# Scattering Statistics of Generalized Spatial Poisson Point Processes

We present a machine learning model for the analysis of randomly generated discrete signals, which we model as the points of a homogeneous or inhomogeneous, compound Poisson point process. Like the wavelet scattering transform introduced by S. Mallat, our construction is a mathematical model of convolutional neural networks and is naturally invariant to translations and reflections. Our model replaces wavelets with Gabor-type measurements and therefore decouples the roles of scale and frequency. We show that, with suitably chosen nonlinearities, our measurements distinguish Poisson point processes from common self-similar processes, and separate different types of Poisson point processes based on the first and second moments of the arrival intensity λ(t), as well as the absolute moments of the charges associated to each point.

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## 1 Introduction

Convolutional neural networks (CNNs) have obtained impressive results for a number of learning tasks in which the underlying signal data can be modelled as a stochastic process, including texture discrimination (Sifre and Mallat, 2013), texture synthesis (Gatys et al., 2015; Antognini et al., 2018), time series analysis (e.g., finance) (Binkowski et al., 2018), and wireless networks (Brochard et al., 2018). However, precise theoretical understanding of these results is lacking. In many scenarios it is natural to model the signal data as the points of a (potentially complex) spatial point process. Furthermore, there are numerous other fields, including stochastic geometry (Haenggi et al., 2009), forestry (Genet et al., 2014), geoscience (Schoenberg, 2016) and genetics (Fromion et al., 2013), in which spatial point processes are used to model the underlying generating process of certain phenomena (e.g., earthquakes). Motivated by these existing empirical results, as well as the potential for numerous others in yet untapped research, we consider the capacity of CNNs to capture the statistical properties of spatial point processes.

In order to facilitate provable statistical guarantees, we leverage a CNN architecture similar to the wavelet scattering transform of Mallat (2012). The wavelet scattering transform is a mathematical model for CNNs that is provably invariant to local (or global) translations of the input signal and is also Lipschitz stable to the actions of diffeomorphisms on the input. It has been used to achieve near state of the art results in the fields of audio signal processing (Andén and Mallat, 2011, 2014; Wolf et al., 2014, 2015; Andén et al., 2018)

(Bruna and Mallat, 2011, 2013; Sifre and Mallat, 2012, 2013, 2014; Oyallon and Mallat, 2015), and quantum chemistry (Hirn et al., 2017; Eickenberg et al., 2017, 2018; Brumwell et al., 2018), amongst others. It consists of an alternating cascade of linear wavelet transforms and complex modulus nonlinearities. In this paper, we examine a generalized scattering transform that utilizes a broader class of filters, but which still includes wavelets as a special case. Our main focus is on scattering architectures constructed with filters that have small spatial support as is the case in most traditional CNNs.

Expected wavelet scattering moments for stochastic processes with stationary increments were introduced in Bruna et al. (2015), where it is shown that such moments capture important statistical information of one-dimensional Poisson processes, fractional Brownian motion, -stable Lévy processes, and a number of other stochastic processes. In this paper, we extend the notion of scattering moments to our generalized architecture, and in the process of doing so, we recover many of the important small scale results in Bruna et al. (2015)

. However, the main contributions contained here consist of new results for more general spatial point processes, including inhomogeneous Poisson point processes, which are not stationary and do not have stationary increments. The collection of expected scattering moments is a non-parametric model for these processes, which we prove captures important summary statistics of inhomogeneous, compound spatial Poisson point processes.

The remainder of this paper is organized as follows. Expected scattering moments are introduced in Section 2. Sections 3 and 4 analyze the first-order and second-order scattering moments of inhomogeneous, compound spatial Poisson point processes. Section 5 compares the scattering moments of one-dimensional Poisson processes to two self-similar processes, fractional Brownian motions and the -stable process. Section 6 presents stylized numerical examples to highlight certain aspects of the presented theory. A short conclusion is given in Section 7. All proofs are in the appendices, in addition to details on the numerical work.

