Deep generative models have emerged as a powerful class of priors for signals in various inverse problems such as compressed sensing, phase retrieval and super-resolution. Here, we assume an unknown signal to lie in the range of some pre-trained generative model. A popular approach for signal recovery is via gradient descent in the low-dimensional latent space. While gradient descent has achieved good empirical performance, its theoretical behavior is not well understood. In this paper, we introduce the use of stochastic gradient Langevin dynamics (SGLD) for compressed sensing with a generative prior. Under mild assumptions on the generative model, we prove the convergence of SGLD to the true signal. We also demonstrate competitive empirical performance to standard gradient descent.

Deep generative models have emerged as a powerful class of priors for signals in various inverse problems such as compressed sensing, phase retrieval and super-resolution. In this work, we consider the compressed sensing problem and assume the unknown signal to lie in the range of some pre-trained generative model. A popular approach for signal recovery is via gradient descent in the low-dimensional latent space. While gradient descent has achieved good empirical performance, its theoretical behavior is not well understood. We introduce the use of stochastic gradient Langevin dynamics (SGLD) for compressed sensing with a generative prior. Under mild assumptions on the generative model, we prove the convergence of SGLD to the true signal. We also demonstrate competitive empirical performance to standard gradient descent.

Traditionally, dynamic range enhancement for images has involved a combination of contrast improvement (via gamma correction or histogram equalization) and a denoising operation to reduce the effects of photon noise. More recently, modulo-imaging methods have been introduced for high dynamic range photography to significantly expand dynamic range at the sensing stage itself. The transformation function for both of these problems is highly non-linear, and the image reconstruction procedure is typically non-convex and ill-posed. A popular recent approach is to regularize the above inverse problem via a neural network prior (such as a trained autoencoder), but this requires extensive training over a dataset with thousands of paired regular/HDR image data samples.In this paper, we introduce a new approach for HDR image reconstruction using neural priors that require no training data. Specifically, we employ deep image priors, which have been successfully used for imaging problems such as denoising, super-resolution, inpainting and compressive sensing with promising performance gains over conventional regularization techniques. In this paper, we consider two different approaches to high dynamic range (HDR) imaging - gamma encoding and modulo encoding - and propose a combination of deep image prior and total variation (TV) regularization for reconstructing low-light images. We demonstrate the significant improvement achieved by both of these approaches as compared to traditional dynamic range enhancement techniques.

We propose and analyze algorithms for training ReLU networks with skipped connections. Skipped connections are the key feature of residual networks (or ResNets) which have been shown to provide superior performance in deep learning applications. We analyze two approaches for training such networks - gradient descent and alternating minimization - and compare convergence criteria of both methods. We show that under typical (Gaussianity) assumptions on the d-dimensional input data, both gradient descent and alternating minimization provably converge in a linearly convergent fashion, assuming any good enough initialization; moreover, we show that a simple “identity” initialization suffices. Furthermore, we provide statistical upper bounds which indicate that n = O(d3) suffice to achieve this convergence rate. To our knowledge, these constitute the first global parameter recovery guarantees for shallow ResNet-type networks with ReLU activations.

Deep neural networks as image priors have been recently introduced for problems such as denoising, super-resolution and inpainting with promising performance gains over hand-crafted image priors such as sparsity. Unlike learned generative priors they do not require any training over large datasets. However, few theoretical guarantees exist in the scope of using untrained network priors for inverse imaging problems. We explore new applications and theory for untrained neural network priors. Specifically, we consider the problem of solving linear inverse problems, such as compressive sensing, as well as non-linear problems, such as compressive phase retrieval. We model images to lie in the range of an untrained deep generative network with a fixed seed. We further present a projected gradient descent scheme that can be used for both compressive sensing and phase retrieval and provide rigorous theoretical guarantees for its convergence. We also show both theoretically as well as empirically that with deep neural network priors, one can achieve better compression rates for the same image quality as compared to when hand crafted priors are used.

