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Researcher profile

Tom Needham

Tom Needham contributes to research discovery and scholarly infrastructure.

ResearcherAffiliation not importedOpen to collaborate
math.MGMachine Learningmath.DGApplicationsComputational Geometrymath.ATmath.FAComputer Visionmath.SGphysics.soc-ph

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Published work

9 published item(s)

preprint2026arXiv

The Observable Wasserstein Distance

We introduce the observable Wasserstein distance, a framework for deriving lower bounds on the Wasserstein distance between probability measures on Polish metric spaces, designed to bypass the computational intractability of exact optimal transport in large-scale, non-Euclidean datasets. Analogous to the sliced Wasserstein distance in $\mathbb{R}^d$, our approach projects measures onto the real line via 1-Lipschitz observables and computes the Wasserstein distances between the resulting pushforward distributions. We define a hierarchy of pseudo-metrics by restricting observables to a nested chain of subspaces. A central theoretical contribution is an injectivity result linking the metric covering dimension of the support of a measure to the specific order in the hierarchy that guarantees unique recovery. This serves as a metric-space analogue to the Cramér-Wold Device for Euclidean distributions. We demonstrate that this hierarchy offers a tunable trade-off between sharpness as a lower bound on the Wasserstein distance and computational efficiency. We also present a discrete computational model for finite grids and numerical experiments validating the efficacy and utility of these approximations.

0 citations0 reviewsNo code linkedOpen access
preprint2022arXiv

Admissibility and Frame Homotopy for Quaternionic Frames

We consider the following questions: when do there exist quaternionic frames with given frame spectrum and given frame vector norms? When such frames exist, is it always possible to interpolate between any two while fixing their spectra and norms? In other words, the first question is the admissibility question for quaternionic frames and the second is a generalization of the frame homotopy conjecture. We give complete answers to both questions. For the first question, the existence criterion is exactly the same as in the real and complex cases. For the second, the non-empty spaces of quaternionic frames with specified frame spectrum and frame vector norms are always path-connected, just as in the complex case. Our strategy for proving these results is based on interpreting equivalence classes of frames with given frame spectrum as adjoint orbits, which is an approach that is also well-suited to the study of real and complex frames.

0 citations0 reviewsNo code linkedOpen access
preprint2022arXiv

Distance distributions and inverse problems for metric measure spaces

Applications in data science, shape analysis and object classification frequently require comparison of probability distributions defined on different ambient spaces. To accomplish this, one requires a notion of distance on a given class of metric measure spaces -- that is, compact metric spaces endowed with probability measures. Such distances are typically defined as comparisons between metric measure space invariants, such as distance distributions (also referred to as shape distributions, distance histograms or shape contexts in the literature). Generally, distances defined in terms of distance distributions are actually pseudometrics, in that they may vanish when comparing nonisomorphic spaces. The goal of this paper is to set up a formal framework for assessing the discrimininative power of distance distributions, i.e., the extend to which these pseudometrics fail to define proper metrics. We formulate several precise inverse problems in terms of these invariants and answer them in several categories of metric measure spaces, including the category of plane curves, where we give a counterexample to the Curve Histogram Conjecture of Brinkman and Olver, the categories of embedded and Riemannian manifolds, where we obtain sphere rigidity results, and the category of metric graphs, where we obtain a local injectivity result along the lines of classical work of Boutin and Kemper on point cloud configurations. The inverse problems are further contextualized by the introduction of a variant of the Gromov-Wasserstein distance on the space of metric measure spaces, which is inspired by the original Monge formulation of optimal transport.

0 citations0 reviewsNo code linkedOpen access
preprint2022arXiv

Statistical Shape Analysis of Brain Arterial Networks (BAN)

Structures of brain arterial networks (BANs) - that are complex arrangements of individual arteries, their branching patterns, and inter-connectivities - play an important role in characterizing and understanding brain physiology. One would like tools for statistically analyzing the shapes of BANs, i.e. quantify shape differences, compare population of subjects, and study the effects of covariates on these shapes. This paper mathematically represents and statistically analyzes BAN shapes as elastic shape graphs. Each elastic shape graph is made up of nodes that are connected by a number of 3D curves, and edges, with arbitrary shapes. We develop a mathematical representation, a Riemannian metric and other geometrical tools, such as computations of geodesics, means and covariances, and PCA for analyzing elastic graphs and BANs. This analysis is applied to BANs after separating them into four components -- top, bottom, left, and right. This framework is then used to generate shape summaries of BANs from 92 subjects, and to study the effects of age and gender on shapes of BAN components. We conclude that while gender effects require further investigation, the age has a clear, quantifiable effect on BAN shapes. Specifically, we find an increased variance in BAN shapes as age increases.