## 2 Expected Scattering Moments for Random Signed Measures

Let be a compactly supported mother wavelet with dilations , let and be a stochastic process with stationary increments defined on the real line. In Bruna et al. (2015), first-order wavelet scattering moments are defined as , where the expectation does not depend on since if has stationary increments, then is stationary so long as is a wavelet. Much of the mathematical analysis of wavelet scattering moments relies on the fact that they can be rewritten as , where is the primitive of , i.e., . This reformulation motivates us to define scattering moments as the integration of a filter, which is not necessarily a wavelet, against a random signed measure

To that end, let be a continuous window function with support contained in the unit cube . Denote by the dilation of , supported on the cube , and set to be the Gabor-type filter with scale and central frequency ,

 (1)

Note that with an appropriately chosen window function (1) includes dyadic wavelet families in the case that one selects and

. However, it also includes many other families of filters, including Gabor filters used in the windowed Fourier transform.

For a random signed measure we define the first-order scattering moments, at location as

 (2)

Note there is no assumption on the stationarity of , which is why these scattering moments a priori depend on . We define invariant (i.e., location independent) first-order scattering coefficients of by

 SY(γ,p)=limR→∞1(2R)d∫|ti|≤RE[|gγ∗Y(t)|p]dt, (3)

if the limit on the right hand side exists.

We call a periodic measure if there exists such that for any Borel set , the family of sets satisfies

 Y(B)d=Y(B+Tei),∀1≤i≤d,

where is the standard orthonormal basis for . In this case one can verify, by approximating with simple functions, that , and therefore

 Sγ,pY(t+Tei)=Sγ,pY(t),∀t∈Rd.

Thus the limit in (3) exists, and

 SY(γ,p)=1Td∫QTE[|gγ∗Y(t)|p]dt. (4)

Note that in the special case when the distribution of depends only on the Lebesgue measure of , then is independent of and the above limit (3) exists with for any .

First-order scattering moments compute summary statistics of the measure based upon its responses against the filters . Higher-order summary statistics can be obtained by computing first-order scattering moments for larger powers , or by cascading lower-order modulus nonlinearities as in a CNN. This leads us to define second-order scattering moments by

 Sγ,p,γ′,p′Y(t)=E[||gγ∗Y|p∗gγ′(t)|p′].

First-order invariant scattering moments collapse additional information by aggregating the variations of the random measure , which removes information related to the intermittency of . Second-order invariant scattering moments augment first-order scattering moments by iterating on the cascade of linear filtering operations and nonlinear operators, thus recovering some of this lost information. They are defined (assuming the limit on the right exists) by

 SY(γ,p,γ′,p′)=limR→∞1(2R)d∫|ti|≤RE[∣∣|gγ∗Y|p∗gγ′(t)∣∣p′]dt.

The collection of (invariant) scattering moments is a set of non-parametric statistical measurements of the random measure . In the following sections, we analyze these moments for arbitrary frequencies and small scales , thus allowing the filters to serve as a model for the learned filters in CNNs. In particular, we will analyze the asymptotic behavior of the scattering moments as the scale parameter decreases to zero.

## 3 First-Order Scattering Moments of Generalized Poisson Processes

We consider the case where is an inhomogeneous, compound spatial Poisson point process. Such processes generalize ordinary Poisson point processes by incorporating variable charges (heights) at the points of the process and a non-uniform intensity for the locations of the points. They thus provide a flexible family of point processes that can be used to model many different phenomena. In this section we consider first-order scattering moments of these generalized Poisson processes. In Sec. 3.1 we provide a review of such processes, and in Sec. 3.2 we show that first-order scattering moments capture a significant amount of statistical information related these processes, particularly when using very localized filters.

### 3.1 Inhomogeneous, Compound Spatial Poisson Point Processes

Let be a continuous function on such that

 0<λmin:=inftλ(t)≤∥λ∥∞<∞, (5)

and let be an inhomogeneous Poisson point process with intensity function . That is,

 N(dt)=∞∑j=1δtj(dt)

is a random measure, concentrated on a countable set of points such that for all Borel sets , the number of points of in , denoted

, is a Poisson random variable with parameter

 Λ(B)=∫Bλ(t)dt, (6)

i.e.,

 P[N(B)=n]=e−Λ(B)(Λ(B))nn!,

and is independent of for all sets that do not intersect . Now let be a sequence of i.i.d. random variables independent of , and let be the random signed measure that gives charge to each point of , i.e.,

 Y(dt)=∞∑j=1Ajδtj(dt). (7)

We refer to as an inhomogeneous, compound Poisson point process. For a Borel set ,

has a compound Poisson distribution and we will (in a slight abuse of notation) write

 Y(B)=N(B)∑j=1Aj.

