Sparse Recovery for Overcomplete Frames: Sensing Matrices and Recovery Guarantees
Xuemei Chen, Christian Kümmerle, Rongrong Wang
TL;DR
This work surveys sparse recovery when signals are $F$-$k$-sparse from undersampled linear measurements $y=Az+w$, focusing on convex, basis-pursuit-like decoders and their guarantees across orthonormal bases and general frames. It delineates how recovery is governed by properties such as RIP, NSP, and their frame analogues (F-NSP, F-RNSP), and extends these guarantees to heavy-tailed and structured sensing matrices via the small-ball method and moment conditions. A key highlight is the frame-specific RIP-like result (Theorem 3.10) and the synthesis/analysis dichotomy, including conditions under which $A$ achieves robust, stable recovery for $z$ and the sparse coefficient $x$. The chapter also discusses practical sensing-matrix designs (sub-Gaussian, heavy-tailed, structured) and outlines proof techniques, along with open questions about extending these results to broader structured designs and non-convex algorithms. Overall, it connects decoder choices, frame structure, and sensing designs to near-optimal measurement complexity in high-dimensional frame-sparse recovery. The results have implications for sensing in imaging, signal processing, and data analysis where redundant representations are essential and measurements are limited.
Abstract
Signal models formed as linear combinations of few atoms from an over-complete dictionary or few frame vectors from a redundant frame have become central to many applications in high dimensional signal processing and data analysis. A core question is, by exploiting the intrinsic low dimensional structure of the signal, how to design the sensing process and decoder in a way that the number of measurements is essentially close to the complexity of the signal set. This chapter provides a survey of important results in answering this question, with an emphasis on a basis pursuit like convex optimization decoder that admits a wide range of random sensing matrices. The results are quite established in the case signals are sparse in an orthonormal basis, while the case with frame sparse signals is much less explored. In addition to presenting the latest results on recovery guarantee and how few random heavier-tailed measurements fulfill these recovery guarantees, this chapter also aims to provide some insights in proof techniques. We also take the opportunity of this book chapter to publish an interesting result (Theorem 3.10) about a restricted isometry like property related to a frame.