Deconvolution with Unknown Error Distribution Interpreted as Blind Isotonic Regression

TL;DR

This paper introduces a matrix-based approach for collective deconvolution with unknown noise distribution, providing an efficient algorithm with optimal error rates and linking it to statistical seriation.

Contribution

It proposes a novel matrix viewpoint for collective deconvolution, offering a simple algorithm with non-asymptotic error bounds and connecting it to statistical seriation.

Findings

Algorithm achieves minimax optimal rate in a restricted sense.

Matrix structure utilization improves deconvolution accuracy.

Collective deconvolution is conjectured to be easier than single distribution deconvolution.

Abstract

Deconvolution is a statistical inverse problem to estimate the distribution of a random variable based on its noisy observations. Despite the extensive studies on the topic, deconvolution with unknown noise distribution remains as a notoriously hard problem. We propose a matrix-based viewpoint for collective deconvolution that subsumes the setup with repeated measurements as a special case. As the main result, we describe a simple algorithm that partially utilizes matrix structure to solve deconvolution problem and provide non-asymptotic error analysis for the algorithm. We show that the proposed algorithm achieves the minimax optimal rate for deconvolution in a restricted sense. We also remark the connection between the collective deconvolution and the so-called statistical seriation as a byproduct or our matrix viewpoint. We conjecture that the link suggests that collective deconvolution, as well as deconvolution with repeated measurements, is intrinsically much easier than usual deconvolution of a single distribution.