arXiv subjects

David M. Mount

Publications and source records attributed to David M. Mount.

At least 19 recordsLinked to original sources

arXiv2026

Duality Bounds for Convexified Packing in Hilbert Geometry

Let $G$ and $K$ be convex bodies in $\mathbb{R}^d$, where $0 \in \text{int} G$ and $G \subset \text{int} K$. Given $\alpha > 0$, the Hilbert packing number $M_H(G, K; \alpha)$ is the maximum cardinality of a set of points in $G$, each pair of which is separated by a distance of at least $\alpha$ in the Hilbert geometry defined by $K$. The Hilbert convexified packing number $\widehat{M}_H(G, K; \alpha)$ is the maximum length of a sequence of points in $G$, such that each point is separated by distance at least $\alpha$ from the convex hull of its predecessors. We prove a dimension-free primal-polar bound for convexified packing in Hilbert geometry. Letting $G^\circ$ and $K^\circ$ denote the polar bodies, we show that there exist absolute constants $C, c > 0$ such that, for every $\alpha > 0$, \[ \widehat{M}_H(G, K; \alpha) ~ \leq ~ C \cdot \widehat{M}_H(K^\circ, G^\circ; c\alpha)^2 M_H(K^\circ, G^\circ; c\alpha). \] As a direct corollary, we have \[ \widehat{M}_H(G, K; \alpha) ~ \leq ~ C \cdot M_H(K^\circ, G^\circ; c\alpha)^3. \] Thus, the primal convexified packing number is bounded by a fixed polynomial in the ordinary packing number for the reversed polar bodies, with absolute constants independent of the dimension. This is motivated by the duality conjecture for packing and covering numbers, which relates covering $G$ by $K$ to covering $K^\circ$ by $G^\circ$. Our results represent a key step in extending the work of Artstein, Milman, Szarek, and Tomczak-Jaegermann from normed spaces to Hilbert geometries.

math.MG↗

arXiv2026

Differentiable Approximations for Distance Queries

The widespread use of gradient-based optimization has motivated the adaptation of various classical algorithms into differentiable solvers compatible with learning pipelines. In this paper, we investigate the enhancement of traditional geometric query problems such that the result consists of both the geometric function as well as its gradient. Specifically, we study the fundamental problem of distance queries against a set of points $P$ in $\mathbb{R}^d$, which also underlies various similarity measures for learning algorithms. The main result of this paper is a multiplicative $(1+\varepsilon)$-approximation of the Euclidean distance to $P$ which is differentiable at all points in $\mathbb{R}^d \setminus P$ with asymptotically optimal bounds on the norms of its gradient and Hessian, from a data structure with storage and query time matching state-of-the-art results for approximate nearest-neighbor searching. The approximation is realized as a regularized distance through a partition-of-unity framework, which efficiently blends multiple local approximations, over a suitably defined covering of space, into a smooth global approximation. In order to obtain the local distance approximations in a manner that facilitates blending, we develop a new approximate Voronoi diagram based on a simple point-location data structure, simplifying away both the lifting transformation and ray shooting.

cs.CG↗

arXiv2026

Proximity Alert: Ipelets for Neighborhood Graphs and Clustering

Neighborhood graphs and clustering algorithms are fundamental structures in both computational geometry and data analysis. Visualizing them can help build insight into their behavior and properties. The Ipe extensible drawing editor, developed by Otfried Cheong, is a widely used software system for generating figures. One particular aspect of Ipe is the ability to add Ipelets, which extend its functionality. Here we showcase a set of Ipelets designed to help visualize neighborhood graphs and clustering algorithms. These include: $\eps$-neighbor graphs, furthest-neighbor graphs, Gabriel graphs, $k$-nearest neighbor graphs, $k^{th}$-nearest neighbor graphs, $k$-mutual neighbor graphs, $k^{th}$-mutual neighbor graphs, asymmetric $k$-nearest neighbor graphs, asymmetric $k^{th}$-nearest neighbor graphs, relative-neighbor graphs, sphere-of-influence graphs, Urquhart graphs, Yao graphs, and clustering algorithms including complete-linkage, DBSCAN, HDBSCAN, $k$-means, $k$-means++, $k$-medoids, mean shift, and single-linkage. Our Ipelets are all programmed in Lua and are freely available.

