Showing papers in "Discrete and Computational Geometry in 2021"

Symmetric Non-Negative Forms and Sums of Squares

Abstract: We study symmetric non-negative forms and their relationship with symmetric sums of squares. For a fixed number of variables n and degree 2d, symmetric non-negative forms and symmetric sums of squares form closed, convex cones in the vector space of n-variate symmetric forms of degree 2d. Using representation theory of the symmetric group we characterize both cones in a uniform way. Further, we investigate the asymptotic behavior when the degree 2d is fixed and the number of variables n grows. Here, we show that, in sharp contrast to the general case, the difference between symmetric non-negative forms and sums of squares does not grow arbitrarily large for any fixed degree 2d. We consider the case of symmetric quartic forms in more detail and give a complete characterization of quartic symmetric sums of squares. Furthermore, we show that in degree 4 the cones of non-negative symmetric forms and symmetric sums of squares approach the same limit, thus these two cones asymptotically become closer as the number of variables grows. We conjecture that this is true in arbitrary degree 2d.

Spatiotemporal Persistent Homology for Dynamic Metric Spaces

Abstract: Characterizing the dynamics of time-evolving data within the framework of topological data analysis (TDA) has been attracting increasingly more attention. Popular instances of time-evolving data include flocking/swarming behaviors in animals and social networks in the human sphere. A natural mathematical model for such collective behaviors is a dynamic point cloud, or more generally a dynamic metric space (DMS). In this paper we extend the Rips filtration stability result for (static) metric spaces to the setting of DMSs. We do this by devising a certain three-parameter “spatiotemporal” filtration of a DMS. Applying the homology functor to this filtration gives rise to multidimensional persistence module derived from the DMS. We show that this multidimensional module enjoys stability under a suitable generalization of the Gromov–Hausdorff distance which permits metrization of the collection of all DMSs. On the other hand, it is recognized that, in general, comparing two multidimensional persistence modules leads to intractable computational problems. For the purpose of practical comparison of DMSs, we focus on both the rank invariant or the dimension function of the multidimensional persistence module that is derived from a DMS. We specifically propose to utilize a certain metric d for comparing these invariants: In our work this d is either (1) a certain generalization of the erosion distance by Patel, or (2) a specialized version of the well-known interleaving distance. In either case, the metric d can be computed in polynomial time.

The Geometry of Synchronization Problems and Learning Group Actions

Abstract: We develop a geometric framework, based on the classical theory of fibre bundles, to characterize the cohomological nature of a large class of synchronization-type problems in the context of graph inference and combinatorial optimization. We identify each synchronization problem in topological group G on connected graph $$\Gamma $$ with a flat principal G-bundle over $$\Gamma $$ , thus establishing a classification result for synchronization problems using the representation variety of the fundamental group of $$\Gamma $$ into G. We then develop a twisted Hodge theory on flat vector bundles associated with these flat principal G-bundles, and provide a geometric realization of the graph connection Laplacian as the lowest-degree Hodge Laplacian in the twisted de Rham–Hodge cochain complex. Motivated by these geometric intuitions, we propose to study the problem of learning group actions—partitioning a collection of objects based on the local synchronizability of pairwise correspondence relations—and provide a heuristic synchronization-based algorithm for solving this type of problems. We demonstrate the efficacy of this algorithm on simulated and real datasets.

Recursive Scheme for Angles of Random Simplices, and Applications to Random Polytopes

