Solving inverse problems via auto-encoders

Compressed sensing (CS) is about recovering a structured signal from its under-determined linear measurements. Starting from sparsity, recovery methods have steadily moved towards more complex structures. Emerging machine learning tools such as generative functions that are based on neural networks are able to learn general complex structures from training data. This makes them potentially powerful tools for designing CS algorithms. Consider a desired class of signals $\cal Q$, ${\cal Q}\subset{R}^n$, and a corresponding generative function $g:{\cal U}^k\rightarrow {R}^n$, ${\cal U}\subset {R}$, such that $\sup_{{\bf x}\in {\cal Q}}\min_{{\bf u}\in{\cal U}^k}{1\over \sqrt{n}}\|g({\bf u})-{\bf x}\|\leq \delta$. A recovery method based on $g$ seeks $g({\bf u})$ with minimum measurement error. In this paper, the performance of such a recovery method is studied, under both noisy and noiseless measurements. In the noiseless case, roughly speaking, it is proven that, as $k$ and $n$ grow without bound and $\delta$ converges to zero, if the number of measurements ($m$) is larger than the input dimension of the generative model ($k$), then asymptotically, almost lossless recovery is possible. Furthermore, the performance of an efficient iterative algorithm based on projected gradient descent is studied. In this case, an auto-encoder is used to define and enforce the source structure at the projection step. The auto-encoder is defined by encoder and decoder (generative) functions $f:{R}^n\to{\cal U}^k$ and $g:{\cal U}^k\to{R}^n$, respectively. We theoretically prove that, roughly, given $m>40k\log{1\over \delta}$ measurements, such an algorithm converges to the vicinity of the desired result, even in the presence of additive white Gaussian noise. Numerical results exploring the effectiveness of the proposed method are presented

Deep Learning Techniques for Inverse Problems in Imaging

Recent work in machine learning shows that deep neural networks can be used to solve a wide variety of inverse problems arising in computational imaging. We explore the central prevailing themes of this emerging area and present a taxonomy that can be used to categorize different problems and reconstruction methods. Our taxonomy is organized along two central axes: (1) whether or not a forward model is known and to what extent it is used in training and testing, and (2) whether or not the learning is supervised or unsupervised, i.e., whether or not the training relies on access to matched ground truth image and measurement pairs. We also discuss the trade-offs associated with these different reconstruction approaches, caveats and common failure modes, plus open problems and avenues for future work

Robust compressed sensing of generative models

The goal of compressed sensing is to estimate a high dimensional vector from an underdetermined system of noisy linear equations. In analogy to classical compressed sensing, here we assume a generative model as a prior, that is, we assume the vector is represented by a deep generative model $G: \mathbb{R}^k \rightarrow \mathbb{R}^n$. Classical recovery approaches such as empirical risk minimization (ERM) are guaranteed to succeed when the measurement matrix is sub-Gaussian. However, when the measurement matrix and measurements are heavy-tailed or have outliers, recovery may fail dramatically. In this paper we propose an algorithm inspired by the Median-of-Means (MOM). Our algorithm guarantees recovery for heavy-tailed data, even in the presence of outliers. Theoretically, our results show our novel MOM-based algorithm enjoys the same sample complexity guarantees as ERM under sub-Gaussian assumptions. Our experiments validate both aspects of our claims: other algorithms are indeed fragile and fail under heavy-tailed and/or corrupted data, while our approach exhibits the predicted robustness

Instance-Optimal Compressed Sensing via Posterior Sampling

We characterize the measurement complexity of compressed sensing of signals drawn from a known prior distribution, even when the support of the prior is the entire space (rather than, say, sparse vectors). We show for Gaussian measurements and \emph{any} prior distribution on the signal, that the posterior sampling estimator achieves near-optimal recovery guarantees. Moreover, this result is robust to model mismatch, as long as the distribution estimate (e.g., from an invertible generative model) is close to the true distribution in Wasserstein distance. We implement the posterior sampling estimator for deep generative priors using Langevin dynamics, and empirically find that it produces accurate estimates with more diversity than MAP