaze tracker. The network's gaze predictions
for each image are hashed into the 2D offset-table, which performs an additive correction
to the network's prediction. The offset table is filled after the network is fully trained. The
user manually moves and visually tracks the cursor to regions in which the ANN is not
performing accurately. The offset table is updated by subtracting the predicted position of
the cursor from the actual position_This procedure can also be automated, with the cursor
moving in a similar manner to the procedure used for gathering testing and training
images. However, manually moving the cursor can help to concentrate effort on areas
where the ANN is not performing well; thereby reducing the time required for offset table
creation.
With the use of the offset table, the current system works at approximately 15 hz. The best
on-line accuracy we have achieved is 1.5 degrees. Although we have not yet matched the
best gaze tracking systems, which have achieved approximately 0.75 degree accuracy, our
system is non-intrusive, and does not require the expensive hardware which many other
systems require. We have used the gaze tracker in several forms; we have used it as an
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Baluja and Pomerleau
input modality to replace the mouse, as a method of selecting windows in an X-Window
environment, and as a tool to report gaze direction, for human-computer interaction studies.
The gaze tracker is currently trained for 260 epochs, using standard back propagation.
Training the 8x2 hidden layer network using the 15x40 input retina, with 2000 images,
takes approximately 30-40 minutes on a Sun SPARC 10 machine.
5 CONCLUSIONS
We have created a non-intrusive gaze tracking system which is based upon a simple ANN.
Unlike other gaze-tracking systems which employ more traditional vision techniques,
such as a edge detection and circle fitting, this system develops its own features for successfully completing the task. The system's average on-line accuracy is 1.7 degrees. It has
successfully been used in HCI studies and as an input device. Potential extensions to the
system, to achieve head-position and user independence, are presented in [Baluja &
Pomerleau, 1994].
Acknowledgments
The authors would like to gratefully acknowledge the help of Kaari Flagstad, Tammy
Carter, Greg Nelson, and Ulrike Harke for letting us scrutinize their eyes, and being "willing" subjects. Profuse thanks are also due to Henry Rowley for aid in revising this paper.
Shumeet Baluja is supported by a National Science Foundation Graduate Fellowship. This
research was supported by the Department of the Navy, Office of Naval Research under
Grant No. NOO014-93-1-0806. The views and conclusions contained in this document are
those of the authors and should not be interpreted as representing the official policies,
either expressed or implied, of the National Science Foundation, ONR, or the U.S. government.
References
Baluja, S. Pomerleau, D.A. (1994) "Non-Intrusive Gaze Tracking Using Artificial Neural Networks"
CMU-CS-94.
Jochem, T.M., D.A. Pomerleau, C.E. Thorpe (1993), "MANIAC: A Next Generation Neurally
Based Autonomous Road Follower". In Proceedings of the International Conference on Intelligent
Autonomous Systems (IAS-3).
Nodine, c.P., H.L. Kundel, L.c. Toto & E.A. Krupinksi (1992) "Recording and analyzing eye-position data using a microcomputer workstation", Behavior Research Methods, Instruments & Computers 24 (3) 475-584.
Pomerleau, D.A. (1991) "Efficient Training of Artificial Neural Networks for Autonomous Navigation," Neural Computation 3: I, Terrence Sejnowski (Ed).
Pomerleau, D.A. (1993) Neural Network Perception for Mobile Robot Guidance. Kluwer Academic
Publishing.
Pomerleau, D.A. (1993) "Input Reconstruction Reliability Estimation", Neural Information Processing Systems 5. Hanson, Cowan, Giles (eds.) Morgan Kaufmann, pp. 270-286.
Starker, I. & R. Bolt (1990) "A Gaze-Responsive Self Disclosing Display", In CHI-90. Addison
Wesley, Seattle, Washington.
Waibel, A., Sawai, H. & Shikano, K. (1990) "Consonant Recognition by Modular Construction of
Large Phonemic Time-Delay Neural Networks". Readings in Speech Recognition. Waibel and Lee.
Ware, C. & Mikaelian, H. (1987) "An Evaluation of an Eye Tracker as a Device for Computer
Input", In 1. Carrol and P. Tanner (ed.) Human Factors in Computing Systems -IV. Elsevier.
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total:1 pas:1 attempted:2 zag:2 arises:1 tested:8 |
7,095 | 864 | Digital Boltzmann VLSI for
constraint satisfaction and learning
Michael Murray t
Ming-Tak Leung t
Kan Boonyanit t
Kong Kritayakirana t
James B. Burrt*
Gregory J. Wolff+
Takahiro Watanabe+
Edward Schwartz+
David G. Storktt
Allen M. Petersont
t Department of Electrical Engineering
Stanford University
Stanford, CA 94305-4055
+Ricoh California Research Center
2882 Sand Hill Road Suite 115
Menlo Park, CA 94025-7022
and
*Sun Mlcrosystems
.
2550 Garcia Ave., MTV-29, room 203
Mountain View, CA 94043
Abstract
We built a high-speed, digital mean-field Boltzmann chip and SBus
board for general problems in constraint satjsfaction and learning.
Each chip has 32 neural processors and 4 weight update processors,
supporting an arbitrary topology of up to 160 functional neurons.
On-chip learning is at a theoretical maximum rate of 3.5 x 108 connection updates/sec; recall is 12000 patterns/sec for typical conditions. The chip's high speed is due to parallel computation of inner
products, limited (but adequate) precision for weights and activations (5 bits), fast clock (125 MHz), and several design insights.
896
Digital Boltzmann VLSI for Constraint Satisfaction and Learning
1
INTRODUCTION
A vast number of important problems can be cast into a form of constraint satisfaction. A crucial difficulty when solving such problems is the fact that there are local
minima in the solution space, and hence simple gradient descent methods rarely suffice. Simulated annealing via the Boltzmann algorithm (BA) is attractive because it
can avoid local minima better than many other methods (Aarts and Korst, 1989).
It is well known that the problem of learning also generally has local minima in
weight (parameter) space; a Boltzmann algorithm has been developed for learning
which is effective at avoiding local minima (Ackley and Hinton, 1985). The BA
has not received extensive attention, however, in part because of its slow operation
which is due to the annealing stages in which the network is allowed to slowly relax
into a state of low error. Consequently there is a great need for fast and efficient
special purpose VLSI hardware for implementing the algorithm. Analog Boltzmann
chips have been described by Alspector, Jayakumar and Luna (1992) and by Arima
et al. (1990); both implement stochastic BA. Our digital chip is the first to implement the deterministic mean field BA algorithm (Hinton, 1989), and although its
raw throughput is somewhat lower than the analog chips just mentioned, ours has
unique benefits in capacity, ease of interfacing and scalability (Burr, 1991, 1992).
2
BOLTZMANN THEORY
The problems of constraint satisfaction and of learning are unified through the
Boltzmann learning algorithm. Given a partial pattern and a set of constraints,
the BA completes the pattern by means of annealing (gradually lowering a computational "temperature" until the lowest energy state is found) - an example
of constraint satisfaction. Over a set of training patterns, the learning algorithm
modifies the constraints to model the relationships in the data.
2.1
CONSTRAINT SATISFACTION
A general constraint satisfaction problem over variables Xi (e.g., neural activations)
is to find the set Xi that minimize a global energy function E = -~ Lij WijXiXj,
where Wij are the (symmetric) connection weights between neurons i and j and
represent the problem constraints.
There are two versions of the BA approach to minimizing E. In one version - the
stochastic BA - each binary neuron Xi E {-I, I} is polled randomly, independently
and repeatedly, and its state is given a candidate perturbation. The probability of
acceptance of this perturbation depends upon the amount of the energy change
and the temperature. Early in the annealing schedule (Le., at high temperature)
the probability of acceptance is nearly independent of the change in energy; late in
annealing (Le., at low temperature), candidate changes that lead to lower energy
are accepted with higher probability.
In the deterministic mean field BA, each continuous valued neuron (-1 < Xi ::;
1) is updated simultaneously and in parallel, its new activation is set to Xi =
I(Lj WijXj), where 10 is a monotonic non-linearity, typically a sigmoid which
corresponds to a stochastic unit at a given temperature (assuming independent
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Murray, Leung, Boonyanit, Kritayakirana, Burr, Wolff, Watanabe, Schwartz, Stork, and Peterson
inputs). The inverse slope of the non-linearity is proportional to the temperature; at
the end of the anneal the slope is very high and f (.) is effectively a step function. It
has been shown that if certain non-restrictive assump'tions hold, and if the annealing
schedule is sufficiently slow, then the final binary states (at 0 temperature) will be
those of minimum E (Hinton, 1989, Peterson and Hartman, 1989).
2.2
LEARNING
The problem of Boltzmann learning is the following: given a network topology
of input and output neurons, interconnected by hidden neurons, and given a set of
training patterns (input and desired output), find a set of weights that leads to high
probability of a desired output activations for the corresponding input activations.
In the Boltzmann algorithm such learning is achieved using two main phases the Teacher phase and the Student phase - followed by the actual Weight update.
During the Teacher phase the network is annealed with the inputs and outputs
clamped (held at the values provided by the omniscient teacher). During the anneal
of the Student phase, only the inputs are clamped - the outputs are allowed to
vary. The weights are updated according to:
D..Wij =
?( (x!x;) - (x:xj))
(1)
where ? is a learning rate and (x~x;) the coactivations of neurons i and j at the end
of the Teacher phase and (x:xj) in at the end of the Student phase (Ackley and
Hinton, 1985). Hinton (1989) has shown that Eq. 1 effectively performs gradient
descent on the cross-entropy distance between the probability of a state in the
Teacher (clamped) and the Student (free-running) phases.
Recent simulations by Galland (1993) have shown limitations of the deterministic
BA for learning in networks having hidden units directly connected to other hidden
units. While his results do not cast doubt on the deterministic BA for constraint
satisfaction, they do imply that the deterministic BA for learning is most successful
in networks of a single hidden layer. Fortunately, with enough hidden units this
topology has the expressive power to represent all but the most pathological inputoutput mappings.
3
FUNCTIONAL DESIGN AND CHIP OPERATION
Figure 1 shows the functional block diagram of our chip. The most important units
are the Weight memory, Neural processors, Weight update processors, Sigmoid and
Rotating Activation Storage (RAS), and their operation are best explained in terms
of constraint satisfaction and learning.
