Support and distribution inference from noisy data

Abstract

We consider noisy observations of a distribution with unknown support. In the deconvolution model, it has been proved recently [19] that, under very mild assumptions, it is possible to solve the deconvolution problem without knowing the noise distribution and with no sample of the noise. We first give general settings where the theory applies and provide classes of supports that can be recovered in this context. We then exhibit classes of distributions over which we prove adaptive minimax rates (up to a log log factor) for the estimation of the support in Hausdorff distance. Moreover, for the class of distributions with compact support, we provide estimators of the unknown (in general singular) distribution and prove maximum rates in Wasserstein distance. We also prove an almost matching lower bound on the associated minimax risk.

References in corpus (8)

• Disentangling Identifiable Features from Noisy Data with Structured Nonlinear ICA
• Structure-adaptive manifold estimation
• Improved rates for Wasserstein deconvolution with ordinary smooth error in dimension one
• Non-Parametric Estimation of Manifolds from Noisy Data
• Local Regularization of Noisy Point Clouds: Improved Global Geometric Estimates and Data Analysis
• Estimating a density near an unknown manifold: a Bayesian nonparametric approach
• Bump Functions With Monotone Fourier Transforms Satisfying Decay Bounds
• Estimation via length-constrained generalized empirical principal curves under small noise