We study compressive sensing with a deep generative network prior. Initial theoretical guarantees for efficient recovery from compressed linear measurements have been developed for signals in the range of a ReLU network with Gaussian weights and logarithmic expansivity: that is when each layer is larger than the previous one by a logarithmic factor. It was later shown that constant expansivity is sufficient for recovery. It has remained open whether the expansivity can be relaxed allowing for networks with contractive layers, as often the case of real generators. In this work we answer this question, proving that a signal in the range of a Gaussian generative network can be recovered from a few linear measurements provided that the width of the layers is proportional to the input layer size (up to log factors). This condition allows the generative network to have contractive layers. Our result is based on showing that Gaussian matrices satisfy a matrix concentration inequality, which we term Range Restricted Weight Distribution Condition (R2WDC), and weakens the Weight Distribution Condition (WDC) upon which previous theoretical guarantees were based. The WDC has also been used to analyze other signal recovery problems with generative network priors. By replacing the WDC with the R2WDC, we are able to extend previous results for signal recovery with expansive generative network priors to non-expansive ones. We discuss these extensions for phase retrieval, denoising, and spiked matrix recovery.

We provide a non-asymptotic analysis of the spiked Wishart and Wigner matrix models with a generative neural network prior. Spiked random matrices have the form of a rank-one signal plus noise and have been used as models for high dimensional Principal Component Analysis (PCA), community detection and synchronization over groups. Depending on the prior imposed on the spike, these models can display a statistical-computational gap between the information theoretically optimal reconstruction error that can be achieved with unbounded computational resources and the sub-optimal performances of currently known polynomial time algorithms. These gaps are believed to be fundamental, as in the emblematic case of Sparse PCA. In stark contrast to such cases, we show that there is no statistical-computational gap under a generative network prior, in which the spike lies on the range of a generative neural network. Specifically, we analyze a gradient descent method for minimizing a nonlinear least squares objective over the range of an expansive-Gaussian neural network and show that it can recover in polynomial time an estimate of the underlying spike with a rate-optimal sample complexity and dependence on the noise level.

Many problems in statistics and machine learning require the reconstruction of a lowrank signal matrix from noisy data. Enforcing additional prior information on the low-rank component is often key to guaranteeing good recovery performance. One such prior on the low-rank component is sparsity, giving rise to the sparse principal component analysis problem.Unfortunately, this problem suffers from a computational-to-statistical gap, which may be fundamental. In this work, we study an alternative prior where the low-rank component is in the range of a trained generative network. We provide a non-asymptotic analysis with optimal sample complexity, up to logarithmic factors, for low-rank matrix recovery under an expansive-Gaussian network prior. Specifically, we establish a favorable global optimization landscape for a mean squared error optimization, provided the number of samples is on the order of the dimensionality of the input to the generative model. As a result, we establish that generative priors have no computational-to-statistical gap for structured low-rank matrix recovery in the finite data, nonasymptotic regime. We present this analysis in the case of both the Wishart and Wigner spiked matrix models.

Sobolev loss is used when training a network to approximate the values and derivatives of a target function at a prescribed set of input points. Recent works have demonstrated its successful applications in various tasks such as distillation or synthetic gradient prediction. In this work we prove that an overparametrized two-layer relu neural network trained on the Sobolev loss with gradient flow from random initialization can fit any given function values and any given directional derivatives, under a separation condition on the input data.

Generative Adversarial Networks (GANs) have been shown to be powerful and flexible priors when solving inverse problems. One challenge of using them is overcoming representation error, the fundamental limitation of the network in representing any particular signal. Recently, multiple proposed inversion algorithms reduce representation error by optimizing over intermediate layer representations. These methods are typically applied to generative models that were trained agnostic of the downstream inversion algorithm. In our work, we introduce a principle that if a generative model is intended for inversion using an algorithm based on optimization of intermediate layers, it should be trained in a way that regularizes those intermediate layers. We instantiate this principle for two notable recent inversion algorithms: Intermediate Layer Optimization and the Multi-Code GAN prior. For both of these inversion algorithms, we introduce a new regularized GAN training algorithm and demonstrate that the learned generative model results in lower reconstruction errors across a wide range of under sampling ratios when solving compressed sensing, inpainting, and super-resolution problems.