arXiv subjects

Henry Adams

Publications and source records attributed to Henry Adams.

At least 37 records · Page 2Linked to original sources

arXiv2024

Efficient Evader Detection in Mobile Sensor Networks

Suppose one wants to monitor a domain with sensors, each sensing a small ball-shaped region, but the domain is hazardous enough that one cannot control the placement of the sensors. A prohibitively large number of randomly placed sensors could be required to obtain static coverage. Instead, one can use fewer sensors by providing mobile coverage, a generalization of the static setup wherein every possible evader is detected by the moving sensors in a bounded amount of time. Here, we use topology in order to implement algorithms certifying mobile coverage that use only local data to solve the global problem. Our algorithms do not require knowledge of the sensors' locations, only their connectivity information. We experimentally study the statistics of mobile coverage in two dynamical scenarios. We allow the sensors to move independently (billiard dynamics and Brownian motion), or to locally coordinate their dynamics (collective animal motion models). Our detailed simulations show, for example, that collective motion can enhance performance: The expected time until the mobile sensor network achieves mobile coverage is lower for the D'Orsogna collective motion model than for the billiard motion model. Further, we show that even when the probability of static coverage is low, all possible evaders can nevertheless be detected relatively quickly by the mobile sensor network.

math.DS↗

arXiv2023

Time-Varying Spaces and Mobile Sensor Networks

Consider a mobile sensor network, in which each sensor covers a ball. Sensors do not know their locations, but can detect if the covered balls overlap. An intruder cannot pass undetected between overlapping sensors. An evasion path exists if it is possible for an intruder to move in the domain without ever entering a covered region. We examine two time-varying topological spaces arising from such mobile sensor networks. These examples were constructed by Adams and Carlsson to show that the time-varying homotopy type of the covered region does not determine whether an evasion path exists or not. One of these spaces has an evasion path and the other does not, which means that the uncovered regions are not time-varying homotopy equivalent. We show that the covered regions of these spaces are not time-varying homeomorphic, even though they are time-varying homotopy equivalent. We then elaborate on this time-varying homotopy equivalence between the covered regions.

math.AT↗

arXiv2023

Čech complexes of hypercube graphs

A Čech complex of a finite simple graph $G$ is a nerve complex of balls in the graph, with one ball centered at each vertex. More precisely, let the Čech complex $\mathcal{N}(G,r)$ be the nerve of all closed balls of radius $\frac{r}{2}$ centered at vertices of $G$, where these balls are drawn in the geometric realization of the graph $G$ (equipped with the shortest path metric). The simplicial complex $\mathcal{N}(G,r)$ is equal to the graph $G$ when $r=1$, and homotopy equivalent to the graph $G$ when $r$ is smaller than half the length of the shortest loop in $G$. For higher values of $r$, the topology of $\mathcal{N}(G,r)$ is not well-understood. We consider the $n$-dimensional hypercube graphs $\mathbb{I}_n$ with $2^n$ vertices. Our main results are as follows. First, when $r=2$, we show that the Čech complex $\mathcal{N}(\mathbb{I}_n,2)$ is homotopy equivalent to a wedge of 2-spheres for all $n\ge 1$, and we count the number of 2-spheres appearing in this wedge sum. Second, when $r=3$, we show that $\mathcal{N}(\mathbb{I}_n,3)$ is homotopy equivalent to a simplicial complex of dimension at most 4, and that for $n\ge 4$ the reduced homology of $\mathcal{N}(\mathbb{I}_n, 3)$ is nonzero in dimensions 3 and 4, and zero in all other dimensions. Finally, we show that for all $n\ge 1$ and $r\ge 0$, the inclusion $\mathcal{N}(\mathbb{I}_n, r)\hookrightarrow \mathcal{N}(\mathbb{I}_n, r+2)$ is null-homotopic, providing a bound on the length of bars in the persistent homology of Čech complexes of hypercube graphs.

