Signal Recovery on Incoherent Manifolds

TL;DR

This paper introduces SPIN, a projected gradient method for recovering signals composed of two components from incoherent nonlinear manifolds, even with noisy and underdetermined measurements, extending current low-dimensional inverse problem solutions.

Contribution

The paper presents SPIN, a novel first-order algorithm with provable guarantees for recovering signals on incoherent manifolds, broadening the scope of inverse problem techniques.

Findings

SPIN successfully recovers signal components under theoretical conditions.

The method outperforms existing algorithms in certain scenarios.

Recovery is guaranteed despite nonconvexity and underdetermined measurements.

Abstract

Suppose that we observe noisy linear measurements of an unknown signal that can be modeled as the sum of two component signals, each of which arises from a nonlinear sub-manifold of a high dimensional ambient space. We introduce SPIN, a first order projected gradient method to recover the signal components. Despite the nonconvex nature of the recovery problem and the possibility of underdetermined measurements, SPIN provably recovers the signal components, provided that the signal manifolds are incoherent and that the measurement operator satisfies a certain restricted isometry property. SPIN significantly extends the scope of current recovery models and algorithms for low dimensional linear inverse problems and matches (or exceeds) the current state of the art in terms of performance.