In many of our proofs, it will be convenient to consider the random measure defined formally by

 |Y|p(dt)\coloneqq∞∑j=1|Aj|pδtj(dt).

For a further overview of these processes, and closely related marked point processes, we refer the reader to Section 6.4 of Daley and Vere-Jones (2003).

### 3.2 First-order Scattering Asymptotics

Computing the convolution of with gives

 (gγ∗Y)(t)=∫Rdgγ(t−u)Y(du)=∞∑j=1Ajgγ(t−tj),

which can be interpreted as a waveform emitting from each location . Invariant scattering moments aggregate the random interference patterns in . The results below show that the expectation of these interferences, for small scale waveforms , encode important statistical information related to the point process.

For notational convenience, we let

 Λs(t):=Λ([t−s,t]d)=∫[t−s,t]dλ(u)du

denote the expected number of points of in the support of . If is a periodic function in each coordinate with period , then for and therefore, the invariant scattering coefficients of may be defined as in (4).

Let and suppose that is an inhomogeneous, compound Poisson point process as defined above, where is an i.i.d. sequence of random variables, and is a continuous intensity function satisfying (5). Then for every every such that and for every

 (8)

where the error term satisfies

 |ϵ(m,s,ξ,t)|≤Cm,p∥λ∥∞λmin∥w∥ppE[|A1|p]∥λ∥m+1∞sd(m+1) (9)

and is an i.i.d. sequence of random variables, independent of the , taking values in the unit cube and with density

 pV(v)=sdΛs(t)λ(t−vs),v∈Q1.

The main idea of the proof of Theorem 3.2 is to condition on , which is the number of points in the support of , and to use the fact that

 P[N([t−s,t]d)>m]=O((sd∥λ∥∞)m+1),∀sd∥λ∥∞<1.

Theorem 3.2 shows that even at small scales the scattering moments depend upon higher-order information related to the distribution of the points, encapsulated by the term , regardless of the scattering moment . However, the influence of the higher-order terms diminishes rapidly as the scale of the filter shrinks, which is indicated by the bound (9) on the error function. Theorem 3.2 also shows that scattering moments depend on the moments of the charges, . The next result uses Theorem 3.2 to examine the behavior of scattering moments for small filters in the asymptotic regime as the scale .

Let and suppose that is an inhomogeneous, compound Poisson point process satisfying the same assumptions as in Theorem 3.2. Let be a sequence of scale and frequency pairs such that . Then

 limk→∞Sγk,pY(t)sdk=λ(t)E[|A1|p]∥w∥pp. (10)

Furthermore, if is periodic with period along each coordinate, then

 limk→∞SY(γk,p)sdk=m1(λ)E[|A1|p]∥w∥pp,wherem1(λ)=1Td∫QTλ(t)dt. (11)

Theorem 3.2

is proved via asymptotic analysis of the

case of Theorem 3.2. The key to the proof, which is similar to the technique used to prove Theorem 2.1 of Bruna et al. (2015), is that in a small cube there is at most one point of

with overwhelming probability. Therefore, when

is very small, with very high probability,

This theorem shows that for small scales the scattering moments encode the intensity function , up to factors depending upon the summary statistics of the charges and the window . Recall that , defined in (6), determines the concentration of events within the set . Thus even a one-layer location dependent scattering network yields considerable information regarding the underlying data generation, at least in the case of inhomogeneous Poisson processes. However, it is often the case, e.g., Bruna and Mallat (2018), that invariant statistics are utilized. In this case (11) shows that invariant scattering statistics mix the mean of and the moment of the charge magnitudes. However, we can decouple these statistics as we now explain.

As a special case, Theorem 3.2 proves that for non-compound inhomogeneous Poisson processes (i.e., for all ), small scale scattering moments recover or , depending on whether one computes invariant or time-dependent scattering moments. For compound processes, we can add an additional nonlinearity, namely the signum function , which when applied to the Poisson point process in (7) yields,

 ¯¯¯¯Y(dt)=sgn[Y(dt)]=∞∑j=1δtj(dt).

