# Constrained Sampling: Optimum Reconstruction in Subspace with Minimax Regret Constraint

## TL;DR

This paper introduces a constrained sampling framework for optimal signal reconstruction that balances minimizing error within a known subspace and controlling maximum regret for all signals, enhancing robustness and practical applicability.

## Contribution

It proposes a novel constrained GSRP framework that combines subspace optimality with minimax regret constraints, addressing limitations of existing methods.

## Key findings

- Achieves near-optimal reconstruction for signals in the subspace
- Provides robustness for signals near the subspace
- Demonstrates effectiveness on Gaussian and speech signals

## Abstract

This paper considers the problem of optimum reconstruction in generalized sampling-reconstruction processes (GSRPs). We propose constrained GSRP, a novel framework that minimizes the reconstruction error for inputs in a subspace, subject to a constraint on the maximum regret-error for any other signal in the entire signal space. This framework addresses the primary limitation of existing GSRPs (consistent, subspace and minimax regret), namely, the assumption that the \emph{a priori} subspace is either fully known or fully ignored. We formulate constrained GSRP as a constrained optimization problem, the solution to which turns out to be a convex combination of the subspace and the minimax regret samplings. Detailed theoretical analysis on the reconstruction error shows that constrained sampling achieves a reconstruction that is 1) (sub)optimal for signals in the input subspace, 2) robust for signals around the input subspace, and 3) reasonably bounded for any other signals with a simple choice of the constraint parameter. Experimental results on sampling-reconstruction of a Gaussian input and a speech signal demonstrate the effectiveness of the proposed scheme.