On Design of Polyhedral Estimates in Linear Inverse Problems

TL;DR

This paper develops a new risk analysis for polyhedral estimates in linear inverse problems, focusing on signals within intersections of ellitopes and polytopes, enabling improved estimate design.

Contribution

It introduces a novel risk analysis framework for polyhedral estimates when the signal set is an intersection of an ellitope and a polytope, enhancing estimate design.

Findings

Provides a new risk analysis method for complex signal sets.

Enables more efficient and accurate polyhedral estimate design.

Improves estimation performance in linear inverse problems.

Abstract

Polyhedral estimate is a generic efficiently computable nonlinear in observations routine for recovering unknown signal belonging to a given convex compact set from noisy observation of signal's linear image. Risk analysis and optimal design of polyhedral estimates may be addressed through efficient bounding of optimal values of optimization problems. Such problems are typically hard; yet, it was shown in Juditsky, Nemirovski 2019 that nearly minimax optimal ("up to logarithmic factors") estimates can be efficiently constructed when the signal set is an ellitope - a member of a wide family of convex and compact sets of special geometry (see, e.g., Juditsky, Nemirovski 2018). The subject of this paper is a new risk analysis for polyhedral estimate in the situation where the signal set is an intersection of an ellitope and an arbitrary polytope allowing for improved polyhedral estimate design in this situation.