cs.CG↗

arXiv2026

On the Duality of Coverings in Hilbert Geometry

We prove polarity duality for covering problems in Hilbert geometry. Let $G$ and $K$ be convex bodies in $\mathbb{R}^d$ where $G \subset \operatorname{int}(K)$ and $\operatorname{int}(G)$ contains the origin. Let $N^H_K(G,\alpha)$ and $S^H_K(G,\alpha)$ denote, respectively, the minimum numbers of radius-$\alpha$ Hilbert balls in the geometry induced by $K$ needed to cover $G$ and $\partial G$. Our main result is a Hilbert-geometric analogue of the K\"{o}nig-Milman covering duality: there exists an absolute constant $c \geq 1$ such that for any $\alpha \in (0,1]$, \[ c^{-d}\,N^H_{G^{\circ}}(K^{\circ},\alpha) ~ \leq ~ N^H_K(G,\alpha) ~ \leq ~ c^{d}\,N^H_{G^{\circ}}(K^{\circ},\alpha), \] and likewise, \[ c^{-d}\,S^H_{G^{\circ}}(K^{\circ},\alpha) ~ \leq ~ S^H_K(G,\alpha) ~ \leq ~ c^{d}\,S^H_{G^{\circ}}(K^{\circ},\alpha). \] We also recover the classical volumetric duality for translative coverings of centered convex bodies, and obtain a new boundary-covering duality in that setting. The Hilbert setting is subtler than the translative one because the metric is not translation invariant, and the local Finsler unit ball depends on the base point. The proof involves several ideas, including $\alpha$-expansions, a stability lemma that controls the interaction between polarity and expansion, and, in the boundary case, a localized relative isoperimetric argument combined with Holmes--Thompson area estimates. In addition, we provide an alternative proof of Faifman's polarity bounds for Holmes--Thompson volume and area in the Funk and Hilbert geometries.

math.MG↗

arXiv2026

On Voronoi diagrams in the Funk Conical Geometry

The forward and reverse Funk weak metrics are fundamental distance functions on convex bodies that serve as the building blocks for the Hilbert and Thompson metrics. In this paper we study Voronoi diagrams under the forward and reverse Funk metrics in polygonal and elliptical cones. We establish several key geometric properties: (1) bisectors consist of a set of rays emanating from the apex of the cone, and (2) Voronoi diagrams in the $d$-dimensional forward (or reverse) Funk metrics are equivalent to additively-weighted Voronoi diagrams in the $(d-1)$-dimensional forward (or reverse) Funk metrics on bounded cross sections of the cone. Leveraging this, we provide an $O\big(n^{\ceil{\frac{d-1}{2}}+1}\big)$ time algorithm for creating these diagrams in $d$-dimensional elliptical cones using a transformation to and from Apollonius diagrams, and an $O(mn\log(n))$ time algorithm for 3-dimensional polygonal cones with $m$ facets via a reduction to abstract Voronoi diagrams. We also provide a complete characterization of when three sites have a circumcenter in 3-dimensional cones. This is one of the first algorithmic studies of the Funk metrics.

cs.CG↗

arXiv2026

Cauchy's Surface Area Formula in the Funk Geometry

Cauchy's surface area formula expresses the surface area of a convex body as the average area of its orthogonal projections over all directions. While this tool is fundamental in Euclidean geometry, with applications ranging from geometric tomography to approximation theory, extensions to non-Euclidean settings remain less explored. In this paper, we establish an analog of Cauchy's formula for the Funk geometry induced by a convex body $K$ in $\mathbb{R}^d$, for the Holmes--Thompson surface area. The formula is based on central projections to boundary points of $K$. We show that when $K$ is a convex polytope, the formula reduces to a weighted sum of contributions associated with the vertices of $K$. Finally, as a consequence of our analysis, we derive a generalization of Crofton's formula for surface areas in the Funk geometry. By viewing Euclidean, Minkowski, Hilbert, and hyperbolic geometries as limiting or special cases of the Funk setting, our results provide a unified framework for these classical surface area formulas.

