New identities for simplex volume in Euclidean space.
problem Calculating the volume of a spherically faced simplex.
method Using Cayley-Menger determinants and hypergeometric integrals.
result Derivation of two identities describing simplex volume.
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48 results for “spherical simplex”
Timelike geometry of spherical simplices is shown to be isometric to vector spaces.
problem Characterizing the geometry of spherical simplices.
method Proved isometry to vector spaces with a timelike norm.
result Timelike spherical Hilbert geometry of simplices is isometric to a union of six copies of vector spaces.
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.
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…
Study of zonal spherical functions on partial flag manifolds using Jacobi polynomials.
problem Understanding zonal spherical functions on partial flag manifolds.
method Matrix variate version of Koornwinder's method for constructing orthogonal polynomials, using Hermitian Jacobi polynomials and multivariate Schur polynomials.
result Conjecture that Hermitian Jacobi polynomials are elementary zonal spherical functions.
The Apollonius theorem is generalized for m-simplices, with applications in geometry and optimization.
problem Generalizing the Apollonius theorem for m-simplices.
method Direct generalization of the theorem to m-simplices in n-dimensional space.
result Applications in geometry and optimization, including minimal surface enclosures, simplex thickness, and root-finding methods.
Algorithm checks if geometrically triangulated manifolds are isometric.
problem Determining if two geometric triangulations of manifolds are isometric.
method Sequence of Pachner moves and barycentric subdivisions with bounds on lengths.
result Bounding the length of transformations between triangulations.
It is known that the space of convex polygons in the Euclidean plane with fixed normals, up to homotheties and translations, endowed with the area form, is isometric to a hyperbolic polyhedron. In this note we show a class of convex polygons in the Lorentzian plane such that their moduli space, if the normals are fixed…
Geometric triangulations can be transformed by bistellar moves.
problem Transforming geometric triangulations of different manifolds.
method Using bistellar moves, a type of local change to triangulations.
result Geometric triangulations of compact manifolds can be connected by bistellar moves.
In his paper "On the Schlafli differential equality", J. Milnor conjectured that the volume of n-dimensional hyperbolic and spherical simplices, as a function of the dihedral angles, extends continuously to the closure of the space of allowable angles. A proof of this has recently been given by F. Luo (see math.GT/0412…
We prove the theorem mentioned in the title, for , where . The case of the simplex was known previously. Also, the case was settled, but there the infimum was some well-defined function of the side lengths. We also consider the cases of spherical and hyperbolic -spaces. There we give s…
Simple Deep LDA models achieve accuracy competitive with softmax baselines.
problem Training Deep LDA models by maximum likelihood estimation leads to overlapping or collapsed class clusters.
method Proposed a constrained Deep LDA formulation with geometric constraints to fix class means and covariance.
result MLE becomes stable under geometric constraints, yielding well-separated class clusters.
Kernel-Gradient Drifting improves generative modeling for non-Euclidean data.
problem Challenges in generative modeling for non-Euclidean data.
method Replaces Euclidean displacement with kernel-induced directions, exposing score-based structure.
result Kernel-gradient drifting enables state-of-the-art one-step generation for non-Euclidean data.
New geometric approach for analyzing compositional data like gut microbiomes.
problem Analyzing non-negative compositional data with relative values only.
method Reinterpret compositional data as quotient topology of a sphere, using spherical harmonics and reflection group actions.
result Construction of Reproducing Kernel Hilbert Space (RKHS) for compositional data.
A new associative memory uses Sinkhorn divergence for efficient pattern retrieval.
problem Efficiently retrieving patterns from large datasets of weighted point clouds.
method Derived retrieval dynamics as a SHK gradient flow, discretized for a deterministic algorithm.
result Proved basin invariance, geometric convergence, and robust recovery from perturbations.