Abstract: Consider a random simplex $$[X_1,\ldots ,X_n]$$ defined as the convex hull of independent identically distributed (i.i.d.) random points $$X_1,\ldots ,X_n$$ in $$\mathbb {R}^{n-1}$$ with the following beta density: Let $$J_{n,k}(\beta )$$ be the expected internal angle of the simplex $$[X_1,\ldots ,X_n]$$ at its face $$[X_1,\ldots ,X_k]$$ . Define $${\tilde{J}}_{n,k}(\beta )$$ analogously for i.i.d. random points distributed according to the beta $$'$$ density $${\tilde{f}}_{n-1,\beta } (x) \propto (1+\Vert x\Vert ^2)^{-\beta }, x\in \mathbb {R}^{n-1}, \beta > ({n-1})/{2}.$$ We derive formulae for $$J_{n,k}(\beta )$$ and $${\tilde{J}}_{n,k}(\beta )$$ which make it possible to compute these quantities symbolically, in finitely many steps, for any integer or half-integer value of $$\beta $$ . For $$J_{n,1}(\pm 1/2)$$ we even provide explicit formulae in terms of products of Gamma functions. We give applications of these results to two seemingly unrelated problems of stochastic geometry: (i) We compute explicitly the expected f-vectors of the typical Poisson–Voronoi cells in dimensions up to 10. (ii) Consider the random polytope $$K_{n,d} := [U_1,\ldots ,U_n]$$ where $$U_1,\ldots ,U_n$$ are i.i.d. random points sampled uniformly inside some d-dimensional convex body K with smooth boundary and unit volume. Reitzner (Adv. Math. 191(1), 178–208 (2005)) proved the existence of the limit of the normalised expected f-vector of $$K_{n,d}$$ : $$ \lim _{n\rightarrow \infty } n^{-{({d-1})/({d+1})}}{\mathbb {E}}{\mathbf {f}}(K_{n,d}) = {\mathbf {c}}_d \cdot \Omega (K),$$ where $$\Omega (K)$$ is the affine surface area of K, and $${\mathbf {c}}_d$$ is an unknown vector not depending on K. We compute $${\mathbf {c}}_d$$ explicitly in dimensions up to $$d=10$$ and also solve the analogous problem for random polytopes with vertices distributed uniformly on the sphere.

Persistent Homology and the Upper Box Dimension

Abstract: We introduce a fractal dimension for a metric space defined in terms of the persistent homology of extremal subsets of that space. We exhibit hypotheses under which this dimension is comparable to the upper box dimension; in particular, the dimensions coincide for subsets of $${\mathbb {R}}^2$$ whose upper box dimension exceeds 1.5. These results are related to extremal questions about the number of persistent homology intervals of a set of n points in a metric space.

Substitutive Structure of Jeandel–Rao Aperiodic Tilings

Abstract: Jeandel and Rao proved that 11 is the size of the smallest set of Wang tiles, i.e., unit squares with colored edges, that admit valid tilings (contiguous edges of adjacent tiles have the same color) of the plane, none of them being invariant under a nontrivial translation. We study herein the Wang shift $$\Omega _0$$ made of all valid tilings using the set $$\mathcal {T}_0$$ of 11 aperiodic Wang tiles discovered by Jeandel and Rao. We show that there exists a minimal subshift $$X_0$$ of $$\Omega _0$$ such that every tiling in $$X_0$$ can be decomposed uniquely into 19 distinct patches of size ranging from 45 to 112 that are equivalent to a set of 19 self-similar aperiodic Wang tiles. We suggest that this provides an almost complete description of the substitutive structure of Jeandel–Rao tilings, as we believe that $$\Omega _0{\setminus } X_0$$ is a null set for any shift-invariant probability measure on $$\Omega _0$$ . The proof is based on 12 elementary steps, 10 of which involve the same procedure allowing one to desubstitute Wang tilings from the existence of a subset of marker tiles, while the other 2 involve addition decorations to deal with fault lines and changing the base of the $$\mathbb {Z}^2$$ -action through a shear conjugacy. Algorithms are provided to find markers, recognizable substitutions, and shear conjugacy from a set of Wang tiles.

Threshold Phenomena for Random Cones

Abstract: We consider an even probability distribution on the d-dimensional Euclidean space with the property that it assigns measure zero to any hyperplane through the origin. Given N independent random vectors with this distribution, under the condition that they do not positively span the whole space, the positive hull of these vectors is a random polyhedral cone (and its intersection with the unit sphere is a random spherical polytope). It was first studied by Cover and Efron. We consider the expected face numbers of these random cones and describe a threshold phenomenon when the dimension d and the number N of random vectors tend to infinity. In a similar way we treat the solid angle, and more generally the Grassmann angles. We further consider the expected numbers of k-faces and of Grassmann angles of index $$d-k$$ when also k tends to infinity.

