Sparse Legendre expansions via $\ell_1$ minimization

TL;DR

This paper demonstrates that sparse Legendre polynomials can be efficiently recovered from few random samples using l1-minimization, with theoretical guarantees based on restricted isometry properties, and extends results to other orthogonal polynomial systems.

Contribution

It provides the first rigorous analysis of sparse Legendre polynomial recovery from limited samples using compressed sensing techniques, including RIP verification and extension to other polynomial systems.

Findings

Sparse Legendre polynomials recoverable from O(s log^4 N) samples

l1-minimization effectively reconstructs sparse polynomials

Results extend to Jacobi and other orthogonal polynomials

Abstract

We consider the problem of recovering polynomials that are sparse with respect to the basis of Legendre polynomials from a small number of random samples. In particular, we show that a Legendre s-sparse polynomial of maximal degree N can be recovered from m = O(s log^4(N)) random samples that are chosen independently according to the Chebyshev probability measure. As an efficient recovery method, l1-minimization can be used. We establish these results by verifying the restricted isometry property of a preconditioned random Legendre matrix. We then extend these results to a large class of orthogonal polynomial systems, including the Jacobi polynomials, of which the Legendre polynomials are a special case. Finally, we transpose these results into the setting of approximate recovery for functions in certain infinite-dimensional function spaces.