Sampling Finite Unit Norm Tight Frames Using Symplectic Geometry

TL;DR

This paper introduces the Eigenlift algorithm for sampling random finite unit norm tight frames by leveraging symplectic geometry, enabling efficient sampling through a geometric fiber bundle structure.

Contribution

It presents a novel geometric approach using symplectic reduction and Hamiltonian torus actions to sample FUNTFs, which was not previously explored.

Findings

Algorithm successfully samples FUNTFs in low dimensions

Utilizes symplectic geometry to structure the sampling process

Validated through Python implementation and rejection sampling

Abstract

Unit-norm tight frames in finite-dimensional Hilbert spaces (FUNTFs) are fundamental in signal processing, offering optimal robustness to noise and measurement loss. In this paper we introduce the Eigenlift algorithm for sampling random FUNTFs. Our approach exploits the symplectic geometry of the FUNTF space, which we characterize as a symplectic reduction of frame space by a symmetry group. We then define a Hamiltonian torus action on this reduced space whose momentum map induces a fiber bundle structure. The algorithm proceeds by sampling a point from the base space, which is a convex polytope, lifting it deterministically to a point on the corresponding fiber, then acting on this point by a random element of the torus to obtain a random FUNTF. We implement the method in Python and validate it in low-dimensional settings where it is computationally feasible to sample the base polytope via rejection sampling.