3.1
CONSTRAINT SATISFACTION
For constraint satisfaction, the weights (constraints) are loaded into the Weight
memory, the form of the transfer function is loaded into the Sigmoid Unit, and
the values and duration of the annealing temperatures (the annealing schedule) are
loaded into the Temperature Unit. Then an input pattern is loaded into a bank
of the RAS to be annealed. Such an anneal occurs as follows: At an initial high
Digital Boltzmann VLSI for Constraint Satisfaction and Learning
temperature, the 32 Neural processors compute Xi = Lj WijXj in parallel for the
hidden units. A 4 x multiplexing here permits networks of up to 128 neurons to
be annealed, with the remaining 32 neurons used as (non-annealed) inputs. Thus
our chip supports networks of up to 160 neurons total. These activations are then
stored in the Neural Processor Latch and then passed sequentially to the Sigmoid
unit, where they are multiplied by the reciprocal of the instantaneous temperature.
This Sigmoid unit employs a lookup table to convert the inputs to neural outputs
by means of non-linearity f(?). These outputs are sequentially loaded back into the
activation store. The temperature is lowered (according to the annealing schedule), and the new activations are calculated as before, and so on. The final set of
activations Xi (i.e., at the lowest temperature) represent the solution.
r-----t.... 4
Rotating
Activation
weight update processors
weight update cache
Weight
memory
1
32 Neural Processors (NP)
Sigmoid
Figure 1: Boltzmann VLSI block diagram. The rotating activation storage (black)
consists of three banks, which for learning problems contain the last pattern (already annealed), the current pattern (being annealed) and the next pattern (to be
annealed) read onto the chip through the external interface.
3.2
LEARNING
When the chip is used for learning, the weight memory is initialized with random
weights and the first, second and third training patterns are loaded into the RAS.
The three-bank RAS is crucial for our chip's speed because it allows a three-fold
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Murray, Leung, Boonyanit, Kritayaldrana, Burr, Wolff, Watanabe, Schwartz, Stork, and Peterson
concurrency: 1) a current pattern of activations is annealed, while 2) the annealed
last pattern is used to update the weights, while 3) the next pattern is being loaded
from off-chip. The three banks form a circular buffer, each with a Student and a
Teacher activation store.
During the Teacher anneal phase (for the current pattern), activations of the input
and output neurons are held at the values given by the teacher, and the values of
the hidden units found by annealing (as described in the previous subsection). After
the last such annealling step (Le., at the lowest temperature), the final activations
are left in the Teacher activation store - the Teacher phase is then complete. The
annealing schedule is then reset to its initial temperature, and the above process is
then repeated for the Student phase; here only the input activations are clamped
to their values and the outputs are free to vary. At the end of this Student anneal,
the final activations are left in the Student activation storage.
In steady state, the MUX then rotates the storage banks of the RAS such that the
next, current, and last banks are now called the current, last, and next, respectively.
To update the weights, the activations in the Student and Teacher storage bank
for the pattern just annealed (now called the "last" pattern) are sent to the four
Weight update processors, along with the weights themselves. The Weight update
processors compute the updated weights according to Eq. 1, and write them back
to the Weight memory. While such weight update is occuring for the last pattern,
the current pattern is annealing and the next pattern is being loaded from off chip.
After the chip has been trained with all of the patterns, it is ready for use in
recall. During recall, a test pattern is loaded to the input units of an activation
bank (Student side), the machine performs a Student anneal and the final output
activations are placed in the Student activation store, then read off the chip to
the host computer as the result. In a constraint satisfaction problem, we merely
download the weights (constraints) and perform a Student anneal.
4
HARDWARE IMPLEMENTATION
Figure 2 shows the chip die. The four main blocks of the Weight memory are at
the top, surrounded by 32 Neural processors (above and below this memory), and
four Weight update processors (between the memory banks). The three banks of
the Rotating Activation Store are at the bottom of the chip. The Sigmoid processor
is at the lower left, and instruction cache and external interface at the lower right.
Most of the rest of the chip consists of clocking and control circuitry.
4.1
VLSI
The chip mixes dynamic and static memory on the same die. The Activation and
Temperature memories are static RAM (which needs no refresh circuitry) while the
Weight memory is dynamic (for area efficiency) . The system clock is distributed to
various local clock drivers in order to reduce the global clock capacitance and to selectively disable the clocks in inactive subsystems for reducing power consumption.
Each functional block has its own finite state machine control which communicates
Digital Boltzmann VLSI for Constraint Satisfaction and Learning
.. "
._ ? ...- .
.. ....
? . -,
"o.t ' .
. IM
.... . .
'7 ","",
Figure 2: Boltzmann VLSI chip die.
asynchronously. For diagnostic purposes, the State Machines and counters are observable through the External Interface. There is a Single Step mode which has
been very useful in verifying sub-system performance. Figure 3 shows the power
dissipation throughout a range of frequencies. Note that the power is less than
2 Watts throughout.
Extensive testing of the first silicon revealed two main classes of chip error: electrical
and circuit. Most of the electrical problems can be traced to fast edge rates on
the DRAM sense-amp equalization control signals, which cause inductive voltage
transients on the power supply rails of roughly 1 Volt. This appears to be at least
partly responsible for the occasional loss of data in dynamic storage nodes. There
also seems to be insufficient latchup protection in the pads, which is aggravated by
the on-chip voltage surges. The circuit problems can be traced to having to modify
the circuits used in the layout for full chip simulation.
In light of these problems, we have simulated the circuit in great detail in order to
explore possible corrective steps. We have modified the design to provide improved
electrical isolation, resized drivers and reduced the logic depth in several components. These corrections solve the problems in simulation, and give us confidence
that the next fab run will yield a fully working chip.
4.2
BOARD AND SBus INTERFACE
An SBus interface board was developed to allow the Boltzmann chip to be used
with a SparcStation host. The registers and memory in the chip can be memory
mapped so that they are directly accessible to user software. The board can support
901
902
Murray, Leung, Boonyanit, Kritayakirana, Burr, Wolff, Watanabe, Schwartz, Stork, and Peterson
Table 1: Boltzmann VLSI chip specifications
Architecture
Size
Neurons
Weight memory
Activation store
Technology
Transistors
Pins
Clock
I/O rate
Learning rate
Recall rate
Power dissipation
n-Iayer, arbitrary intercoItnnections
9.5 mm x 9.8 mm
32 processors --+ 160 virtual
20,480 5-bit weights (on chip)
3 banks, 160 teacher & 160 student values in each
1. 211m CMOS
400,000
84
125 MHz (on chip)
3 x 107 activations/sec (sustained)
3.5 x 108 connection updates/sec (on chip)
12000 patterns/sec
:::;2 Watts (see Figure 3)
20-bit transfers to the chip at a sustained rate in excess of 8 Mbytes/second. The
board uses reconfigurable Xilinx FPGAs (field-programmable gate arrays) to allow
flexibility for testing with and without the chip installed.
4.3
SOFTWARE
The chip control program is written in C (roughly 1,500 lines of code) and communicates to the Boltzmann interface card through the virtual memory. The user can
read/write to all activation and weight memory locations and all functions of the
chip (learning, recall, annealing, etc.) can thus be specified in software.
5
CONCLUSIONS AND FUTURE WORK
The chip was designed so that interchip communications could be easily incorporated by means of high-speed parallel busses. The SBus board, interface and software described above will require only minor changes to incorporate a multi-chip
module (MCM) containing several such chips (for instance 16). There is minimal
,--
2
1. 75
til
.w 1.5
.w 1. 25
111
2:
1
~
Q)
0.75
~
0
0.5
0.
0.25
0
i
i,
I
f--T;
!
i
-
,
----
i
,i
I
i
i
I
i
,
i
50
60
70
80
90
I
I
100 110
frequency, MHz
Figure 3: Power dissipation of the chip during full operation at 5 Volts.
Digital Boltzmann VLSI for Constraint Satisfaction and Learning
inter chip communication delay ? 3% overhead), and thus MCM versions of our
system promise to be extremely powerful learning systems for large neural network
problems (Murrayet al., 1992).
Acknowledgements
Thanks to Martin Boliek and Donald Wynn for assistance in design and construction of the SBus board. Research support by NASA through grant NAGW419 is
gratefully acknowledged; VLSI fabrication by MOSIS. Send reprint requests to Dr.
Stor k: stor [email protected].
References
E. Aarts & J. Korst. (1989) Simulated Annealing and Boltzmann Machines: A
stochastic approach to combinatorial optimization and neural computing. New York:
Wiley.
D. H. Ackley & G. E. Hinton. (1985) A learning algorithm for Boltzmann machines.
Cognitive Science 9, 147-169.
J. Alspector, A. Jayakumar & S. Luna. (1992) ExpeJimental evaluation of learning
in a neural microsystem. Advances in Neural Information Processing Systems-4,
J. E. Moody, S. J. Hanson & R. P. Lippmann (eds.), San Mateo, CA: Morgan
Kaufmann, 871-878.
Y. Arima, K. Mashiko, K. Okada, T. Yamada, A. Maeda, H. Kondoh & S. Kayano.
(1990) A self-learning neural network chip with 125 neurons and 10K selforganization synapses. In Symposium on VLSI Circuits, Solid State Circuits Council
Staff, Los Alamitos, CA: IEEE Press, 63-64.
J. B. Burr. (1991) Digital Neural Network Implementations. Neural Networks:
Concepts, Applications, and Implementations, Volume 2, P. Antognetti & V. Milutinovic (eds.) 237-285, Englewood Cliffs, NJ: Prentice Hall.
J. B. Burr. (1992) Digital Neurochip Design. Digital Parallel Implementations of
Neural Networks. K. Wojtek Przytula & Viktor K. Prasanna (eds.), Englewood
Cliffs, N J: Prentice Hall.
C. C. Galland. (1993) The limitations of deterministic Boltzmann machine learning.
Network 4, 355-379.
G. E. Hinton. (1989) Deterministic Boltzmann learning performs steepest descent
in weight-space. Neural Computation 1, 143-150.
C. Peterson & E. Hartman. (1989) Explorations of the mean field theory learning
algorithm. Neural Networks 2, 475-494.
M. Murray, J. B. Burr, D. G. Stork, M.-T. Leung, K. Boonyanit, G. J. Wolff
& A. M. Peterson. (1992) Deterministic Boltzmann machine VLSI can be scaled
using multi-chip modules. Proc. of the International Conference on Application
Specific Array Processors. Berkeley, CA (August 4-7) Los Alamitos, CA: IEEE
Press, 206-217.
