Optimization over finite frame varieties and structured dictionary design

A finite (μ,S)-frame variety consists of real or complex matrices F=[f1⋯fN] satisfying FF⁎=S and ‖fn‖=μn for all n=1,…,N. This paper introduces an approximate geometric gradient descent procedure over these varieties, which is powered by minimization algorithms for the frame operator distance and recent characterizations of these varietiesʼ tangent spaces. For almost all compatible pairings (μ,S), we demonstrate that minimization of the frame operator distance converges linearly under a threshold, we derive a process for constructing the orthogonal projection onto these varietiesʼ tangent spaces, and finally demonstrate that the approximate gradient descent procedure converges. Finally, we apply this procedure to numerically construct Grassmannian frames and Welch bound equality sequences with low mutual coherence.