cs.CG↗

arXiv2026

Classifiers in High Dimensional Hilbert Metrics

Classifying points in high dimensional spaces is a fundamental geometric problem in machine learning. In this paper, we address classifying points in the $d$-dimensional Hilbert polygonal metric. The Hilbert metric is a generalization of the Cayley-Klein hyperbolic distance to arbitrary convex bodies and has a diverse range of applications in machine learning and convex geometry. We first present an efficient LP-based algorithm in the metric for the large-margin SVM problem. Our algorithm runs in time polynomial to the number of points, bounding facets, and dimension. This is a significant improvement on previous works, which either provide no theoretical guarantees on running time, or suffer from exponential runtime. We also consider the closely related Funk metric. We also present efficient algorithms for the soft-margin SVM problem and for nearest neighbor-based classification in the Hilbert metric.

cs.CG↗

arXiv2025

French Onion Soup, Ipelets for Points and Polygons

There are many structures, both classical and modern, involving point-sets and polygons whose deeper understanding can be facilitated through interactive visualizations. The Ipe extensible drawing editor, developed by Otfried Cheong, is a widely used software system for generating geometric figures. One of its features is the capability to extend its functionality through programs called Ipelets. In this media submission, we showcase a collection of new Ipelets that construct a variety of geometric based structures based on point sets and polygons. These include quadtrees, trapezoidal maps, beta skeletons, floating bodies of convex polygons, onion graphs, fractals (Sierpi\'nski triangle and carpet), simple polygon triangulations, and random point sets in simple polygons. All of our Ipelets are programmed in Lua and are freely available.

cs.CG↗

arXiv2025

Software for the Thompson and Funk Polygonal Geometry

Metric spaces defined within convex polygons, such as the Thompson, Funk, reverse Funk, and Hilbert metrics, are subjects of recent exploration and study in computational geometry. This paper contributes an educational piece of software for understanding these unique geometries while also providing a tool to support their research. We provide dynamic software for manipulating the Funk, reverse Funk, and Thompson balls in convex polygonal domains. Additionally, we provide a visualization program for traversing the Hilbert polygonal geometry.

cs.CG↗

arXiv2024

On The Heine-Borel Property and Minimum Enclosing Balls

In this paper, we contribute a proof that minimum radius balls over metric spaces with the Heine-Borel property are always LP type. Additionally, we prove that weak metric spaces, those without symmetry, also have this property if we fix the direction in which we take their distances from the centers of the balls. We use this to prove that the minimum radius ball problem is LP type in the Hilbert and Thompson metrics and Funk weak metric. In doing so, we contribute a proof that the topology induced by the Thompson metric coincides with the Hilbert. We provide explicit primitives for computing the minimum radius ball in the Hilbert metric.

cs.CG↗

arXiv2024

Evolving Distributions Under Local Motion

Geometric data sets arising in modern applications are often very large and change dynamically over time. A popular framework for dealing with such data sets is the evolving data framework, where a discrete structure continuously varies over time due to the unseen actions of an evolver, which makes small changes to the data. An algorithm probes the current state through an oracle, and the objective is to maintain a hypothesis of the data set's current state that is close to its actual state at all times. In this paper, we apply this framework to maintaining a set of $n$ point objects in motion in $d$-dimensional Euclidean space. To model the uncertainty in the object locations, both the ground truth and hypothesis are based on spatial probability distributions, and the distance between them is measured by the Kullback-Leibler divergence (relative entropy). We introduce a simple and intuitive motion model where with each time step, the distance that any object can move is a fraction of the distance to its nearest neighbor. We present an algorithm that, in steady state, guarantees a distance of $O(n)$ between the true and hypothesized placements. We also show that for any algorithm in this model, there is an evolver that can generate a distance of $\Omega(n)$, implying that our algorithm is asymptotically optimal.