math.CO↗

arXiv2023

Lower bounds on the homology of Vietoris-Rips complexes of hypercube graphs

We provide novel lower bounds on the Betti numbers of Vietoris-Rips complexes of hypercube graphs of all dimensions, and at all scales. In more detail, let $Q_n$ be the vertex set of $2^n$ vertices in the $n$-dimensional hypercube graph, equipped with the shortest path metric. Let $VR(Q_n;r)$ be its Vietoris--Rips complex at scale parameter $r \ge 0$, which has $Q_n$ as its vertex set, and all subsets of diameter at most $r$ as its simplices. For integers $r<r'$ the inclusion $VR(Q_n;r)\hookrightarrow VR(Q_n;r')$ is nullhomotopic, meaning no persistent homology bars have length longer than one, and we therefore focus attention on the individual spaces $VR(Q_n;r)$. We provide lower bounds on the ranks of homology groups of $VR(Q_n;r)$. For example, using cross-polytopal generators, we prove that the rank of $H_{2^r-1}(VR(Q_n;r))$ is at least $2^{n-(r+1)}\binom{n}{r+1}$. We also prove a version of \emph{homology propagation}: if $q\ge 1$ and if $p$ is the smallest integer for which $rank H_q(VR(Q_p;r))\neq 0$, then $rank H_q(VR(Q_n;r)) \ge \sum_{i=p}^n 2^{i-p} \binom{i-1}{p-1} \cdot rank H_q(VR(Q_p;r))$ for all $n \ge p$. When $r\le 3$, this result and variants thereof provide tight lower bounds on the rank of $H_q(VR(Q_n;r))$ for all $n$, and for each $r \ge 4$ we produce novel lower bounds on the ranks of homology groups. Furthermore, we show that for each $r\ge 2$, the homology groups of $VR(Q_n;r)$ for $n \ge 2r+1$ contain propagated homology not induced by the initial cross-polytopal generators.

math.CO↗

arXiv2022

Vietoris thickenings and complexes have isomorphic homotopy groups

We study the relationship between metric thickenings and simplicial complexes associated to coverings of metric spaces. Let $\mathcal{U}$ be a cover of a separable metric space $X$ by open sets with a uniform diameter bound. The Vietoris complex contains all simplices with vertex set contained in some $U \in \mathcal{U}$, and the Vietoris metric thickening is the space of probability measures with support in some $U \in \mathcal{U}$, equipped with an optimal transport metric. We show that the Vietoris metric thickening and the Vietoris complex have isomorphic homotopy groups in all dimensions. In particular, by choosing the cover $\mathcal{U}$ appropriately, we get isomorphisms between the homotopy groups of Vietoris--Rips metric thickenings and simplicial complexes, where both spaces are defined using the convention ``diameter $< r$'' (instead of $\le r$). Similarly, we get isomorphisms between the homotopy groups of Čech metric thickenings and simplicial complexes, where both spaces are defined using open balls (instead of closed balls).

math.MG↗

arXiv2022

Support vector machines and Radon's theorem

A support vector machine (SVM) is an algorithm that finds a hyperplane which optimally separates labeled data points in $\mathbb{R}^n$ into positive and negative classes. The data points on the margin of this separating hyperplane are called support vectors. We connect the possible configurations of support vectors to Radon's theorem, which provides guarantees for when a set of points can be divided into two classes (positive and negative) whose convex hulls intersect. If the convex hulls of the positive and negative support vectors are projected onto a separating hyperplane, then the projections intersect if and only if the hyperplane is optimal. Further, with a particular type of general position, we show that (a) the projected convex hulls of the support vectors intersect in exactly one point, (b) the support vectors are stable under perturbation, (c) there are at most $n+1$ support vectors, and (d) every number of support vectors from 2 up to $n+1$ is possible. Finally, we perform computer simulations studying the expected number of support vectors, and their configurations, for randomly generated data. We observe that as the distance between classes of points increases for this type of randomly generated data, configurations with fewer support vectors become more likely.

cs.LG↗

arXiv2022

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↗

arXiv2022

A Primer on Topological Data Analysis to Support Image Analysis Tasks in Environmental Science

