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Error analysis for denoising smooth modulo signals on a graph

Hemant Tyagi 1 
1 MODAL - MOdel for Data Analysis and Learning
LPP - Laboratoire Paul Painlevé - UMR 8524, Université de Lille, Sciences et Technologies, Inria Lille - Nord Europe, METRICS - Evaluation des technologies de santé et des pratiques médicales - ULR 2694, Polytech Lille - École polytechnique universitaire de Lille
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.
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Submitted on : Monday, January 10, 2022 - 1:36:51 PM
Last modification on : Thursday, September 1, 2022 - 11:13:45 AM

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Hemant Tyagi. Error analysis for denoising smooth modulo signals on a graph. Applied and Computational Harmonic Analysis, Elsevier, In press. ⟨hal-03101720v2⟩

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