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At least 19 recordsLinked to original sources

arXiv2026

On the Power Distance Representation of Bregman Divergences

We express Bregman divergences between parameters as power distances between corresponding weighted points, and investigate several applications of this representation. First, we show that the Bregman closest pair and diameter problems can be reformulated as bichromatic weighted point problems under the power distance. This reformulation enables Bregman nearest and farthest neighbor queries under power distances to be handled in practice through corresponding Euclidean neighbor queries in one additional dimension, after offsetting the weights. Next, we revisit the problem of computing minimum enclosing Bregman balls (Bregman MEBs) and establish their equivalence to double-sided weight power MEBs. We further show that Bregman MEBs can be approximated by relaxing the double-sided weight power MEB problem to its single-sided weight counterpart. We then present two approaches to solving general power MEB problems: An exact algorithm based on the LP-type framework and an arbitrarily accurate approximation scheme based on iterative Frank-Wolfe optimization. Finally, we show that the potential-function lifting transformations used to construct Bregman Voronoi diagrams can be reinterpreted as classical paraboloid lifting transformations applied to the corresponding weighted point sets. This establishes a direct connection between Bregman Voronoi diagrams and power diagrams, further illustrating how Bregman divergences can be represented as power distances.

cs.IT↗

arXiv2026

Connectivity of Districting Metagraphs

In this article, we prove irreducibility results for a family of Markov chains arising in the study of redistricting and detecting gerrymandering. These chains use ReCom moves as their transition mechanism and are commonly employed in Markov chain Monte Carlo methods to generate ensembles of districting plans. Such ensembles are frequently used for outlier analysis, in which a proposed districting map is compared against the ensemble to determine whether it behaves atypically; this methodology often appears in expert testimony in redistricting litigation. We show that when the underlying dual graph is a triangular subset of the triangular lattice with side length 5 or larger, and each district consists of two merged geographic regions, the associated ReCom chain is irreducible. This provides another entry in the very small list of known classes of ReCom chains for which irreducibility has been established. We also demonstrate the fragility of this phenomenon by constructing an infinite family of maps for which the corresponding ReCom chain is not irreducible. Indeed, we produce a districting map that, after implementing a single ReCom move, always yields the same original map. These examples remain structurally close to the triangular lattice: they arise as subdivisions of the triangular lattice, and the resulting graphs have maximum degree at most 8. Finally, we prove irreducibility for a further special case: the ReCom chain on a 3 x n grid graph partitioned into three districts of size n.

math.CO↗

arXiv2026

Degenerating orbits of the Longest Edge Bisection process

We study the Longest Edge Bisection (LEB) process as a dynamical system on the projective shape space of simplices. A long-standing conjecture going back to Adler and Rivara-Levin and motivated by finite-element mesh refinement, often taken as a standing assumption, is that this procedure is non-degenerate and, in fact, in a certain way periodic. We prove: \begin{itemize} \item There are 3-dimensional simplices such that the longest edge-bisection algorithm degenerates. \item There is an open set of 4-dimensional simplices on which the longest edge-bisection algorithm degenerates. \item If parametrizing the space of $d$-dimensional simplices by independent standard Gaussian vectors, then as $d$ increases, a random simplex degenerates asymptotically almost surely. \end{itemize} This is realized through exhibiting hyperbolic behaviour of the LEB process. We also exhibit elliptic behaviour that is nonperiodic.

math.DS↗

arXiv2026

A Sublinear Approximation Algorithm for Minimum Dilation Trees in the Plane

The dilation of a geometric graph measures how much longer the path between pairs of points becomes when restricted to graph edges, rather than following the direct path through the ambient space. The minimum dilation tree of a point set is the spanning tree with minimum dilation, where edge lengths in the tree are given by distances in the ambient space. In the Euclidean plane, computing the minimum dilation tree is NP-hard, but no hardness of approximation result is known. On the other hand, the minimum spanning tree is an $(n-1)$-approximation to the minimum dilation tree, but no asymptotically-better approximation algorithm is known for general point sets in the Euclidean plane. We give the first sublinear approximation algorithm for the minimum dilation tree in the Euclidean plane. Our approximation ratio is $\tilde{O}(n^{14/15})$ and our algorithm runs in polynomial time. This resolves an open problem proposed by Eppstein in 1996.

