Quotient Manifold Optimization for Spectral Compressed Sensing

TL;DR

This paper introduces a quotient-manifold optimization approach for spectral compressed sensing that exploits Riemannian geometry and matrix structures to improve computational efficiency and accuracy.

Contribution

It develops a novel Riemannian conjugate gradient algorithm on quotient manifolds for spectral compressed sensing, leveraging Hankel-Toeplitz structures and FFTs for efficiency.

Findings

Outperforms existing methods in speed and accuracy

Efficiently exploits matrix structures with FFTs

Provides rigorous geometric derivations

Abstract

Spectral compressed sensing involves reconstructing a spectral-sparse signal from a subset of uniformly spaced samples, with applications in radar imaging and wireless channel estimation. By fully exploiting the signal structures, this problem is formulated as a rank-constrained semidefinite program subject to Hankel-Toeplitz structural constraints in our previous work. To further enhance computational efficiency, this paper proposes a quotient-manifold-based optimization framework that leverages the underlying Riemannian geometry in a matrix factorization space. Specifically, we establish an equivalence between spectral-sparse signals and matrix equivalence classes under the action of the real orthogonal group, where each class member corresponds to a rank-constrained positive-semidefinite Hankel-Toeplitz structured matrix. The associated quotient manifold geometry--including the Riemannian metric, horizontal space, retraction, and vector transport--is rigorously derived. Based on these results, we develop a Riemannian conjugate gradient descent algorithm, where each iteration is efficiently implemented using fast Fourier transforms (FFTs) by exploiting the Hankel and Toeplitz structures. Extensive numerical experiments demonstrate the superior performance of the proposed algorithm in both computational speed and accuracy compared to state-of-the-art methods.