Title: DEEP DENOISING: RATE-OPTIMAL RECOVERY OF STRUCTURED SIGNALS WITH A DEEP PRIOR

Abstract: Deep neural networks provide state-of-the-art performance for image denoising, where the goal is to recover a near noise-free image from a noisy image. The underlying principle is that neural networks trained on large datasets have empirically been shown to be able to generate natural images well from a lowdimensional latent representation of the image. Given such a generator network, or prior, a noisy image can be denoised by finding the closest image in the range of the prior. However, there is little theory to justify this success, let alone to predict the denoising performance as a function of the networks parameters. In this paper we consider the problem of denoising an image from additive Gaussian noise, assuming the image is well described by a deep neural network with ReLu activations functions, mapping a k-dimensional latent space to an n-dimensional image. We state and analyze a simple gradient-descent-like iterative algorithm that minimizes a non-convex loss function, and provably removes a fraction of p1 ´Opk{nqq of the noise energy. We also demonstrate in numerical experiments that this denoising performance is, indeed, achieved by generative priors learned from data.

1. INTRODUCTION:

We consider the image or signal denoising problem, where the goal is to remove noise from an unknown image or signal. In more detail, our goal is to obtain an estimate of an image or signal y ˚P R n from y " y ˚`η,

where η is unknown noise, often modeled as a zero-mean white Gaussian random variable with covariance matrix σ 2 {nI.

Image denoising relies on modeling or prior assumptions on the image y ˚. For example, suppose that the image y ˚lies in a k-dimensional subspace of R n denoted by Y. Then we can estimate the original image by finding the closest point in 2 -distance to the noisy observation y on the subspace Y. The corresponding estimate, denoted by ŷ, obeys

EQUATION

with high probability (throughout, }¨} denotes the 2 -norm). Thus, the noise energy is reduced by a factor of k{n over the trivial estimate ŷ " y which does not use any prior knowledge of the signal. The denoising rate (1) shows that the more concise the image prior or image representation (i.e., the smaller k), the more noise can be removed. If on the other hand the prior (the subspace, in this example) does not include the original image y ˚, then the error bound (1) increases as we would remove a significant part of the signal along with noise when projecting onto the range of the signal prior. Thus a concise and accurate prior is crucial for denoising.

Real world signals rarely lie in a priori known subspaces, and the last few decades of image denoising research have developed sophisticated and accurate image models or priors and algorithms.

Examples include models based on sparse representations in overcomplete dictionaries such as wavelets (Donoho, 1995) and curvelets (Starck et al., 2002) , and algorithms based on exploiting self-similarity within images (Dabov et al., 2007) . A prominent example of the former class of algorithms is the BM3D (Dabov et al., 2007) algorithm, which achieves state-of-the-art performance for certain denoising problems. However, the nuances of real world images are difficult to describe with handcrafted models. Thus, starting with the paper (Elad & Aharon, 2006 ) that proposes to learn sparse representation based on training data, it has become common to learn concise representation for denoising (and other inverse problems) from a set of training images.

In 2012, Burger et al. (Burger et al., 2012) applied deep networks to the denoising problem, by training a deep network on a large set of images. Since then, deep learning based denoisers (Zhang et al., 2017) have set the standard for denoising. The success of deep network priors can be attributed to their ability to efficiently represent and learn realistic image priors, for example via autodecoders (Hinton & Salakhutdinov, 2006) and generative adversarial models (Goodfellow et al., 2014) . Over the last few years, the quality of deep priors has significantly improved (Karras et al., 2017; Ulyanov et al., 2017) . As this field matures, priors will be developed with even smaller latent code dimensionality and more accurate approximation of natural signal manifolds. Consequently, the representation error from deep priors will decrease, and thereby enable even more powerful denoisers.

As the influence of deep networks in inverse problems grows, it becomes increasingly important to understand their performance at a theoretical level. Given that most optimization approaches for deep learning are first order gradient methods, a justification is needed for why they do not get stuck in local minima. The closest theoretical work to this question is Bora et al. (2017) , which solves a noisy compressive sensing problem with generative priors by minimizing empirical risk. Under the assumption that the network is Lipschitz, they show that if the global optimizer can be found, which is in principle NP-hard, then a signal estimate is recovered to within the noise level. While the Lipschitzness assumption is quite mild, the resulting theory does not provide justification for why global optimality can be reached.

The most related work that establishes theoretical reasons for why gradient methods would not get stuck in local minima, when using deep generative priors for solving inverse problems, is Hand & Voroninski (2018) . In it, the authors establish global favorability for optimization of the noiseless empirical risk function. Specifically, they show existence of a descent direction outside a ball around the global optimizer and a negative multiple of it in the latent space of the generative model. This work does not provide a specific algorithm which provably estimates the global minimizer, nor does it provide an analysis of the robustness of the problem with respect to noise.

In this paper, we propose the first algorithm for solving denoising with deep generative priors that provably finds an approximation of the underlying image. Following the lead of Hand & Voroninski (2018) , we assume an expansive Gaussian model for the deep generative network in order to establish this result.

Contributions::

The goal of this paper is to analytically quantify the denoising performance of deep-prior based denoisers. Specifically, we characterize the performance of a simple and efficient algorithm for denoising based on a d-layer generative neural network G : R k Ñ R n , with k ă n, and random weights. In more detail, we propose a gradient method with a tweak that attempts to minimize the least-squares loss f pxq " 1 2 }Gpxq ´y} 2 between the noisy image y and an image in the range of the prior, Gpxq. While f is non-convex, we show that the gradient method yields an estimate x obeying

}Gpxq ´y˚} 2 À σ 2 k n ,

with high probability, where the notation À absorbs a constant factor depending on the number of layers of the network, and its expansitivity, as discussed in more detail later. Our result shows that the denoising rate of a deep prior based denoiser is determined by the dimension of the latent representation.

We also show in numerical experiments, that this rate-shown to be analytically achieved for random priors-is also experimentally achieved for priors learned from real imaging data. 

2. PROBLEM FORMULATION:

We consider the problem of estimating a vector y ˚P R n from a noisy observation y " y ˚`η. We assume that the vector y ˚belongs to the range of a d-layer generative neural network G : R k Ñ R n , with k ă n. That is, y ˚" Gpx ˚q for some x ˚P R k . We consider a generative network of the form

Gpxq " relupW d . . . relupW 2 relupW 1 x ˚qq . . .q,

where relupxq " maxpx, 0q applies entrywise, W i P R niˆni´1 , are the weights in the i-th layer, n i is the number of neurons in the ith layer, and the network is expansive in the sense that k " n 0 ă n 1 ă ¨¨¨ă n d " n. The problem at hand is: Given the weights of the network W 1 . . . W d and a noisy observation y, obtain an estimate ŷ of the original image y ˚such that }ŷ ´y˚} is small and ŷ is in the range of G.

3. DENOISING VIA EMPIRICAL RISK MINIMIZATION:

As a way to solve the above problem, we first obtain an estimate of x ˚, denoted by x, and then estimate y ˚as Gpxq. In order to estimate x ˚, we minimize the empirical risk objective

f pxq :" 1 2 }Gpxq ´y} 2 .

Since this objective is nonconvex, there is no a priori guarantee of efficiently finding the global minimum. Approaches such as gradient methods could in principle get stuck in local minima, instead of finding a global minimizer that is close to x ˚.

However, as we show in this paper, under appropriate conditions, a gradient method with a tweakintroduced next-finds a point that is very close to the original latent parameter x ˚, with the distance to the parameter x ˚controlled by the noise. In order to state the algorithm, we first introduce a useful quantity. For analyzing which rows of a matrix W are active when computing relupW xq, we let

W `,x " diagpW x ą 0qW.