903
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7,096 | 865 | Monte Carlo Matrix Inversion and
Reinforcement Learning
Andrew Barto and Michael Duff
Computer Science Department
University of Massachusetts
Amherst, MA 01003
Abstract
We describe the relationship between certain reinforcement learning (RL) methods based on dynamic programming (DP) and a class
of unorthodox Monte Carlo methods for solving systems of linear
equations proposed in the 1950's. These methods recast the solution of the linear system as the expected value of a statistic suitably
defined over sample paths of a Markov chain. The significance of
our observations lies in arguments (Curtiss, 1954) that these Monte
Carlo methods scale better with respect to state-space size than do
standard, iterative techniques for solving systems of linear equations. This analysis also establishes convergence rate estimates.
Because methods used in RL systems for approximating the evaluation function of a fixed control policy also approximate solutions
to systems of linear equations, the connection to these Monte Carlo
methods establishes that algorithms very similar to TD algorithms
(Sutton, 1988) are asymptotically more efficient in a precise sense
than other methods for evaluating policies. Further, all DP-based
RL methods have some of the properties of these Monte Carlo algorithms, which suggests that although RL is often perceived to
be slow, for sufficiently large problems, it may in fact be more efficient than other known classes of methods capable of producing
the same results.
687
688
Barto and Duff
1
Introduction
Consider a system whose dynamics are described by a finite state Markov chain with
transition matrix P, and suppose that at each time step, in addition to making a
transition from state Xt = i to XHI = j with probability Pij, the system produces
a randomly determined reward, rt+1! whose expected value is R;. The evaluation
junction, V, maps states to their expected, infinite-horizon discounted returns:
It is well known that V uniquely satifies a linear system of equations describing
local consistency:
V = R + -yPV,
or
(I - -yP)V = R.
( 1)
The problem of computing or estimating V is interesting and important in its
own right, but perhaps more significantly, it arises as a (rather computationallyburdensome) step in certain techniques for solving Markov Decision Problems. In
each iteration of Policy-Iteration (Howard, 1960), for example, one must determine
the evaluation function associated with some fixed control policy, a policy that
improves with each iteration.
Methods for solving (1) include standard iterative techniques and their variantssuccessive approximation (Jacobi or Gauss-Seidel versions), successive overrelaxation, etc. They also include some of the algorithms used in reinforcement
learning (RL) systems, such as the family of TD algorithms (Sutton, 1988). Here
we describe the relationship between the latter methods and a class of unorthodox
Monte Carlo methods for solving systems of linear equations proposed in the 1950's.
These methods recast the solution of the linear system as the expected value of a
statistic suitably defined over sample paths of a Markov chain.
The significance of our observations lies in arguments (Curtiss, 1954) that these
Monte Carlo methods scale better with respect to state-space size than do standard, iterative techniques for solving systems of linear equations. This analysis also
establishes convergence rate estimates. Applying this analysis to particular members of the family of TD algorithms (Sutton, 1988) provides insight into the scaling
properties of the TD family as a whole and the reasons that TD methods can be
effective for problems with very large state sets, such as in the backgammon player
of Tesauro (Tesauro, 1992).
Further, all DP-based RL methods have some of the properties of these Monte
Carlo algorithms, which suggests that although RL is often slow, for large problems
(Markov Decision Problems with large numbers of states) it is in fact far more practical than other known methods capable of producing the same results. First, like
many RL methods, the Monte Carlo algorithms do not require explicit knowledge
of the transition matrix, P. Second, unlike standard methods for solving systems
of linear equations, the Monte Carlo algorithms can approximate the solution for
some variables without expending the computational effort required to approximate
Monte Carlo Matrix Inversion and Reinforcement Learning
the solution for all of the variables. In this respect, they are similar to DP-based
RL algorithms that approximate solutions to Markovian decision processes through
repeated trials of simulated or actual control, thus tending to focus computation
onto regions of the state space that are likely to be relevant in actual control (Barto
et. al., 1991).
This paper begins with a condensed summary of Monte Carlo algorithms for solving systems of linear equations. We show that for the problem of determining an
evaluation function, they reduce to simple, practical implementations. Next, we
recall arguments (Curtiss, 1954) regarding the scaling properties of Monte Carlo
methods compared to iterative methods. Finally, we conclude with a discussion of
the implications of the Monte Carlo technique for certain algorithms useful in RL
systems.
2
Monte Carlo Methods for Solving Systems of Linear
Equations
The Monte Carlo approach may be motivated by considering the statistical evaluation of a simple sum, I:k ak. If {Pk} denotes a set of values for a probability mass
function that is arbitrary (save for the requirement that ak =P 0 imply Pk =P 0), then
I:k ak = I:k (~) Pk,
which may be interpreted as the expected value of a random
variable Z defined by Pr { Z
= ~ } = Pk.
From equation (1) and the Neumann series representation of the inverse it is is clear
that
V
(1 - -yp)-l R R + -yP R + -y2 p2 R + ...
whose ith component is
=
=
Vi = R; + -y
L P"l R;l + -y2 L P"lP'1'2 R;2 + ...
. . . + -yk
L
Pii 1 ... P,/o-li/oR;/o
+ ...
(2)
and it is this series that we wish to evaluate by statistical means.
A technique originated by Ulam and von-Neumann (Forsythe & Leibler, 1950) utilizes an arbitrarily defined Markov chain with transition matrix P and state set
{I, 2, "., n} (V is assumed to have n components). The chain begins in state i and
is allowed to make k transitions, where k is drawn from a geometric distribution
with parameter Pdep; i.e., Pr{k state transitions} = P~tep(1 - P,tep)' The Markov
chain, governed by P and the geometrically-distributed stopping criterion, defines
a mass function assigning probability to every trajectory of every length starting in
state i, Xo = io = i --+ Zl = i l --+ ... --+ Zk = ik, and to each such trajectory there
corresponds a unique term in the sum (2).
For the
cas/~
of value estimation, "Z" is defined by
689
690
Barto and Duff
which for j> = P and
P,tep
Pr
= 'Y becomes
{z = 1~"
k
= 'Yk(1 -
}
'Y)
'Y
IT Pij_li;-
;=1
The sample average of sampled values of Z is guaranteed to converge (as the number
of samples grows large) to state i's expected, infinite-horizon discounted return.
In Wasow's method (Wasow, 1952), the truncated Neumann series
~ = R; + 'Y LPiilR;l + 'Y2 LPii l Pi l i 2R;2 + ... + 'YN L
Pii l ?? ?PiN_liNR;N
is expressed as R; plus the expected value of the sum of N random variables
ZlI Z2, ... , ZN, the intention being that
E(Zk) = 'Yk
L
PihPi l i2" ?pi"_d,,R;,,?
i 1 ???i"
Let trajectories of length N be generated by the Markov chain governed by P. A
given term 'Y"Pii 1Pi li 2 ?? 'Pi"_li"R;" is associated with all trajectories i -+ i1 -+ i2 -+
... -+ i k -+ i k +1 -+ ... -+ iN whose first k + 1 states are i, ill ... , i k . The measure
of this set of trajectories is just Pii 1Pi l i 2 ... Pi"_li". Thus, the random variables Zk,
k = 1, N are defined by
If P = P, then the estimate becomes an average of sample truncated, discounted
returns: ~ = R; + 'YR;1 + 'Y2 R;.2 + ... + 'YN R;N.
The Ulam/von Neumann approach may be reconciled with that of Wasow by processing a given trajectory a posteriori, converting it into a set of terminated paths
consistent with any choice of stopping-state transition probabilities. For example,
for a stopping state transition probability of 1 - 'Y, a path of length k has probability 'Yk(1 - 'Y). Each "prefix" of the observed path x(O) -+ x(1) -+ z(2) -+ ... can
be weighted by the probability of a path of corresponding length, resulting in an
estimate, V, that is the sampled, discounted return:
00
V =
L
-rk RZ(k).
k=O
3
Complexity
In (Curtiss, 1954) Curtiss establishes a theoretical comparison of the complexity
(number of multiplications) required by the Ulam/von Neumann method and a
stationary linear iterative process for computing a single component of the solution
to a system of linear equations. Curtiss develops an analytic formula for bounds
on the conditional mean and variance of the Monte-Carlo sample estimate, V, and
mean and variance of a sample path's time to absorption, then appeals to the
Monte Carlo Matrix Inversion and Reinforcement Learning
n 1000
900
800
700
600
500
)"=.5
),,=.7
)"=.9
400
300
200
100
O~----------~----~--~--~--~
a 100 200 300 400 500 600 700 800 900 1000
1/~
Figure 1: Break-even size of state space versus accuracy.
Central Limit Theorem to establish a 95%-confidence interval for the complexity of
his method to reduce the initial error by a given factor,
1
e.
For the case of. value-estimation, Curtiss' formula for the Monte-Carlo complexity
may be written as
WORKMonte-Carlo
=
1~ "'; (1 + e
(3)
22 ) .
This is compared to the complexity of the iterative method, which for the valueestimation problem takes the form of the classical dynamic programming recursion,
v(n+l) = R + ",;pv(n):
WORKiterati'lle
lOge) 2
= ( 1 + log",; n
+ n.
The iterative methodts complexity has the form an 2 + n, with a > It while the
Monte-Carlo complexity is independent of n-it is most sensitive to the amount of
error reduction desired, signified bye. Thus, given a fixed amount of computation,
for large enough n, the Monte-Carlo method is likely (with 95% confidence level) to
produce better estimates. The theoretical "break-even" points are plotted in Figure
It and Figure 2 plots work versus state-space size for example values of",; and
e.
IThat is, for the iterative method, e is defined via IIV(oo) - yen) II
while for the Monte Carlo method, e is defined via IV(OD)(i) - VMI
where VM is the average over M sample V's.
< eIlV(oo)
< eIlV(OD)
- yeO) II,
- V(O)II,
691
692
Barto and Duff
.::&.50000
....
o
I
I
~45000
I
I
40000~------------~/------~-------I
35000
I
30000
I
I
25000
,
I
20000
15000
Iterative
Monte Carlo
Gauss
10000
5000
O~~--~~~~--~~--~--~~--~
o
10
20
30
40
50
60
70
80
90 100
n
Figure 2: Work versus number of states for"Y = .5 and
4
e=
.01.