The $h^*$-polynomials of locally anti-blocking lattice polytopes and their $γ$-positivity

Abstract: A lattice polytope $$\mathscr {P} \subset \mathbb {R}^d$$ is called a locally anti-blocking polytope if for any closed orthant $${\mathbb R}^d_{\varepsilon }$$ in $$\mathbb {R}^d$$ , $$\mathscr {P} \cap \mathbb {R}^d_{\varepsilon }$$ is unimodularly equivalent to an anti-blocking polytope by reflections of coordinate hyperplanes. We give a formula for the $$h^*$$ -polynomials of locally anti-blocking lattice polytopes. In particular, we discuss the $$\gamma $$ -positivity of $$h^*$$ -polynomials of locally anti-blocking reflexive polytopes.

Packing Disks by Flipping and Flowing

Abstract: We provide a new type of proof for the Koebe–Andreev–Thurston (KAT) planar circle packing theorem based on combinatorial edge-flips. In particular, we show that starting from a disk packing with a maximal planar contact graph G, one can remove any flippable edge $$e^-$$ of this graph and then continuously flow the disks in the plane, so that at the end of the flow, one obtains a new disk packing whose contact graph is the graph resulting from flipping the edge $$e^-$$ in G. This flow is parameterized by a single inversive distance.

Enumeration of Lattice Polytopes by Their Volume

Abstract: A well-known result by Lagarias and Ziegler states that there are finitely many equivalence classes of d-dimensional lattice polytopes having volume at most K, for fixed constants d and K. We describe an algorithm for the complete enumeration of such equivalence classes for arbitrary constants d and K. The algorithm, which gives another proof of the finiteness result, is implemented for small values of K, up to dimension six. The resulting database contains and extends several existing ones, and has been used to correct mistakes in other classifications. When specialized to three-dimensional smooth polytopes, it extends previous classifications by Bogart et al., Lorenz, and Lundman. Moreover, we give a structure theorem for smooth polytopes with few lattice points that proves that they have a quadratic triangulation and which we use, together with the classification, to describe smooth polytopes having small volume in arbitrary dimension. In dimension three we enumerate all the simplices having up to 11 interior lattice points and we use them to conjecture a set of sharp inequalities for the coefficients of the Ehrhart $$h^*$$ -polynomials, unifying several existing conjectures. Finally, we extract and discuss some interesting minimal examples from the classification, and study the frequency of properties such as being spanning, very ample, IDP, and having a unimodular cover or triangulation. In particular, we find the smallest polytopes that are very ample but not IDP, and with a unimodular cover but without a unimodular triangulation.

The typical cell of a Voronoi tessellation on the sphere

Abstract: The typical cell of a Voronoi tessellation generated by $$n+1$$ uniformly distributed random points on the d-dimensional unit sphere $$\mathbb {S}^d$$ is studied. Its f-vector is identified in distribution with the f-vector of a beta’ polytope generated by n random points in $$\mathbb {R}^d$$ . Explicit formulas for the expected f-vector are provided for any d and the low-dimensional cases $$d\in \{2,3,4\}$$ are studied separately. This implies an explicit formula for the total number of k-dimensional faces in the spherical Voronoi tessellation as well.

Quantitative Simplification of Filtered Simplicial Complexes

Abstract: We introduce a new invariant defined on the vertices of a given filtered simplicial complex, called codensity, which controls the impact of removing vertices on the persistent homology of this filtered complex. We achieve this control through the use of an interleaving type of distance between filtered simplicial complexes. We study the special case of Vietoris–Rips filtrations and show that our bounds offer a significant improvement over the immediate bounds coming from considerations related to the Gromov–Hausdorff distance. Based on these ideas we give an iterative method for the practical simplification of filtered simplicial complexes. As a byproduct of our analysis we identify a notion of core of a filtered simplicial complex which admits the interpretation as a minimalistic simplicial filtration which retains all the persistent homology information.

