Geometric Structures and Optimization on Spaces of Finite Frames
Geometric Structures and Optimization on Spaces of Finite Frames
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Date
2011
Authors
Strawn, Nathaniel
Advisor
Benedetto, John J
Balan, Radu V
Balan, Radu V
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DRUM DOI
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.