We consider the problem of recovering a signal x* ∈ Rn, from magnitude-only measurements, yi= |〈ai, x*〉| for i = {1, 2, ... , m}. This is a stylized version of the classical phase retrieval problem, and is a fundamental challenge in nanoand bio-imaging systems, astronomical imaging, and speech processing. It is well-known that the above problem is ill-posed, and therefore some additional assumptions on the signal and/or the measurements are necessary. In this paper, we consider the case where the underlying signal x* is s-sparse. For this case, we develop a novel recovery algorithm that we call Compressive Phase Retrieval with Alternating Minimization, or CoPRAM. Our algorithm is simple and is obtained via a natural combination of the classical alternating minimization approach for phase retrieval with the CoSaMP algorithm for sparse recovery. Despite its simplicity, we prove that our algorithm achieves a sample complexity of O (s2log n) with Gaussian measurements ai, which matches the best known existing results; moreover, it also demonstrates linear convergence in theory and practice. An appealing feature of our algorithm is that it requires no extra tuning parameters other than the signal sparsity level s. Moreover, we show that our algorithm is robust to noise. The quadratic dependence of sample complexity on the sparsity level is suboptimal, and we demonstrate how to alleviate this via additional assumptions beyond sparsity. First, we study the (practically) relevant case where the sorted coefficients of the underlying sparse signal exhibit a power law decay. In this scenario, we show that the CoPRAM algorithm achieves a sample complexity of O (s log n), which is close to the information-theoretic limit. We then consider the case where the underlying signal x* arises from structured sparsity models. We specifically examine the case of block-sparse signals with uniform block size of b and block sparsity k = s/b. For this problem, we design a recovery algorithm that we call Block CoPRAM that further reduces the sample complexity to O (ks logn). For sufficiently large block lengths of b = O(s), this bound equates to O (slogn). To our knowledge, our approach constitutes the first family of linearly convergent algorithms for signal recovery from magnitude-only Gaussian measurements that exhibit a sub-quadratic dependence on the signal sparsity level.

In this paper, we introduce a principled algorithmic approach for Fourier ptychographic imaging of dynamic, time-varying targets. To the best of our knowledge, this setting has not been explicitly addressed in the ptychography literature. We argue that such a setting is very natural, and that our methods provide an important first step towards helping reduce the sample complexity (and hence acquisition time) of imaging dynamic scenes to managaeble levels. With significantly reduced acquisition times per image, it is conceivable that dynamic ptychographic imaging of fast changing scenes indeeed becomes practical in the near future.

We consider the problem of super-resolution for sub-diffraction imaging. We adapt conventional Fourier ptychographic approaches, for the case where the images to be acquired have an underlying structured sparsity. We propose some sub-sampling strategies which can be easily adapted to existing ptychographic setups. We then use a novel technique called CoPRAM with some modifications, to recover sparse (and block sparse) images from sub-sampled pty-chographic measurements. We demonstrate experimentally that this algorithm performs better than existing phase retrieval techniques, in terms of quality of reconstruction, using fewer number of samples.

In this paper, we introduce a novel algorithmic framework for sub-diffractive super-resolution imaging of dynamic, time-varying targets. We extend recent works in low rank Fourier ptychographic imaging, to incorporate model-correction schemes, which correct for errors propagated due to inaccuracies in fitting an exact low rank model to the target video acquired. Through our algorithm, we are able to demonstrate superior reconstruction quality of video from phaseless Fourier ptychographic measurements, at low sample complexities, as compared to conventional ptychographic setups.

We consider the problem of estimating a structured high-dimensional parameter vector using random Gaussian quadratic samples. This problem is a generalization of the classical problem of phase retrieval and impacts numerous problems in computational imaging. We provide a generic algorithm based on alternating minimization that, if properly initialized, achieves information-theoretically optimal sample complexity. In essence, we show that solving a system of random quadratic equations with structural constraints is (nearly) as easy as solving the corresponding linear system with the same constraints, if a proper initial guess of the solution is available. As an immediate consequence, our approach improves upon the best known existing sample complexity results for phase retrieval (structured or otherwise). We support our theory via several numerical experiments. A full version of this paper is accessible at: https://gaurijagatap.github.io/assets/ISIT18.pdf.

We consider the problem of recovering a signal x in R^n, from magnitude-only measurements, yi = |ai^T x| for i={1,2...m}. Also known as the phase retrieval problem, it is a fundamental challenge in nano-, bio- and astronomical imaging systems, astronomical imaging, and speech processing. The problem is ill-posed, and therefore additional assumptions on the signal and/or the measurements are necessary. In this paper, we first study the case where the underlying signal x is s-sparse. We develop a novel recovery algorithm that we call Compressive Phase Retrieval with Alternating Minimization, or CoPRAM. Our algorithm is simple and can be obtained via a natural combination of the classical alternating minimization approach for phase retrieval, with the CoSaMP algorithm for sparse recovery. Despite its simplicity, we prove that our algorithm achieves a sample complexity of O(s^2 log n) with Gaussian samples, which matches the best known existing results. It also demonstrates linear convergence in theory and practice and requires no extra tuning parameters other than the signal sparsity level s. We then consider the case where the underlying signal x arises from to structured sparsity models. We specifically examine the case of block-sparse signals with uniform block size of b and block sparsity k=s/b. For this problem, we design a recovery algorithm that we call Block CoPRAM that further reduces the sample complexity to O(ks log n). For sufficiently large block lengths of b=Theta(s), this bound equates to O(s log n). To our knowledge, this constitutes the first end-to-end linearly convergent family of algorithms for phase retrieval where the Gaussian sample complexity has a sub-quadratic dependence on the sparsity level of the signal.