Topological data analysis (TDA) is a tool from data science and mathematics that is beginning to make waves in environmental science. In this work, we seek to provide an intuitive and understandable introduction to a tool from TDA that is particularly useful for the analysis of imagery, namely persistent homology. We briefly discuss the theoretical background but focus primarily on understanding the output of this tool and discussing what information it can glean. To this end, we frame our discussion around a guiding example of classifying satellite images from the Sugar, Fish, Flower, and Gravel Dataset produced for the study of mesocale organization of clouds by Rasp et. al. in 2020 (arXiv:1906:01906). We demonstrate how persistent homology and its vectorization, persistence landscapes, can be used in a workflow with a simple machine learning algorithm to obtain good results, and explore in detail how we can explain this behavior in terms of image-level features. One of the core strengths of persistent homology is how interpretable it can be, so throughout this paper we discuss not just the patterns we find, but why those results are to be expected given what we know about the theory of persistent homology. Our goal is that a reader of this paper will leave with a better understanding of TDA and persistent homology, be able to identify problems and datasets of their own for which persistent homology could be helpful, and gain an understanding of results they obtain from applying the included GitHub example code.

cs.LG↗

arXiv2022

The Persistent Topology of Optimal Transport Based Metric Thickenings

A metric thickening of a given metric space $X$ is any metric space admitting an isometric embedding of $X$. Thickenings have found use in applications of topology to data analysis, where one may approximate the shape of a dataset via the persistent homology of an increasing sequence of spaces. We introduce two new families of metric thickenings, the $p$-Vietoris-Rips and $p$-Čech metric thickenings for all $1\le p\le \infty$, which include all measures on $X$ whose $p$-diameter or $p$-radius is bounded from above, equipped with an optimal transport metric. The $p$-diameter (resp. $p$-radius) of a measure is a certain $\ell_p$ relaxation of the usual notion of diameter (resp. radius) of a subset of a metric space. These families recover the previously studied Vietoris-Rips and Čech metric thickenings when $p=\infty$. As our main contribution, we prove a stability theorem for the persistent homology of $p$-Vietoris-Rips and $p$-Čech metric thickenings, which is novel even in the case $p=\infty$. In the specific case $p=2$, we prove a Hausmann-type theorem for thickenings of manifolds, and we derive the complete list of homotopy types of the $2$-Vietoris-Rips thickenings of the $n$-sphere as the scale increases.

math.MG↗

arXiv2022

Geometric Approaches on Persistent Homology

We introduce several geometric notions, including the width of a homology class, to the theory of persistent homology. These ideas provide geometric interpretations of persistence diagrams. Indeed, we give quantitative and geometric descriptions of the "life span" or "persistence" of a homology class. As a case study, we analyze the power filtration on unweighted graphs, and provide explicit bounds for the life spans of homology classes in persistence diagrams in all dimensions.

math.AT↗

arXiv2021

On Vietoris--Rips complexes of hypercube graphs

We describe the homotopy types of Vietoris-Rips complexes of hypercube graphs at small scale parameters. In more detail, let $Q_n$ be the vertex set of the hypercube graph with $2^n$ vertices, equipped with the shortest path metric. Equivalently, $Q_n$ is the set of all binary strings of length $n$, equipped with the Hamming distance. The Vietoris-Rips complex of $Q_n$ at scale parameter zero is $2^n$ points, and the Vietoris-Rips complex of $Q_n$ at scale parameter one is the hypercube graph, which is homotopy equivalent to a wedge sum of circles. We show that the Vietoris-Rips complex of $Q_n$ at scale parameter two is homotopy equivalent to a wedge sum of 3-spheres, and furthermore we provide a formula for the number of 3-spheres. Many questions about the Vietoris-Rips complexes of $Q_n$ at larger scale parameters remain open.

math.CO↗

arXiv2021

Counting Rocks! An Introduction to Combinatorics

This textbook, "Counting Rocks!", is the written component of an interactive introduction to combinatorics at the undergraduate level. Throughout the text, we link to videos where we describe the material and provide examples. The major topics in this text are counting problems (Chapters 1-4), proof techniques (Chapter 5), recurrence relations and generating functions (Chapters 6-7), and an introduction to graph theory (Chapters 8-12). The material and the problems we include are standard for an undergraduate combinatorics course. In addition to the linked videos, most chapters contain an investigation section, where students are led through a series of deeper problems on a topic. In several sections, we show students how to use the free, open source computing software SAGE in order to solve problems. We have included many illustrative figures throughout the text, and we end each section and chapter with a list of exercises of varying difficulty.

