113 research outputs found
Gromov-Monge quasi-metrics and distance distributions
Applications in data science, shape analysis and object classification
frequently require maps between metric spaces which preserve geometry as
faithfully as possible. In this paper, we combine the Monge formulation of
optimal transport with the Gromov-Hausdorff distance construction to define a
measure of the minimum amount of geometric distortion required to map one
metric measure space onto another. We show that the resulting quantity, called
Gromov-Monge distance, defines an extended quasi-metric on the space of
isomorphism classes of metric measure spaces and that it can be promoted to a
true metric on certain subclasses of mm-spaces. We also give precise
comparisons between Gromov-Monge distance and several other metrics which have
appeared previously, such as the Gromov-Wasserstein metric and the continuous
Procrustes metric of Lipman, Al-Aifari and Daubechies. Finally, we derive
polynomial-time computable lower bounds for Gromov-Monge distance. These lower
bounds are expressed in terms of distance distributions, which are classical
invariants of metric measure spaces summarizing the volume growth of metric
balls. In the second half of the paper, which may be of independent interest,
we study the discriminative power of these lower bounds for simple subclasses
of metric measure spaces. We first consider the case of planar curves, where we
give a counterexample to the Curve Histogram Conjecture of Brinkman and Olver.
Our results on plane curves are then generalized to higher dimensional
manifolds, where we prove some sphere characterization theorems for the
distance distribution invariant. Finally, we consider several inverse problems
on recovering a metric graph from a collection of localized versions of
distance distributions. Results are derived by establishing connections with
concepts from the fields of computational geometry and topological data
analysis.Comment: Version 2: Added many new results and improved expositio
Fusion Frame Homotopy and Tightening Fusion Frames by Gradient Descent
Finite frames, or spanning sets for finite-dimensional Hilbert spaces, are a
ubiquitous tool in signal processing. There has been much recent work on
understanding the global structure of collections of finite frames with
prescribed properties, such as spaces of unit norm tight frames. We extend some
of these results to the more general setting of fusion frames -- a fusion frame
is a collection of subspaces of a finite-dimensional Hilbert space with the
property that any vector can be recovered from its list of projections. The
notion of tightness extends to fusion frames, and we consider the following
basic question: is the collection of tight fusion frames with prescribed
subspace dimensions path connected? We answer (a generalization of) this
question in the affirmative, extending the analogous result for unit norm tight
frames proved by Cahill, Mixon and Strawn. We also extend a result of Benedetto
and Fickus, who defined a natural functional on the space of unit norm frames
(the frame potential), showed that its global minimizers are tight, and showed
that it has no spurious local minimizers, meaning that gradient descent can be
used to construct unit-norm tight frames. We prove the analogous result for the
fusion frame potential of Casazza and Fickus, implying that, when tight fusion
frames exist for a given choice of dimensions, they can be constructed via
gradient descent. Our proofs use techniques from symplectic geometry and
Mumford's geometric invariant theory
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