arXiv subjects

Johnathan Bush

Publications and source records attributed to Johnathan Bush.

9 recordsLinked to original sources

arXiv2024

The connectivity of Vietoris-Rips complexes of spheres

We survey what is known and unknown about Vietoris-Rips complexes and thickenings of spheres. Afterwards, we show how to control the homotopy connectivity of Vietoris-Rips complexes of spheres in terms of coverings of spheres and projective spaces. Let $S^n$ be the $n$-sphere with the geodesic metric, and of diameter $\pi$, and let $\delta > 0$. Suppose that the first nontrivial homotopy group of the Vietoris-Rips complex $\mathrm{VR}(S^n;\pi-\delta)$ of the $n$-sphere at scale $\pi-\delta$ occurs in dimension $k$, i.e., suppose that the connectivity is $k-1$. Then $\mathrm{cov}_{S^n}(2k+2) \le \delta < 2\cdot \mathrm{cov}_{\mathbb{R}P^n}(k)$. In other words, there exist $2k+2$ balls of radius $\delta$ that cover $S^n$, and no set of $k$ balls of radius $\frac{\delta}{2}$ cover the projective space $\mathbb{R}P^n$. As a corollary, the homotopy type of $\mathrm{VR}(S^n;r)$ changes infinitely many times as the scale $r$ increases.

math.AT↗

arXiv2023

Topological feature selection for time series data

We use tools from applied topology for feature selection on time series data. We develop a method for scoring the variables in a multivariate time series that reflects their contributions to the topological features of the corresponding point cloud. Our approach produces a piecewise-linear Lipschitz gradient path in the standard geometric simplex that starts at the barycenter, which weights the variables equally, and ends at the score. Taking the mean of stochastic gradient descent results in a mean gradient path that satisfies a strong law of large numbers and central limit theorem. Our theory is motivated by the analysis of the neuronal activities of the nematode C. elegans, and our method selects an informative subset of the neurons that optimizes the coordinated dynamics.

math.AT↗

arXiv2022

Gromov-Hausdorff distances, Borsuk-Ulam theorems, and Vietoris-Rips complexes

We explore emerging relationships between the Gromov--Hausdorff distance, Borsuk--Ulam theorems, and Vietoris--Rips simplicial complexes. The Gromov--Hausdorff distance between two metric spaces $X$ and~$Y$ can be lower bounded by the distortion of (possibly discontinuous) functions between them. The more these functions must distort the metrics, the larger the Gromov--Hausdorff distance must be. Topology has few tools to obstruct the existence of discontinuous functions. However, an arbitrary function $f\colon X\to Y$ induces a continuous map between their Vietoris--Rips simplicial complexes, where the allowable choices of scale parameters depend on how much the function $f$ distorts distances. We can then use equivariant topology to obstruct the existence of certain continuous maps between Vietoris--Rips complexes. With these ideas we bound how discontinuous an odd map between spheres $S^k\to S^n$ with $k>n$ must be, generalizing a result by Dubins and Schwarz (1981), which is the case $k=n+1$. As an application, we recover or improve upon all of the lower bounds from Lim, M{\'e}moli, and Smith (2022) on the Gromov--Hausdorff distances between spheres of different dimensions. We also provide new upper bounds on the Gromov--Hausdorff distance between spheres of adjacent dimensions.

math.MG↗

arXiv2022

Toroidal Coordinates: Decorrelating Circular Coordinates With Lattice Reduction

The circular coordinates algorithm of de Silva, Morozov, and Vejdemo-Johansson takes as input a dataset together with a cohomology class representing a $1$-dimensional hole in the data; the output is a map from the data into the circle that captures this hole, and that is of minimum energy in a suitable sense. However, when applied to several cohomology classes, the output circle-valued maps can be "geometrically correlated" even if the chosen cohomology classes are linearly independent. It is shown in the original work that less correlated maps can be obtained with suitable integer linear combinations of the cohomology classes, with the linear combinations being chosen by inspection. In this paper, we identify a formal notion of geometric correlation between circle-valued maps which, in the Riemannian manifold case, corresponds to the Dirichlet form, a bilinear form derived from the Dirichlet energy. We describe a systematic procedure for constructing low energy torus-valued maps on data, starting from a set of linearly independent cohomology classes. We showcase our procedure with computational examples. Our main algorithm is based on the Lenstra--Lenstra--Lov\'asz algorithm from computational number theory.