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preprint2022arXiv

Toric Symplectic Geometry and Full Spark Frames

The collection of $d \times N$ complex matrices with prescribed column norms and prescribed (nonzero) singular values forms a compact algebraic variety, which we refer to as a frame space. Elements of frame spaces -- i.e., frames -- are used to give robust representations of complex-valued signals, so that geometrical and measure-theoretic properties of frame spaces are of interest to the signal processing community. This paper is concerned with the following question: what is the probability that a frame drawn uniformly at random from a given frame space has the property that any subset of $d$ of its columns gives a basis for $\mathbb{C}^d$? We show that the probability is one, generalizing recent work of Cahill, Mixon and Strawn. To prove this, we first show that frame spaces are related to highly structured objects called toric symplectic manifolds. This relationship elucidates the geometric meaning of eigensteps -- certain spectral invariants of a frame -- and should be a more broadly applicable tool for studying probabilistic questions about the structure of frame spaces. As another application of our symplectic perspective, we completely characterize the norm and spectral data for which the corresponding frame space has singularities, answering some open questions in the frame theory literature.

0 citations0 reviewsNo code linkedOpen access
preprint2021arXiv

Generalized Spectral Clustering via Gromov-Wasserstein Learning

We establish a bridge between spectral clustering and Gromov-Wasserstein Learning (GWL), a recent optimal transport-based approach to graph partitioning. This connection both explains and improves upon the state-of-the-art performance of GWL. The Gromov-Wasserstein framework provides probabilistic correspondences between nodes of source and target graphs via a quadratic programming relaxation of the node matching problem. Our results utilize and connect the observations that the GW geometric structure remains valid for any rank-2 tensor, in particular the adjacency, distance, and various kernel matrices on graphs, and that the heat kernel outperforms the adjacency matrix in producing stable and informative node correspondences. Using the heat kernel in the GWL framework provides new multiscale graph comparisons without compromising theoretical guarantees, while immediately yielding improved empirical results. A key insight of the GWL framework toward graph partitioning was to compute GW correspondences from a source graph to a template graph with isolated, self-connected nodes. We show that when comparing against a two-node template graph using the heat kernel at the infinite time limit, the resulting partition agrees with the partition produced by the Fiedler vector. This in turn yields a new insight into the k-cut graph partitioning problem through the lens of optimal transport. Our experiments on a range of real-world networks achieve comparable results to, and in many cases outperform, the state-of-the-art achieved by GWL.

0 citations0 reviewsNo code linkedOpen access
preprint2020arXiv

Gromov-Wasserstein Averaging in a Riemannian Framework

We introduce a theoretical framework for performing statistical tasks---including, but not limited to, averaging and principal component analysis---on the space of (possibly asymmetric) matrices with arbitrary entries and sizes. This is carried out under the lens of the Gromov-Wasserstein (GW) distance, and our methods translate the Riemannian framework of GW distances developed by Sturm into practical, implementable tools for network data analysis. Our methods are illustrated on datasets of letter graphs, asymmetric stochastic blockmodel networks, and planar shapes viewed as metric spaces. On the theoretical front, we supplement the work of Sturm by producing additional results on the tangent structure of this "space of spaces", as well as on the gradient flow of the Fréchet functional on this space.

0 citations0 reviewsNo code linkedOpen access
preprint2020arXiv

The (homological) persistence of gerrymandering

We apply persistent homology, the dominant tool from the field of topological data analysis, to study electoral redistricting. Our method combines the geographic information from a political districting plan with election data to produce a persistence diagram. We are then able to visualize and analyze large ensembles of computer-generated districting plans of the type commonly used in modern redistricting research (and court challenges). We set out three applications: zoning a state at each scale of districting, comparing elections, and seeking signals of gerrymandering. Our case studies focus on redistricting in Pennsylvania and North Carolina, two states whose legal challenges to enacted plans have raised considerable public interest in the last few years. To address the question of robustness of the persistence diagrams to perturbations in vote data and in district boundaries, we translate the classical stability theorem of Cohen--Steiner et al. into our setting and find that it can be phrased in a manner that is easy to interpret. We accompany the theoretical bound with an empirical demonstration to illustrate diagram stability in practice.

0 citations0 reviewsNo code linkedOpen access
preprint2020arXiv

The Weighted Euler Curve Transform for Shape and Image Analysis

The Euler Curve Transform (ECT) of Turner et al.\ is a complete invariant of an embedded simplicial complex, which is amenable to statistical analysis. We generalize the ECT to provide a similarly convenient representation for weighted simplicial complexes, objects which arise naturally, for example, in certain medical imaging applications. We leverage work of Ghrist et al.\ on Euler integral calculus to prove that this invariant---dubbed the Weighted Euler Curve Transform (WECT)---is also complete. We explain how to transform a segmented region of interest in a grayscale image into a weighted simplicial complex and then into a WECT representation. This WECT representation is applied to study Glioblastoma Multiforme brain tumor shape and texture data. We show that the WECT representation is effective at clustering tumors based on qualitative shape and texture features and that this clustering correlates with patient survival time.

0 citations0 reviewsNo code linkedOpen access