Thus computing and the ratio at small scales decouples the mean of from the moment of

. We remark that the signum function is a simple perceptron and is closely related to the sigmoid nonlinearity, which is used in many neural networks. We further remark that the computation of

constitutes a small two-layer network, consisting of the nonlinear function, the linear filtering by the collection of filters , the nonlinear modulus , and the linear integration operator.

If is a homogeneous Poisson process, then is constant, meaning that (10) and (11) are equivalent. In the case of ordinary (non-compound) Poisson processes, Theorem 3.2 recovers the constant intensity. For periodic and invariant scattering moments, the effect of higher-order moments of can be partially isolated by considering higher-order expansions (e.g., ) in (8). The next theorem considers second-order expansions and illustrates their dependence on the second moment of .

Let and suppose is an inhomogeneous, compound Poisson point process satisfying the same assumptions as in Theorem 3.2. If is periodic with period in each coordinate, and if , is a sequence of scale and frequency pairs such that and , then

 limk→∞ (SY(γk,p)s2dkE[|A1|p]E[|Vk|p]−1Td∫QTΛsk(t)s2dkdt) = m2(λ)(E[∣∣A1w(U1)eiL⋅U1+A2w(U2)eiL⋅U2∣∣p]2∥w∥ppE[|A1|p]), (12)

where ; , are independent uniform random variables on ; and is a sequence of random variables independent of the taking values in the unit cube and with respective densities,

 pVk(v)=sdkΛsk(t)λ(t−vsk),v∈Q1.

We first remark that the scale normalization on the left hand side of (12) is , compared to a normalization of in Theorem 3.2. Thus even though (12) is written as a small scale limit, intuitively Theorem 3.2 is capturing information at moderately small scales that are larger than the scales considered in Theorem 3.2. This is further indicated by the term multiplied against on the right hand side of (12), which depends on two points of the process (as indicated by the presence of two charges and ).

Unlike Theorem 3.2, which gives a way to compute , Theorem 3.2 does not allow one to compute since it would require knowledge of in addition to the distribution from which the charges are drawn. However, Theorem 3.2 does show that at moderately small scales the invariant scattering coefficients depend non-trivially on the second moment of . This behavior at moderately small scales can be used to distinguish between, for example, an inhomogeneous Poisson point process with intensity function and a homogeneous Poisson point process with constant intensity , whereas Theorem 3.2 indicates that at very small scales the two processes will have the same invariant scattering moments.

## 4 Second-Order Scattering Moments of Generalized Poisson Processes

We prove that second-order scattering moments, in the small scale regime, encode higher-order moment information about the charges

Let and . Suppose that is an inhomogeneous Poisson point process satisfying the same assumptions as in Theorem 3.2 as well as the additional assumption that Let and be two sequences of scale and frequency pairs such that for some fixed constant and . Then,

 limk→∞Sγk,p,γ′k,p′Y(t)sd(p′+1)k=Kλ(t)E[|A1|q], (13)

where

 K\coloneqq∥∥gc,L/c∗|g1,0|p∥∥p′p′,

is a constant depending on and Furthermore, if is periodic with period along each coordinate, then

 limk→∞SY(γk,p,γ′k,p′)sd(p′+1)k=Km1(λ)E[|A1|q]. (14)

Note that the scaling factor depends on but not . Intuitively this corresponds to the behavior as . Theorem 4 proves that second-order scattering moments capture higher-order moments of the charges via two pairs of lower-order filtering and modulus operators. If , then will be larger than either or and the result above will give us information about the higher order moment

It is also useful to consider the case. Indeed, in Sec. 5 below it is shown that first-order invariant scattering moments can distinguish Poisson point processes from fractional Brownian motion and -stable processes, if but may fail to do so for larger values of However, Theorem 3.2 shows that first-order invariant scattering moments for will not be able to distinguish between the various different types of Poisson point processes with a one-layer network at very small scales. Theorem 4 shows that a second-order calculation that augments the first-order calculation with and , will capture a higher-order moment of the charges .