Semi-regular Tilings of the Hyperbolic Plane

Abstract: A semi-regular tiling of the hyperbolic plane is a tessellation by regular geodesic polygons with the property that each vertex has the same vertex-type, which is a cyclic tuple of integers that determine the number of sides of the polygons surrounding the vertex. We determine combinatorial criteria for the existence, and uniqueness, of a semi-regular tiling with a given vertex-type, and pose some open questions.

Approximate Carathéodory’s Theorem in Uniformly Smooth Banach Spaces

Abstract: We study the ‘no-dimension’ analogue of Caratheodory’s theorem in Banach spaces. We prove such a result together with its colorful version for uniformly smooth Banach spaces. It follows that uniform smoothness leads to a greedy de-randomization of Maurey’s classical lemma Pisier (in: Seminaire Analyse fonctionnelle (dit “Maurey-Schwartz”), 1980), which is itself a ‘no-dimension’ analogue of Caratheodory’s theorem with a probabilistic proof. We find the asymptotically tight upper bound on the deviation of the convex hull from the k-convex hull of a bounded set in $$L_p$$ with $$1 < p \le 2$$ and get asymptotically the same bound as in Maurey’s lemma for $$L_p$$ with $$1< p < \infty $$ .

Polarization problem on a higher-dimensional sphere for a simplex

Abstract: We study the problem of maximizing the minimal value over the sphere $$S^{d-1}\subset {\mathbb {R}}^d$$ of the potential generated by a configuration of $$d+1$$ points on $$S^{d-1}$$ (the maximal discrete polarization problem). The points interact via the potential given by a function f of the Euclidean distance squared, where $$f:[0,4]\rightarrow (-\infty ,\infty ]$$ is continuous (in the extended sense), decreasing on [0, 4], and finite and convex on (0, 4], with a concave or convex derivative $$f'$$ . We prove that the configuration of the vertices of a regular d-simplex inscribed in $$S^{d-1}$$ is optimal. This result is new for $$d>3$$ (certain special cases for $$d=2$$ and $$d=3$$ are also new). As a byproduct, we find a simpler proof for the known optimal covering property of the vertices of a regular d-simplex inscribed in $$S^{d-1}$$ .

On the Regularity Radius of Delone Sets in $${\mathbb {R}}^3$$

Abstract: We complete the proof of the upper bound $${\hat{\rho }}_3\le 10R$$ for the regularity radius of Delone sets in three-dimensional Euclidean space. Namely, summing up the results obtained earlier, and adding the missing cases, we show that if all $$10R$$ -clusters of a Delone set X in $${\mathbb {R}}^3$$ with parameters (r, R) are equivalent, then X is regular (has a transitive symmetry group).

On extremal sections of subspaces of $L_p$

Abstract: Let $$m,n\in {\mathbb {N}}$$ and $$p\in (0,\infty )$$ . For a finite dimensional quasi-normed space $$X=({\mathbb {R}}^m, \Vert \cdot \Vert _X)$$ , let $$\begin{aligned} B_p^n(X) = \left\{ (x_1,\ldots ,x_n)\in \big ({\mathbb {R}}^{m}\big )^n: \sum _{i=1}^n \Vert x_i\Vert _X^p \leqslant 1\right\} . \end{aligned}$$ We show that for every $$p\in (0,2)$$ and X which admits an isometric embedding into $$L_p$$ , the function $$\begin{aligned} S^{n-1} i \uptheta = (\uptheta _1,\ldots ,\uptheta _n) \longmapsto \left| B_p^n(X) \cap \left\{ (x_1,\ldots ,x_n)\in \big ({\mathbb {R}}^{m}\big )^n: \sum _{i=1}^n \uptheta _i x_i=0 \right\} \right| \end{aligned}$$ is a Schur convex function of $$(\uptheta _1^2,\ldots ,\uptheta _n^2)$$ , where $$|\cdot |$$ denotes Lebesgue measure. In particular, it is minimized when $$\uptheta =\big (\frac{1}{\sqrt{n}},\ldots ,\frac{1}{\sqrt{n}}\big )$$ and maximized when $$\uptheta =(1,0,\ldots ,0)$$ . This is a consequence of a more general statement about Laplace transforms of norms of suitable Gaussian random vectors which also implies dual estimates for the mean width of projections of the polar body $$(B_p^n(X))^\circ $$ if the unit ball $$B_X$$ of X is in Lewis’ position. Finally, we prove a lower bound for the volume of projections of $$B_\infty ^n(X)$$ , where $$X=({\mathbb {R}}^m,\Vert \cdot \Vert _X)$$ is an arbitrary quasi-normed space.