Discussion
It was noted that the analytic complexity Curtiss develops is for the work required
to compute one component of a solution vector. In the worst case, all components
could be estimated by constructing n separate, independent estimators. This would
multiply the Monte-Carlo complexity by a factor of n, and its scaling supremacy
would be only marginally preserved. A more efficient approach would utilize data
obtained in the course of estimating one component to estimate other components
as well; Rubinstein (Rubinstein, 1981) decribes one way of doing this, using the
notion of "covering paths." Also, it should be mentioned that substituting more
sophisticated iterative methods, such as Gauss-Seidel, in place of the simple successive approximation scheme considered here, serves only to improve the condition
number of the underlying iterative operator-the amount of computation required
by iterative methods remains an 2 + n, for some a> 1.
An attractive feature of the the analysis provided by Curtiss is that, in effect, it
yields information regarding the convergence rate of the method; that is, Equation
4 can be re-arranged in terms of
Figure 3 plots versus work for example values
of"Y and n.
e.
e
The simple Monte Carlo scheme considered here is practically identical to the
limiting case of TD-A with A equal to one (TD-l differs in that its averaging of
sampled, discounted returns is weighted with recency). Ongoing work (Duff) explores the connection between TD-A (Sutton, 1988), for general values of A, and
Monte Carlo methods augmented by certain variance reduction techniques. Also,
Barnard (Barnard) has noted that TD-O may be viewed as a stochastic approxima-
Monte Carlo Matrix Inversion and Reinforcement Learning
~ 1.0
...
0.9
0.8
Iterative
Monte Carlo
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
0
10000
20000
30000
40000
50000
Work
Figure 3: Error reduction versus work for
"y
= .9 and n = 100.
tion method for solving (1).
On-line RL methods for solving Markov Decision Problems, such as Real-Time
Dynamic Programming (RTDP)(Barto et. al., 1991), share key features with the
Monte Carlo method. As with many algorithms, RTDP does not require explicit
knowledge of the transition matrix, P, and neither, of course, do the Monte Carlo
algorithms. RTDP approximates solutions to Markov Decision Problems through
repeated trials of simulated or actual control, focusing computation upon regions of
the state space likely to be relevant in actual control. This computational "focusing"
is also a feature of the Monte Carlo algorithms. While it is true that a focusing
of sorts is exhibited by Monte Carlo algorithms in an obvious way by virtue of
the fact that they can compute approximate solutions for single components of
solution vectors without exerting the computational labor required to compute all
solution components, a more subtle form of computational focusing also occurs.
Some of the terms in the Neumann series (2) may be very unimportant and need
not be represented in the statistical estimator at all. The Monte Carlo method's
stochastic estimation process achieves this automatically by, in effect, making the
appearance of the representative of a non-essential term a very rare event.
These correspondences-between TD-O and stochastic approximation, between TD). and Monte Carlo methods with variance reduction, between DP-based RL algorithms for solving Markov Decision Problems and Monte Carlo algorithms together with the comparatively favorable scaling and convergence properties enjoyed by the simple Monte Carlo method discussed in this paper, suggest that DPbased RL methods like TD/stochastic-approximation or RTDP, though perceived
to be slow, may actually be advantageous for problems having a sufficiently large
693
694
Barto and Duff
number of states.
Acknowledgement
This material is based upon work supported by the National Science Foundation
under Grant ECS-9214866.
References
E. Barnard.
publication.
Temporal-Difference Methods and Markov Models.
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A. Barto, S. Bradtke, & S. Singh. (1991) Real-Time Learning and Control Using
Asynchronous Dynamic Programming. Computer Science Department, University
of Massachusetts, Tech. Rept. 91-57.
1. Curtiss. (1954) A Theoretical Comparison of the Efficiencies of Two Classical
Methods and a Monte Carlo Method for Computing One Component of the Solution
of a Set of Linear Algebraic Equations. In H. A. Mayer (ed.), Symposium on Monte
Carlo Methods, 191-233. New york, NY: Wiley.
M. Duff. A Control Variate Perspective for the Optimal Weighting of Truncated,
Corrected Returns. In Preparation.
S. Forsythe & R. Leibler. (1950) Matrix Inversion by a Monte Carlo Method. Math.
Tables Other Aids Comput., 4:127-129.
R. Howard. (1960) Dynamic Programming and Markov Proceses. Cambridge, MA:
MIT Press.
R. Rubinstein. (1981) Simulation and the Monte Carlo Method. New York, NY:
Wiley.
R. Sutton. (1988) Learning to Predict by the Method of Temporal Differences.
Machine Learning 3:9-44.
G. Tesauro. (1992) Practical Issues in Temporal Difference Learning.
Learning 8:257-277.
Machine
W. Wasow. (1952) A Note on the Inversion of Matrices by Random Walks. Math.
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| 865 |@word trial:2 version:1 inversion:6 advantageous:1 suitably:2 simulation:1 reduction:4 initial:1 series:4 prefix:1 z2:1 od:2 assigning:1 must:1 written:1 analytic:2 plot:2 stationary:1 yr:1 ith:1 provides:1 math:2 successive:2 symposium:1 ik:1 expected:7 discounted:5 td:12 automatically:1 actual:4 considering:1 becomes:2 begin:2 estimating:2 underlying:1 provided:1 mass:2 interpreted:1 rtdp:4 temporal:3 every:2 control:8 zl:1 grant:1 yn:2 producing:2 local:1 rept:1 limit:1 io:1 sutton:5 ak:3 path:8 plus:1 suggests:2 practical:3 unique:1 differs:1 significantly:1 intention:1 confidence:2 suggest:1 onto:1 operator:1 recency:1 applying:1 map:1 starting:1 insight:1 estimator:2 his:1 notion:1 limiting:1 suppose:1 programming:5 observed:1 worst:1 region:2 yk:4 mentioned:1 complexity:9 reward:1 dynamic:6 iiv:1 singh:1 solving:12 upon:2 efficiency:1 represented:1 describe:2 effective:1 monte:40 rubinstein:3 whose:4 statistic:2 relevant:2 convergence:4 requirement:1 neumann:6 ulam:3 produce:2 oo:2 andrew:1 approxima:1 p2:1 proceses:1 pii:4 stochastic:4 material:1 require:2 absorption:1 practically:1 sufficiently:2 considered:2 predict:1 substituting:1 achieves:1 perceived:2 estimation:3 favorable:1 condensed:1 sensitive:1 pdep:1 establishes:4 weighted:2 mit:1 forsythe:2 rather:1 barto:8 publication:1 focus:1 backgammon:1 tech:1 sense:1 posteriori:1 stopping:3 signified:1 i1:1 issue:1 ill:1 equal:1 having:1 identical:1 develops:2 xhi:1 randomly:1 national:1 multiply:1 evaluation:5 chain:7 implication:1 capable:2 iv:1 loge:1 desired:1 plotted:1 re:1 walk:1 theoretical:3 markovian:1 zn:1 rare:1 explores:1 amherst:1 vm:1 michael:1 together:1 von:3 central:1 return:6 yp:3 li:4 yeo:1 ithat:1 vi:1 tion:1 break:2 doing:1 sort:1 yen:1 accuracy:1 variance:4 yield:1 marginally:1 carlo:41 trajectory:6 submitted:1 ed:1 obvious:1 associated:2 jacobi:1 sampled:3 massachusetts:2 recall:1 knowledge:2 improves:1 exerting:1 subtle:1 sophisticated:1 actually:1 focusing:4 arranged:1 though:1 just:1 defines:1 perhaps:1 grows:1 effect:2 y2:4 true:1 leibler:2 i2:2 attractive:1 uniquely:1 covering:1 noted:2 criterion:1 tep:3 bradtke:1 tending:1 rl:13 discussed:1 approximates:1 cambridge:1 enjoyed:1 consistency:1 etc:1 own:1 perspective:1 tesauro:3 certain:4 arbitrarily:1 converting:1 determine:1 converge:1 ii:3 expending:1 seidel:2 iteration:3 preserved:1 addition:1 interval:1 unlike:1 exhibited:1 unorthodox:2 member:1 dpbased:1 enough:1 variate:1 reduce:2 regarding:2 motivated:1 effort:1 algebraic:1 york:2 useful:1 clear:1 unimportant:1 amount:3 estimated:1 key:1 drawn:1 neither:1 utilize:1 asymptotically:1 overrelaxation:1 geometrically:1 sum:3 inverse:1 zli:1 place:1 family:3 utilizes:1 decision:6 scaling:4 bound:1 guaranteed:1 correspondence:1 argument:3 department:2 wasow:4 vmi:1 lp:1 making:2 satifies:1 pr:3 xo:1 equation:13 remains:1 describing:1 serf:1 junction:1 save:1 rz:1 denotes:1 include:2 establish:1 approximating:1 classical:2 comparatively:1 occurs:1 rt:1 dp:5 separate:1 simulated:2 reason:1 length:4 relationship:2 implementation:1 policy:5 observation:2 markov:13 howard:2 finite:1 truncated:3 precise:1 duff:7 arbitrary:1 required:5 connection:2 mayer:1 recast:2 event:1 recursion:1 scheme:2 improve:1 imply:1 geometric:1 acknowledgement:1 multiplication:1 determining:1 interesting:1 versus:5 foundation:1 pij:1 consistent:1 pi:6 share:1 course:2 summary:1 supported:1 bye:1 asynchronous:1 lle:1 distributed:1 evaluating:1 transition:9 reinforcement:6 far:1 ec:1 approximate:5 conclude:1 assumed:1 iterative:13 table:2 zk:3 ca:1 curtis:10 constructing:1 significance:2 pk:4 reconciled:1 terminated:1 whole:1 repeated:2 allowed:1 augmented:1 representative:1 slow:3 ny:2 wiley:2 aid:2 originated:1 explicit:2 wish:1 pv:1 comput:2 lie:2 governed:2 weighting:1 rk:1 formula:2 theorem:1 xt:1 appeal:1 virtue:1 essential:1 horizon:2 likely:3 appearance:1 expressed:1 labor:1 corresponds:1 ma:2 conditional:1 viewed:1 barnard:3 determined:1 infinite:2 corrected:1 averaging:1 gauss:3 player:1 latter:1 arises:1 preparation:1 ongoing:1 evaluate:1 |
7,097 | 866 | Two-Dimensional Object Localization by
Coarse-to-Fine Correlation Matching
Chien-Ping Lu and Eric Mjolsness
Department of Computer Science
Yale University
New Haven, CT 06520-8285
Abstract
We present a Mean Field Theory method for locating twodimensional objects that have undergone rigid transformations.
