Structure-Blind Signal Recovery

TL;DR

This paper introduces a new family of convex optimization-based estimators for recovering signals in Gaussian noise without prior knowledge of the signal set, achieving near-optimal risk bounds.

Contribution

It proposes a novel, structure-blind estimator framework that does not require prior signal set specification, with proven oracle inequalities and efficient computation methods.

Findings

Estimators perform close to the oracle risk in numerical tests.

The approach is computationally efficient via convex optimization.

Numerical illustrations demonstrate the method's potential.

Abstract

We consider the problem of recovering a signal observed in Gaussian noise. If the set of signals is convex and compact, and can be specified beforehand, one can use classical linear estimators that achieve a risk within a constant factor of the minimax risk. However, when the set is unspecified, designing an estimator that is blind to the hidden structure of the signal remains a challenging problem. We propose a new family of estimators to recover signals observed in Gaussian noise. Instead of specifying the set where the signal lives, we assume the existence of a well-performing linear estimator. Proposed estimators enjoy exact oracle inequalities and can be efficiently computed through convex optimization. We present several numerical illustrations that show the potential of the approach.