Estimating Linear and Quadratic forms via Indirect Observations

TL;DR

This paper introduces a computationally efficient method for estimating linear and quadratic forms of signals from noisy indirect observations, broadening applicability beyond traditional models with near-optimal risk guarantees.

Contribution

It develops an operational convex programming approach for estimation that is computationally feasible and applicable to a wider range of observation schemes compared to traditional statistical methods.

Findings

Estimation routines are provably near-optimal under certain conditions.

The approach applies to a broader class of observation schemes.

Estimates are obtained via convex programming with guaranteed risks.

Abstract

In this paper, we further develop the approach, originating in [14 (arXiv:1311.6765),20 (arXiv:1604.02576)], to "computation-friendly" hypothesis testing and statistical estimation via Convex Programming. Specifically, we focus on estimating a linear or quadratic form of an unknown "signal," known to belong to a given convex compact set, via noisy indirect observations of the signal. Most of the existing theoretical results on the subject deal with precisely stated statistical models and aim at designing statistical inferences and quantifying their performance in a closed analytic form. In contrast to this descriptive (and highly instructive) traditional framework, the approach we promote here can be qualified as operational -- the estimation routines and their risks are yielded by an efficient computation. All we know in advance is that under favorable circumstances to be specified below, the risk of the resulting estimate, whether high or low, is provably near-optimal under the circumstances. As a compensation for the lack of "explanatory power," this approach is applicable to a much wider family of observation schemes than those where "closed form descriptive analysis" is possible. The paper is a follow-up to our paper [20 (arXiv:1604.02576)] dealing with hypothesis testing, in what follows, we apply the machinery developed in this reference to estimating linear and quadratic forms.