The paper triangulates Heisenberg groups with horizontal and straight simplexes.
problem Triangulating Heisenberg groups with specific regularity properties.
method Constructing triangulations with horizontal and straight simplexes on a polyhedral structure and extending to the whole Heisenberg group.
result Explicit examples of grid and triangulations provided.
Study classifies submanifolds in probability simplex.
problem Classifying submanifolds in the probability simplex.
method Complete classification through geometric analysis.
result Doubly totally-umbilical submanifolds identified and classified.
Study optimal transport on simplex boundary, proving transport map and potential regularity.
problem Regularity of transport map and potential on simplex boundary.
method Boundary regularity results for optimal transport maps, exploiting simplex symmetries.
result Regularity properties of transport map and its convex potential.
Paper analyzes Bezier simplex fitting risks and optimal sampling.
problem Analyzing risks and optimal sampling for Bezier simplex fitting.
method Two fitting methods: inductive skeleton and all-at-once.
result Optimal subsample ratio for inductive skeleton fitting reduces risk.
The study explores special submanifolds in the probability simplex with unique geometric properties.
problem Exploring special submanifolds in the probability simplex.
method Algebraic characterization and classification of doubly autoparallel submanifolds.
result Characterization and classification of doubly autoparallel submanifolds on the probability simplex.
PRISM identifies simplex vertices from noisy data.
problem Identifying vertices of a simplex from noisy data.
method Probabilistic simplex model with maximum likelihood inference.
result Vertices are identifiable under certain assumptions.
A new optimization method for probability simplex problems.
problem Optimizing convex problems over the probability simplex.
method Cauchy-Simplex iteration scheme, mapping to sphere, gradient descent, and back-mapping.
result Convergence results and faster convergence in high dimensions.
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…
New convex body defined to prove Busemann Random Simplex Inequality.
problem Proving Busemann Random Simplex Inequality for general p.
method Defining and proving isoperimetric inequality for its dual.
result Equivalent to -Busemann-Petty centroid inequality for p=1,2.
Paper calculates Gromov-Hausdorff distance between simplexes and 2-distance spaces.
problem Calculating Gromov-Hausdorff distance between simplexes and 2-distance spaces.
method Formulas derived for clique covering number and chromatic number of graphs.
result Complete solution to generalized Borsuk problem for 2-distance spaces.
The study proves which circle bundles can be triangulated over a 3-simplex boundary.
problem Which circle bundles can be triangulated over a 3-simplex boundary?
method Proof of which circle bundles can be triangulated over a 3-simplex boundary.
result Only trivial and Hopf circle bundles can be triangulated over a 3-simplex boundary.
A method for diffusion on probability simplex for generative models.
problem Tension between continuous and discrete data in diffusion models.
method Proposes using softmax function applied to Ornstein-Uhlenbeck Process on probability simplex.
result Method extends to bounded image generation.
Concrete distribution properties examined on simplex.
problem Properties of Concrete distribution on simplex.
method Reflection and location-scale transformation of uniform distribution; explicit parameterization to Poincaré half-space.
result Fisher information and information metric are hyperbolic space; Fisher-Rao geodesic distance computed.
Subgradient algorithm achieves optimal regret for both adversarial and i.i.d. costs on the simplex.
problem Achieving optimal regret for both adversarial and i.i.d. costs on the simplex.
method Demonstrates the universality of the Subgradient algorithm for online learning on the simplex.
result Shows simultaneous regret for adversarial costs and pseudo-regret for i.i.d. costs.
A new method for SSMF improves upon existing algorithms.
problem Identify identifiable solutions in simplex-structured matrix factorization.
method Dual simplex volume maximization approach.
result The proposed method outperforms state-of-the-art SSMF algorithms.
A faster Wasserstein k-means algorithm for histogram data reduces computation and maintains clustering quality.
problem Efficiently clustering histogram data with reduced computation time.
method Sparse simplex projection to reduce data samples, centroids, and ground cost matrix, dynamically removing lower-valued samples.
result Significant reduction in computational complexity without compromising clustering quality.