For a fixed weight matrix W , the matrix W `,x zeros out the rows of W that do not have a positive dot product with x. Alternatively put, W `,x contains weights from only the neurons that are active for the input x. We also define W 1,`,x " pW 1 q `,x " diagpW 1 x ą 0qW 1 and

W i,`,x " diagpW i W i´1,`,x ¨¨¨W 2,`,x W 1,`,x x ą 0qW i .

The matrix W i,`,x consists only of the weights of the neurons in the ith layer that are active if the input to the first layer is x.

We are now ready to state our algorithm: a gradient method with a tweak informed by the loss surface of the function to be minimized. Given a noisy observation y, the algorithm starts with an arbitrary initial point x 0 ‰ 0. At each iteration i " 0, 1, . . ., the algorithm computes the step direction ṽxi " pΠ 1 i"d W i,`,xi q t pGpx i q ´yq, which is equal to the gradient of f if f is differentiable at x i . It then takes a small step opposite to ṽxi . The tweak is that before each iteration, the algorithm checks whether f p´x i q is smaller than f px i q, and if so, negates the sign of the current iterate x i . This tweak is informed by the loss surface. To understand this step, it is instructive to examine the loss surface for the noiseless case in Figure 1 . It can be seen that while the loss function has a global minimum at x ˚, it is relatively flat close to ´x˚. In expectation, there is a critical point that is a negative multiple of x ˚with the property that the curvature in the ˘x˚d irection is positive, and the curvature in the orthogonal directions is zero. Further, around approximately ´x˚, the loss function is larger than around the optimum x ˚. As a simple gradient descent method (without the tweak) could potentially get stuck in this region, the negation check provides a way to avoid converging to this region. Our algorithm is formally summarized as Algorithm 1 below.

Algorithm 1 Gradient method Require: Weights of the network W i , noisy observation y, and step size α ą 0 1: Choose an arbitrary initial point x 0 P R k zt0u 2: for i " 0, 1, . . . do 3:

if f p´x i q ă f px i q then 4:

x i Ð ´xi ; 5: end if 6:

Compute ṽxi " pΠ 1 i"d W i,`,xi q t pGpx i q ´yq 7:

x i`1 " x i ´αṽ xi 8: end for Other variations of the tweak are also possible. For example, the negation check in Step 2 could be performed after a convergence criterion is satisfied, and if a lower objective is achieved by negating the latent code, then the gradient descent can be continued again until a convergence criterion is again satisfied.

4. MAIN RESULTS:

For our analysis, we consider a fully-connected generative network G : R k Ñ R n with Gaussian weights and no bias terms. Specifically, we assume that the weights W i are independently and identically distributed as N p0, 2{n i q, but do not require them to be independent across layers. Moreover, we assume that the network is sufficiently expansive: Expansivity condition. We say that the expansivity condition with constant ą 0 holds if

n i ě c ´2 logp1{ qn i´1 log n i´1 , for all i,

where c is a particular numerical constant.

In a real-world generative network the weights are learned from training data, and are not drawn from a Gaussian distribution. Nonetheless, the motivation for selecting Gaussian weights for our analysis is as follows:

1. The empirical distribution of weights from deep neural networks often have statistics consistent with Gaussians. AlexNet is a concrete example (Arora et al., 2015) .

2. The field of theoretical analysis of recovery guarantees for deep learning is nascent, and Gaussian networks can permit theoretical results because of well developed theories for random matrices.

3. It is not clear which non-Gaussian distribution for weights is superior from the joint perspective of realism and analytical tractability.

4. Truly random nets, such as in the Deep Image Prior (Ulyanov et al., 2017) , are increasingly becoming of practical relevance. Thus, theoretical advances on random nets is of independent interest.

We are now ready to state our main result. Theorem 1. Consider a network with the weights in the i-th layer, W i P R niˆni´1 , i.i.d. N p0, 2{n i q distributed, and suppose that the network satisfies the expansivity condition for some ď K{d 90 . Also, suppose that the noise variance obeys

ω ď }x ˚}K 1 d 16 , ω :" c 18σ 2 k n logpn d 1 n d´1 2 . . . n d q.

Consider the iterates of Algorithm 1 with stepsize α " K 4 1 d 2 . Then, there exists a number of steps N upper bounded by

N ď K 2 d 4

f px 0 q }x ˚} such that after N steps, the iterates of Algorithm 1 obey

EQUATION

with probability at least 1´2e ´2k log n ´řd i"2 8n i e ´K7ni´2 ´8n 1 e ´K7 2 logp1{ qk . Here, K 1 , K 2 , .. are numerical constants, and x 0 is the initial point in the optimization.

The error term in the bound (2) consists of two terms-the first is controlled by , and the second depends on the noise. The first term is negligible if is chosen sufficiently small, but that comes at the expense of the expansivity condition being more stringent. The second term in the bound (2) is more interesting and controls the effect of noise. Specifically, for sufficiently small, our result guarantees that after sufficiently many iterations,

}x i ´x˚} 2 À σ 2 k n ,

where the notation À absorbs a factor logarithmic in n and polynomial in d. One can show that G is Lipschitz in a region around x ˚1,

}Gpx i q ´Gpx ˚q} 2 À σ 2 k n .

Thus, the theorem guarantees that our algorithm yields the denoising rate of σ 2 k{n, and, as a consequence, denoising based on a generative deep prior provably reduces the energy of the noise in the original image by a factor of k{n. We note that the intention of this paper is to show rate-optimality of recovery with respect to the noise power, the latent code dimensionality, and the signal dimensionality. As a result, no attempt was made to establish optimal bounds with respect to the scaling of constants or to powers of d. The bounds provided in the theorem are highly conservative in the constants and dependency on the number of layers, d, in order to keep the proof as simple as possible. Numerical experiments shown later reveal that the parameter range for successful denoising are much broader than the constants suggest. As this result is the first of its kind for rigorous analysis of denoising performance by deep generative networks, we anticipate the results can be improved in future research, as has happened for other problems, such as sparsity-based compressed sensing and phase retrieval.

4.1. THE WEIGHT DISTRIBUTION CONDITION (WDC):

To prove our main result, we make use of a deterministic condition on G, called the Weight Distribution Condition (WDC), and then show that Gaussian W i , as given by the statement of Theorem 1 are such that W i { ? 2 satisfies the WDC with the appropriate probability for all i, provided the expansivity condition holds. Our main result, Theorem 1, continues to hold for any weight matrices such that W i { ? 2 satisfy the WDC.

The condition is on the spatial arrangement of the network weights within each layer. We say that the matrix W P R nˆk satisfies the Weight Distribution Condition with constant if for all nonzero x, y P R k ,

EQUATION

where

w i P R k is the ith row of W ; M xØŷ P R kˆk is the matrix 2 such that x Þ Ñ ŷ, ŷ Þ Ñ x,

and z Þ Ñ 0 for all z P spanptx, yuq K ; x " x{}x} 2 and ŷ " y{}y} 2 ; θ 0 " =px, yq; and 1 S is the indicator function on S. The norm in the left hand side of (3) is the spectral norm. Note that an elementary calculation 3 gives that Q x,y " Er ř n i"1 1 wi,x ą0 1 wi,y ą0 ¨wi w t i s for w i " N p0, I k {nq. As the rows w i correspond to the neural network weights of the ith neuron in a layer given by W , the WDC provides a deterministic property under which the set of neuron weights within the layer given by W are distributed approximately like a Gaussian. The WDC could also be interpreted as a deterministic property under which the neuron weights are distributed approximately like a uniform random variable on a sphere of a particular radius. Note that if x " y, Q x,y is an isometry up to a factor of 1{2.

5. APPLICATIONS TO COMPRESSED SENSING:

In this section we briefly discuss another important scenario to which our results apply to, namely regularizing inverse problems using deep generative priors. Approaches that regularize inverse problems using deep generative models (Bora et al., 2017) have empirically been shown to improve over sparsity-based approaches, see (Lucas et al., 2018) for a review for applications in imaging, and (Mardani et al., 2017) for an application in Magnetic Resonance Imaging showing a significant performance improvement over conventional methods.

