Biologically Plausible Online Principal Component Analysis Without Recurrent Neural Dynamics
  Artificial neural networks that learn to perform Principal Component Analysis
(PCA) and related tasks using strictly local learning rules have been
previously derived based on the principle of similarity matching: similar pairs
of inputs should map to similar pairs of outputs. However, the operation of
these networks (and of similar networks) requires a fixed-point iteration to
determine the output corresponding to a given input, which means that dynamics
must operate on a faster time scale than the variation of the input. Further,
during these fast dynamics such networks typically "disable" learning, updating
synaptic weights only once the fixed-point iteration has been resolved. Here,
we derive a network for PCA-based dimensionality reduction that avoids this
fast fixed-point iteration. The key novelty of our approach is a modification
of the similarity matching objective to encourage near-diagonality of a
synaptic weight matrix. We then approximately invert this matrix using a Taylor
series approximation, replacing the previous fast iterations. In the offline
setting, our algorithm corresponds to a dynamical system, the stability of
which we rigorously analyze. In the online setting (i.e., with stochastic
gradients), we map our algorithm to a familiar neural network architecture and
give numerical results showing that our method converges at a competitive rate.
The computational complexity per iteration of our online algorithm is linear in
the total degrees of freedom, which is in some sense optimal.