cs.CG↗

arXiv2023

Delaunay Triangulations in the Hilbert Metric

The Hilbert metric is a distance function defined for points lying within the interior of a convex body. It arises in the analysis and processing of convex bodies, machine learning, and quantum information theory. In this paper, we show how to adapt the Euclidean Delaunay triangulation to the Hilbert geometry defined by a convex polygon in the plane. We analyze the geometric properties of the Hilbert Delaunay triangulation, which has some notable differences with respect to the Euclidean case, including the fact that the triangulation does not necessarily cover the convex hull of the point set. We also introduce the notion of a Hilbert ball at infinity, which is a Hilbert metric ball centered on the boundary of the convex polygon. We present a simple randomized incremental algorithm that computes the Hilbert Delaunay triangulation for a set of $n$ points in the Hilbert geometry defined by a convex $m$-gon. The algorithm runs in $O(n (\log n + \log^3 m))$ expected time. In addition we introduce the notion of the Hilbert hull of a set of points, which we define to be the region covered by their Hilbert Delaunay triangulation. We present an algorithm for computing the Hilbert hull in time $O(n h \log^2 m)$, where $h$ is the number of points on the hull's boundary.

cs.CG↗

arXiv2023

Linear-Sized Spectral Sparsifiers and the Kadison-Singer Problem

The Kadison-Singer Conjecture, as proved by Marcus, Spielman, and Srivastava (MSS) [Ann. Math. 182, 327-350 (2015)], has been informally thought of as a strengthening of Batson, Spielman, and Srivastava's theorem that every undirected graph has a linear-sized spectral sparsifier [SICOMP 41, 1704-1721 (2012)]. We formalize this intuition by using a corollary of the MSS result to derive the existence of spectral sparsifiers with a number of edges linear in their number of vertices for all undirected, weighted graphs. The proof consists of two steps. First, following a suggestion of Srivastava [Asia Pac. Math. Newsl. 3, 15-20 (2013)], we show the result in the special case of graphs with bounded leverage scores by repeatedly applying the MSS corollary to partition the graph, while maintaining an appropriate bound on the leverage scores of each subgraph. Then, we extend to the general case by constructing a recursive algorithm that repeatedly (i) divides edges with high leverage scores into multiple parallel edges and (ii) uses the bounded leverage score case to sparsify the resulting graph.

cs.DS↗

arXiv2023

Smooth Distance Approximation

Traditional problems in computational geometry involve aspects that are both discrete and continuous. One such example is nearest-neighbor searching, where the input is discrete, but the result depends on distances, which vary continuously. In many real-world applications of geometric data structures, it is assumed that query results are continuous, free of jump discontinuities. This is at odds with many modern data structures in computational geometry, which employ approximations to achieve efficiency, but these approximations often suffer from discontinuities. In this paper, we present a general method for transforming an approximate but discontinuous data structure into one that produces a smooth approximation, while matching the asymptotic space efficiencies of the original. We achieve this by adapting an approach called the partition-of-unity method, which smoothly blends multiple local approximations into a single smooth global approximation. We illustrate the use of this technique in a specific application of approximating the distance to the boundary of a convex polytope in $\mathbb{R}^d$ from any point in its interior. We begin by developing a novel data structure that efficiently computes an absolute $\varepsilon$-approximation to this query in time $O(\log (1/\varepsilon))$ using $O(1/\varepsilon^{d/2})$ storage space. Then, we proceed to apply the proposed partition-of-unity blending to guarantee the smoothness of the approximate distance field, establishing optimal asymptotic bounds on the norms of its gradient and Hessian.

cs.CG↗

arXiv2023

Approximate Nearest Neighbor Searching with Non-Euclidean and Weighted Distances

We present a new approach to approximate nearest-neighbor queries in fixed dimension under a variety of non-Euclidean distances. We are given a set $S$ of $n$ points in $\mathbb{R}^d$, an approximation parameter $\varepsilon > 0$, and a distance function that satisfies certain smoothness and growth-rate assumptions. The objective is to preprocess $S$ into a data structure so that for any query point $q$ in $\mathbb{R}^d$, it is possible to efficiently report any point of $S$ whose distance from $q$ is within a factor of $1+\varepsilon$ of the actual closest point. Prior to this work, the most efficient data structures for approximate nearest-neighbor searching in spaces of constant dimensionality applied only to the Euclidean metric. This paper overcomes this limitation through a method called convexification. For admissible distance functions, the proposed data structures answer queries in logarithmic time using $O(n \log (1 / \varepsilon) / \varepsilon^{d/2})$ space, nearly matching the best known bounds for the Euclidean metric. These results apply to both convex scaling distance functions (including the Mahalanobis distance and weighted Minkowski metrics) and Bregman divergences (including the Kullback-Leibler divergence and the Itakura-Saito distance).