Consider an inverse problem, where the goal is to reconstruct an unknown vector y ˚P R n from m ă n noisy linear measurements:

z " Ay ˚`η P R m ,

where A P R mˆn is called the measurement matrix and η is zero mean Gaussian noise with covariance matrix σ2 {nI, as before. As before, assume that y ˚lies in the range of a generative prior G, i.e., y ˚" Gpx ˚q for some x ˚. As a way to recover x ˚, consider minimizing the empirical risk objective f pxq " 1 2 }AGpxq ´z}, using Algorithm 1, with Step 6 substituted by ṽxi " pAΠ 1 i"d W i,`,xi q t pAGpx i q ´yq, to account for the fact that measurements were taken with the matrix A.

Suppose that A is a random projection matrix, for concreteness assume that A has i.i.d. Gaussian entries with variance 1{m. One could prove an analogous result as Theorem 1, but with ω " b 18σ 2 k m logpn d 1 n d´1 2 . . . n d q, (note that n has been replaced by m). This extension shows that, provided is chosen sufficiently small, that our algorithm yields an iterate x i obeying

}Gpx i q ´Gpx ˚q} 2 À σ 2 k m ,

where again À absorbs factors logarithmic in the n i 's, and polynomial in d. Proving this result would be analogous to the proof of Theorem 1, but with the additional assumption that the sensing matrix A acts like an isometry on the union of the ranges of Π 1 i"d W i,`,xi , analogous to the proof in (Hand & Voroninski, 2018) . This extension of our result shows that Algorithm 1 enables solving inverse problems under noise efficiently, and quantifies the effect of the noise. map to e1 and cos θ0 ¨e1 `sin θ0 ¨e2 respectively, then M xØŷ " R t ¨cos θ0 sin θ0 0 sin θ0 ´cos θ0 0 0 0 0 k´2 ‚R, where 0 k´2 is a k ´2 ˆk ´2 matrix of zeros. If θ0 " 0 or π, then M xØŷ " xx t or ´xx t , respectively. 3 To do this calculation, take x " e1 and y " cos θ0 ¨e1 `sin θ0 ¨e2 without loss of generality. Then each entry of the matrix can be determined analytically by an integral that factors in polar coordinates.

We hasten to add that the paper (Bora et al., 2017) also derived an error bound for minimizing empirical loss. However, the corresponding result (for example Lemma 4.3) differs in two important aspects to our result. First, the result in (Bora et al., 2017) only makes a statement about the minimizer of the empirical loss and does not provide justification that an algorithm can efficiently find a point near the global minimizer. As the program is non-convex, and as non-convex optimization is NP-hard in general, the empirical loss could have local minima at which algorithms get stuck. In contrast, the present paper presents a specific practical algorithm and proves that it finds a solution near the global optimizer regardless of initialization. Second, the result in (Bora et al., 2017) considers arbitrary noise η and thus can not assert denoising performance. In contrast, we consider a random model for the noise, and show the denoising behavior that the resulting error is no more than Opk{nq, as opposed to }η} 2 « Op1q, which is what we would get from direct application of the result in (Bora et al., 2017) .

6. EXPERIMENTAL RESULTS:

In this section we provide experimental evidence that corroborates our theoretical claims that denoising with deep priors achieves a denoising rate proportional to σ 2 k{n. We consider both a synthetic, random prior, as studied theoretically in the paper, as well as a prior learned from data. All our results are reproducible with the code provided in the supplement.

6.1. DENOISING WITH A SYNTHETIC PRIOR:

We start with a synthetic generative network prior with ReLu-activation functions, and draw its weights independently from a Gaussian distribution. We consider a two-layer network with n " 1500 neurons in the output layer, 500 in the middle layer, and vary the number of input neurons, k, and the noise level, σ. We next present simulations showing that if k is sufficiently small, our algorithm achieves a denoising rate proportional to σk{n as guaranteed by our theory.

Towards this goal, we generate Gaussian inputs x ˚to the network and observe the noisy image y " Gpx ˚q `η, η " N p0, σ 2 {nIq. From the noisy image, we first obtain an estimate x of the latent representation by running Algorithm 1 until convergence, and second we obtain an estimate of the image as ŷ " Gpxq. In the left and middle panel of Figure 3 , we depict the normalized mean squared error of the latent representation, MSEpx, x ˚q, and the mean squared error in the image domain, MSEpGpxq, Gpx ˚qq, where we defined MSEpz, z 1 q " }z ´z1 } 2 . For the left panel, we fix the noise variance to σ 2 " 0.25, and vary k, and for the middle panel we fix k " 50 and vary the noise variance. The results show that, if the network is sufficiently expansive, guaranteed by k being sufficiently small, then in the noiseless case (σ 2 " 0), the latent representation and image are perfectly recovered. In the noisy case, we achieve a MSE proportional to σ 2 k{n, both in the representation and image domains.

We also observed that for the problem instances considered here, the negation trick in step 3-4 of Algorithm 1 is often not necessary, in that even without that step the algorithm typically converges to the global minimum. Having said this, in general the negation step is necessary, since there exist problem instances that have a local minimum opposite of x ˚.

6.2. DENOISING WITH A LEARNED PRIOR:

We next consider a prior learned from data. Technically, for such a prior our theory does not apply since we assume the weights to be chosen at random. However, the numerical results presented in this section show that even for the learned prior we achieve the rate predicted by our theory pertaining to a random prior. Towards this goal, we consider a fully-connected autoencoder parameterized by k, consisting of an decoder and encoder with ReLu activation functions and fully connected layers. We choose the number of neurons in the three layers of the encoder as 784, 400, k, and those of the decoder as k, 400, 784. We set k " 10 and k " 20 to obtain two different autoencoders. We train both autoencoders on the MNIST (Lecun et al., 1998) training set.

We then take an image y ˚from the MNIST test set, add Gaussian noise to it, and denoise it using our method based on the learned decoder-network G for k " 10 and k " 20. Specifically, we estimate noisy denoised noise variance 0 0.3 0.6 0.9 1.2 1.5 1.8 2.1 2.4 2.7 As suggested by the theory pertaining to decoders with random weights, if k is sufficiently small, and thus the network is sufficiently expansive, the denoising rate is proportional to σ 2 k{n. Right panel: Denoising of handwritten digits based on a learned decoder with k " 10 and k " 20, along with the least-squares fit as dotted lines. The learned decoder with k " 20 has more parameters and thus represents the images with a smaller error; therefore the MSE at σ " 0 is smaller. However, the denoising rate for the decoder with k " 20, which is the slope of the curve is larger as well, as suggested by our theory.

the latent representation x by running Algorithm 1, and then set ŷ " Gpxq. See Figure 2 for a few examples demonstrating the performance of our approach for different noise levels.

We next show that this achieves a mean squared error (MSE) proportional to σ 2 k{n, as suggested by our theory which applies for decoders with random weights. We add noise to the images with noise variance ranging from σ 2 " 0 to σ 2 " 6. In the right panel of Figure 3 we show the MSE in the image domain, MSEpGpxq, Gpx ˚qq, averaged over a number of images for the learned decoders with k " 10 and k " 20. We observe an interesting tradeoff: The decoder with k " 10 has fewer parameters, and thus does not represent the digits as well, therefore the MSE is larger than that for k " 20 for the noiseless case (i.e., for σ " 0). On the other hand, the smaller number of parameters results in a better denoising rate (by about a factor of two), corresponding to the steeper slope of the MSE as a function of the noise variance, σ 2 .

