Sparse Harmonic Transforms II: Best $s$-Term Approximation Guarantees for Bounded Orthonormal Product Bases in Sublinear-Time

TL;DR

This paper introduces a sublinear-time compressive sensing algorithm for high-dimensional functions in bounded orthonormal product bases, providing strong approximation guarantees and practical efficiency for extremely large basis sets.

Contribution

It develops a novel sublinear-time support identification algorithm and extends CoSaMP variants to achieve robust, fast sparse approximation in high-dimensional settings.

Findings

Achieves best $s$-term recovery guarantees in BOPB

Works efficiently with basis sets up to size ~10^{230}

Provides a robust, sublinear-time algorithm for high-dimensional sparse approximation

Abstract

In this paper, we develop a sublinear-time compressive sensing algorithm for approximating functions of many variables which are compressible in a given Bounded Orthonormal Product Basis (BOPB). The resulting algorithm is shown to both have an associated best $s$-term recovery guarantee in the given BOPB, and also to work well numerically for solving sparse approximation problems involving functions contained in the span of fairly general sets of as many as $\sim10^{230}$ orthonormal basis functions. All code is made publicly available. As part of the proof of the main recovery guarantee new variants of the well known CoSaMP algorithm are proposed which can utilize any sufficiently accurate support identification procedure satisfying a {Support Identification Property (SIP)} in order to obtain strong sparse approximation guarantees. These new CoSaMP variants are then proven to have both runtime and recovery error behavior which are largely determined by the associated runtime and error behavior of the chosen support identification method. The main theoretical results of the paper are then shown by developing a sublinear-time support identification algorithm for general BOPB sets which is robust to arbitrary additive errors. Using this new support identification method to create a new CoSaMP variant then results in a new robust sublinear-time compressive sensing algorithm for BOPB-compressible functions of many variables.