cs.CG↗

arXiv2026

Coresets for Continuous $k$-Center in Hyperbolic Space

We study the continuous $k$-center problem in fixed-dimensional hyperbolic space. The input is a finite point set $P \subset \mathbb H^D$, while the centers may be placed anywhere in the ambient space. A nonempty subset $P_\varepsilon\subseteq P$ is an $\varepsilon$-coreset if every exact optimal continuous $k$-center solution for $P_\varepsilon$ is a $(1+\varepsilon)$-approximate solution for $P$. Our construction uses Gonzalez's farthest-first traversal, whose radius $R_G$ gives a constant-factor estimate of the optimum. When $R_G$ is bounded, we map each cluster to the origin of the Poincaré ball model and apply a local Euclidean grid. For larger $R_G$, we partition each cluster into radial shells and project them onto representative hyperbolic spheres, avoiding a discretization of the ambient ball. By the hyperbolic law of cosines, the intersection of a hyperbolic ball with each sphere corresponds to a spherical cap. A recursive $k$-cap coverage certificate then replaces each shell with a radius-independent set of input witnesses that preserves its farthest-point behavior for every set of at most $k$ centers. The resulting coreset $P_\varepsilon$ has size $|P_\varepsilon| = \left({k}/{\varepsilon}\right)^{O(kD)}$ and satisfies $Φ_P(C_{P_\varepsilon}) \le (1+\varepsilon)\operatorname{OPT}_k(P)$ for every optimal continuous $k$-center solution $C_{P_\varepsilon}$ of $P_\varepsilon$. It can be constructed in $O\left( nkD\left({k}/{\varepsilon}\right)^{O(kD)} \right)$ time. Consequently, for fixed $D$, $k$, and $\varepsilon$, the coreset has constant size and can be computed in linear time. We complement this upper bound by showing that, for every fixed $D \ge 1$, every $k \ge 2$, and all sufficiently small $\varepsilon$, there are instances for which every $\varepsilon$-coreset has size $Ω\left(k\varepsilon^{-(D-1)/2}\right)$.

cs.CG↗

arXiv2026

Statistical Inference for Persistence Diagrams via Landmark Embeddings: Minimax Theory and Finite Approximation

Hilbert-space embeddings enable inference for populations of persistence diagrams, but separation between individual diagrams need not survive population averaging. We develop a framework for inference on population mean embeddings, with particular attention to the additive landmark representations PLACE and PALACE. Treating each diagram as one independent observation, we apply Hilbert-space limit theory to obtain covariance estimators, two-sample tests, and confidence balls under suitable moment conditions, without requiring a lower-distortion bound. For additive embeddings, we identify the population mean as an embedding of the mean counting measure and show that geometric separation of these measures alone cannot guarantee uniform testing power. We then introduce a model with latent template diagrams, missing features, and location perturbations. Under common or feature-specific prevalence conditions, a diagram-level lower-distortion certificate yields explicit lower bounds on population mean separation. These margins provide finite-sample uniform power guarantees, and an additional information-divergence comparison gives matching sample-complexity bounds over restricted scale ranges. Confidence sets yield lower bounds on transport separation of population mean measures and exclusion guarantees for specified structured alternatives. We also quantify how orthogonal truncation changes the certified signal and the approximation allowance needed for confidence sets targeting the full embedding, relating sample size, retained coordinates, and template separation. Simulations examine calibration, power, and coverage, and an analysis of resting-state connectivity from the Autism Brain Imaging Data Exchange illustrates the procedures.

math.ST↗

arXiv2026

An explicit mono-monostatic polyhedron

A convex body is mono-monostatic if, resting under gravity on a horizontal plane, it has exactly one stable and one unstable equilibrium position. Smooth mono-monostatic homogeneous bodies exist (the Gömböc of Domokos and Várkonyi), and Lángi proved that (homogeneous) mono-monostatic polyhedra exist; although no explicit example appears to have been published, to the author's knowledge. We construct explicitly two such mono-monostatic polytopes, the smaller one having $56946$ faces; importantly, we certify them: the polytope is presented as an intersection of half-spaces with rational data, and a certifying verification establishes, using exact rational arithmetic for every decisive comparison, that the body has equilibrium signature $(S,H,U)=(1,0,1)$ with respect to its own exact centroid, with explicit nondegeneracy margins. We describe the geometric obstructions that make naive discretisations of smooth mono-monostatic bodies fail, the adaptive construction that overcomes them, and the certification strategy. While the result itself is not strikingly novel, we emphasise the non-standard (but increasingly more common) methodology: the entire programme, i.e. experiments, constructions and the verifier itself, was implemented by AI agents under human mathematical direction. We argue that exact certification of numerically discovered objects is the natural contract between such AI-assisted workflows and mathematical standards of rigour.