A PROOFS:

In this section we prove our main result, Theorem 1. Instead of proving Theorem 1 as stated, we will prove the following equivalent rescaled statement for when W i have i.i.d. N p0, 1{n i q entries. Because of this rescaling, Gpxq scales like 2 ´d{2 }x}, the noise ω is assumed to scale like 2 ´d{2 , ∇f scales like 2 d , and α scales like 2 d . Theorem 2. Consider a network with the weights in the i-th layer, W i P R niˆni´1 , i.i.d. N p0, 1{n i q distributed, and suppose that the network satisfies the expansivity condition for some ď K{d 90 . Also, suppose that the noise variance obeys

ω ď }x ˚}K 1 2 ´d{2 d 16

, ω :"

c 18σ 2 k n logpn d 1 n d´1 2 . . . n d q.

Consider the iterates of Algorithm 1 with stepsize α " K 4

2 d d 2 .

Then, there exists a number of steps N upper bounded by

N ď K 2 d 4

f px 0 q2 d }x ˚} such that after N steps, the iterates of Algorithm 1 obey

EQUATION

with probability at least 1´2e ´2k log n ´řd i"2 8n i e ´K7ni´2 ´8n 1 e ´K7 2 logp1{ qk . Here, K 1 , K 2 , .. are numerical constants, and x 0 is the initial point in the optimization.

As mentioned in Section 4.1, our proof makes use of a deterministic condition, called the Weight Distribution Condition (WDC), formally defined in Section 4.1. The following proposition establishes that the expansivity condition ensures that the WDC holds: Lemma 3 (Lemma 9 in (Hand & Voroninski, 2018) ). Fix P p0, 1q. If the entires of W i P R niˆni´1 are i.i.d. N p0, 1{n i q and the expansivity condition n i ą c ´2 logp1{ qn i´1 log n i´1 holds, then W i satisfies the WDC with constant with probability at least 1 ´8n i e ´K 2 ni´1 . Here, c and K are numerical constants.

We note that the form of dependence of n i on can be read off the proofs of Lemma 10 in (Hand & Voroninski, 2018) . It follows from Lemma 3, that the WDC holds for all W i with probability at least 1 ´řd i"2 8n i e ´K7ni´2 ´8n 1 e ´K7 2 logp1{ qk . In the remainder of the proof we work on the event that the WDC holds for all W i .

A.1 PRELIMINARIES:

Recall that the goal of our algorithm is to minimize the empirical risk objective

f pxq " 1 2 }Gpxq ´y} 2 ,

where y :" Gpx ˚q `η, with η " N p0, σ 2 {nIq.

Our results rely on the fact that outside of two balls around x " x ˚and x " ´ρd x ˚, with ρ d a constant defined below, the direction chosen by the algorithm is a descent direction, with high probability. Towards this goal, we use a concentration argument, similar to the arguments used in (Hand & Voroninski, 2018) . First, define Λ x :" Π 1 i"d W i,`,x (with W i,`,x defined in Section 3) for notational convenience, and note that the step direction of our algorithm can be written as ṽx " v x `q x , with v x :" Λ t

x Λ x x ´pΛ x q t pΛ x˚q x ˚, and qx :" Λ t x η.

(5) Note that at points x where G (and hence f ) is differentiable, we have that ṽx " ∇f pxq.

The proof is based on showing that ṽx concentrates around a particular h x P R k , defined below, that is a continuous function of nonzero x, x ˚and is zero only at x " x ˚and x " ´ρd x ˚. The definition of h x depends on a function that is helpful for controlling how the operator x Þ Ñ W `,x x distorts angles, defined as:

gpθq :" cos ´1´p π ´θq cos θ `sin θ π

¯. (6):

With this notation, we define

h x :" ´1 2 d ´d´1 ź i"0 π ´θi π ¯x˚`1 2 d « x ´d´1 ÿ i"0 sin θ i π ´d´1 ź j"i`1 π ´θj π ¯}x ˚}2 }x} 2 x ff ,

where θ 0 " =px, x ˚q and θ i " gpθ i´1 q. Note that h x is deterministic and only depends on x, x ˚, and the number of layers, d.

In order to bound the deviation of ṽx from h x we use the following two lemmas, bounding the deviation controlled by the WDC and the deviation from the noise: Lemma 4 (Lemma 6 in (Hand & Voroninski, 2018) ). Suppose that the WDC holds with ă 1{p16πd 2 q 2 . Then, for all nonzero x, x ˚P R k ,

EQUATION

EQUATION

EQUATION

Proof. Equation ( 7) and ( 8) are Lemma 6 in (Hand & Voroninski, 2018) . Regarding ( 9), note that the WDC implies that }W i,`,x } 2 ď 1{2 ` . It follows that

› › Π 1 i"d W i,`,x › › 2 ď 1 2 d p1 `2 q d " 1 2 d e d logp1`2 q ď 1 `4 d 2 d ď 13 12 2 ´d,

where the last inequalities follow by our assumption on .

Lemma 5. Suppose the WDC holds with ă 1{p16πd 2 q 2 , that any subset of n i´1 rows of W i are linearly independent for each i, and that η " N p0, σ 2 {nIq. Then the event

E noise :" ! › › pΠ 1 i"d W i,`,x q t η › › ď ω 2 d{2 , for all x )

, ω :"

c 16σ k n logpn d 1 n d´1 2 . . . n d q (10)

holds with probability at least 1 ´2e ´2k log n .

As the cost function f is not differentiable everywhere, we will make use of the generalized subdifferential in order to reference the subgradients at nondifferentiable points. For a Lipschitz function f defined from a Hilbert space X to R, the Clarke generalized directional derivative of f at the point x P X in the direction u, denoted by f o px; uq, is defined by f o px; uq " lim sup yÑx,tÓ0

f py`tuq´f pyq t

, and the generalized subdifferential of f at x, denoted by B f pxq, is defined by B f pxq " tv P R k | xv, uy ď f o px; uq, for all u P X u. Since f pxq is a piecewise quadratic function, we have

EQUATION

)

where conv denotes the convex hull of the vectors v 1 , . . . , v t , t is the number of quadratic functions adjoint to x, and v i is the gradient of the i-th quadratic function at x. Lemma 6. Under the assumption of Lemma 5, and assuming that E noise holds, we have that, for any x ‰ 0 and any v x P Bf pxq,

}v x ´hx } ď K d 3 ? 2 d maxp}x} 2 , }x ˚}2 q `ω 2 d{2 .

In particular, this holds for the subgradient v x " ṽx .

Proof. By (11), Bf pxq " convpv 1 , . . . v t q for some finite t, and thus v x " a 1 v 1 `. . . a t v t for some a 1 , . . . , a t ě 0, ř i a i " 1. For each v i , there exists a w such that v i " lim tÓ0 ṽx`tw . On the event E noise , we have that for any x ‰ 0, for any ṽx P Bf pxq }ṽ x ´hx } " }v x `q x ´hx } ď }v

x ´hx } `}q x } ď K d 3 ? 2 d maxp}x} 2 , }x ˚}2 q `ω 2 d{2 ,

where the last inequality follows from Lemmas 4 and 5 above. The proof is concluded by appealing to the continuity of h x with respect to nonzero x, and by noting that

}v x ´hx } ď ÿ i a i }v i ´hx } ď K d 3 ? 2 d maxp}x} 2 , }x ˚}2 q `ω 2 d{2 ,

where we used the inequality above and that ř i a i " 1.

We will also need an upper bound on the norm of the step direction of our algorithm: Lemma 7. Suppose that the WDC holds with ă 1{p16πd 2 q 2 and that the event E noise holds with ω ď 2 ´d{2 }x˚} 8π

. Then, for all x,

EQUATION

)

where K is a numerical constant.

Proof. Define for convenience ζ j " ś d´1 i"j π´θj,x,x π . We have

}ṽ x } ď}h x } `}h x ´ṽ x } ď › › › › › 1 2 d x ´1 2 d ζ 0 x ˚´1 2 d d´1 ÿ i"0 sin θi,x π ζ i`1 }x ˚} }x} x › › › › › `K1 d 3 ? 2 d maxp}x} 2 , }x ˚}2 q `ω 2 d{2 ď 1 2 d }x} `ˆ1 2 d `d π2 d ˙}x ˚} `K1 d 3 ? 2 d maxp}x}, }x ˚}q `ω 2 d{2

ď dK 2 d maxp}x}, }x ˚}q, where the second inequality follows from the definition of h x and Lemma 6, the third inequality uses |ζ j | ď 1, and the last inequality uses the assumption ω ď 2 ´d{2 }x˚} 8π

.