The resulting algorithm is a form of coarse-to-fine correlation
matching. We first consider problems of matching synthetic point
data, and derive a point matching objective function. A tractable
line segment matching objective function is derived by considering
each line segment as a dense collection of points, and approximating it by a sum of Gaussians. The algorithm is tested on real images
from which line segments are extracted and matched.
1
Introduction
Assume that an object in a scene can be viewed as an instance of the model placed
in space by some spatial transformation, and object recognition is achieved by discovering an instance of the model in the scene. Two tightly coupled subproblems
need to be solved for locating and recognizing the model: the correspondence problem (how are scene features put into correspondence with model features?), and the
localization problem (what is the transformation that acceptably relates the model
features to the scene features?). If the correspondence is known, the transformation
can be determined easily by least squares procedures. Similarly, for known transformation, the correspondence can be found by aligning the model with the scene,
or the problem becomes an assignment problem if the scene feature locations are
jittered by noise.
985
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Lu and Mjolsness
Several approaches have been proposed to solve this problem. Some tree-pruning
methods [1, 3] make hypotheses concerning the correspondence by searching over
a tree in which each node represents a partial match. Each partial match is then
evaluated through the pose that best fits it. In the generalized Hough transform or
equivalently template matching approach [7, 3], optimal transformation parameters
are computed for each possible pairing of a model feature and a scene feature, and
these "optimal" parameters then "vote" for the closest candidate in the discretized
transformation space.
By contrast with the tree-pruning methods and the generalized Hough transform, we
propose to formulate the problem as an objective function and optimize it directly
by using Mean Field Theory (MFT) techniques from statistical physics, adapted as
necessary to produce effective algorithms in the form of analog neural networks.
2
Point Matching
Consider the problem of locating a two-dimensional "model" object that is believed
to appear in the "scene". Assume first that both the model and the scene are
represented by a set of "points" respectively, {xd and {Ya}. The problem is to
recover the actual transformation (translation and rotation) that relates the two
sets of points. It can be solved by minimizing the following objective function
Ematch(Mia, 0, t) = L Miallxi - ReYa - tll 2
ia
(1)
=
where {Mia}
M is a Ofl-valued "match matrix" representing the unknown correspondence, Re is a rotation matrix with rotation angle 0, and t is a translation
vector.
2.1
Constraints on match variables
We need to enforce some constraints on correspondence (match) variables Mia;
otherwise all Mia = a in (1). Here, we use the following constraint
LMia = N, 'iMia ~ 0;
(2)
ia
implying that there are exactly N matches among all possible matches, where N
is the number of the model features. Summing over permutation matrices obeying
this constraint, the effective objective function is approximately [5]:
F(O, t, (3)
= -.!. L
13
e-.8l1 x .- R8 y .. -tIl 2 ,
(3)
ia
which has the same fixed points as
Epenalty(M, 0, t) = Ematch(M, 0, t)
1
+-
13
L
Mia (log Mia - 1),
(4)
ia
where Mia is treated as a continuous variable and is subject to the penalty function
x(logx-l).
Two-Dimensional Object Localization by Coarse-ta-Fine Correlation Matching
Figure 1: Assume that there is only translation between the model and the scene,
each containing 20 points. The objective functions at at different temperatures
(,8- 1 ): 0.0512 (top left), 0.0128 (top right) , 0.0032 (bottom left) and 0.0008 (bottom right), are plotted as energy surfaces of x and y components of translation .
Now, let j3 = 1/2u2 and write
Epoint(O, t) =
L
e-~lIx,-R8y(1-tIl2.
(5)
ia
The problem then becomes that of maximizing Epoint , which in turn can be interpretated as minimizing the Euclidean distance between two Gaussian-blurred images
containing the scene points Xi and a transformed version of the model points Ya.
Tracking the local maximum of the objective function from large u to small u, as in
deterministic annealing and other continuation methods, corresponds to a coarseto-fine correlation matching. See Figure 1 for a demonstration of a simpler case in
which only translation is applied to the model.
2.2
The descent dynamics
A gradient descent dynamics for finding the saddle point of the effective objective
function F is
ia
o
-I\,
L
ia
mia(Xi -
R 9Ya - t)t(R9+~Ya) ,
(6)
987
988
Lu and Mjolsness
=
=
where mia
(Mia}/3
e-/3ll x .- R 8y,,-tIl 2 is the "soft correspondence" associated
with Mia. Instead of updating t by descent dynamics, we can also solve for t
directly.
3
The Vernier Network
Though the effective objective is non-convex over translation at low temperatures,
its dependence on rotation is non-convex even at relatively high temperatures.
3.1
Hierachical representation of variables
We propose overcoming this problem by applying Mean Field Theory (M FT) to a
hierachical representation of rotation resulting from the change of variables [4]
B-1
o
L
Xb(Ob
+ (h),
(h E [-te, te],
(7)
b=O
where te = 7r /2B, Ob = (b + l)~ are the constant centers of the intervals, and (h
are fine-scale "vernier" variabfes. The Xb'S are binary variables (so Xb E {O, I}) that
satisfy the winner-take-all (WTA) constraint Lb Xb = 1.
The essential reason that this hierarchical representation of 0 has fewer spurious
local minima than the conventional analog representation is that the change of
variables also changes the connectivity of the network's state space: big jumps in 0
can be achieved by local variations of X.
3.2
Epoint
Vernier optimization dynamics
can be transformed as (see [6, 4])
Epoint(O,
t)
~vbl
~
1
E(LXb(Ob +Ob),2:XV t b)
b
LXbE(Ov
b
+ Ob, tb)
b
1 Notation:
Coordinate descent with 2-phase clock 'IlIa(t):
(8)
a
? EB for clocked sum
? x for
a clamped variable
? x A for a set of variables to be optimized analytically
? (v, u)H for Hopfield/Grossberg dynamics
? E(x, y)fJJ for coordinate descent/ascent on x, then y, iterated if necessary. Nested
angle brackets correspond to nested loops.
Two-Dimensional Object Localization by Coarse-to-Fine Correlation Matching
? . e.. .
' ...... .
.,
. '.
0 . ? ? -.
,
,
,
00?
o
"
?
?
??
?
?
?
0
0
0?
o
?
?
,0
'0,4),
? ? I ??
,
.0
o
I
,
0
o?
????
<f> () 0
0
-0'
COO
??
.
,
?
?
o?
.
? c;P~
'.
0
,
I
?? -.
0
~
~ ~
? e,
00 .? 0
:
.????
0
,
. ..
Q, ?
? ? ~Q
~f)
Q:)q ??
.
I
[)
0'??
?) -??
,
..
Figure 2: Shown here is an example of matching a 20-point model to a scene with
66.7% spurious outliers. The model is represented by circles. The set of square dots
is an instance of the model in the scene. All other dots are outliers. From left to
right are configurations at the annealing steps 1, 10, and 51, respectively.
MFT
~
1~
sinh(tub)
[~
~ XbE(th + Vb, tb) + ,8 ~(UbVb -log
t
)
A
b
b
+WTA(x,,8)]
(((v, u)H, t A ), XA)$
(9)
Each bin-specific rotation angle Vb can be found by the following fixed point equations
a
ia
(10)
The algorithm is illustrated in Figure 2.
4
Line Segment Matching
In many vision problems, representation of images by line segments has the advantage of compactness and subpixel accuracy along the direction transverse to the
line. However, such a representation of an object may vary substantially from image
to image due to occlusions and different illumination conditions.
4.1
Indexing points on line segements
The problem of matching line segments can be thought of as a point matching
problem in which each line segment is treated as a dense collection of points. Assume
now that both the scene and the model are represented by a set of line segments
respectively, {sil and {rna} . Both the model and the scene line segments are
989
990
Lu and Mjolsness
'!
o
1!io
'J
...../
\
(
"
f
\
D.'
1.2\
1 S
-e .lS
Figure 3: Approximating e(t) by a sum of 3 Gaussians.
represented by their endpoints as Si = (pi, p~) and rna = (qa, q~), where Pi, p~,
and qa, q~ are the endpoints of the ith scene segment and the ath model segment,
respectively. The locations of the points on each scene segment and model segments
can be parameterized as
+ u(p~ Ya = IDa(v) = qa + v(q~ Xi
= Si(U) =
Pi
Pi),
U
(ll)
E [0,1] and
(12)
qa), v E [0,1].
Now the model points and the scene points can be though of as indexed by i =
(i, u) and a
(a, v). Using this indexing, we have Li ex Li Ii Jol du and La ex
La1aJoi dv, where Ii = Ilpi-P~II andla = IIqa-q~ll? The point matching objective
function (5) can be specialized to line segment matching as [5]
=
Eseg((}, t)
=L
hla
ia
t
(I e- ~IIS.(u)-Rem,,(v)-tIl2 du dv.
(13)
Jo Jo
As a special case of point matching objective function, (13) can readily be transformed to the vernier network previously developed for point matching problem.
4.2
Gaussian sum approximation
Note that, as in Figure 3 and [5],
e (t) -_ {Io
if t E [0: 1]
~
1 (Ck - t)2
otherWIse ~ ~ Ak exp -"2
(72
k~I
(14)
k
where by numerical minimization of the Euclidean distance between these two functions of t, the parameters may be chosen as Al = A3 = 0.800673, A2 = 1.09862,
(71 = (73 = 0.0929032, (72 = 0.237033, C1 = 1 - C3 = 0.1l6807, and C2 = 0.5.
Using this approximation, each finite double integral in (13) can be replaced by
3
k~l AkAl
1+ 1+
00
-00
00
-00
_ _1_(Ck_U)2
e 2"'~
-
1
e ~
(cr-v)2
1
e- 2,;2l1 s .(u)+
R
t
em,,(v)-
2
II du dv. (15)
Each of these nine Gaussian integrals can be done exactly. Defining
= Si(Ck) - Rema(cl) - t
Pi = pi - Pi, qa = Re(q~ - qa),
Viakl
(16)
(17)
Two-Dimensional Object Localization by Coarse-to-Fine Correlation Matching
Figure 4: The model line segments, which are transformed with the optimal parameter found by the matching algorithm, are overlayed on the scene image. The
algorithm has successfully located the model object in the scene.
