Error analysis for denoising smooth modulo signals on a graph

TL;DR

This paper analyzes and improves the theoretical understanding of denoising algorithms for smooth modulo signals on graphs, providing noise regimes where these methods effectively recover original signals.

Contribution

It offers a refined analysis of trust region and unconstrained relaxations for denoising modulo signals, establishing conditions under which they provably reduce noise.

Findings

Derived noise regimes where denoising algorithms are effective

Provided refined $ ext{L}_2$ error bounds for the relaxations

Applicable to general connected graphs

Abstract

In many applications, we are given access to noisy modulo samples of a smooth function with the goal being to robustly unwrap the samples, i.e., to estimate the original samples of the function. In a recent work, Cucuringu and Tyagi proposed denoising the modulo samples by first representing them on the unit complex circle and then solving a smoothness regularized least squares problem -- the smoothness measured w.r.t the Laplacian of a suitable proximity graph $G$ -- on the product manifold of unit circles. This problem is a quadratically constrained quadratic program (QCQP) which is nonconvex, hence they proposed solving its sphere-relaxation leading to a trust region subproblem (TRS). In terms of theoretical guarantees, $\ell_2$ error bounds were derived for (TRS). These bounds are however weak in general and do not really demonstrate the denoising performed by (TRS). In this work, we analyse the (TRS) as well as an unconstrained relaxation of (QCQP). For both these estimators we provide a refined analysis in the setting of Gaussian noise and derive noise regimes where they provably denoise the modulo observations w.r.t the $\ell_2$ norm. The analysis is performed in a general setting where $G$ is any connected graph.