## 5 Poisson Point Processes Compared to Self Similar Processes

For one-dimensional processes (i.e., ), we show that first-order invariant scattering moments can distinguish between inhomogeneous, compound Poisson point processes and certain self-similar processes. In particular, we show that if is either an -stable process or a fractional Brownian motion (fBM), then the corresponding first-order scattering moments will have different asymptotic behavior for infinitesimal scales than in the case of a Poisson point process. Similar results were initially reported in Bruna et al. (2015); here we generalize those results to the non-wavelet filters defined in (1) and for general scattering moments, and further clarify their usefulness in the context of the new results presented in Sec. 3 and Sec. 4. As in Bruna et al. (2015), the proof will be based on the scaling relationships of these processes and therefore will not be able to distinguish between -stable processes and fBM111We note that Bruna et al. (2015) proves that second-order scattering moments defined with wavelet filters do distinguish between -stable processes and fBM, but we do not pursue this direction in this paper as we are concerned primarily with point processes.. The key will be proving a lemma that says if a stochastic process has a scaling relation, then that scaling relation is inherited by integrals of deterministic functions against .

More precisely, for a stochastic process , we consider the convolution of the filter with the noise defined by

 gγ∗dX(t)=∫Rgγ(t−u)dX(u),

and define (in a slight abuse of notation) the first-order scattering moments at time by

 Sγ,pX(t)=E[|gγ∗dX(t)|p]. (15)

In the case where is a compound, inhomogeneous Poisson (counting) process, will be a compound Poisson random measure and the scattering moments defined in (15) will coincide with the first-order scattering moments defined in (2).

The following two theorems analyze the small scale first-order scattering moments when is either an -stable process, for , or fractional Brownian motion. Thus will be stable Lévy noise or fractional Gaussian noise, respectively. These results show that the asymptotic decay of the corresponding scattering moments is guaranteed to differ from Poisson point processes, in the case We also note that both -stable processes and fBM have stationary increments; therefore the scattering moments do not depend on time and

 Sγ,pX(t)=SX(γ,p)=limR→∞12R∫|u|≤RE[|gγ∗dX(u)|p]du,∀t∈R.

Let and suppose is a symmetric -stable process for some . Let be a sequence of scale and frequency pairs such that and . Then,

 limk→∞SX(γk,p)sp/αk=E[∣∣∣∫10w(u)eiLudX(u)∣∣∣p].

Let suppose is a fractional Brownian motion with Hurst parameter . Assume that the window function has bounded variation on and let be a sequence of scale and frequency pairs such that and . Then,

 limk→∞SX(γk,p)spHk=E[∣∣∣∫10w(u)eiLudX(u)∣∣∣p].

The key to proving Theorem 5 and Theorem 5 is the following lemma stated in Appendix E, which shows that is a self-similar process, then, then stochastic integrals against satisfy an identity corresponding to the scaling relation of .

Together, these two theorems indicate that first-order invariant scattering moments distinguish inhomogeneous, compound Poisson processes from both -stable processes and fractional Brownian motion except in the cases where or . In particular, if is a Brownian motion, then will distinguish from a Poisson point process except in the case that . For this reason, it appears that is the best choice of the parameter for the purposes of distinguishing a Poisson point process from a self-similar process. In the case of a multi-layer network, it is advisable to set . Larger values of in the second layer can then allow us to determine the higher moments of the arrival heights .

## 6 Numerical Illustrations

We carry out several experiments to numerically validate the previously stated results and to illustrate their capacity for distinguishing between different types of random processes. In all of the experirments below, we will hold the frequency constant while we let the scale decrease to zero.

#### Homogeneous, compound Poisson point processes with the same intensities:

We generated three different types of homogeneous compound Poisson point processes, all with the same intensity . The three point processes are (ordinary), (Gaussian), and (Rademacher), where the charges are sampled according to , , and Rademacher distribution (i.e., with equal probability). The charges of the three signals have the same first moment and different second moment with and . Theorem 3.2 thus predicts that invariant first-order scattering moments will not be able to distinguish between the three processes, but invariant first-order scattering moments will distinguish the Gaussian Poisson point process from the other two. Figure 1 illustrates this point by plotting the normalized invariant scattering moments for and . Figure 1: First-order invariant scattering moments for three types of homogeneous compound Poisson point processes with the same intensity λ0. Left: Top: ordinary Poisson point process. Middle:Gaussian compound Poisson point process with normally distributed charges. Bottom: Rademacher compound Poisson point process with charges drawn from the Rademacher distribution. Middle: Normalized invariant scattering moments \nicefracSY(s,ξ,1)s∥w∥1 (i.e., p=1), which all converge to 0.01 as s→0 (up to numerical errors) since λ0E[|A1|] is the same for all three point processes. Right: Normalized invariant scattering moments \nicefracSY(s,ξ,2)s∥w∥22 (i.e., p=2). In this case the ordinary Poisson point process and the Rademacher Poisson point process still converge to the same value as s→0 since E[|A1|2]=1 for both of them. However, the Gaussian Poisson point process converges to a different value since E[|A1|2]=\nicefracπ2 for this process.

