Average-case Hardness of RIP Certification

The restricted isometry property (RIP) for design matrices gives guarantees for optimal recovery in sparse linear models. It is of high interest in compressed sensing and statistical learning. This property is particularly important for computationally efficient recovery methods. As a consequence, even though it is in general NP-hard to check that RIP holds, there have been substantial efforts to find tractable proxies for it. These would allow the construction of RIP matrices and the polynomial-time verification of RIP given an arbitrary matrix. We consider the framework of average-case certifiers, that never wrongly declare that a matrix is RIP, while being often correct for random instances. While there are such functions which are tractable in a suboptimal parameter regime, we show that this is a computationally hard task in any better regime. Our results are based on a new, weaker assumption on the problem of detecting dense subgraphs

Construction of a Large Class of Deterministic Sensing Matrices that Satisfy a Statistical Isometry Property

Compressed Sensing aims to capture attributes of $k$-sparse signals using very few measurements. In the standard Compressed Sensing paradigm, the \m\times \n measurement matrix \A is required to act as a near isometry on the set of all $k$-sparse signals (Restricted Isometry Property or RIP). Although it is known that certain probabilistic processes generate \m \times \n matrices that satisfy RIP with high probability, there is no practical algorithm for verifying whether a given sensing matrix \A has this property, crucial for the feasibility of the standard recovery algorithms. In contrast this paper provides simple criteria that guarantee that a deterministic sensing matrix satisfying these criteria acts as a near isometry on an overwhelming majority of $k$-sparse signals; in particular, most such signals have a unique representation in the measurement domain. Probability still plays a critical role, but it enters the signal model rather than the construction of the sensing matrix. We require the columns of the sensing matrix to form a group under pointwise multiplication. The construction allows recovery methods for which the expected performance is sub-linear in \n, and only quadratic in \m; the focus on expected performance is more typical of mainstream signal processing than the worst-case analysis that prevails in standard Compressed Sensing. Our framework encompasses many families of deterministic sensing matrices, including those formed from discrete chirps, Delsarte-Goethals codes, and extended BCH codes.Comment: 16 Pages, 2 figures, to appear in IEEE Journal of Selected Topics in Signal Processing, the special issue on Compressed Sensin

Non-convex Optimization for Machine Learning

A vast majority of machine learning algorithms train their models and perform inference by solving optimization problems. In order to capture the learning and prediction problems accurately, structural constraints such as sparsity or low rank are frequently imposed or else the objective itself is designed to be a non-convex function. This is especially true of algorithms that operate in high-dimensional spaces or that train non-linear models such as tensor models and deep networks. The freedom to express the learning problem as a non-convex optimization problem gives immense modeling power to the algorithm designer, but often such problems are NP-hard to solve. A popular workaround to this has been to relax non-convex problems to convex ones and use traditional methods to solve the (convex) relaxed optimization problems. However this approach may be lossy and nevertheless presents significant challenges for large scale optimization. On the other hand, direct approaches to non-convex optimization have met with resounding success in several domains and remain the methods of choice for the practitioner, as they frequently outperform relaxation-based techniques - popular heuristics include projected gradient descent and alternating minimization. However, these are often poorly understood in terms of their convergence and other properties. This monograph presents a selection of recent advances that bridge a long-standing gap in our understanding of these heuristics. The monograph will lead the reader through several widely used non-convex optimization techniques, as well as applications thereof. The goal of this monograph is to both, introduce the rich literature in this area, as well as equip the reader with the tools and techniques needed to analyze these simple procedures for non-convex problems.Comment: The official publication is available from now publishers via http://dx.doi.org/10.1561/220000005