Theoretical Perspectives on Deep Learning Methods in Inverse Problems

In recent years, there have been significant advances in the use of deep learning methods in inverse problems such as denoising, compressive sensing, inpainting, and super-resolution. While this line of works has predominantly been driven by practical algorithms and experiments, it has also given rise to a variety of intriguing theoretical problems. In this paper, we survey some of the prominent theoretical developments in this line of works, focusing in particular on generative priors, untrained neural network priors, and unfolding algorithms. In addition to summarizing existing results in these topics, we highlight several ongoing challenges and open problems

Inferring Rankings Using Constrained Sensing

We consider the problem of recovering a function over the space of permutations (or, the symmetric group) over $n$ elements from given partial information; the partial information we consider is related to the group theoretic Fourier Transform of the function. This problem naturally arises in several settings such as ranked elections, multi-object tracking, ranking systems, and recommendation systems. Inspired by the work of Donoho and Stark in the context of discrete-time functions, we focus on non-negative functions with a sparse support (support size $\ll$ domain size). Our recovery method is based on finding the sparsest solution (through $\ell_0$ optimization) that is consistent with the available information. As the main result, we derive sufficient conditions for functions that can be recovered exactly from partial information through $\ell_0$ optimization. Under a natural random model for the generation of functions, we quantify the recoverability conditions by deriving bounds on the sparsity (support size) for which the function satisfies the sufficient conditions with a high probability as $n \to \infty$. $\ell_0$ optimization is computationally hard. Therefore, the popular compressive sensing literature considers solving the convex relaxation, $\ell_1$ optimization, to find the sparsest solution. However, we show that $\ell_1$ optimization fails to recover a function (even with constant sparsity) generated using the random model with a high probability as $n \to \infty$. In order to overcome this problem, we propose a novel iterative algorithm for the recovery of functions that satisfy the sufficient conditions. Finally, using an Information Theoretic framework, we study necessary conditions for exact recovery to be possible.Comment: 19 page

Information-Driven Adaptive Structured-Light Scanners

Sensor planning and active sensing, long studied in robotics, adapt sensor parameters to maximize a utility function while constraining resource expenditures. Here we consider information gain as the utility function. While these concepts are often used to reason about 3D sensors, these are usually treated as a predefined, black-box, component. In this paper we show how the same principles can be used as part of the 3D sensor. We describe the relevant generative model for structured-light 3D scanning and show how adaptive pattern selection can maximize information gain in an open-loop-feedback manner. We then demonstrate how different choices of relevant variable sets (corresponding to the subproblems of locatization and mapping) lead to different criteria for pattern selection and can be computed in an online fashion. We show results for both subproblems with several pattern dictionary choices and demonstrate their usefulness for pose estimation and depth acquisition.United States. Office of Naval Research (Grant N00014-09-1-1051)United States. Army Research Office (Grant W911NF-11- 1-0391)United States. Office of Naval Research (Grant N00014- 11-1-0688