A.2 PROOF OF THEOREM 2

We are now ready to prove Theorem 2. The logic of the proof is illustrated in Figure 4 . Recall that x i is the ith iterate of x as per Algorithm 1. We first ensure that we can assume throughout that x i is bounded away from zero: Lemma 8. Suppose that WDC holds with ă 1{p16πd 2 q 2 and that E noise holds with ω in (10)

obeying ω ď 2 ´d{2 }x˚} 8π

. Moreover, suppose that the step size in Algorithm 1 satisfies 0 ă α ă K2 d d 2 , where K is a numerical constant. Then, after at most N " p 38πK02 d α q 2 steps, we have that for all i ą N that x i R Bp0, K 0 }x ˚}q, K 0 " 1 32π .

In particular, if α " K2 d {d 2 , then N is bounded by a constant times d 4 .

We can therefore assume throughout this proof that x i R Bp0, K 0 }x ˚}q, K 0 " 1 32π . We prove Theorem 2 by showing that if }h x } is sufficiently large, i.e., if the iterate x i is outside of set

S β " ! x P R k | }h x } ď 1 2 d β maxp}x}, }x ˚}q ) , x ρd x ˚0 S β S β

Figure 4 : Logic of the proof: Starting at an arbitrary point, Algorithm 1 moves away from 0, at least till its iterates are outside the gray ring, as 0 is a local maximum; and once an iterate x i leaves the gray ring around 0, all subsequent iterates will never be in the white circle around 0 again (see Lemma 8). Then the algorithm might move towards ´ρd x ˚, but once it enters the dashed ball around ´ρd x ˚, it enters a region where the function value is strictly larger than that of the dashed ball around x ˚, by Lemma 10. Thus steps 3-5 of the algorithm will ensure that the next iterate x i is in the dashed ball around x ˚. From there, the iterates will move into the region S β , since outside of S β Y S β the algorithm chooses a descent direction in each step (see the argument around equation ( 16)). The region S β is covered by a ball of radius r, by Lemma 9, determined by the noise and .

with

EQUATION

then the algorithm makes progress in the sense that f px i`1 q´f px i q is smaller than a certain negative value. The set S β is contained in two balls around x ˚and ´ρx ˚, whose radius is controlled by β:

Lemma 9. For any β ď 1 64 2 d 12 ,

EQUATION

)

Here, ρ d ą 0 is defined in the proof and obeys ρ d Ñ 1 as d Ñ 8.

Note that by the assumption ω ď

}x˚}K12 ´d{2 d 16

and Kd 45 ? ď 1, our choice of β in (13) obeys β ď 1 64 2 d 12 for sufficiently small K 1 , K, and thus Lemma 9 yields:

S β Ă Bpx ˚, rq Y Bp´ρ d x ˚, b r}x ˚}d 8 q.

were we define the radius r " K 2 d 9 ? }x ˚} `K3 d 6 ω2 d{2 , where K 2 , K 3 are numerical constants. Note that hat the radius r is equal to the right hand side in the error bound (4) in our theorem. In order to guarantee that the algorithm converges to a ball around x ˚, and not to that around ´ρd x ˚, we use the following lemma:

Lemma 10. Suppose that the WDC holds with ă 1{p16πd 2 q 2 . Moreover suppose that E noise holds, and that ω in the event E noise obeys ω 2 ´d{2 }x˚}2 ď K 9 {d 2 , where K 9 ă 1 is a universal constant. Then for any φ d P rρ d , 1s, it holds that

f pxq ă f pyq (15)

for all x P Bpφ d x ˚, K 3 d ´10 }x ˚}q and y P Bp´φ d x ˚, K 3 d ´10 }x ˚}q, where K 3 ă 1 is a universal constant.

In order to apply Lemma 10, define for convenience the two sets:

S β :"S β X Bpx ˚, rq, and

S β :"S β X Bp´ρ d x ˚, b r}x ˚}d 8 q.

By the assumption that Kd 45 ? ď 1 and ω ď K 1 d ´16 2 ´d{2 }x ˚}, we have that for sufficiently small K 1 , K,

S β Ď Bpx ˚, K 3 d ´10 }x ˚}q and S β Ď Bp´ρ d x ˚, K 3 d ´10 }x ˚}q.

Thus, the assumptions of Lemma 10 are met, and the lemma implies that for any x P S β and y P S β , it holds that f pxq ą f pyq. We now show that the algorithm converges to a point in S β . This fact and the negation step in our algorithm (line 3-5) establish that the algorithm converges to a point in S β if we prove that the objective is nonincreasing with iteration number, which will form the remainder of this proof. Consider i such that x i R S β . By the mean value theorem (Clason, 2017, Theorem 8.13 ), there is a t P r0, 1s such that for xi " x i ´tαṽ xi there is a v xi P Bf px i q, where Bf is the generalized subdifferential of f , obeying f px i ´αṽ xi q ´f px i q "xv xi , ´αṽ xi y "xṽ xi , ´αṽ xi y `xv xi ´ṽ xi , ´αṽ xi y ď ´α}ṽ xi } 2 `α}v xi ´ṽ xi }}ṽ xi } " ´α}ṽ xi }p}ṽ xi } ´}v xi ´ṽ xi }q.

(16)

In the next subsection, we guarantee that for any t P r0, 1s, v xi with xi " x i ´tαṽ xi is close to ṽxi :

EQUATION

Applying ( 17) to ( 16) yields

f px i ´αṽ xi q ´f px i q ď ´1 12 α}ṽ xi } 2 2 ,

where we used that αK 7

d 2 2 d ď 1

12 , by our assumption on the stepsize α being sufficiently small. Thus, the maximum number of iterations for which x i R S β is f px 0 q12{pα min i }ṽ xi } 2 q. We next lower-bound }ṽ xi }. We have that on E noise , for all x R S β , with β given by (13).

}ṽ x } 2 ě }h x } ´}h x ´ṽ x } ě 2 ´d maxp}x}, }x ˚}q ´β ´K1 d 3 ? ´ω 2 d{2 }x ˚} ě 2 ´d maxp}x}, }x ˚}q ˆ3Kd 3 ? `12ω 2 d{2 }x ˚} ˙(18) ě 2 ´d}x ˚}3K d 3 ?

where the second inequality follows by the definition of S β and Lemma 6, and the third inequality follows from our definition of β in (13). Thus,

f px i ´αṽ xi q ´f px i q ď ´αK 5 2 ´2d d 6 }x ˚}2 ď ´2´d d 4 K 6 }x ˚}2

where we used α " K 4 2 d d 2 . Hence, there can be at most f px0q2 d K6d 4 }x˚} 2 iterations for which x i R S β . In order to conclude our proof, we remark that once x i is inside a ball of radius r around x ˚, the iterates do not leave a ball of radius 2r around x ˚. To see this, note that by (12) and our choice of stepsize,

α}ṽ xi } ď K d maxp}x i }, }x ˚}q.

This concludes our proof.

The remainder of the proof is devoted to prove the lemmas used in this section.

A.3 PROOF OF EQUATION (17):

Our proof relies on h x being Lipschitz, as formalized by the lemma below, which is proven in Section A.9:

Lemma 11. For any x, y R Bp0, K 0 }x ˚}q, where K 0 and K 4 are numerical constants,

}h x ´hy } ď K 4 d 2 2 d }x ´y}.