(15) becomes
1 vlaklu2
X
exp -"2
+ (Viakl
pd2u~
+ (Viakl
Qa)2uf
(u2 + f>;un(u2 + Q~uf) - U~U;(f>i . Qa)2
X
X
(18)
as was calculated by Garrett [2, 5]. From the Gaussian sum approximation, we get
a closed form objective function which can be readily optimized to give a solution
to the line segment matching problem.
5
Results and Discussion
The line segment matching algorithm described in this paper was tested on scenes
captured by a CCD camera producing 640 x 480 images, which were then processed
by an edge detector. Line segments were extracted using a polygonal approximation
to the edge images. The model line segments were extracted from a scene containing
a canonically positioned model object (Figure 4 left). They were then matched to
that extracted from a scene containing differently positioned and partially occluded
model object (Figure 4 nght). The result of matching is shown in Figure 5.
Our approach is based on a scale-space continuation scheme derived from an application of Mean Field Theory to the match variables. It provides a means to avoid
trapping by local extrema and is more efficient than stochastic searches such as
simulated annealing. The estimation of location parameters based on continuously
improved "soft correspondences" and scale-space is often more robust than that
based on crisp (but usually inaccurate) correspondences.
The vernier optimization dynamics arises from an application of Mean Field Theory to a hierarchical representation of the rotation, which turns the original unconstrained optimization problem over rotation into several constrained optimization
problems over smaller intervals. Such a transformation results in a Hopfield-style
e
e
991
992
Lu and Mjolsness
Figure 5: Shows how the model line segments (gray) and the scene segments (black)
are matched. The model line segments, which are transformed with the optimal parameter found by the matching algorithm, are overlayed on the scene line segments
with which they are matched. Most of the the endpoints and the lengths of the line
segments are different. Furthermore, one long segment frequently corresponds to
several short ones. However, the matching algorithm is robust enough to uncover
the underlying rigid transformation from the incomplete and ambiguous data.
dynamics on rotation 0, which effectively coordinates the dynamics of rotation and
translation during the optimization. The algorithm tends to find a roughly correct
translation first, and then tunes up the rotation.
6
Acknowledgements
This work was supported under grant NOOOl4-92-J-4048 from ONRjDARPA.
References
[1] H. S. Baird. Model-Based Image Matching Using Location . The MIT Press,
Cambridge, Massachusetts, first edition, 84.
[2] C. Garrett, 1990. Private communication to Eric Mjolsness.
[3] W. E. L. Grimson and T. Lozano-Perez. Localizing overlapping parts by searching the interpretation tree. IEEE Transaction on Pattern Analysis and Machine
Int elligence, 9 :469-482, 1987.
[4] C.-P. Lu and E. Mjolsness. Mean field point matching by vernier network and
by generalized Hough transform. In World Congress on Neural Networks, pages
674-684, 1993.
[5] E. Mjolsness. Bayesian inference on visual grammars by neural nets that optimize. In SPIE Science of Artificial Neural Networks, pages 63-85, April 1992.
[6] E. Mjolsness and W. L. Miranker.
Greedy Lagrangians for neural networks: Three levels of optimization in relaxation dynamics. Technical Report
YALEUjDCSjTR-945, Yale Computer Science Department, January 1993.
[7] G. Stockman. Object recognition and localization via pose clustering. Computer
Vision, Graphics, and Image Processing, (40), 1987.
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7,098 | 867 | The Role of MT Neuron Receptive Field
Surrounds in Computing Object Shape from
Velocity Fields
G.T.Buracas & T.D.Albright
Vision Center Laboratory, The Salk Institute,
P.O.Box 85800, San Diego, California 92138-9216
Abstract
The goal of this work was to investigate the role of primate
MT neurons in solving the structure from motion (SFM)
problem. Three types of receptive field (RF) surrounds
found in area MT neurons (K.Tanaka et al.,1986; Allman et
al.,1985) correspond, as our analysis suggests, to the oth, pt
and 2 nd order fuzzy space-differential operators. The large
surround/center radius ratio (;::: 7) allows both
differentiation of smooth velocity fields and discontinuity
detection at boundaries of objects.
The model is in
agreement with recent psychophysical data on surface
interpolation involvement in SFM. We suggest that area
MT partially segregates information about object shape
from information about spatial relations necessary for
navigation and manipulation.
1 INTRODUCTION
Both neurophysiological investigations [8] and lesioned human patients'
data show that the Middle Temporal (MT) cortical area is crucial to
perceiving three-dimensional shape in moving stimuli. On the other hand,
969
970
Buracas and Albright
a solid body of data (e.g. [1]) has been gathered about functional properties
of neurons in the area MT. Hoever, the relation between our ability to
perceive structure in stimuli, simulating 3-D objects, and neuronal
properties has not been addressed up to date. Here we discuss a
possibility, that area MT RF surrounds might be involved in shape-frommotion perception. We introduce a simplifying model of MT neurons and
analyse the implications to SFM problem solving.
2 REDEFINING THE SFM PROBLEM
2.1 RELATIVE MOTION AS A CUE FOR RELATIVE DEPTH
Since Helmholtz motion parallax is known to be a powerful cue providing
information about both the structure of the surrounding environment and
the direction of self-motion. On the other hand, moving objects also induce
velocity fields allowing judgement about their shapes. We can capture both
cases by assuming that an observer is tracking a point on a surface of
interest. The velocity field of an object then is (fig. 1): V = t z + W x (R - Ro)
=-tz+wxz, where w=[wx,wy,O] is an effective rotation vector of a surface
z=[x,y,z(x,y)]; Ro=[O,O,zo] is a positional vector of the fixation point; t z is a
translational component along Z axis.
z
Fig.l: The coordinate system assumed in this paper. The origin is set at
the fixation point. The observer is at Zo distance from a surface.
The Role of MT Neuron Receptive Field Surrounds in Computing Object Shape
The component velocities of a retinal velocity field under perspective
projection can be calculated from:
-xt z - WxXY+WyX 2
WxZ
V=-"--
(Zo + Z)2
-yt z +WyXY-W xy 2
Zo +Z
(Zo + Z)2
In natural viewing conditions the distance to the surface Zo is usually much
larger than variation in distance on the surface z : zo?z. In such the
second term in the above equations vanishes. In the case of translation
tangential to the ground, to which we confine our analysis, w=[O,wy,O] =
[O,w,O], and the retinal velocity reduces to
u = -wz/(zo+z) : : : -wz/zo '
v=O
(1).
The latter relation allows the assumption of orthographic projection, which
approximates the retinal velocity field rather well within the central 20 deg
of the visual field.
2.2 SFM PERCEPTION INVOLVES SURFACE INTERPOLATION
Human SFM perception is characterized by an interesting peculiarity -surface interpolation [7]. This fact supports the hypothesis that an
assumption of surface continuity is embedded in visual system. Thus, we
can redefine the SFM problem as a problem of characterizing the
interpolating surfaces. The principal normal curvatures are a local
measure of surface invariant with respect to translation and rotation of the
coordinate system. The orientation of the surface (normal vector) and its
distance to the observer provide the information essential for navigation
and object manipulation. The first and second order differentials of a
surface function allow recovery of both surface curvature and orientation.
3 MODEL OF AREA MT RECEPTIVE FIELD SURROUNDS
3.1 THREE TYPES OF RECEPTIVE FIELD SURROUNDS
The Middle Temporal (MT) area of monkeys is specialized for the
systematic representation of direction and velocity of visual motion [1,2].
MT neurons are known to posess large, silent (RFS, the "nonclassical RF".
Born and Tootell [4] have very recently reported that the RF surrounds of
neurons in owl monkey MT can be divided into antagonistic and synergistic
types (Fig.2a).
971
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Buracas and Albright
a)
25
~2O
~ 15
~10
enc. 5
o~~----~------~
o
10
20
AnnlJus diameter deg
Fig.2: Top left (a): an example of a
synergistic RF surround, redrawn from
[4] (no velocity tuning known). Bottom
left (b): a typical V-shaped tuning curve
for RF surround The horizontal axis
represents the logarithmic scale of ratio
between stimulus speeds in the RF
center and surround, redrawn from [9].
Bottom (c,d): monotonically increasing
and decreasing tuning curves for RF
surrounds, redrawn from [9].
b)
1
III Ql 0.8
> (II
,.. cc: 06
.
?I Q. 4
Gi
(II
a: 2!
O.
0.2
0
0.1
c)
1
V
1
Rotio of CIS speeds
Ql Ql 0.8
> til
'' ; ~ 0.6
?I Q.
Qj til 0.4
a: 2! 0.2
10
0
0 .1
c:t
1
J
1
Q8
06
04
02
10
R otIo of CIS speeds
0
01
~
1
10
RatootCS speeds
About 44% of the owl monkey neuron RF8s recorded by Allman et al. [3]
showed antagonistic properties. Approximately 33% of these demonstrated
V(or U)-shaped (Fig.2b), and 66% - quasi-linear velocity tuning curves
(Fig.2c,d). One half of Macaca fuscata neurons with antagonistic RF8
found by Tanaka et al [9] have had V(U)-shaped velocity tuning curves,
and 50% monotonically increasing or decreasing velocity tuning curves.
The RF8 were tested for symmetry [9] and no asymmetrical surrounds
were found in primate MT.
3.2 CONSTRUCTING IDEALIZED MT FILTERS
The surround (8) and center (C) responses seem to be largely independent
(except for the requirement that the velocity in the center must be nonzero)
and seem to combine in an additive fashion [5]. This property allows us to
combine C and 8 components in our model independently. The resulting
filters can be reduced to three types, described below.
3.2.1 Discrete Filters
The essential properties of the three types of RF8s in area MT can be
captured by the following difference equations. We choose the slopes of
velocity tuning curves in the center to be equal to the ones in the surround;
this is essential for obtaining the desired properties for 12 but not 10 , The 0order (or low-pass) and the 2nd order (or band-pass) filters are defined by:
The Role of MT Neuron Receptive Field Surrounds in Computing Object Shape
i
j
i
j
where g is gain, Wij =1, ije [-r,r] (r = radius of integration). Speed scalars
u(iJ) at points [ij] replace the velocity vectors V due to eq. (1). Constants
correspond to spontaneous activity levels.
In order to achieve the V(U) -shaped tuning for the surround in Fig.2b, a
nonlinearity has to be introduced:
II = gl
L L (u e i
Us
(i,j))2
+ Constl. (3)
j
The responses of 11 and 12 filters to standard mapping stimuli used in [3,9]
are plotted together with their biological correlates in Fig.3.