On Recovery Guarantees for One-Bit Compressed Sensing on Manifolds

Abstract: This paper studies the problem of recovering a signal from one-bit compressed sensing measurements under a manifold model; that is, assuming that the signal lies on or near a manifold of low intrinsic dimension. We provide a convex recovery method based on the Geometric Multi-Resolution Analysis and prove recovery guarantees with a near-optimal scaling in the intrinsic manifold dimension. Our method is the first tractable algorithm with such guarantees for this setting. The results are complemented by numerical experiments confirming the validity of our approach.

Compact Packings of Space with Two Sizes of Spheres

Abstract: Compact sphere packings are sphere packings which can be seen as tilings. They are usually good candidates to maximize the density. We show that the compact packings of Euclidean three-dimensional space with two sizes of spheres are exactly those obtained by filling with spheres of size $$\sqrt{2}-1$$ the octahedral holes of a close-packing of spheres of size 1.

On the Stability of Interval Decomposable Persistence Modules

Abstract: The algebraic stability theorem for persistence modules is a central result in the theory of stability for persistent homology. We introduce a new proof technique which we use to prove a stability theorem for n-dimensional rectangle decomposable persistence modules up to a constant $$2n-1$$ that generalizes the algebraic stability theorem, and give an example showing that the bound cannot be improved for $$n=2$$ . We then apply the technique to prove stability for block decomposable modules, from which novel results for zigzag modules and Reeb graphs follow. These results are improvements on weaker bounds in previous work, and the bounds we obtain are optimal.

Combinatorial Modifications of Reeb Graphs and the Realization Problem

Abstract: We prove that, up to homeomorphism, any graph subject to natural necessary conditions on orientation and the cycle rank can be realized as the Reeb graph of a Morse function on a given closed manifold M. Along the way, we show that the Reeb number $$\mathcal {R}(M)$$ , i.e., the maximum cycle rank among all Reeb graphs of functions on M, is equal to the corank of fundamental group $$\pi _1(M)$$ , thus extending a previous result of Gelbukh to the non-orientable case.

Topological Complexity of Configuration Spaces of Fully Articulated Graphs and Banana Graphs

Abstract: We determine the topological complexity of configuration spaces of graphs that are not necessarily trees, which was a crucial assumption in previous results. We do this for two very different classes of graphs: fully articulated graphs and banana graphs. We also complete the computation in the case of trees to include configuration spaces with any number of points, extending a proof of Farber. At the end we show that an unordered configuration space on a graph does not always have the same topological complexity as the corresponding ordered configuration space (not even when they are both connected). Surprisingly, in our counterexamples the topological complexity of the unordered configuration space is in fact smaller than for the ordered one.

Computing Min-Convex Hulls in the Affine Building of $$\hbox {SL}_d$$

Abstract: We describe an algorithm for computing the min-convex hull of a finite collection of points in the affine building of $$\hbox {SL}_d(K)$$ , for K a field with discrete valuation. These min-convex hulls describe the relations among a finite collection of invertible matrices over K. As a consequence, we bound the dimension of the tropical projective space needed to realize the min-convex hull as a tropical polytope.

Classification of Triples of Lattice Polytopes with a Given Mixed Volume

Abstract: We present an algorithm for the classification of triples of lattice polytopes with a given mixed volume m in dimension 3. It is known that the classification can be reduced to the enumeration of so-called irreducible triples, the number of which is finite for fixed m. Following this algorithm, we enumerate all irreducible triples of normalized mixed volume up to 4 that are inclusion-maximal. This produces a classification of generic trivariate sparse polynomial systems with up to 4 solutions in the complex torus, up to monomial changes of variables. By a recent result of Esterov, this leads to a description of all generic trivariate sparse polynomial systems that are solvable by radicals.

