Approximation by partial isometries and symmetric approximation of finite frames

TL;DR

This paper addresses the problem of best approximation of matrices by partial isometries using unitarily invariant norms, providing parametrizations, conditions for uniqueness, and applications to frame approximation.

Contribution

It introduces a comprehensive solution for approximating matrices with partial isometries, extending symmetric approximation of frames, and characterizing solutions in subspaces.

Findings

Provided parametrization of solutions for strictly convex norms.

Established necessary and sufficient conditions for uniqueness.

Extended symmetric approximation concepts to subspace frames.

Abstract

We solve the problem of best approximation by partial isometries of given rank to an arbitrary rectangular matrix, when the distance is measured in any unitarily invariant norm. In the case where the norm is strictly convex, we parametrize all the solutions. In particular, this allow us to give a simple necessary and sufficient condition for uniqueness. We then apply these results to solve the global problem of approximation by partial isometries, and to extend the notion of symmetric approximation of frames introduced in M. Frank, V. Paulsen, T. Tiballi, Symmetric Approximation of frames and bases in Hilbert Spaces, Trans. Amer. Math. Soc. 354 (2002), 777-793. In addition, we characterize symmetric approximations of frames belonging to a prescribed subspace.