math.HO↗

arXiv2021

Lions and contamination, triangular grids, and Cheeger constants

Suppose each vertex of a graph is originally occupied by contamination, except for those vertices occupied by lions. As the lions wander on the graph, they clear the contamination from each vertex they visit. However, the contamination simultaneously spreads to any adjacent vertex not occupied by a lion. How many lions are required in order to clear the graph of contamination? We give a lower bound on the number of lions needed in terms of the Cheeger constant of the graph. Furthermore, the lion and contamination problem has been studied in detail on square grid graphs by Brass et al. and Berger et al., and we extend this analysis to the setting of triangular grid graphs.

math.CO↗

arXiv2021

Capturing Dynamics of Time-Varying Data via Topology

One approach to understanding complex data is to study its shape through the lens of algebraic topology. While the early development of topological data analysis focused primarily on static data, in recent years, theoretical and applied studies have turned to data that varies in time. A time-varying collection of metric spaces as formed, for example, by a moving school of fish or flock of birds, can contain a vast amount of information. There is often a need to simplify or summarize the dynamic behavior. We provide an introduction to topological summaries of time-varying metric spaces including vineyards [19], crocker plots [56], and multiparameter rank functions [37]. We then introduce a new tool to summarize time-varying metric spaces: a crocker stack. Crocker stacks are convenient for visualization, amenable to machine learning, and satisfy a desirable continuity property which we prove. We demonstrate the utility of crocker stacks for a parameter identification task involving an influential model of biological aggregations [58]. Altogether, we aim to bring the broader applied mathematics community up-to-date on topological summaries of time-varying metric spaces.

cs.LG↗

arXiv2021

An Adaptation for Iterative Structured Matrix Completion

The task of predicting missing entries of a matrix, from a subset of known entries, is known as \textit{matrix completion}. In today's data-driven world, data completion is essential whether it is the main goal or a pre-processing step. Structured matrix completion includes any setting in which data is not missing uniformly at random. In recent work, a modification to the standard nuclear norm minimization (NNM) for matrix completion has been developed to take into account \emph{sparsity-based} structure in the missing entries. This notion of structure is motivated in many settings including recommender systems, where the probability that an entry is observed depends on the value of the entry. We propose adjusting an Iteratively Reweighted Least Squares (IRLS) algorithm for low-rank matrix completion to take into account sparsity-based structure in the missing entries. We also present an iterative gradient-projection-based implementation of the algorithm that can handle large-scale matrices. Finally, we present a robust array of numerical experiments on matrices of varying sizes, ranks, and level of structure. We show that our proposed method is comparable with the adjusted NNM on small-sized matrices, and often outperforms the IRLS algorithm in structured settings on matrices up to size $1000 \times 1000$.

math.NA↗

arXiv2021

Topology Applied to Machine Learning: From Global to Local

Through the use of examples, we explain one way in which applied topology has evolved since the birth of persistent homology in the early 2000s. The first applications of topology to data emphasized the global shape of a dataset, such as the three-circle model for $3 \times 3$ pixel patches from natural images, or the configuration space of the cyclo-octane molecule, which is a sphere with a Klein bottle attached via two circles of singularity. In these studies of global shape, short persistent homology bars are disregarded as sampling noise. More recently, however, persistent homology has been used to address questions about the local geometry of data. For instance, how can local geometry be vectorized for use in machine learning problems? Persistent homology and its vectorization methods, including persistence landscapes and persistence images, provide popular techniques for incorporating both local geometry and global topology into machine learning. Our meta-hypothesis is that the short bars are as important as the long bars for many machine learning tasks. In defense of this claim, we survey applications of persistent homology to shape recognition, agent-based modeling, materials science, archaeology, and biology. Additionally, we survey work connecting persistent homology to geometric features of spaces, including curvature and fractal dimension, and various methods that have been used to incorporate persistent homology into machine learning.

math.AT↗

arXiv2021

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↗

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↗