Mathematics Theses and Dissertations

Permanent URI for this collectionhttp://hdl.handle.net/1903/2793

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    Geometric Structures and Optimization on Spaces of Finite Frames
    (2011) Strawn, Nathaniel; Benedetto, John J; Balan, Radu V; Mathematics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)
    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.
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    Quantum Detection and Finite Frames
    (2005-04-12) Kebo, Andrew Kei; Benedetto, John J; Mathematics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)
    In quantum mechanics, the definition of a Von Neumann measurement can be generalized using positive-operator-valued measures. This modified definition of a quantum measurement allows one to better distinguish between a set of nonorthogonal quantum states. In this thesis we examine a quantum detection problem, where we have a physical system whose state is limited to be in only one of a finite number of possibilities. These possible states are not necessarily orthogonal. We want to find the best method of measuring the system in order to distinguish which state the system is in. Mathematically, we want to find a positive-operator-valued measure that minimizes the probability of a detection error. It is shown that all tight-frames with frame constant 1 correspond to positive-operator-valued measures. We reformulate the problem in terms of tight-frames that minimize the detection error. In the finite dimensional case, the problem of finding the tight-frame that minimizes the error can be converted into a Hamiltonian system on the group $SO(N)$. The minimum energy solutions of this Hamiltonian system correspond exactly to the tight-frames that minimize the detection error. In this setting, several numerical methods can be applied to give numerical constructions of the desired tight-frames.