math.MG↗

arXiv2026

A Sub-4 Approximation for Fair $k$-Means

Fairness in clustering has attracted sustained research interest, motivated by the need to ensure equitable representation of protected groups in machine learning applications. We study fair $k$-means clustering in Euclidean space, where the proportion of each protected group in every cluster must lie within specified lower and upper bounds. These constraints make it challenging to determine both cluster centers and point assignments. We propose an approximation algorithm that combines a linear programming relaxation with geometric transformations of the input to construct candidate center sets. Given a $ρ$-approximate algorithm for weighted $k$-means and any $ε>0$, our algorithm returns a fractional solution whose cost is at most $1+(3-1/Γ)ρ+O(ε)$ times the optimal integral fair cost, where $Γ\approx6.357$ is an upper bound on the integrality gap of the standard Euclidean $k$-means LP. With a PTAS as the subroutine, the approximation ratio becomes $3.8427+O(ε)$, improving the previous factor of $5+O(ε)$ to below $4$. The solution satisfies all fairness constraints exactly and can be rounded to an integral assignment with a bounded additive violation of fairness and no increase in cost. The same approximation guarantee extends to the $k$-sparse Wasserstein barycenter problem.

cs.CG↗

arXiv2026

Quantum Query Complexity of Persistence Statistics in Graph Zigzags

We study the query complexity of estimating scalar summaries of zigzag bar lifetimes from snapshot-adjacency bits. For graphs $G_1,\ldots,G_m$ on $n$ labeled vertices, let $\ell_b$ be the snapshot lifetime of a degree-one bar $b$ of the intersection zigzag. For a probability generating function $ϕ(x)=\mathbb{E}[x^R]$, the statistic $F_ϕ=\sum_bϕ(\ell_b/m)$ includes normalized degree-$r$ total persistence and the mean generalized rank over a uniform time window. An exact identity underlies our algorithm: sample $R$ uniform times; the expected generalized rank between their minimum and maximum equals $F_ϕ$. For graphs that rank is the circuit rank of an intersection graph, so a nonlinear barcode functional becomes an average of edge and component counts, and no barcode is computed. Without spectral-gap, homology-state, or QRAM assumptions, this gives a quantum estimator with additive error $\varepsilon n$ and $\widetilde O(\sqrt{m(K+n)}/\varepsilon)$ queries when a bound $K\ge F_ϕ$ is supplied, against $\widetilde O(m\min\{n^2,(K+n)/\varepsilon^2\})$ classically, and an adaptive quantum variant with the same instance dependence. These estimators are optimal in two regimes. For every fixed power weight $x^r$, $r\ge2$, and for the uniform-window mean, the worst-case complexities are $\widetildeΘ(n\sqrt m/\varepsilon)$ quantum and $Θ(n^2m)$ classical. On sparse instances, under an explicit split-leakage promise met by power and binomial weights of logarithmic degree and the promise $F_ϕ\le K$, they are $\widetildeΘ(\sqrt{mK}/\varepsilon)$ and $\widetildeΘ(m\min\{n^2,K/\varepsilon^2\})$. The classical lower bounds hold against fully adaptive algorithms, and fewer than $m$ such statistics cannot determine the positive-lifetime histogram. All bounds concern snapshot access; with an explicit update stream, near-linear full-barcode algorithms are known.

quant-ph↗

arXiv2026

Proximity3D: Shape from Capacitive Proximity on Sensing Manifold

Most shape reconstruction methods assume measurements defined over planar sensing domains, such as RGB images or depth maps. In this paper, we use a curved capacitive textile as a shape sensor, treating its surface as a non-planar sensing manifold. Each scan is represented as a capacitive proximity field on this manifold, induced by the interaction between the curved electrode layout and nearby object geometry. We introduce a multi-view feedforward reconstruction model that aggregates these fields across known sensor views and recovers the observed object shape. Simulated and physical experiments demonstrate robust reconstruction from capacitive proximity signals acquired on curved sensing surfaces, pointing toward a new route to robotic near-field geometric awareness via embodied sensing.

