arXiv subjects

Clare Wickman

Publications and source records attributed to Clare Wickman.

2 recordsLinked to original sources

arXiv2022

Gradient Flows for Frame Potentials on the Wasserstein Space

In this paper we bring together some of the key ideas and methods of two disparate fields of mathematical research, frame theory and optimal transport, using the methods of the second to answer questions posed in the first. In particular, we construct gradient flows in the Wasserstein space $P_2(\mathbb{R}^d)$ for a new potential, the tightness potential, which is a modification of the probabilistic frame potential. It is shown that the potential is suited for the application of a gradient descent scheme from optimal transport that can be used as the basis of an algorithm to evolve an existing frame toward a tight probabilistic frame.

math.FA↗

arXiv2017

Duality and Geodesics for Probabilistic Frames

Probabilistic frames are a generalization of finite frames into the Wasserstein space of probability measures with finite second moment. We introduce new probabilistic definitions of duality, analysis, and synthesis and investigate their properties. In particular, we formulate a theory of transport duals for probabilistic frames and prove certain properties of this class. We also investigate paths of probabilistic frames, identifying conditions under which geodesic paths between two such measures are themselves probabilistic frames. In the discrete case, this is related to ranks of convex combinations of matrices, while, in the continuous case, this is related to the continuity of the optimal transport plan.

math.FA↗