By Lemma 11, for all t P r0, 1s and i ą N (recall that by Lemma 8, after at most N steps,

EQUATION

where xi " x i ´tαṽ xi . Thus, we have that on E noise , for any v xi P Bf px i q by Lemma 6,

EQUATION

where the second inequality is from Lemma 6 and Equation 19, and the fourth inequality is from ( 12) and the assumption ω 2 ´d{2 }x˚}2 ď K 9 {d 2 . Combining (20) and ( 18), we get that

}v xi ´ṽ xi } ď ˆ5 6 `αK 7 d 2 2 d ˙}ṽ xi },

with the appropriate constants chosen sufficiently small. This concludes the proof of Equation (17).

A.4 PROOF OF LEMMA 8

First suppose that x i P Bp0, 2K 0 }x ˚}q. We show that after a polynomial number of iterations N , we have that x i`N R Bp0, 2K 0 }x ˚}q. Below, we prove that

x, ṽx ă 0 and }ṽ x } ě 1 2 d 16π }x ˚} for all x P Bp0, 2K 0 }x ˚}q.

It follows that for any x i P Bp0, 2K 0 }x ˚}q, x i and the next iterate produced by the algorithm, x i`1 " x i ´αṽ xi , form an obtruse triangle. As a consequence,

}x i`1 } 2 ě }x i } 2 `α2 }ṽ xi } 2 ě }x i } 2 `α2 1 p2 d 16πq 2 }x ˚}2 ,

where the last inequality follows from (21). Thus, the norm of the iterates x i will increase until after `2K02 d 16π α ˘2 iterations, we have x i`N R Bp0, 2K 0 }x ˚}q.

The proof of the lemma is concluded by showing that

EQUATION

)

As a consequence, in a polynomial number N of steps, for each iterate, we have that x i R Bp0, K 0 }x ˚}q, for all i ě N , as claimed.

We next prove the implication (22). Consider x i R Bp0, 2K 0 }x ˚}q, and note that

}x i`1 } " }x i ´αṽ xi } ě }x i } ´α}ṽ xi } ě }x i } ´α dK 2 d maxp}x i }, }x ˚}q ě }x i } ´α dK 2 d }x i } 2K 0 ě }x i } ´1 2 }x i }

where the second inequality follows from ( 12), the third inequality from }x i } ě 2K 0 }x ˚}, and finally the last inequality from our assumption on the stepsize α. This concludes the proof of ( 22).

Proof of (21): It remains to prove (21). We start with proving x, ṽx ă 0. For brevity of notation, let Λ z " ś 1 i"d W i,`,z . We have

x T ṽx " Λ T x Λ x x ´ΛT x Λ x˚x˚`Λ T x η, x ď 13 12 2 ´d}x} 2 ´1 4π 1 2 d }x}}x ˚} `}x} ω 2 d{2 ď}x} ˆ13 12 2 ´d}x} `1{p8πq 2 d }x ˚} ´1 4π 1 2 d }x ˚}ď }x} 1 2 d ˆ2}x} ´1 8π }x ˚}˙.

The first inequality follows from ( 8) and ( 9), and the second inequality follows from our assumption on ω. Therefore, for any x P Bp0, 1 16π }x ˚}q, x, ṽx ă 0, as desired. We next show that, for any x P Bp0, 1 16π }x ˚}q }ṽ x } "}Λ T

x Λ x x ´ΛT x Λ x˚x˚`Λ T x η} ě }Λ T x Λ x˚x˚} ´}Λ T x Λ x x} ´}Λ T x η} ě 1 4π 1 2 d }x ˚} ´13 12 1 2 d }x} ´w 2 d{2 ě 1 2 d ˆ1 8π ´1 16π ˙}x ˚}.

where the second inequality is from ( 8) and ( 9). This concludes the proof of (21).

A.5 PROOF OF LEMMA 5

Let Λ x " Π 1 i"d W i,`,x . We have that

}q x } 2 " › › Λ t x η › › 2 ď }Λ x } 2 }P Λx η} 2 ,

where P Λx is a projector onto the span of Λ x . As a consequence, }P Λx η} 2 is χ 2 -distributed random variable with k-degrees of freedom scaled by σ{n. A standard tail bound (see (?, p. 43)) yields that, for any β ě k,

P " }P Λx η} 2 ě 4β ı ď 2e ´β .

Next, we note that by applying Lemmas 13-14 from (Hand & Voroninski, 2018, Proof of Lem. 15 ))4 , with probability one, that the number of different matrices Λ x can be bounded as

| tΛ x |x ‰ 0u | " | Π 1 i"d W i,`,x |x ‰ 0 ( | ď 10 d 2 pn d 1 n d´1 2 . . . n d q k ď pn d 1 n d´1 2 . . . n d q 2k ,

where the second inequality holds for logp10q ď k{4 logpn 1 q. To see this, note that pn d 1 n d´1 2 . . . n d q k ě 10 d 2 is implied by kpd logpn 1 q `pd ´1q logpn 2 q `. . . logpn d qq ě kd 2 {4 logpn 1 q ě d 2 logp10q. We now show that h x is away from zero outside of a neighborhood of x ˚and ´ρd x ˚. We prove Lemma 9 by establishing the following: Lemma 12. Suppose 64d 6 ? β ď 1. Define

ρ d :" d´1 ÿ i"0 sin q θ i π ˜d´1 ź j"i`1 π ´q θ j π ¸,

where q θ 0 " π and q θ i " gp q θ i´1 q. If x P S β , then we have that either 

|θ 0 | ď 32d 4 β

EQUATION

)

Additionally, ρ d Ñ 1 as d Ñ 8.

Proof. Without loss of generality, let }x ˚} " 1, x ˚" e 1 and x " r cos θ 0 ¨e1 `r sin θ 0 ¨e2 for θ 0 P r0, πs. Let x P S β .

First we introduce some notation for convenience. Let

ξ " d´1 ź i"0 π ´θi π , ζ " d´1 ÿ i"0 sin θ i π d´1 ź j"i`1

π ´θj π , r " }x} 2 , M " maxpr, 1q.

Thus, h x " ´1 2 d ξ x0 `1 2 d pr ´ζqx. By inspecting the components of h x , we have that x P S β implies | ´ξ `cos θ 0 pr ´ζq| ď βM (24)

| sin θ 0 pr ´ζq| ď βM (25)

Now, we record several properties. We have:

EQUATION

We now establish (28). Observe 0 ă gpθq ď `1 3π `1 θ ˘´1 ": gpθq for θ P p0, πs. As g and g are monotonic increasing, we have q θ i " g ˝ip q θ 0 q " g ˝ipπq ď g˝i pπq " `i 3π `1 π ˘´1 " 3π i`3 . Similarly, gpθq ě p 1 π `1 θ q ´1 implies that q θ i ě π i`1 , establishing (29).

We now establish (30). Using (28) and θ i ď q θ i , we have

d´1 ź i"1 ´1 ´θi π ¯ě d´1 ź i"1 ´1 ´3 i `3 ¯ě d ´3,

where the last inequality can be established by showing that the ratio of consecutive terms with respect to d is greater for the product in the middle expression than for d ´3.

We establish (31) by using the fact that |g 1 pθq| ď 1 for all θ P r0, πs and using the same logic as for (Hand & Voroninski, 2018, Eq. 17) .

We now establish (33). As θ 0 " π `O1 pδq, we have θ i " q θ i `O1 piδq. Thus, if

EQUATION

Thus (33) holds.

Next, we establish that x P S β ñ r ď 4d, and thus M ď 4d. 1´?2β ď 4d. Thus, we have x P S β ñ r ď 4d ñ M ď 4d.

Next, we establish that we only need to consider the small angle case (θ 0 « 0) and the large angle case (θ 0 « π), by considering the following three cases: ? βq, as θ 0 ě π{2 and as β ă 1.

EQUATION

At least one of the Cases I,II, or III hold. Thus, we see that it suffices to consider the small angle case θ 0 " O 1 p32d 4 βq or the large angle case θ 0 " π `O1 p8πd 4 ? βq.