cs.CV↗

arXiv2026

Subset selection problems in planar point sets

Given a finite point set satisfying condition $\mathcal{A}$, the subset selection problem asks, how large of a subset satisfying condition $\mathcal{B}$ can be extracted? In this paper, we make progress on three instances of subset selection problems in planar point sets. Let $n,s\in\mathbb{N}$ with $n\geq s$, and let $P\subseteq\mathbb{R}^2$ be a set of $n$ points, where at most $s$ points lie on the same line. Firstly, we select a general position subset of $P$. This problem was proposed by Erdős under the regime when $s$ is a constant. For $s$ being non-constant, we give new lower and upper bounds on the maximum size of such a subset. In particular, we show that in the worst case such a set can have size at most $O(n^{5/6+o(1)}/\sqrt{s})$ when $3\leq s\leq n^{1/3}$ and $O(n/s)$ when $n^{1/3}\leq s\leq n$. Secondly, we select a monotone general position subset of $P$, that is, a subset in general position where the points are ordered from left to right and their $y$-coordinates are either non-decreasing or non-increasing. We present bounds on the maximum size of such a subset. In particular, when $s=Ω(\sqrt{n})$, our upper and lower bounds differ at most by a logarithmic factor. Lastly, we select a subset of $P$ with pairwise distinct slopes. This problem was initially studied by Erdős, Graham, Ruzsa, and Taylor on the grid. We show that for $s=O(\sqrt{n})$ such a subset of size $Ω((n/\log{s})^{1/3})$ can always be found in $P$. When $s=Θ(\sqrt{n})$, this matches a lower bound given by Zhang on the grid. As for the upper bound, we show that in the worst case such a subset has size at most $O(\sqrt{n})$ for $2\leq s\leq n^{3/8}$ and $O((n/s)^{4/5})$ for $n^{3/8}\leq s=O(\sqrt{n})$. The proofs use a wide range of tools such as incidence geometry, probabilistic methods, the hypergraph container method, and additive combinatorics.

math.CO↗

arXiv2026

Algorithmic exploration of the unit distance problem in the rational plane

This paper presents reproducible experimental evidence on unit-distance graph density. Our approach is based on a novel algorithmic exploration of the rational plane for the generation of unit-distance graphs. An efficient algorithm for this utility must perform a local-breadth search on a bounded and finite set of elements and generate a graph that potentially encompasses the general properties of a unit-distance graph, not affected by restrictions on its generation. To this end, we show that our approach accomplishes this purpose by overcoming the limitations of grid-based structures, used in the literature for generating unit-distance graphs. When applied to large unit-distance graphs, our method generates large graphs with a scaling exponent of 1.17, that surpasses recent theoretical predictions on graph density growth.

cs.CG↗

arXiv2026

Optimal exponential memory for sequential Euclidean connections: edge-power costs and phase transitions

We study the edge-power cost of the labelled tree generated by the $γ$-strategy, a constant-gain rule for sequential Euclidean connections. Starting with $x_0=p_0$, each input point $p_i$ is attached to $x_{i-1}$, and the state is updated by $x_i=γx_{i-1}+(1-γ)p_i$. Retaining $x_i$ subdivides the insertion segment into a spine edge and a leaf edge. The memory parameter $γ$ controls how long earlier points influence subsequent attachment points. We minimize the sum of the $α$-powers of the edge lengths under independent uniform input and arbitrary input sequences. For uniform points in the unit ball, the stationary problem has a transition at $α=1$. Its continuous extension is minimized at the boundary for $0<α\leq1$, while every global minimizer is interior for $α>1$. We determine the finite optimizer in the joint window $α_N=1+\varepsilon_N$, $\varepsilon_N\log N\toλ$. Below an explicit threshold it lies on the $N^{-1/2}$ scale, at the threshold its scale is $\sqrt{\log N/(N\log\log N)}$, and above the threshold it approaches an explicit stationary root with two computable corrections. A second threshold identifies the governing correction, and differentiated estimates prove eventual uniqueness. At $α=3d+8$, the linear coefficient at the stationary endpoint changes sign and a branch of strict local maxima enters the parameter interval. For arbitrary input sequences, the optimal parameter and asymptotic worst-case edge-power cost per point are explicit for $0<α\leq3$. At high powers, periodic antipodal block inputs give explicit lower bounds which, with a separation argument, show that the optimized cost is asymptotic to $2\log2/\logα$. Exact results for powers two and four, a rational recursion for every even power, and a high-dimensional expansion complete the analysis.

cs.CG↗

arXiv2026

Helly-Type Theorems for Splitting Point Sets

Let $0 < α\leq 1/2$. We say that a finite point set $P$ in $\mathbb{R}^d$ is $α$-split by a hyperplane $h$ if each of the closed half-spaces determined by $h$, contains at least $α|P|$ of the points of $P$. We further say $P$ is $α$-split by a $k$-dimensional flat $τ$ if $P$ is $α$-split by any hyperplane through $τ$. In the standard notation (which coincides with Tukey depth for $k= 0$), the $k$-flat $τ$ has depth $α$ with respect to $P$. We establish interesting Helly-type theorems for splitting families of finite point sets in $\mathbb{R}^d$. Unlike the classical sufficient Helly-type criteria for transversals to families of compact convex sets, which exist only for point and hyperplanes, our results extend to splitting families of point sets by collections of $k$-flats of arbitrary dimensionality $ 0 \leq k \leq d-1$.

