arXiv research

Paper studies weighted Fermat-Frechet problem for simplex edge lengths.

We study a natural intrinsic definition of geometric simplices in Riemannian manifolds of arbitrary dimension $n$, and exploit these simplices to obtain criteria for triangulating compact Riemannian manifolds. These geometric simplices are defined using Karcher means. Given a finite set of vertices in a convex set on t…

A new method normalizes flow mixtures for better inference across different data types.

We define barycentric coordinates on a Riemannian manifold using Karcher's center of mass technique applied to point masses for n+1 sufficiently close points, determining an n-dimensional Riemannian simplex defined as a "Karcher simplex." Specifically, a set of weights is mapped to the Riemannian center of mass for the…

AdaBoost cycles in probability simplex dynamics.

CAST predicts distribution-valued time series by stabilizing and transporting simplex-supported successors.

A new method for approximating softmax and Gaussian kernels with reduced error.

Positive weights improve kernel quadrature's accuracy.

Defines circumcenter of mass for polytopes without triangulation.

New compression methods handle biased input sequences for more accurate posterior summaries.

The paper triangulates Heisenberg groups with horizontal and straight simplexes.

Study classifies submanifolds in probability simplex.

New algorithm for optimizing statistical utilities in bandits.

Study optimal transport on simplex boundary, proving transport map and potential regularity.

We analyze neural collapse in neural networks, showing that features collapse to vertices of a Simplex ETF.

This paper introduces a new nonlinear dictionary learning method for histograms in the probability simplex. The method leverages optimal transport theory, in the sense that our aim is to reconstruct histograms using so-called displacement interpolations (a.k.a. Wasserstein barycenters) between dictionary atoms; such at…

The Bezier simplex fitting is a novel data modeling technique which exploits geometric structures of data to approximate the Pareto front of multi-objective optimization problems. There are two fitting methods based on different sampling strategies. The inductive skeleton fitting employs a stratified subsampling from e…

PRISM identifies simplex vertices from noisy data.

A new optimization method for probability simplex problems.

We establish the second part of Milnor's conjecture on the volume of simplexes in hyperbolic and spherical spaces. A characterization of the closure of the space of the angle Gram matrices of simplexes is also obtained.

A hyperbolic 3-simplex reflection group is a Coxeter group arising as a lattice in the isometry group of hyperbolic 3-space, with fundamental domain a geodesic simplex (possibly with some ideal vertices). The classification of these groups is known, and there are exactly 9 cocompact examples, and 23 non-cocompact examp…

We introduce the non-pure versions of simplicial balls and spheres with minimum number of vertices. These are a special type of non-homogeneous balls and spheres (NH-balls and NH-spheres) satisfying a minimality condition on the number of maximal simplices. The main result is that minimal NH-balls and NH-spheres are pr…

In 1973, J. Cheeger and J. Simons raised the following question that still remains open and is known as the Rational Simplex Problem: Given a geodesic simplex in the spherical 3-space so that all of its interior dihedral angles are rational multiples of $π$, is it true that its volume is a rational multiple of the volu…

PolyILR: A Tree-Structured Orthonormal Decomposition of Compositional Data

Improved regret bounds for adversarial linear contextual bandits.

A method for diffusion on probability simplex for generative models.

Study shows neural collapse is invariant to class imbalances under certain conditions.

Concrete distribution properties examined on simplex.

A new method for SSMF improves upon existing algorithms.

A faster Wasserstein k-means algorithm for histogram data reduces computation and maintains clustering quality.

Proposes an accuracy-preserving calibration method for DNNs.

We formulate the Riemannian calculus of the probability set embedded with $L^2$-Wasserstein metric. This is an initial work of transport information geometry. Our investigation starts with the probability simplex (probability manifold) supported on vertices of a finite graph. The main idea is to embed the probability m…

On the probability simplex, we can consider the standard information geometric structure with the e- and m-affine connections mutually dual with respect to the Fisher metric. The geometry naturally defines submanifolds simultaneously autoparallel for the both affine connections, which we call {\em doubly autoparallel s…

Given $L$ a convex body, the $L_p$-Busemann Random Simplex Inequality is closely related to the centroid body $Γ_p L$ for $p=1$ and $2$, and only in these cases it can be proved using the $L_p$-Busemann-Petty centroid inequality. We define a convex body $N_p L$ and prove an isoperimetric inequality for $(N_p L)^\circ$ …

Associated to any finite flag complex L there is a right-angled Coxeter group W_L and a contractible cubical complex Sigma_L (the Davis complex) on which W_L acts properly and cocompactly, and such that the link of each vertex is L. It follows that if L is a generalized homology sphere, then Sigma_L is a contractible h…

Improved convergence rates for saddle-point optimization algorithms.

The paper is devoted to modeling optimal exercise strategies of the behavior of investors and issuers working with convertible bonds. This implies solution of the problems of stock price modeling, payoff computation and min-max optimization. Stock prices (underlying asset) were modeled under the assumption of the geome…

We show an efficient algorithm for the following problem: Given uniformly random points from an arbitrary n-dimensional simplex, estimate the simplex. The size of the sample and the number of arithmetic operations of our algorithm are polynomial in n. This answers a question of Frieze, Jerrum and Kannan [FJK]. Our resu…

Algorithm learns latent simplex from perturbed points in input-sparsity time.

Geometric approach clusters intersecting manifolds with high probability.

We provide an elementary proof of a simple, efficient algorithm for computing the Euclidean projection of a point onto the probability simplex. We also show an application in Laplacian K-modes clustering.

Annealed Entropic Allocation improves ranking and selection by mitigating hard switching and improving finite-budget discrimination.

New framework estimates staged tree models using hierarchical clustering on the probability simplex.

Timelike geometry of spherical simplices is shown to be isometric to vector spaces.

We study regularized deep neural networks (DNNs) and introduce a convex analytic framework to characterize the structure of the hidden layers. We show that a set of optimal hidden layer weights for a norm regularized DNN training problem can be explicitly found as the extreme points of a convex set. For the special cas…

Paper solves graph matching problem using convex relaxation to the simplex.

In the present paper we calculate the Gromov-Hausdorff distance between an arbitrary simplex (a metric space all whose non-zero distances are the same) and a finite metric space whose non-zero distances take two distinct values (so-called $2$-distance spaces). As a corollary, a complete solution to generalized Borsuk p…

New algorithm for online optimization over symmetric cones, unifying previous methods.