3.2.2 Continuous analogues of MT filters
We now develop continuous, more biologicaly plausible, versions of our
three MT filters. We assume that synaptic weights for both center and
surround regions fall off with distance from the RF center as a Gaussian
function G(x,y,O'), and 0' is different for center and surround: O'c 7; O's. Then,
by convolving with Gaussians equation (2) can be rewritten:
Lo (i,j) = u(i, j)* G( 0' e) + u(i,j)* G( 0' s ),
L~ (i, j ) = ? [u ( i , j ) * G ( 0' e
)-
U( i, j) * G ( 0' s )].
The continuous nonlinear Ll filter can be defined if equivalence to 11 (eq. 3)
is observed only up to the second order term of power series for u(ij):
LI (i, j) = U2 (i, j ) * G ( 0' e ) + U2 (i, j ) * G ( 0' s ) - C . [ u ( i , j ) * G ( 0' e )]. [u ( i , j ) * G ( 0' s )];
u 2(ij) corresponds to full-wave rectification and seems to be common in
area VI complex neurons; C = 2IErf2(nl2 112 ) is a constant, and Erf() is an
error function.
3.3 THE ROLE OF MT NEURONS IN SFM PERCEPTION.
Expanding z(x,y) function in (1) into power series around an arbitrary
point and truncating above the second order term yields:
u(x,y)=w(ax2+by2+cxy+dx+ey+Olzo, where a,b,c,d,e,f are expansion
coefficients. We assume that w is known (from proprioceptive input) and
=1. Then Zo remans an unresolved scaling factor and we omit it for
simplicity.
973
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Buracas and Albright
0.5
0
0.5
0
0.5
0
J L,
/ L+,
V
J
~
1/4 112 I
Fig. 3: The comparison between data
[9] and model velocity tuning curves
for RF surrounds.
The standard
mapping stimuli (optimaly moving bar
in the center of RF, an annulus of
random dots with varying speed) were
applied to L1 and L2 filters. Thee
output of the filters was passed
through a sigmoid transfer function to
accout for a logarithmic compresion in
the data.
MODEL
DATA
~
1/4 112 I
2 4
Fig. 4: Below, left: the response profile
of the L1 filter in orientation space (x
and y axes represent the components of
normal vector). Right: the response
profile of the L2 filter in curvature
space. x and y axes represent the two
normal principal curvatures.
L2
2 4
Surround/Center speed ratio
L2 response in curvature space
-15
-10
-5
o
5
10
15
-15
-10
?5
o
5
10
15
Applying Lo on u(x,y), high spatial frequency information is filtered out,
but otherwise u(x,y) does not change, i.e. Lo*u covaries with lower
frequencies ofu(x,y). L2 applied on u(x,y) yields:
L2 * U = (2 a + 2 b ) C2 (0' ~ - 0'; ) = C2 ( 0' ~ - 0'; ) V 2 U , (4)
that is, L2 shows properties of the second order space-differential operator Laplacian; C2(O'c 2 - 0'82) is a constant depending only on the widths of the
center and surround Gaussians. Note that L2*u == 1<:1 + 1<:2 ' (1<:12 are
principal normal curvatures) at singular points of surface z(x,y).
'
The Role of MT Neuron Receptive Field Surrounds in Computing Object Shape
When applied on planar stimuli up(x,y) = d x + e y, L1 has properties of a
squared first order differential operator:
~ *up = (d 2 +e 2 )C, (a~ -a;) = C, (a~ -a; >( (!)2 +( ~)2 )up,
(5)
where C2(O'e 2 - O's2) is a function of O'e and O's only. Thus the output of L1 is
monotonically related to the norm of gradient vector. It is straightforward
to calculate the generic second order surface based on outputs of three Lo,
four L1 and one L2 filters.
Plotting the responses of L1 and L2 filters in orientation and curvature
space can help to estimate the role they play in solving the SFM problem
(FigA). The iso-response lines in the plot reflect the ambiguity of MT filter
responses.
However, these responses covary with useful geometric
properties of surfaces -- norm of gradient (L 1) and mean curvature (L 2).
3.4 EXTRACTING VECTOR QUANTITIES
Equations (4) and (5) show, that only averaged scalar quantities can be
extracted by our MT operators. The second order directional derivatives
for estimating vectorial quantities can be computed using an oriented RFs
with the following profile: 02=G(x,O's) [G(y,O's) - G(y'O'e)). 01 then can be
defined by the center - surround relationship of L1 filter. The outputs of
MT filters L1 and L2 might be indispensible in normalizing responses of
oriented filters. The normal surface curvature can be readily extracted
using combinations of MT and hypothetical filters. The oriented spatial
differential operators have not been found in primate area MT so far.
However, preliminary data from our lab indicate that elongated RFs may
be present in areas FST or MST [6).
?
3.5 L2: LAPLACIAN VS. NAKAYAMA'S CONVEXITY OPERATOR
The physiologically tested ratio of standard deviations for center and surround Gaussians O'/O'e ;::: 7. Thus, besides performing the second order
differentiation in the low frequency domain, L2 can detect discontinuities
in optic flow.
4. CONCLUSIONS
We propose that the RF surrounds in MT may enable the neurons to
function as differential operators. The described operators can be thought
of as providing a continuous interpolation of cortically represented
surfaces.
Our model predicts that elongated RFs with flanking surrounds will be
found (possibly in areas FST or MST [6]). These RFs would allow extraction
975
976
Buracas and Albright
of the directional derivatives necessary to estimate the principal curvatures
and the normal vector of surfaces.
From velocity fields, area MT extracts information relevant to both the
"where" stream (motion trajectory, spatial orientation and relative distance
of surfaces) and the "what" stream (curvature of surfaces).
Acknowledgements
Many thanks to George Carman, Lisa Croner, and Kechen Zhang for
stimulating discussions and Jurate Bausyte for helpful comments on the
poster. This project was sponsored by a grant from the National Eye
Institute to TDA and by a scholarship from the Lithuanian Foundation to
GTB. The presentation was supported by a travel grant from the NIPS
foundation.
References
[1] Albright, T.D. (1984) Direction and orientation selectivity of neurons in
visual area MT of the macaque. J. Neurophysiol., 52: 1106-1130.
[2] Albright, T.D., R.Desimone. (1987) Local precision of visuotopic
organization in the middle temporal area (MT) of the macaque. Exp.Brain
Res., 65, 582-592.
[3] Allman, J., Miezin, F., McGuinnes. (1985) Stimulus specific responses
from beyond the classical receptive field. Ann.Rev.Neurosci., 8, 407-430.
[4] Born R.T. & Tootell R.B.H. (1992) Segregation of global and local motion
processing in primate middle temporal visual area. Nature, 357, 497-499.
[5] Born R.T. & Tootell R.B.H. (1993) Center - surround interactions in
direction - selective neurons of primate visual area MT. Neurosci. Abstr.,
19,315.5.
[6] Carman G.J., unpublished results.
[7] Hussain M., Treue S. & Andersen R.A. (1989) Surface interpolation in
three-dimensional Structure-from-Motion perception. Neural Computation,
1,324-333.
[8] Siegel, R.M. and R.A. Andersen. (1987) Motion perceptual deficits
following ibotenic acid lesions of the middle temporal area in the behaving
rhesus monkey. Soc.Neurosci.Abstr., 12, 1183.
[9]Tanaka, K., Hikosaka, K., Saito, H.-A., Yukie, M., Fukada, Y., Iwai, E.
(1986) Analysis of local and wide-field movements in the superior temporal
visual areas of the macaque monkey. J.Neurosci., 6,134-144.
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7,099 | 868 | Development of Orientation and Ocular
Dominance Columns in Infant Macaques
Klaus Obermayer
Howard Hughes Medical Institute
Salk-Institute
La Jolla, CA 92037
Lynne Kiorpes
Center for Neural Science
New York University
New York, NY 10003
Gary G. Blasdel
Department of Neurobiology
Harvard Medical School
Boston, MA 02115
Abstract
Maps of orientation preference and ocular dominance were recorded
optically from the cortices of 5 infant macaque monkeys, ranging in
age from 3.5 to 14 weeks. In agreement with previous observations,
we found that basic features of orientation and ocular dominance
maps, as well as correlations between them, are present and robust
by 3.5 weeks of age. We did observe changes in the strength of
ocular dominance signals, as well as in the spacing of ocular dominance bands, both of which increased steadily between 3.5 and 14
weeks of age. The latter finding suggests that the adult spacing
of ocular dominance bands depends on cortical growth in neonatal
animals. Since we found no corresponding increase in the spacing
of orientation preferences, however, there is a possibility that the
orientation preferences of some cells change as the cortical surface
expands. Since correlations between the patterns of orientation
selectivity and ocular dominance are present at an age, when the
visual system is still immature, it seems more likely that their development may be an innate process and may not require extensive
visual experience.
543
544
Obennayer, Kiorpes, and Blasdel
1
INTRODUCTION
Over the past years, high-resolution images of the simultaneous representation of
orientation selectivity and ocular dominance have been obtained in large areas of
macaque striate cortex using optical techniques [3, 4, 5, 6, 12, 18]. These studies
confirmed that ocular dominance and orientation preference are organized in large
parts in slabs. While optical recordings of ocular dominance are in accordance with
previous findings, it turned out that iso-orientation slabs are much shorter than
expected, and that the orientation map contains several other important elements
of organization - singularities, fractures, and saddle-points.
A comparison between maps of orientation preference and ocular dominance, which
were derived from the same region of adult monkey striate cortex, showed a pronounced relationship between both patterns [5, 12, 13, 15, 17]. Fourier analyses,
for example, reveal that orientation preferences repeat at closer intervals along the
ocular dominance slabs than they do across them. Singularities were found to align
with the centers of ocular dominance bands, and the iso-orientation bands, which
connect them, intersect the borders of ocular dominance bands preferably at angles
close to 90?.