Small Angle Case. Assume θ 0 " O 1 pδq with δ " 32d 4 β. As θ i ď θ 0 ď δ for all i, we have 1 ě ξ ě p1 ´δ π q d " 1 `O1 p 2δd π q provided δd{π ď 1{2 (which holds by our choice δ " 32d 4 β by assumption 64d 6 ? β ď 1). By ( 27), we also have ζ " O 1 p d π δq. By (24), we have | ´ξ `cos θ 0 pr ´ζq| ď βM.

Thus, as cos θ 0 " 1 `O1 pθ

2 0 {2q " 1 `O1 pδ 2 {2q, ´´1 `O1 p 2δd π q ¯`p1 `O1 p 2δd π qqpr `O1 p δd π qq " O 1 p4dβq,

and r ď M ď 4d (shown above) provides,

EQUATION

)

By plugging in that δ " 32d 4 β, we have that r ´1 " O 1 p132d 6 βq, where we have used that

32d 5 β π ď 1{2.

Large Angle Case. Assume θ 0 " π `O1 pδq where δ " 8πd 4 ? β. By ( 32) and (33), we have ξ " O 1 pδ{πq, and we have ζ " ρ d `O1 p3d 3 δq if 8d 6 ? β ď 1. By (24), we have | ´ξ `cos θ 0 pr ´ζq| ď βM, so, as cos θ 0 " 1 ´O1 pθ 2 0 {2q, O 1 pδ{πq `p1 `O1 pδ 2 {2qqpr ´ρd `O1 p3d 3 δqq " O 1 pβM q, and thus, using r ď 4d, ρ d ď d, and δ " 8πd 4 ? β ď 1,

EQUATION

To conclude the proof of (23), we use the fact that

}x ´x˚}2 ď ˇˇ}x} 2 ´}x ˚}2 ˇˇ`p}x ˚}2 `ˇ} x} 2 ´}x ˚}2 ˇˇqθ 0 .

This fact simply says that if a 2d point is known to have magnitude within ∆r of some r and is known to be within angle ∆θ from 0, then its Euclidean distance to the point of polar coordinates pr, 0q is no more than ∆r `pr `∆rq∆θ.

Finally, we establish that ρ d Ñ 1 as d Ñ 8. Note that ρ d`1 " p1 ´q θ d π qρ d `sin q θ d π and ρ 0 " 0. It suffices to show ρd Ñ 0, where ρd :" 1 ´ρd . The following recurrence relation holds: ρd " p1 ´q θ d´1 π qρ d´1 `q θ d´1 ´sin q θ d´1 π , with ρ0 " 1. Using the recurrence formula (Hand & Voroninski, 2018, Eq. ( 15)) and the fact that q θ 0 " π, we get that ρd " Using (28) and the fact that q θ i´1 ´sin q θ i´1 ď q θ 3 i´1 {6, we have that ρd ď

d ÿ i"1 q θ i´1 ´sin q θ i´1 π d ź j"i`

ř d i"1 q θ 3 i´1 6π

¨i`1 d`1 Ñ 0 as d Ñ 8.

A.8 PROOF OF LEMMA 13:

It holds that }x ´y} ě 2 sinpθ x,y {2q minp}x}, }y}q, @x, y (46) sinpθ{2q ě θ{4, @θ P r0, πs (47) d dθ gpθq P r0, 1s @θ P r0, πs

logp1 `xq ď x @x P r´0.5, 1s (49) logp1 ´xq ě ´2x @x P r0, 0.75s (50) where θ x,y " =px, yq. We recall the results (36), (37), and ( 50) in (Hand & Voroninski, 2018) :

θi ď 3π i `3

and θi ě π i `1 @i ě 0 1 ´ρd "

d´1 ź i"1 ˆ1 ´θ i π ˙`d´1 ÿ i"1 θi ´sin θi π d´1 ź j"i`1 ˆ1 ´θ j π ˙.

Therefore, we have for all 0 ď i ď d ´2, d´1 ź j"i`1 ˆ1 ´θ j π ˙ď d´1 ź j"i`1 ˆ1 ´1 j `1 ˙" e ř d´1 j"i`1 logp1´1 j`1 q ď e ´řd´1 j"i`1

1 j`1 ď e ´şd i`1 1 s`1 ds " i `2 d `1 , d´1 ź j"i`1

ˆ1:

´θ j π ˙ě d´1 ź j"i`1 ˆ1 ´3 j `3 ˙" e ř d´1 j"i`1 logp1´3 j`3 q ě e ´řd´1 j"i`1 6 j`3 ě e where the second and the fifth inequalities follow from ( 49) and ( 50) respectively. Since π 3 {p12pi 1q 3 q ď θ3 i {12 ď θi ´sin θi ď θ3 i {6 ď 27π 3 {p6pi `3q 3 q, we have that for all d ě 3 

1 ´ρd ď 2 d `1 `d´1 ÿ i"1 27π 3 6pi `3q 3 i `2 d `

:

The proof of Lipschitzness follows from applying the Weight Distribution Condition in Section 4.1.

A formula for M xØŷ is as follows. If θ0 " =px, ŷq P p0, πq and R is a rotation matrix such that x and ŷ

The proof in that argument only uses the assumption of independence of subsets of rows of the weight matrices.

Figures and Tables:

Figure1: Loss surface f pxq " }Gpxq ´Gpx ˚q}, x ˚" r1, 0s, of an expansive network G with ReLu activation functions with k " 2 nodes in the input layer and n 2 " 300 and n 3 " 784 nodes in the hidden and output layers, respectively, with random Gaussian weights in each layer. The surface has a critical point near ´x˚, a global minimum at x ˚, and a local maximum at 0.

Figure 2: Denosing with a learned generative prior: Even when the number is barely visible, the denoiser recovers a sharp image.

Thus, by the union bound, where n " n d . Recall from (9) that }Λ x } ď 13 12 . Combining this inequality with }q x }

Table content: <table><tr><td/><td/><td/><td/><td/><td>2 ď</td></tr><tr><td>}Λ x }</td><td colspan="2">2 }P Λx η} 2 concludes the proof.</td><td/><td/></tr><tr><td colspan="3">A.6 PROOF OF LEMMA 9</td><td/><td/></tr><tr><td/><td>P</td><td>" }P Λx η} 2 ď 16k logpn d 1 n d´1 2</td><td>. . . n d q, for all x</td><td>ı</td><td>ě 1 ´2e ´2k logpnq ,</td></tr></table>

and |}x} 2 ´}x ˚}2 | ď 132d 6 β}x ˚}2 or |θ 0 ´π| ď 8πd 4 a β and |}x} 2 ´}x ˚}2 ρ d | ď 200d 7 ? }x ˚}2 . In particular, we have

Table content: <table/>

Lemma 11 follows by noting that if x, y R Bp0, r}x ˚}q, then }h x ´hy } ď ´1 2 d `6d`4d 2 πr2 d ¯}x ´y}. ´θ 0,y | ď 4 max ´1 }x} , 1 }y} ¯}x ´y}. Inequality (48) implies | θi,x ´θ i,y | ď | θj,x ´θ j,y | for all i ě j. It follows that Lemma 15. Suppose a i , b i P r0, πs for i " 1, . . . , k, and |a i ´bi | ď |a j ´bj |, @i ě j.Proof. Prove by induction. It is easy to verify that the inequality holds if k " 1. Suppose the inequality holds with k " t ´1. Then

