Geometric Structures and Optimization on Spaces of Finite Frames

Abstract

A finite (μ, Ѕ)-frame variety consists of the real or complex matrices F = [f1 … fn] with frame operator FF* = S, and which also satisfies ||fi|| = μi for all i = 1,...,N. Here, S is a fixed Hermitian positive definite matrix and μ = [μ1...μN] is a fixed list of lengths. These spaces generalize the well-known spaces of finite unit-norm tight frames. We explore the local geometry of these spaces and develop geometric optimization algorithms based on the resulting insights. We study the local geometric structure of the (μ, Ѕ)-frame

varieties by viewing them as intersections of generalized tori (the length constraints) with distorted Stiefel manifolds (the frame operator constraint). Exploiting this perspective, we characterize the nonsingular points of these varieties by determining where this intersection is transversal in a Hilbert-Schmidt sphere. A corollary of this characterization is a characterization of the tangent spaces of (μ, Ѕ)-frame varieties, which is in turn leveraged to validate explicit local coordinate systems. Explict bases for the tangent spaces are also constructed. Geometric optimization over a (μ, Ѕ)-frame variety is performed by combining

knowledge of the tangent spaces with geometric optimization of the frame operator distance over a product of spheres. Given a differentiable objective function, we project the full gradient onto the tangent space and then minimize the frame operator distance to obtain an approximate gradient descent algorithm. To partially validate this procedure, we demonstrate that the induced flow converges locally. Using Sherman-Morrision type formulas, we also describe a technique for constructing points on these varieties that can be used to initialize the optimization procedure. Finally, we apply the approximate gradient descent procedure to numerically construct equiangular tight frames, Grassmannian frames, and Welch bound equality sequences with low mutual coherence.