Proposes an accuracy-preserving calibration method for DNNs.
problem Calibration of deep neural networks (DNNs) to measure prediction reliability.
method Uses Concrete distribution on the probability simplex to calibrate DNNs without accuracy loss.
result The proposed method outperforms previous methods in accuracy-preserving calibration tasks.
Proposes Dirichlet Simplex Nest for probabilistic modeling of various data types.
problem Modeling and inference for diverse data types.
method Probabilistic models based on Dirichlet distribution and Voronoi tessellation, with fast and accurate inference algorithms exploiting convex geometry and simplicial structure.
result Inference algorithms achieve consistency and strong error bounds across various settings and data distributions.
Improved sampling for sparse simplex spaces in Bayesian models.
problem Discretization error in SGMCMC methods for simplex spaces.
method Proposed SCIR process to remove discretization error.
result Samples from SCIR process are asymptotically unbiased.
Introduces new geometric framework for probability densities on manifolds.
problem Developing a new geometric framework for probability densities on manifolds.
method Introduces -information geometry and defines -probability simplex with -root transform.
result Explicit solution of gradient flow and geodesic completeness of -connection.
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 distribution on simplex for auto-encoding tasks.
problem Developing a new distribution for sparsity in auto-encoding models.
method Using Kumaraswamy distribution and ordered stick-breaking process.
result The new distribution has an exact and closed form reparameterization.
Algorithm learns latent simplex from perturbed points in input-sparsity time.
problem Learning a latent -vertex simplex from noisy data.
method Input-sparsity time algorithm using low-rank approximation and adaptive selection.
result Algorithm achieves time complexity, avoiding .
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…
CAST predicts distribution-valued time series by stabilizing and transporting simplex-supported successors.
problem Forecasting distribution-valued time series with structural failure modes.
method CAST (Causal Anchored Simplex Transport) uses successors retrieved from causal context, stabilized with a persistence anchor, and locally transported on ordered supports.
result CAST outperforms baselines on eleven public and simulated benchmarks, achieving best average rank on both one-step KL and autoregressive rollout JSD.
New framework estimates staged tree models using hierarchical clustering on the probability simplex.
problem Estimating staged tree models with context-specific dependencies.
method Hierarchical clustering on the probability simplex, using simplex-based divergences and linkage methods.
result Total Variation divergence with Ward.D2 linkage produces staged trees with better model fit, structure recovery, and computational efficiency.
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.
Random concave functionsmath.PR
The paper constructs random concave functions on the unit simplex.
problem Understanding probability measures on spaces of concave functions.
method Constructing random concave functions via a scaled minimum of random hyperplanes.
result There is a transition from deterministic to non-trivial limiting distributions as the number of hyperplanes increases.
Novel simplex-valued distribution improves on existing models.
problem Limitations of existing simplex-valued distributions like Dirichlet.
method Introducing continuous categorical distribution.
result Continuous categorical resolves limitations of Dirichlet.
Paper solves graph matching problem using convex relaxation to the simplex.
problem Finding the best alignment between two graphs.
method Introduces a new convex relaxation onto the unit simplex and uses mirror descent scheme.
result Shows exact recovery of ground truth permutation with high probability.
A new topic modeling method that minimizes topic simplex volume.
problem Topic modeling efficiency and accuracy.
method Reformulates LDA as minimizing topic simplex volume, uses convex relaxation and ADMM.
result Relaxed problem has same global minimum as original under assumptions.
Duality structures and discrete conformal variations of piecewise constant curvature surfacesmath.GT
A piecewise constant curvature manifold is a triangulated manifold that is assigned a geometry by specifying lengths of edges and stipulating that for a chosen background geometry (Euclidean, hyperbolic, or spherical), each simplex has an isometric embedding into the background geometry with the chosen edge lengths. Ad…