cs.CG↗

arXiv2021

The topology of projective codes and the distribution of zeros of odd maps

We show that the size of codes in projective space controls structural results for zeros of odd maps from spheres to Euclidean space. In fact, this relation is given through the topology of the space of probability measures on the sphere whose supports have diameter bounded by some specific parameter. Our main result is a generalization of the Borsuk--Ulam theorem, and we derive four consequences of it: (i) We give a new proof of a result of Simonyi and Tardos on topological lower bounds for the circular chromatic number of a graph; (ii) we study generic embeddings of spheres into Euclidean space and show that projective codes give quantitative bounds for a measure of genericity of sphere embeddings; and we prove generalizations of (iii) the Ham Sandwich theorem and (iv) the Lyusternik--Shnirel'man--Borsuk covering theorem for the case where the number of measures or sets in a covering, respectively, may exceed the ambient dimension.

math.GT↗

arXiv2021

Operations on Metric Thickenings

Many simplicial complexes arising in practice have an associated metric space structure on the vertex set but not on the complex, e.g. the Vietoris-Rips complex in applied topology. We formalize a remedy by introducing a category of simplicial metric thickenings whose objects have a natural realization as metric spaces. The properties of this category allow us to prove that, for a large class of thickenings including Vietoris-Rips and Cech thickenings, the product of metric thickenings is homotopy equivalent to the metric thickenings of product spaces, and similarly for wedge sums.

math.CT↗

arXiv2020

Representations of Energy Landscapes by Sublevelset Persistent Homology: An Example With n-Alkanes

Encoding the complex features of an energy landscape is a challenging task, and often chemists pursue the most salient features (minima and barriers) along a highly reduced space, i.e. 2- or 3-dimensions. Even though disconnectivity graphs or merge trees summarize the connectivity of the local minima of an energy landscape via the lowest-barrier pathways, there is more information to be gained by also considering the topology of each connected component at different energy thresholds (or sublevelsets). We propose sublevelset persistent homology as an appropriate tool for this purpose. Our computations on the configuration phase space of n-alkanes from butane to octane allow us to conjecture, and then prove, a complete characterization of the sublevelset persistent homology of the alkane $C_m H_{2m+2}$ potential energy landscapes, for all $m$, and in all homological dimensions. We further compare both the analytical configurational potential energy landscapes and sampled data from molecular dynamics simulation, using the united and all-atom descriptions of the intramolecular interactions. In turn, this supports the application of distance metrics to quantify sampling fidelity and lays the foundation for future work regarding new metrics that quantify differences between the topological features of high-dimensional energy landscapes.

physics.chem-ph↗

arXiv2019

Metric Thickenings, Borsuk-Ulam Theorems, and Orbitopes

Thickenings of a metric space capture local geometric properties of the space. Here we exhibit applications of lower bounding the topology of thickenings of the circle and more generally the sphere. We explain interconnections with the geometry of circle actions on Euclidean space, the structure of zeros of trigonometric polynomials, and theorems of Borsuk-Ulam type. We use the combinatorial and geometric structure of the convex hull of orbits of circle actions on Euclidean space to give geometric proofs of the homotopy type of metric thickenings of the circle. Homotopical connectivity bounds of thickenings of the sphere allow us to prove that a weighted average of function values of odd maps $S^n \to \mathbb{R}^{n+2}$ on a small diameter set is zero. We prove an additional generalization of the Borsuk-Ulam theorem for odd maps $S^{2n-1} \to \mathbb{R}^{2kn+2n-1}$. We prove such results for odd maps from the circle to any Euclidean space with optimal quantitative bounds. This in turn implies that any raked homogeneous trigonometric polynomial has a zero on a subset of the circle of a specific diameter; these results are optimal.

math.GT↗

arXiv2018

A torus model for optical flow

We propose a torus model for high-contrast patches of optical flow. Our model is derived from a database of ground-truth optical flow from the computer-generated video \emph{Sintel}, collected by Butler et al.\ in \emph{A naturalistic open source movie for optical flow evaluation}. Using persistent homology and zigzag persistence, popular tools from the field of computational topology, we show that the high-contrast $3\times 3$ patches from this video are well-modeled by a \emph{torus}, a nonlinear 2-dimensional manifold. Furthermore, we show that the optical flow torus model is naturally equipped with the structure of a fiber bundle, related to the statistics of range image patches.

cs.CV↗