A locally-built, LLM-digested index of recent arXiv papers in quant finance, geometry/topology, and statistical ML — keyword search served straight from SQLite on this machine.
Study optimal transport on simplex boundary, proving transport map and potential regularity.
The paper triangulates Heisenberg groups with horizontal and straight simplexes.
PRISM identifies simplex vertices from noisy data.
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…
The article studies crystallizations of small covers over simple polytopes and finds unique crystallizations for the -simplex.
Study shows neural collapse is invariant to class imbalances under certain conditions.
In this paper we discuss a novel framework for multiclass learning, defined by a suitable coding/decoding strategy, namely the simplex coding, that allows to generalize to multiple classes a relaxation approach commonly used in binary classification. In this framework, a relaxation error analysis can be developed avoid…
Neural networks exhibit simplex symmetry in their final and penultimate layers.
Researchers correct earlier work on surgeries of Gieseking's hyperbolic simplex manifold.
Study classifies submanifolds in probability simplex.
Geometry-aware KDE model improves multiclass quantification.
Locality regularized reconstruction finds sparse coefficients for sparse and structured data.
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…
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.
Proposes a new method for multi-class classification with well-calibrated predictions.
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 , is it true that its volume is a rational multiple of the volu…
High dimensional data often contain multiple facets, and several clustering patterns can co-exist under different variable subspaces, also known as the views. While multi-view clustering algorithms were proposed, the uncertainty quantification remains difficult --- a particular challenge is in the high complexity of es…
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.
In this paper, we find lower bounds for volumes of hyperbolic 3-manifolds with various topological conditions. Let V_3 = 1.01494 denote the volume of a regular ideal simplex in hyperbolic 3-space. As a special case of the main theorem, if a hyperbolic manifold M contains an acylindrical surface S, then Vol(M)>= -2 V_3 …
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…
This work shows that supervised contrastive learning achieves similar results to cross-entropy but requires more iterations.
Simple Deep LDA models achieve accuracy competitive with softmax baselines.
Given a convex body, the -Busemann Random Simplex Inequality is closely related to the centroid body for and , and only in these cases it can be proved using the -Busemann-Petty centroid inequality. We define a convex body and prove an isoperimetric inequality for …
Wasserstein distance plays increasingly important roles in machine learning, stochastic programming and image processing. Major efforts have been under way to address its high computational complexity, some leading to approximate or regularized variations such as Sinkhorn distance. However, as we will demonstrate, regu…
Deep learning method proves convergence for high-dimensional PDEs.
Regular variation provides a convenient theoretical framework to study large events. In the multivariate setting, the dependence structure of the positive extremes is characterized by a measure - the spectral measure - defined on the positive orthant of the unit sphere. This measure gathers information on the localizat…
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 investigate polyhedral -manifolds as subcomplexes of the boundary complex of a regular polytope. We call such a subcomplex {\it -Hamiltonian} if it contains the full -skeleton of the polytope. Since the case of the cube is well known and since the case of a simplex was also previously studied (these are so…
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.
While neural networks have achieved high performance in different learning tasks, their accuracy drops significantly in the presence of small adversarial perturbations to inputs. Defenses based on regularization and adversarial training are often followed by new attacks to defeat them. In this paper, we propose attack-…
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.
Timelike geometry of spherical simplices is shown to be isometric to vector spaces.
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…
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 -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.
We study a natural intrinsic definition of geometric simplices in Riemannian manifolds of arbitrary dimension , 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…