#### Homogeneous, compound Poisson point processes with different intensities and charges:

We consider two homogeneous, compound Poisson point processes with different intensities and different charge distributions, but which nevertheless have the same first-order invariant scattering moments with due to the mixing of intensity and charge information in (11). The first compound Poisson point process has constant intensity and charges , whereas the second has intensity and . In this way, , but and . Figure 2 plots the normalized invariant scattering moments for and . Figure 2: First-order invariant scattering moments for two homogeneous, Gaussian compound Poisson point processes with different intensity and variance. Left: Top: Homogeneous compound Poisson point process with intensity λ1=0.01 and charges A1,j∼N(0,1). Bottom: Homogeneous compound Poisson point process with intensity λ2=\nicefrac0.01√2 and charges A2,j∼N(0,2). The two point processes are difficult to distinguish, visually. Middle: Normalized invariant scattering moments \nicefracSY(s,ξ,1)s∥w∥1 (i.e., p=1), which both converge to approximately 0.08 up to numerical error, thus indicating that these moments cannot distinguish the two processes. Right: Normalized invariant scattering moments \nicefracSY(s,ξ,2)s∥w∥22 (i.e., p=2). The two process are distinguished as s→0 since the values λ1E[|A1,j|2]=0.01 and λ2E[|A2,j|2]≈0.014 differ by a significant margin.

#### Inhomogeneous, non-compound Poisson point processes:

We also consider inhomogeneous Poisson point processes. We use the intensity function

to generate inhomogeneous process. To estimate

, we average the modulus of the scattering transform at time over 1000 realizations. Figure 3 plots the scattering moments for inhomogeneous process at different time. Figure 3: First-order invariant scattering moments for inhomogeneous non-compound Poisson point processes. Left: Inhomogeneous non-compound Poisson point process with intensity λ(t)=0.01(1+0.5sin(\nicefrac2πtN)). Right: Scattering moments \nicefracSγ,pY(t)s∥w∥pp for inhomegeneous non-compound Poisson point process at t1=\nicefracN4, t2=\nicefracN2, t3=\nicefrac3N4. Note that λ(t1)=0.015, λ(t2)=0.01, λ(t3)=0.005. The plots show that for inhomogeneous process, scattering coefficients at time t converges to the intensity at that time.

#### Homogeneous, non-compound Poisson point process and self similar process:

We consider Brownian motion with Hurst parameter and compare it with Poisson point process with intensity and charges . Figure 4 shows that the 2nd moments cannot distinguish between Brownian motion and Poisson point process while the 1st moments can. Figure 4: First-order invariant scattering moments for Brownian motion and Poisson point process. Left: Top: Brownian motion with Hurst parameter H=\nicefrac12. Bottom: Ordinary Poisson point process. Middle: Normalized scattering moments for Brownian Motion (\nicefracSXBM(s,ξ,1)∥w∥2E|Z|) and Poisson point process (\nicefracSYpoisson(s,ξ,1)λE|A|∥w∥1) at p=1. This shows the normalized scattering to √s for Brownian motion converge while to s for Poisson process, indicating the 1st moment can distinguish Brownian motion and Poisson point process. Right: Normalized scattering moments for Brownian Motion (\nicefracSXBM(s,ξ,2)∥w∥22) and Poisson point process (\nicefracSYpoisson(s,ξ,2)∥w∥22) at p=2. Both normalized scattering moments converge to s, so the 2nd moment scattering cannot distinguish the two processes.

