# On a class of optimization-based robust estimators

## TL;DR

This paper introduces a general optimization framework for robust parameter estimation in the presence of both sparse, arbitrarily large noise and bounded dense noise, providing verifiable bounds and empirical validation.

## Contribution

It proposes a new optimization-based approach for robust regression that offers verifiable bounds on noise tolerance and estimation error, improving upon previous isometry-based methods.

## Key findings

- Bound on the number of sparse noise elements that can be tolerated
- Convex optimization problem for bound computation
- Empirical evidence suggests the bounds are tight

## Abstract

We consider in this paper the problem of estimating a parameter matrix from observations which are affected by two types of noise components: (i) a sparse noise sequence which, whenever nonzero can have arbitrarily large amplitude (ii) and a dense and bounded noise sequence of "moderate" amount. This is termed a robust regression problem. To tackle it, a quite general optimization-based framework is proposed and analyzed. When only the sparse noise is present, a sufficient bound is derived on the number of nonzero elements in the sparse noise sequence that can be accommodated by the estimator while still returning the true parameter matrix. While almost all the restricted isometry-based bounds from the literature are not verifiable, our bound can be easily computed through solving a convex optimization problem. Moreover, empirical evidence tends to suggest that it is generally tight. If in addition to the sparse noise sequence, the training data are affected by a bounded dense noise, we derive an upper bound on the estimation error.