# Near-Optimality of Linear Recovery from Indirect Observations

## TL;DR

This paper develops computationally efficient linear estimators for recovering linear images of signals from noisy, indirect observations, proving near-optimality under broad conditions and extending previous results to more general settings.

## Contribution

It introduces near-optimal linear recovery methods for a wide class of convex sets and norms, extending prior work to less restrictive assumptions and different noise models.

## Key findings

- Linear estimators are near-optimal under Gaussian noise.
- Methods apply to various convex sets and norms, including spectral and nuclear norms.
- Results hold without restrictions on the observation and recovery matrices.

## Abstract

We consider the problem of recovering linear image $Bx$ of a signal $x$ known to belong to a given convex compact set ${\cal X}$ from indirect observation $\omega=Ax+\xi$ of $x$ corrupted by random noise $\xi$ with finite covariance matrix. It is shown that under some assumptions on ${\cal X}$ (satisfied, e.g., when ${\cal X}$ is the intersection of $K$ concentric ellipsoids/elliptic cylinders, or the unit ball of the spectral norm in the space of matrices) and on the norm $\|\cdot\|$ used to measure the recovery error (satisfied, e.g., by $\|\cdot\|_p$-norms, $1\leq p\leq 2$, on ${\mathbf{R}}^m$ and by the nuclear norm on the space of matrices), one can build, in a computationally efficient manner, a "presumably good" linear in observations estimate, and that in the case of zero mean Gaussian observation noise, this estimate is near-optimal among all (linear and nonlinear) estimates in terms of its worst-case, over $x\in {\cal X}$, expected $\|\cdot\|$-loss. These results form an essential extension of those in our paper arXiv:1602.01355, where the assumptions on ${\cal X}$ were more restrictive, and the norm $\|\cdot\|$ was assumed to be the Euclidean one. In addition, we develop near-optimal estimates for the case of "uncertain-but-bounded" noise, where all we know about $\xi$ is that it is bounded in a given norm by a given $\sigma$. Same as in arXiv:1602.01355, our results impose no restrictions on $A$ and $B$. This arXiv paper slightly strengthens the journal publication Juditsky, A., Nemirovski, A. "Near-Optimality of Linear Recovery from Indirect Observations," Mathematical Statistics and Learning 1:2 (2018), 171-225.