Error analysis for denoising smooth modulo signals on a graph

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 [11] 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, ℓ 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 ℓ 2 norm. The analysis is performed in a general setting where G is any connected graph.

A Unified Approach to Synchronization Problems over Subgroups of the Orthogonal Group

“…The proof of Theorem 1 requires the following Lemma 2 which asserts that the projection map Π satisfies a certain contraction-type property and is a generalization of [29, Proposition 3.3] from SO(2) to arbitrary closed subgroups of the orthogonal group. Since [29, Proposition 3.3] has already been applied in a number of works to study phase synchronization problems [49] or even other estimation problems [18,44], we believe that Lemma 2 will find further applications in synchronization problems or other estimation or optimization problems over general subgroups of the orthogonal group. Furthermore, the proof of Lemma 2 is different from and much simpler than that of [29,Proposition 3.3].…”

Recovering Hölder smooth functions from noisy modulo samples

“…The output of Algorithm 1 is illustrated in Figure 1. The following other methods are considered: kNN denoising [10] (kNN), an unconstrained quadratic program [15] (UCQP) and a trust region subproblem [16] (TRS). A comparison between these methods is given in Figure 2, where the Root Mean Square Error (RMSE) between the ground truth and the recovered samples is displayed 2 .…”