Robust shape inference from a sparse approximation of the Gaussian trimmed loglikelihood

Given a noisy sample of points lying around some shape M, with possibly outliers or clutter noise, we focus on the question of recovering M, or at least geometric and topological information about M.
Often, such inference is based on the sublevel sets of distance-like functions such as the function distance to M, the distance-to-measure (DTM) [2] or the k-witnessed distance [4]. In this talk, we firstly widespread the concept of trimmed log-likelihood to probability distributions. This trimmed log-likelihood can be
considered as a generalisation of the DTM.
A sparse approximation of the DTM, the m-power distance-to-measure (m-PDTM), has been introduced and studied by Brecheteau and Levrard in 2017 [1]. Its sublevel sets are unions of m balls, with m possibly much smaller than the sample size. By miming the construction of the m-PDTM from the DTM, we propose an approximation of the trimmed log-likelihood associated to the family of Gaussian distributions on Rd. This approximation is sparse is the sense that its sublevel sets are unions of m ellipsoids.