arXiv research

A new method for SSMF improves upon existing algorithms.

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…

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 $(N_p L)^\circ$ …

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

Estimates dimensions of maximal simplices for rational and irrational trees in Outer space.

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…

Unfolding paths in Outer space accumulate on a simplex, not converge.

Moment polytope of toric exponential families is a projection of a simplex.

This work generalizes a geometric Laplacian determinant description to higher dimensions.

Geometrically interprets cup products and defines combinatorial Pin structures.

High dimensional sparse learning has imposed a great computational challenge to large scale data analysis. In this paper, we are interested in a broad class of sparse learning approaches formulated as linear programs parametrized by a {\em regularization factor}, and solve them by the parametric simplex method (PSM). O…

Dual explanation method using convex hulls and example-based vectors.

The abstract proves a conjecture about geometric structures in Calabi-Yau orbifolds.

We introduce canonical measures on a locally finite simplicial complex $K$ and study their asymptotic behavior under infinitely many barycentric subdivisions. We also compute the face polynomial of the asymptotic link and dual block of a simplex in the $d^{th}$ barycentric subdivision $Sd^d(K)$ of $K$, $d\gg0$. It is a…

This reports on the fundamental objects revealed by Ross Street, which he called `orientals'. Street's work was in part inspired by Robert's attempts to use N-category ideas to construct nets of C*-algebras in Minkowski space for applications to relativistic quantum field theory: Roberts' additional challenge was that …

The paper triangulates Heisenberg groups with horizontal and straight simplexes.

Study classifies submanifolds in probability simplex.

A function is exponentially concave if its exponential is concave. We consider exponentially concave functions on the unit simplex. In a previous paper we showed that gradient maps of exponentially concave functions provide solutions to a Monge-Kantorovich optimal transport problem and give a better gradient approximat…

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…

New geometric structures defined on SPD matrices for better understanding.

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…

Solves non-Archimedean Calabi-Yau equation on complex log pairs.

A method for diffusion on probability simplex for generative models.

Concrete distribution properties examined on simplex.

Deep nets exhibit 'Neural Collapse' during training's final phase, simplifying decision-making.

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

Proposes an accuracy-preserving calibration method for DNNs.

The paper proves a new discrete Laplacian for 3D meshes and shows its superiority over primal construction.

A new geometry-preserving method for interpreting compositional data.

We propose a new approach to graph compression by appeal to optimal transport. The transport problem is seeded with prior information about node importance, attributes, and edges in the graph. The transport formulation can be setup for either directed or undirected graphs, and its dual characterization is cast in terms…

A generalized cusp $C$ is diffeomorphic to $[0,\infty)$ times a closed Euclidean manifold. Geometrically $C$ is the quotient of a properly convex domain by a lattice, $Γ$, in one of a family of affine groups $G(ψ)$, parameterized by a point $ψ$ in the (dual closed) Weyl chamber for $SL(n+1,\mathbb{R})$, and $Γ$ determi…

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.

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.

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

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

New algorithm solves saddle point problems in Banach spaces.

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

Novel simplex-valued distribution improves on existing models.

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…

We present two identities (contiguity relation and variation formula) concerning the volume of a spherically faced simplex in the Euclidean space. These identities are described in terms of Cayley-Menger determinants and their differentials involved with hypersphere arrangements. They are derived as a limit of fundamen…

Investigates VaR behavior for sums of one-sided random variables, showing impossibilities and conditions for super-additivity.