The Six Cylinders Problem: $\mathbb{D}_{3}$-symmetry Approach

Abstract: Motivated by a question of W. Kuperberg, we study the 18-dimensional manifold of configurations of six non-intersecting infinite cylinders of radius r, all touching the unit ball in $$\mathbb {R}^{3}$$ . We find a configuration with $$\begin{aligned} r=\frac{1}{8}\Big ( 3+\sqrt{33}\Big ) \approx 1.093070331. \end{aligned}$$ We believe that this value is the maximum possible.

Triangulating Submanifolds: An Elementary and Quantified Version of Whitney’s Method

Abstract: We quantise Whitney’s construction to prove the existence of a triangulation for any $$C^2$$ manifold, so that we get an algorithm with explicit bounds. We also give a new elementary proof, which is completely geometric.

Reconstruction of Convex Bodies from Moments

Abstract: We investigate how much information about a convex body can be retrieved from a finite number of its geometric moments. We give a sufficient condition for a convex body to be uniquely determined by a finite number of its geometric moments, and we show that among all convex bodies, those which are uniquely determined by a finite number of moments form a dense set. Further, we derive a stability result for convex bodies based on geometric moments. It turns out that the stability result is improved considerably by using another set of moments, namely Legendre moments. We present a reconstruction algorithm that approximates a convex body using a finite number of its Legendre moments. The consistency of the algorithm is established using the stability result for Legendre moments. When only noisy measurements of Legendre moments are available, the consistency of the algorithm is established under certain assumptions on the variance of the noise variables.

Smallest k -Enclosing Rectangle Revisited

Abstract: Given a set of n points in the plane, and a parameter $$k$$ , we consider the problem of computing the minimum (perimeter or area) axis-aligned rectangle enclosing $$k$$ points. We present the first near quadratic time algorithm for this problem, improving over the previous near- $$O(n^{5/2})$$ -time algorithm by Kaplan et al. (25th European Symposium on Algorithms. Leibniz Int Proc Inform, vol. 87, # 52. Leibniz-Zent Inform, Wadern, 2017). We provide an almost matching conditional lower bound, under the assumption that $$(\min ,+)$$ -convolution cannot be solved in truly subquadratic time. Furthermore, we present a new reduction (for both perimeter and area) that can make the time bound sensitive to $$k$$ , giving near $$O(nk)$$ time. We also present a near linear time $$(1+\varepsilon )$$ -approximation algorithm to the minimum area of the optimal rectangle containing $$k$$ points. In addition, we study related problems including the 3-sided, arbitrarily oriented, weighted, and subset sum versions of the problem.

Thom Isotopy Theorem for Nonproper Maps and Computation of Sets of Stratified Generalized Critical Values

Abstract: Let $$X\subset {\mathbb {C}}^n$$ be an affine variety and $$f:X\rightarrow {\mathbb {C}}^m$$ be the restriction to X of a polynomial map $${\mathbb {C}}^n\rightarrow {\mathbb {C}}^m$$ . We construct an affine Whitney stratification of X. The set K(f) of stratified generalized critical values of f can also be computed. We show that K(f) is a nowhere dense subset of $${\mathbb {C}}^m$$ which contains the set B(f) of bifurcation values of f by proving a version of the Thom isotopy lemma for nonproper polynomial maps on singular varieties.

A Polynomial Time Algorithm to Compute Geodesics in CAT(0) Cubical Complexes

Abstract: This paper presents the first polynomial time algorithm to compute geodesics in a CAT(0) cubical complex in general dimension. The algorithm is a simple iterative method to update breakpoints of a path joining two points using Owen and Provan’s algorithm (IEEE/ACM Trans Comput Biol Bioinform 8(1):2–13, 2011) as a subroutine. Our algorithm is applicable to any path in any CAT(0) space in which geodesics between two close points can be computed, not limited to CAT(0) cubical complexes.