Auto-tuning unit norm frames

TL;DR

This paper introduces a simple iterative gradient descent algorithm to enhance the tightness of finite unit norm frames, addressing the challenge of constructing such frames and providing convergence guarantees.

Contribution

The authors develop a new gradient descent method for improving the tightness of finite unit norm frames, with convergence analysis and solutions to the Paulsen problem.

Findings

Algorithm converges linearly when the number of elements is relatively prime to the dimension.

Preserves group structures in initial frames during optimization.

Provides explicit bounds on how close almost tight, almost unit norm frames are to true tight frames.

Abstract

Finite unit norm tight frames provide Parseval-like decompositions of vectors in terms of redundant components of equal weight. They are known to be exceptionally robust against additive noise and erasures, and as such, have great potential as encoding schemes. Unfortunately, up to this point, these frames have proven notoriously difficult to construct. Indeed, though the set of all unit norm tight frames, modulo rotations, is known to contain manifolds of nontrivial dimension, we have but a small finite number of known constructions of such frames. In this paper, we present a new iterative algorithm---gradient descent of the frame potential---for increasing the degree of tightness of any finite unit norm frame. The algorithm itself is trivial to implement, and it preserves certain group structures present in the initial frame. In the special case where the number of frame elements is relatively prime to the dimension of the underlying space, we show that this algorithm converges to a unit norm tight frame at a linear rate, provided the initial unit norm frame is already sufficiently close to being tight. By slightly modifying this approach, we get a similar, but weaker, result in the non-relatively-prime case, providing an explicit answer to the Paulsen problem: "How close is a frame which is almost tight and almost unit norm to some unit norm tight frame?"