math.CO↗

arXiv2026

AnyGS2Mesh: Feed-Forward Mesh Reconstruction from 3D Gaussian Splatting with Arbitrary-Resolution Views

Existing 3D mesh reconstruction methods from Gaussian scene representations predominantly rely on iterative optimization, resulting in slow inference and limited scalability to high-resolution inputs. In this paper, we present AnyGS2Mesh, the first feed-forward framework for directly reconstructing 3D meshes from 3D Gaussian Splatting representations with support for arbitrary input image resolutions. Our approach incorporates a Gaussian-Guided Transformer architecture that exploits explicit 3D geometric priors for efficient mesh generation. We introduce three key components: (1) a Gaussian-Guided Spatial Reasoning Transformer represents Gaussian primitives as structured 3D tokens and jointly reasons over Gaussian and image features; (2) a Streaming and Patchwise Geometry Encoder processes native-resolution views sequentially and aggregates information across variable-length view sets; (3) a Scale-Aligned Hybrid Depth Refiner uses a PatchFusion-style encoder--decoder to fuse RGB-conditioned predicted depth with Gaussian-rendered metric depth, combining fine local structures with globally consistent metric scale. The refined depth maps are integrated through TSDF fusion, followed by Marching Cubes for deterministic mesh extraction. Extensive experiments show that AnyGS2Mesh achieves state-of-the-art reconstruction quality while significantly reducing inference time compared with optimization-based baselines, enabling near-real-time, high-quality mesh reconstruction. Our results demonstrate the potential of combining Gaussian representations and feed-forward Transformer architectures for scalable 3D geometry reconstruction. The code will be made publicly available upon acceptance.

cs.CG↗

arXiv2026

Approximating CDTW Distance of Piecewise Algebraic Curves

Curves as input data naturally arise in a variety of fields including finance, seismology, medicine, spatio-temporal data mining, malicious activity detection, and more. A common way to analyze these data sets is to do similarity matching or clustering. The most common metrics used for measuring similarity of curves are Dynamic Time Warping (DTW) and Fréchet distance. These metrics are sensitive to sampling rate and outliers respectively, and do not yield robust outcomes. Continuous Dynamic Time Warping (CDTW) is a more robust distance metric that improves upon DTW and Fréchet distances. Existing algorithms for CDTW are either exact algorithms that focus on non-Euclidean norms and piecewise linear curves, or approximation algorithms limited to piecewise linear curves. We present an approximation algorithm for computing the CDTW distance under Euclidean norm between piecewise (higher degree) algebraic curves. That is, we present a fully polynomial-time approximation scheme (FPTAS) of multiplicative error $\varepsilon$, with $O \left( (m+n)^{\frac{19}{6}} (\frac{1}{\varepsilon})^{\frac{10}{3}} \log \left( \frac{ (m+n) }{\varepsilon^2} \right) \right)$ complexity, where $m$ and $n$ are the number of pieces of the two input curves.

cs.CG↗

arXiv2026

Curves, points, incidences and covering

Given a point set, mostly a grid in our case, we seek upper and lower bounds on the number of curves that are needed to cover the point set. We say a curve covers a point if the curve passes through the point. We consider such coverings by monotonic curves, lines, orthoconvex curves, circles, etc. We also study a problem that is converse of the covering problem -- if a set of $n^2$ points in the plane is covered by $n$ lines then can we say something about the configuration of the points?

math.CO↗

arXiv2026

On the volumes of the elliptope, metric polytope, and cut polytope

In this paper, we investigate the relationships between the volumes of four convex bodies: the cut polytope, metric polytope, rooted metric polytope, and elliptope, defined on graphs with n vertices. After an affine change of coordinates for the elliptope, the cut polytope is contained in each of the other three, which, for optimization purposes, provide polynomial-time relaxations. It is therefore of interest to see how tight these relaxations are. Worst-case ratio bounds are well known, but these are limited to objective functions with non-negative coefficients. Volume ratios, pioneered by Jon Lee with several co-authors, give global bounds and are the subject of this paper. For the rooted metric polytope over the complete graph, we show that for large n its volume is much greater than that of the elliptope. For the metric polytope, for small values of n, we show that its volume is smaller than that of the elliptope; however, for large values, we prove that the converse is true. Volume estimates place the crossover near n = 13. We also give exact formulae for the volumes of several families of sparse cut polytopes. In particular, we give an exact formula for the volume of the elliptope of every cycle and show that its volume ratio with the corresponding cut polytope converges rapidly to one.

cs.DM↗