Given the fact that these relationships between the maps of orientation and ocular
dominance are present in all maps recorded from adult macaques, one naturally wonders how this organization matures. If the ocular dominance slabs were to emerge
initially, for example, the narrower slabs of iso-orientation might later develop in
between. This might seem likely given the anatomical segregation which is apparent
for ocular dominance but not for orientation [9]. However, this possibility is contradicted by physiological studies that show normal, adult-like sequences of orientation
preference in the early postnatal weeks in macaque when ocular dominance slabs
are still immature [19]. The latter findings suggest a different developmental hypothesis; that the organization into regions of different orientation preferences may
precede or even guide ocular dominance formation. A third possibility, consistent
with both previous results, is that orientation and ocular dominance maps form
independently and align in later stages of development.
In order to provide evidence for one or the other hypothesis, we investigated the
relationship between ocular dominance and orientation preference in very young
macaque monkeys. Results are presented in the remainder of this paper. Section 2
contains an overview about the experimental data, and section 3 relates the data
to previous modelling efforts.
2
2.1
ORIENTATION AND OCULAR DOMINANCE
COLUMNS IN INFANT MACAQUES
THE OVERALL STRUCTURE
Figure 1 shows the map of orientation preference (Fig. 1a) and ocular dominance
(Fig. 1b) recorded from area 17 of a 3.5 week old macaque. 1 Both maps look similar
1 For all animals orientation and ocular dominance were recorded from a region close to
the border to area 18 and close to midline.
Development of Orientation and Ocular Dominance Columns in Infant Macaques
a
b
C
Figure 1: Spatial pattern of orientation preference and ocular dominance recorded
from area 17 of a macaque, 3.5 weeks of age. Figures (a) and (b) show orientation
preferences and ocular dominance bands within the same 3.1 mm x 4.3 mm large region of striate cortex. Brightness values in Fig. (a) indicate orientation preferences,
where the interval of 180? is represented by the progression in colors from black
to white. Brightness values in Fig . (b) indicate ocular dominance, where bright
and dark denote ipsi- and contralateral eye-preference. respectively. The data was
recorded from a region close to the border to area 18 and close to midline. Figure
(c) shows an enlarged section of this map in the preference (left) and the in contour
plot (right) representations. Iso-orientation lines on the right indicate intervals of
11.25?. Letters indicate linear zones (L), saddle points (H), singularities (S), and
fractures (F).
to maps which have been recorded from adults. The orientation map exhibits all of
the local elements which have been described [12, 13]: linear zones, saddle points,
545
546
Obermayer, Kiorpes, and Blasdel
a
b
-0
41)
.~
:::
')..r = 741pm
....
')..7; =612pm
00
A. =724f.1m
1.0
(ij
0.8
E
....
0
0.6
....
0.4
c:
?
1.2
41)
~
0
a.
0.2
0.0
0
1
2
3
4
5
spatial frequency [l/mm]
c
1.0 - - - - - - - - - - - ,
c:
o
-.:::: ...-.
0-0
c:
41)
.a.~
0.5
cCU
.2
E
100
-4 1c:
)-
...
o
0.0
o
-0.5
o
200
400
distance
600
[~m]
800
Figure 2: Fourieranalysis of the orientation map shown in Figure la.
(a) Complex 2D-Fouriertransform.
Each pixel corresponds to one Fouriermode and its blackness indicates
the corresponding energy. A distance of one pixel corresponds to
O.23/mm. (b) Power as a function
of radial spatial frequency. (c) Autocorrelations Sij as a function of
distance. The indices i, j E {3,4}
denote the two cartesian coordinates
of the orientation preference vector.
For details on the calculation see
[13, 15].
singularities, and fractures (Fig. lc). The ocular dominance map shows its typical
pattern of alternating bands.
Figure 2a shows the result of a complex 2D Fourier transform of the orientation
map shown in Figure la. Like for maps recorded from adult monkeys [13] the
spectrum is characterized by a slightly elliptical band of modes which is centered
at the origin. The major axis approximately aligns with the axis parallel to the
border to area 18 as well as with the ocular dominance bands. Therefore, like in
the adults, the orientatiQn map is stretched perpendicular to the ocular dominance
bands, apparently to adjust to the wider spacing.
When one neglects the slight anisotropy of the Fourier spectra one can estimate a
power spectrum by averaging the squared Fourier amplitudes for similar frequencies. The result is a pronounced peak whose location is given by the characteristic
frequency of the orientation map (Fig. 2b). As a consequence, autocorrelation functions have a Mexican-hat shape (Fig. 2c), much like it has been reported for adults
[13, 15].
In summary, the basic features of the patterns of orientation and ocular dominance
are established as early as 3.5 weeks of age. Data which were recorded from four
Development of Orientation and Ocular Dominance Columns in Infant Macaques
Table 1: Characteristic wavelengths (AOD) and signal strengths ?TOD) for the ocular
dominance pattern, as well as characteristic wavelengths p.op), density of +180?singularities (p+), density of -180?-singularities (p_), total density of singularities
(p), and percentage of area covered by linear zones (alin) for the orientation pattern
as a function of age.
age
(weeks)
UOD
3.5
5.5
7.5
14
adult
0.92
0.96
0.66
1.23
1.36
>'OD
>'OP
(}jm)
(}jm)
p+
(mm- 2 )
p(mm- 2 )
P
(mm- 2 )
alin
(% area)
686
730
870
917
950
660
714
615
700
768
3.9
3.7
4.5
3.9
3.9
3.9
3.7
4.5
3.8
3.8
7.8
7.4
9.0
7.7
7.7
47
49
45
36
43
other infants ranging from 5.5 to 14 weeks (not shown) confirm the above findings.
2.2
CHARACTERISTIC WAVELENGTHS AND SIGNAL
STRENGTH
A more detailled analysis of the recorded patterns, however, reveals changes of
certain features with age. Table 1 shows the changes in the typical wavelength of
the orientation and ocular dominance patterns as well as the (normalized) ocular
dominance signal strength with age. The strength of the ocular dominance signal
increases by a factor of 1.5 between 3.5 weeks and adulthood, a fact, which could
be explained by the still ongoing segregation of fibers within layer IV c.
The spacing of the ocular dominance columns increases by approximately 30% between 3.5 weeks and adulthood. This change in spacing would be consistent with
the growth of cortical surface area during this period [16] if one assumes that cortex grows anisotropically in the direction perpendicular to the ocular dominance
bands. Interestingly, the characteristic wavelengths of the orientation patterns do
not exhibit such an increase. The wavelengths for the patterns recorded from the
different infants are close to the "adult" values. More evidence for a stable orientation pattern is provided by the fact, that the density of'singularities is approximately
constant with age 2 and that the percentage of cortical area covered by linear zones
does neither increase nor decrease. Hence we are left with the puzzle that at least
the pattern of orientation does not follow cortical growth.
2.3
CORRELATIONS BETWEEN THE ORIENTATION AND
OCULAR DOMINANCE MAPS
Figure 3 shows a contour plot representation of the pattern of orientation preference
in overlay with the borders of the ocular dominance bands for the 3.5 week old
animal. Iso-orientation contours (thin lines) indicate intervals of 15?. Thick lines
indicate the border of the ocular dominance bands. From visual inspection it is
2Note that both types of singularities appear in equal numbers.
547
S48
Obennayer. Kiorpes. and Blasdel
Figure 3: Contour plot representation of t.he orient.a.t.ion map shown in
Figure la in overlay with the borders
of the ocular dominance bands taken
from Figure 1b. Iso-orient.ation lines
(thin lines) indicate intervals of 15?.
The borders of the ocular dominance
bands are indicated by thick lines.
already apparent that singularities have a strong tendency to align with the center
of the ocular dominance bands (arrow 1) and that in the linear zones (arrow 2),
where iso-orientation bands exist, these bands intersect ocular dominance bands at
angles close to 90? most of the time.
Table 2 shows a quantitative analysis of the local intersection angle. Percentage
of area covered by linear zones (cf. [12] for details of the calculation) is given for
regions, where orientation bands intersect ocular dominance bands within 18? of
perpendicular, and regions where they intersect within 18? of parallel. For all of
the animals investigated the percentages are two to four times higher for regions,
where orientation bands intersect ocular dominance bands at angles close to 90?,
much like it has been observed in adults [12]. In particular there is no consistent
trend with age: the correlations between the orientation and ocular dominance
maps are established as early as 3.5 weeks of age.
aperp
Un
par
age
(weeks)
(%area)
(% area)
3.5
5.5
7.5
14
adult
15.9
12.2
13.3
12.4
18.0
4.1
6.8
6.2
3.7
2.7
a'in
Table 2: Percentage of area covered by
linear zones as a function of age for regions, where orientation bands intersect ocular dominance bands within
18? of perpendicular (af/::'P) , and regions where they intersect within 18?
of parallel (afi~) (cf. [12] for details of
the calculation).
Development of Orientation and Ocular Dominance Columns in Infant Macaques
3
CONCLUSIONS AND RELATION TO MODELLING
In summary, our results provide evidence that the pattern of orientation is established at a time when the pattern of ocular dominance is still developing. However,
they provide also evidence for the fact that the pattern of orientation is not linked
to cortical growth. This latter finding still needs to be firmly established in studies
where the development of orientation is followed in one and the same animal. But if
it is taken seriously the consequence would be that orientation preferences may shift.
and that pairs of singularities are formed. The early presence of strong correlations
between both maps indicate that the development of orientation and ocular dominance are not independent processes. Both patterns have to adjust. to each other
while cortex is growing. It, therefore, seems as if the third hypothesis is true (see
Introduction) which states that both patterns develop independently and adjust to
each other in the late stages of development. As has been shown in [13, 15] and is
suggested in [7, 14] these processes are certainly in the realm of models based on
Hebbian learning.
Many features of the orientation and ocular dominance maps are present at an age
when the visual system of the monkey is still immature [8, 11]. In particular, they
are present at a time when spatial vision is strongly impared. Consequently, it
seems unlikely that the development of these features as well as of the correlations
between both patterns requires high acuity form vision, and models which try to
predict the structure of these maps from the structure of visual images [1, 2, 10]
have to take this fact into account. The early development of orientation preference and its correlations with ocular dominance make it also seem more likely that
their development may me an innate process and may not require extensive visua.l
experience. Further experiments, however, are needed to settle these questions.
Acknowledgements
This work was funded in part by the Klingenstein Foundation, the McKnight Foundation, the New England Primate Research Center (P51RR0168-31), the Seaver
Institute, and the Howard Hughes Medical Institute. We thank Terry Sejnowski,
Peter Dayan, and Rich Zemel for useful comments on the manuscript. Linda Ascomb, Jaqueline Mack, and Gina Quinn provided excellent technical assistance.
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