arXiv research

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

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…

PRISM identifies simplex vertices from noisy data.

A new optimization method for probability simplex problems.

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

We introduce a new distance metric for non-linear embeddings of Tempered Exponential Measures.

Consider a structured matrix factorization model where one factor is restricted to have its columns lying in the unit simplex. This simplex-structured matrix factorization (SSMF) model and the associated factorization techniques have spurred much interest in research topics over different areas, such as hyperspectral u…

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

We show that the problem of tiling the Euclidean plane with a finite set of polygons (up to translation) boils down to prove the existence of zeros of a non-negative convex function defined on a finite-dimensional simplex. This function is a generalisation, in the framework of branched surfaces, of the Thurston semi-no…

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

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

We present a novel method named Latent Semantic Imputation (LSI) to transfer external knowledge into semantic space for enhancing word embedding. The method integrates graph theory to extract the latent manifold structure of the entities in the affinity space and leverages non-negative least squares with standard simpl…

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

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…

The paper triangulates Heisenberg groups with horizontal and straight simplexes.

Study classifies submanifolds in probability simplex.

Spectral embedding is a procedure which can be used to obtain vector representations of the nodes of a graph. This paper proposes a generalisation of the latent position network model known as the random dot product graph, to allow interpretation of those vector representations as latent position estimates. The general…

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

Spaces of convex and concave functions appear naturally in theory and applications. For example, convex regression and log-concave density estimation are important topics in nonparametric statistics. In stochastic portfolio theory, concave functions on the unit simplex measure the concentration of capital, and their gr…

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…

We show that a complete hyperbolic n-manifold has a geodesic triangulation such that the tetrahedra contained in the thick part are L-bilipschitz diffeomorphic to the standard Euclidean n-simplex, for some constant L depending only on the dimension and the constant used to define the thick-thin decomposition of M.

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.

The paper explores the relationship between joint mixability and negative dependence structures.

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 a 1967 paper, Banchoff stated that a certain type of polyhedral curvature, that applies to all finite polyhedra, was zero at all vertices of an odd-dimensional polyhedral manifold; one then obtains an elementary proof that odd-dimensional manifolds have zero Euler characteristic. In a previous paper, the author defi…

Non-negative matrix factorization (NMF) is a fundamental non-convex optimization problem with numerous applications in Machine Learning (music analysis, document clustering, speech-source separation etc). Despite having received extensive study, it is poorly understood whether or not there exist natural algorithms that…

This paper studies the addition of linear constraints to the Support Vector Regression (SVR) when the kernel is linear. Adding those constraints into the problem allows to add prior knowledge on the estimator obtained, such as finding probability vector or monotone data. We propose a generalization of the Sequential Mi…

A method for diffusion on probability simplex for generative models.

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.

Study on pairwise counter-monotonicity, a type of negative dependence.

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

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…

The minimization of some multivariate risk indicators may be used as an allocation method, as proposed in Cénac et al. [6]. The aim of capital allocation is to choose a point in a simplex, according to a given criterion. In a previous paper [17] we proved that the proposed allocation technique satisfies a set of cohere…

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 $(N_p L)^\circ$ …

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…

New DKPP family controls positive and negative dependence in random subsets.

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…

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.

Negative dependence improves machine learning performance.

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

Characterizes symmetric Bernoulli distributions with minimal convex sums.

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…

Improved convergence rates for saddle-point optimization algorithms.