## 7 Conclusion

We have constructed Gabor-filter scattering transforms for random measures on and stochastic processes on Our construction is closely related to Bruna et al. (2015), but extends their work in several important ways. First, while our Gabor-type filters include dyadic wavelets as a special case, they also include many other families of filters. We also do not assume that the random measure is stationary, and consider compound, possibly inhomogeneous, Poisson random measures on in addition to ordinary Poisson processes on We do note however, that Bruna et al. (2015) provides a detailed analysis of self-similar processes and multifractal random measures, whereas we have primarily focused on models of random sparse signals. We believe the results presented here open up several avenues of future research. Firstly, we have assumed throughout most of this paper that the points of our random measures were distributed according to a possibly inhomogenous Poisson process. It would be interesting to discover if our measurements can distinguish these signals from other point processes. Secondly, it would be interesting to explore the use of these measurements for a variety of machine learning tasks such as synthesizing new signals.

J.H. and M.H. are partially supported by NSF grant #1620216. M.H. is also partially supported by Alfred P. Sloan Fellowship #FG-2016-6607 and DARPA YFA #D16AP00117.

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## Appendix A Proof of Theorem 3.2

To prove Theorem 3.2 we will need the following lemma.

Let be a Poisson random variable with parameter . Then for all , ,

 E[Zα1{Z>m}]=∞∑k=m+1e−λλkk!kα≤Cm,αλm+1,∀0<λ<1.

For and , . Therefore,

 E[Zα1{Z>m}] =∞∑k=m+1e−λλkk!kα =λm+1∞∑k=0e−λλk(k+m+1)!(k+m+1)α ≤λm+1∞∑k=0(k+m+1)α(k+m+1)! =Cα,mλm+1.

[Theorem 3.2] Recalling the definitions of and , and setting , we see

 Sγ,pY(t) =E[∣∣∣∫[s−t,t]dw(t−us)eiξ⋅(t−u)Y(du)∣∣∣p] =E⎡⎣∣∣ ∣∣Ns(t)∑j=1Ajw(t−tjs)eiξ⋅(t−tj)∣∣ ∣∣p⎤⎦,

where are the points in . Conditioned on the event that , the locations of the points on are distributed as i.i.d. random variables taking values in with density

 pZ(z)=λ(z)Λs(t),z∈[t−s,t]d.

Therefore, the random variables

 Vi:=t−Zis

take values in the unit cube and have density

 pV(v)=sdΛs(t)λ(t−vs),v∈Q1.

Note that in the special case that is homogeneous, i.e. is constant, the are uniform random variables on .

Our proof will be based off of conditioning on . For ,

 E⎡⎣∣∣ ∣∣Ns(t)∑j=1Ajw(t−tjs)eiξ⋅(t−tj)∣∣ ∣∣p:Ns(t)=k⎤⎦ =E⎡⎣∣∣ ∣∣k∑j=1Ajw(Vj)eisξ⋅Vj∣∣ ∣∣p⎤⎦ ≤∥λ∥∞λminkpE[|A1|p]∥w∥pp, (16)

where (16) follows from (i) the independence of the random variables and ; (ii) the fact that for any sequence of i.i.d. random variables ,

and (iii) the fact that

 E[|w(Vi)|p]=∫Q1|w(v)|ppV(v)dv≤∥λ∥∞λmin∥w∥pp.

Therefore, since ,

 E ⎡⎣∣∣ ∣∣Ns(t)∑j=1Ajw(t−tjs)eiξ⋅(t−tj)∣∣ ∣∣p⎤⎦= =∞∑k=0e−Λs(t)(Λs(t))kk!E⎡⎣∣∣ ∣∣Ns(t)∑j=1Ajw(t−tjs)eiξ⋅(t−tj)∣∣ ∣∣p:Ns(t)=k⎤⎦ =∞∑k=1e−Λs(t)(Λs(t))kk!E⎡⎣∣∣ ∣∣k∑j=1Ajw(Vj)eisξ⋅Vj∣∣ ∣∣p⎤⎦ =m∑k=1e−Λs(t)(Λs(t))kk!E⎡⎣∣∣ ∣∣k∑j=1Ajw(Vj)eisξ⋅Vj∣∣ ∣∣p⎤⎦+ϵ(m,s,ξ,t),

where

 ϵ(m,s,t,ξ):=∞∑k=m+1e−Λs(t)(Λs(t))kk!E⎡⎣∣∣ ∣∣k∑j=1Ajw(Vj)eisξ
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