On robustness of &#x2113;<inf>1</inf>-regularization methods for spectral estimation

Abstract

The use of 1-regularization in sparse estimation methods has received huge attention during the last decade, and applications in virtually all fields of applied mathematics have benefited greatly. This interest was sparked by the recovery results of Candès, Donoho, Tao, Tropp, et al. and has resulted in a framework for solving a set of combinatorial problems in polynomial time by using convex relaxation techniques.In this work we study the use of 1-regularization methods for high-resolution spectral estimation. In this problem, the dictionary is typically coherent and existing theory for robust/exact recovery does not apply. In fact, the robustness cannot be guaranteed in the usual strong sense. Instead, we consider metrics inspired by the Monge-Kantorovich transportation problem and show that the magnitude can be robustly recovered if the original signal is sufficiently sparse and separated. We derive both worst case error bounds as well as error bounds based on assumptions on the noise distribution.

Generalized Sinkhorn Iterations for Regularizing Inverse Problems Using Optimal Mass Transport

“…An important property of this distance is that it does not only compare objects point by point, as standard L p metrics, but instead quantifies the length with which that mass is moved. This property makes the distance natural for quantifying uncertainty and modelling deformations [41,46,47]. More specifically geodesics (in, e.g., the associated Wasserstein-2 metric [67]) preserve "lumpiness," and when linking objects via geodesics of the metric there is a natural deformation between the objects.…”

Automatic target recognition using discrimination based on optimal transport

“…In this paper we focus on the Monge-Kantorovich distance [1], also known as the earth movers distance in the computer science community; a distance which is rooted in optimal transport and which has shown promise for both tracking and classification [2,3,4,5,6,7] and is a distance that is robust with respect to measurement error [8,9]. In particular, for data-direct high resolution spectral estimation methods such as sparse methods based on 1 -regularization [10,11] the magnitude of the true solution can be robustly recovered if the error is quantified using the Monge-Kantorovich distance and the support of the true signal is sparse and with separated components [9]. For these problems, the so-called dictionary is by necessity highly coherent and no useful bounds can be obtained in terms of the p norms [12].…”