Table content: <table><tr><td colspan="8">By Lemma 15, we have</td><td/><td/><td/><td/></tr><tr><td/><td/><td/><td/><td/><td/><td/><td colspan="2">T 1 ď</td><td>d π</td><td/><td colspan="2">| θ0,x ´θ 0,y | ď</td><td>4d π</td><td>max</td><td>ˆ1 }x}</td><td>,</td><td>1 }y}</td><td>˙}x ´y}.</td><td>(52)</td></tr><tr><td colspan="9">Additionally, it holds that</td><td/><td/><td/></tr><tr><td>T 2 "</td><td colspan="3">ˇˇˇˇd ´1 ÿ i"0</td><td colspan="4">sin θi,x π</td><td colspan="4">ζ i`1,x x</td><td>´sin θi,x π</td><td>ζ i`1,x ŷ</td><td>`sin θi,x π</td><td>ζ i`1,x ŷ</td><td>´d´1 ÿ i"0</td><td>sin θi,y π</td><td>ζ i`1,y</td><td>ŷˇˇˇˇď</td></tr><tr><td/><td>d π</td><td colspan="5">}x ´ŷ}</td><td colspan="3">`ˇˇˇˇd´1 ÿ i"0</td><td colspan="3">sin θi,x π</td><td>ζ i`1,x</td><td>´d´1 ÿ i"0</td><td>sin θi,y π</td><td>ζ i`1,y</td><td>ˇˇˇľ . (53)</td></tr><tr><td/><td/><td/><td/><td/><td/><td/><td/><td/><td/><td/><td/><td>jh</td><td>n</td></tr><tr><td/><td/><td/><td/><td/><td/><td/><td/><td/><td/><td/><td/><td>T3</td></tr><tr><td>We have</td><td/><td/><td/><td/><td/><td/><td/><td/><td/><td/><td/></tr><tr><td colspan="3">T 3 ď</td><td colspan="2">d´1 ÿ i"0</td><td colspan="4">"ˇˇˇˇs in θi,x π</td><td colspan="4">ζ i`1,x</td><td>´sin θi,x π</td><td>ζ i`1,y ˇˇˇ`ˇˇˇs in θi,x π</td><td>ζ i`1,y</td><td>´sin θi,y π</td><td>ζ i`1,y ˇˇˇ</td></tr><tr><td/><td colspan="2">ď</td><td colspan="2">d´1 ÿ i"0</td><td>"</td><td>1 π</td><td colspan="4">ˆd ´i π</td><td colspan="2">´1</td><td>ˇˇθ i´1,x ´θ i´1,y</td><td>ˇˇ˙`1 π</td><td>| sin θi,x ´sin θi,y |</td><td></td></tr><tr><td/><td colspan="2">ď</td><td>d 2 π</td><td colspan="8">| θ0,x ´θ 0,y | ď</td><td>4d 2 π</td><td>max</td><td>ˆ1 }x}</td><td>,</td><td>1 }y}</td><td>˙}x ´y}.</td><td>(54)</td></tr><tr><td colspan="13">Using (46) and (47) and noting }x ´ŷ} ď θ x,y yield</td></tr><tr><td/><td/><td/><td/><td/><td/><td/><td/><td colspan="5">}x ´ŷ} ď θ x,y ď 2 max</td><td>1 ď ˆ1 }x} d ,</td><td>2 `1 1 }y} ˙}x ´y}. `3π 5 4pd `1q</td><td>ď</td><td>250 d `1</td><td>(55)</td></tr><tr><td colspan="13">and Finally, combining (51), (52), (53), (54) and (55) yields the result.</td></tr><tr><td colspan="2">where we use that</td><td colspan="11">1 ´ρd ě ř 8 i"4 1 i 2 ď π 2 ˆ3 pd `2q 6 and ř n ˙6 i"1 i 3 " Opn 4 q. `d´1 ÿ i"1 π 3 12pi `3q 3 ˇˇˇˇk ź i"1 π ´ai π ´k ź i"1 π ´bi π ˇˇˇˇď k ˆi d π</td><td>`3 `2 ˙6 ě |a 1 ´b1 |.</td><td>1 K 1 pd `2q 2 ,</td><td>Then it holds</td></tr><tr><td colspan="10">A.9 PROOF OF LEMMA 11</td><td/><td/></tr><tr><td colspan="13">To establish Lemma 11, we prove the following:</td></tr><tr><td colspan="13">Lemma 14. For all x, y ‰ 0, }h x ´hy } ď ˇˇˇˇt ź i"1 π ´ai π ´t ź ˆ1 2 d i"1 π ´bi `6d `4d 2 π2 d max π ˇˇˇˇď ˇˇˇˇt ź i"1 π ´ai π</td><td>ˆ1 }x} ´π ´at , 1 }y} π</td><td>˙}x ˚}˙} x ´y} i"1 π ź π ´bi t´1 ˇˇˇπ</td></tr><tr><td/><td/><td/><td/><td/><td/><td/><td/><td/><td/><td/><td/><td>´at π</td><td>ź i"1 t´1</td><td>π ´bi π</td><td>i"1 ´t ź</td><td>π π ´bi</td><td>ˇˇˇď</td></tr><tr><td colspan="13">Proof of Lemma 14. For brevity of notation, let ζ j,z " t ´1 π |a 1 ´b1 |</td><td>ś d´1 i"j `1 π |a t ´bt | ď π´θi,z π . Combining (46) and (47) t π |a 1 ´b1 |.</td></tr><tr><td colspan="10">gives | θ0,x }h x ´hy } ď</td><td colspan="3">1 2 d }x ´y}</td><td>l jh n `1 2 d |ζ 0,x ´ζ0,y |</td><td>}x ˚}</td></tr><tr><td/><td/><td/><td/><td/><td/><td/><td/><td/><td/><td/><td/><td>T1</td></tr><tr><td/><td/><td/><td/><td/><td/><td/><td/><td colspan="4">`1 2 d</td><td>ˇˇˇˇd ´1 ÿ i"0</td><td>sin θi,x π</td><td>ζ i`1,x x</td><td>´d´1 ÿ i"0</td><td>sin θi,y π</td><td>ζ i`1,y</td><td>ŷˇˇˇˇľ }x ˚}. (51)</td></tr><tr><td/><td/><td/><td/><td/><td/><td/><td/><td/><td/><td/><td/><td>jh</td><td>n</td></tr><tr><td/><td/><td/><td/><td/><td/><td/><td/><td/><td/><td/><td/><td>T2</td></tr></table>

Peer review (iclr2019):
 The paper analyzes the recovery accuracy of a "tweaked" gradient descent algorithm for imaging denoising and compressive sensing under deep generative priors. In particular, when assuming Gaussian randomness of the network weights and extremely stringent conditions of network sizes, they demonstrate a specific denoising rate of O(k/n), with k and n being the input and output dimension of the generative network. This is seemingly optimal in terms of the dependence on the latent code dimensionality and the signal dimensionality and is the first result of this kind. Two papers are closely related, but are not sufficiently discussed in the introduction. [Bora et al., 2017] does not require Gaussian randomness of the network weights, but achieves only O(1) error bound assuming the empirical risk minimization problem can be solved to optimality. [Hand & Voroninski, 2018] showed that under same assumptions as in this paper, the nonconvex empirical risk minimization problem exhibits a nice geometric landscape - no spurious stationary points. This implies that virtually anything reasonable would converge to global optimum. Combing both facts, it is not surprising to arrive at the results in this paper. While the paper makes some novel theoretical contributions, two concerns stand out. First, there is a lack of intuition or justification of the tweak in gradient descent - flipping the sign of the iterate at times. The author argued that around approximately -x*, the loss function is larger than around the optimum x* . So simple gradient descent is likely to get stuck in this region, so the negation check is needed. I am not so convinced by the argument. There could be other critical points that are not necessarily in the negative regime of true optimal, right? So why would this be sufficient or necessary for global convergence? Second, even ignoring the unrealistic Gaussian assumption on the network weights, the theorem requires very narrow regimes for the expansivity condition and the noise variance bound. It*s hard to verify whether these conditions can be satisfied at all. The experiment on denoising with learned prior from MNIST data is interesting, as it suggests that the theoretical assumptions are not necessary in practice to observe the optimal recovery rate. It would be more convincing if more experiments are provided, especially for the compressive sensing application.