cs.CG↗

arXiv2023

Optimal Area-Sensitive Bounds for Polytope Approximation

Approximating convex bodies is a fundamental problem in geometry. Given a convex body $K$ in $\mathbb{R}^d$ for a fixed dimension $d$, the objective is to minimize the number of facets of an approximating polytope for a given Hausdorff error $\varepsilon$. The best known uniform bound, due to Dudley (1974), shows that $O((\text{diam}(K)/\varepsilon)^{(d-1)/2})$ facets suffice. Although this bound is optimal for fat objects, such as Euclidean balls, it is far from optimal for ``skinny'' convex bodies. Skinniness can be characterized relative to the Euclidean ball. Given a convex body $K$, define its area radius, $\text{arad}(K)$, to be the radius of the Euclidean ball having the same surface area as $K$. It follows from generalizations of the isoperimetric inequality that $\text{diam}(K) \geq 2 \cdot \text{arad}(K)$. We show that, given a convex body whose minimum width is at least $\varepsilon$, it is possible to approximate the body by a polytope having $O((\text{arad}(K)/\varepsilon)^{(d-1)/2})$ facets. Our approach works by first reducing the problem of approximating convex bodies to that of approximating convex functions. We employ a classical concept from convexity, called Macbeath regions. We demonstrate that there is a polar relationship between the Macbeath regions of a function and the Macbeath regions of its Legendre dual. This is combined with known bounds on the Mahler volume to bound the total size of the approximation.

cs.CG↗

arXiv2023

Analysis of Dynamic Voronoi Diagrams in the Hilbert Metric

The Hilbert metric is a projective metric defined on a convex body which generalizes the Cayley-Klein model of hyperbolic geometry to any convex set. In this paper we analyze Hilbert Voronoi diagrams in the Dynamic setting. In addition we introduce dynamic visualization software for Voronoi diagrams in the Hilbert metric on user specified convex polygons.

cs.CG↗

arXiv2023

Optimal Volume-Sensitive Bounds for Polytope Approximation

Approximating convex bodies is a fundamental question in geometry, which has a wide variety of applications. Given a convex body $K$ in $\textbf{R}^d$ for fixed $d$, the objective is to minimize the number of facets of an approximating polytope for a given Hausdorff error $\varepsilon$. It is known that $O((\text{diam}(K)/\varepsilon)^{(d-1)/2})$ facets suffice and are necessary for many instances, such as the Euclidean ball. However, this bound is far from optimal for ``skinny'' convex bodies. A natural way to characterize the skinniness of a convex object is in terms of its relationship to the Euclidean ball. Given a convex body $K$, its \emph{volume diameter} $\Delta_d(K)$ is defined to be the diameter of a Euclidean ball of the same volume as $K$. The \emph{surface diameter} $\Delta_{d-1}(K)$ is defined analogously for surface area. It follows from generalizations of the isoperimetric inequality that $\text{diam}(K) \geq \Delta_{d-1}(K) \geq \Delta_d(K)$. Arya, da Fonseca, and Mount proved that the diameter-based bound could be made sensitive to the surface diameter, improving the above bound to $O((\Delta_{d-1}(K)/\varepsilon)^{(d-1)/2})$. In this paper, we strengthen this by proving the existence of an approximation with $O((\Delta_d(K)/\varepsilon)^{(d-1)/2})$ facets. As a function of volume alone, this bound is tight up to constant factors. Our improvements arise from a combination of new ideas. We exploit known properties of the original body and its polar dual. In order to obtain a volume-sensitive bound, we explore the problem of computing a low-complexity polytope that is sandwiched between two given convex bodies. We show that this problem can be reduced to a covering problem involving a natural intermediate body based on the harmonic mean. Our proof relies on a geometric analysis of a relative notion of fatness involving these bodies.

cs.CG↗