Date of Award
8-2024
Document Type
Dissertation
Degree Name
Doctor of Philosophy (PhD)
Department
School of Mathematical and Statistical Sciences
Committee Chair/Advisor
Dr. Martin Schmoll
Committee Member
Dr. Mishko Mitkovski
Committee Member
Dr. Shitao Liu
Committee Member
Dr. Cody Stockdale
Abstract
As the generalization of frames in the Euclidean space $\mathbb{R}^n$, a probabilistic frame is a probability measure on $\mathbb{R}^n$ that has a finite second moment and whose support spans $\mathbb{R}^n$. The p-Wasserstein distance with $p \geq 1$ from optimal transport is often used to compare probabilistic frames. It is particularly useful to compare frames of various cardinalities in the context of probabilistic frames. We show that the 2-Wasserstein distance appears naturally in the fundamental objects of frame theory and draws consequences leading to a geometric viewpoint of probabilistic frames.
We convert the classic lower bound estimates of 2-Wasserstein distance \cite{Gelbrich90, cuesta1996lower} from covariance operators to frame operators. As a consequence, we show that the sets of probabilistic frames with given frame operators are all homeomorphic, and the homeomorphism is an optimal transport map given by push-forward with a unique symmetric positive definite map that depends only on the two given frame operators. As an application, we generalize several recent results in probabilistic frames, such as finding the closest frame of a certain kind and showing the connectedness of the set of probabilistic frames with any given frame operator.
Furthermore, we consider the perturbation of probabilistic frames by generalizing the Paley--Wiener theorem to probabilistic frames. The Paley--Wiener Theorem is a classical result about the stability of a basis in a Banach space, claiming that if a sequence is close to a basis, then this sequence is also a basis. Similar results are extended to frames in Hilbert spaces. In this work, we generalize the Paley--Wiener theorem to probabilistic frames and claim that if a probability measure is close to a probabilistic frame in some sense, this probability measure is also a probabilistic frame.
In the end, we discuss some open problems about probabilistic frames. We first mention the minimization problem of probabilistic p-frames in the $p$-Wasserstein metric. Then we introduce probabilistic frames on the unit sphere $\mathbb{S}^{n-1}$. We show the existence of the closest probabilistic Parseval frame on $\mathbb{S}^{n-1}$ and give a lower bound estimate about the minimizing distance.
Recommended Citation
Chen, Dongwei, "Probabilistic Frames and Concepts from Optimal Transport" (2024). All Dissertations. 3677.
https://open.clemson.edu/all_dissertations/3677
Author ORCID Identifier
https://orcid.org/0000-0003-1956-7683
Comments
NA