Identify Equivalent Frames

TL;DR

This paper introduces a polynomial-time algorithm to determine whether two frames, which are overcomplete sets representing signals, are equivalent under permutations, sign changes, or orthogonal transformations, with theoretical guarantees for certain cases.

Contribution

The paper presents the first efficient algorithm for checking frame equivalence, overcoming the combinatorial complexity of previous methods.

Findings

Algorithm can verify frame equivalence in polynomial time.

Theoretical guarantees are established for specific cases.

Provides a practical tool for signal representation analysis.

Abstract

A frame is an overcomplete set that can represent vectors(signals) faithfully and stably. Two frames are equivalent if signals can be essentially represented in the same way, which means two frames differ by a permutation, sign change or orthogonal transformation. Since these operations are combinatorial in nature, it is infeasible to check whether two frames are equivalent by exhaustive search. In this note, we present an algorithm that can check this equivalence in polynomial time. Theoretical guarantees are provided for special cases.