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 “simplex volume”
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.
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 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.
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…
New framework inscribes maximum volume ellipsoid for structured matrix factorization.
problem Structured matrix factorization with columns in unit simplex.
method Maximum volume inscribed ellipsoid (MVIE) via facet enumeration and convex optimization.
result MVIE framework guarantees exact recovery under certain conditions.
This work generalizes a geometric Laplacian determinant description to higher dimensions.
problem Defining and understanding the Laplacian determinant in higher dimensions with non-Delaunay triangulations.
method Geometric description of the Laplacian determinant in higher dimensions, relating it to volume quantities derived from simplex geometry.
result Generalizes geometric Laplacian determinant description to higher dimensions, showing negative semidefiniteness and kernel of constants.
Defines circumcenter of mass for polytopes without triangulation.
problem Defining circumcenter of mass for polytopes without triangulation.
method Investigates how volumes of polytopes change under Möbius transformations.
result Provides a definition of circumcenter of mass independent of triangulation.
In the context of Synthetic Differential Geometry, we describe the square volume of a ``second-infinitesimal simplex'', in terms of square-distance between its vertices. The square-volume function thus described is symmetric in the vertices. The square-volume gives rise to a characterization of the volume form in the t…
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.
Minimal Gaussian surface area is achieved by cones over a regular simplex for sets partitioning .
problem Finding the minimal Gaussian surface area of sets partitioning .
method Volume-preserving variations of the sets, avoiding matrix-valued partial differential inequalities.
result Strengthened Milman-Neeman Gaussian multi bubble theorem and first known dimension-independent bounds for the Plurality is Stablest Conjecture.
Geodesic simplices in pseudo-hyperbolic space get a cohomological treatment.
problem Understanding geodesic simplices in pseudo-hyperbolic space.
method Cohomological interpretation and necessary/sufficient condition formulation.
result Every ideal geodesic polytope in pseudo-hyperbolic space has finite volume.
Built the smallest non-commensurable hyperbolic 4-manifold.
problem Finding the smallest non-commensurable hyperbolic 4-manifold.
method Gluing copies of a polytope to build the manifold.
result The constructed manifold has twice the minimal volume.
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 …
Paper discovers simplicial complexes connecting trained models for improved ensembling.
problem Improving robustness and accuracy of deep learning ensembles.
method Identifies mode-connecting simplicial complexes on loss surfaces.
result Efficiently builds simplicial complexes for ensembling, outperforming independent ensembles.
Geometric calculus on probability simplex using Wasserstein metric.
problem Calculus on the probability simplex with Wasserstein metric.
method Embedding probability simplex in positive measure space with nonlinear metric tensor, deriving Christoffel symbols, connections, curvature tensors, and operators.
result Established geometric computations on probability manifold and density space, connecting Fisher-Rao and optimal transport metrics.
In blind hyperspectral unmixing (HU), the pure-pixel assumption is well-known to be powerful in enabling simple and effective blind HU solutions. However, the pure-pixel assumption is not always satisfied in an exact sense, especially for scenarios where pixels are heavily mixed. In the no pure-pixel case, a good blind…
Sharp reverse affine isoperimetric inequalities for asymmetric Wulff shapes and their polars are established, along with the characterization of all extremals. These new inequalities have as special cases previously obtained simplex inequalities by Ball, Barthe and Lutwak, Yang, and Zhang. In particular, they provide t…
Researchers correct earlier work on surgeries of Gieseking's hyperbolic simplex manifold.
problem Incorrectly identified Gieseking's manifold as orbifolds, leading to a conflict with known theorems.
method Revised and completed the analysis of Dehn surgeries on Gieseking's manifold, identifying them as cone manifolds.
result Corrected the understanding of Gieseking's manifold, identifying it as cone manifolds and derived new orbifold series.
A generalized hyperbolic tetrahedra is a polyhedron (possibly non-compact) with finite volume in hyperbolic space, obtained from a tetrahedron by the polar truncation at the vertices lying outside the space. In this paper it is proved that a volume formula for ordinary hyperbolic tetrahedra devised by J. Murakami and M…
Let M be a hyperbolic n-manifold whose cusps have torus cross-sections. In arXiv:0901.0056, the authors constructed a variety of nonpositively and negatively curved spaces as "2π-fillings" of M by replacing the cusps of M with compact "partial cones" of their boundaries. These 2π-fillings are closed pseudomanifolds, an…
If a hyperbolic link has a prime alternating diagram D, then we show that the link complement's volume can be estimated directly from D. We define a very elementary invariant of the diagram D, its twist number t(D), and show that the volume lies between v_3(t(D) - 2)/2 and v_3(16t(D) - 16), where v_3 is the volume of a…
Proves Seidel's conjectures about ideal tetrahedra in hyperbolic 3-space.
problem Determining the volume of ideal hyperbolic tetrahedra using algebraic maps.
method Analyzes the doubly stochastic Gram matrix of tetrahedron vertices.
result Volume is a monotonic function of the permanent and determinant of the Gram matrix.
We propose a solution to the problem of estimating a Riemannian metric associated with a given differentiable manifold. The metric learning problem is based on minimizing the relative volume of a given set of points. We derive the details for a family of metrics on the multinomial simplex. The resulting metric has appl…
Paper proves min-vol NMF robust to noise under expanded condition.
problem Robustness of min-vol NMF to noise.
method Proved robustness under expanded sufficiently scattered condition.
result Proves min-vol NMF identifies groundtruth factors in noise.
We consider a volume maximization program to construct hyperbolic structures on triangulated 3-manifolds, for which previous progress has lead to consider angle assignments which do not correspond to a hyperbolic metric on each simplex. We show that critical points of the generalized volume are associated to geometric …
We compute volumes of complex convex shapes to predict financial crises.
problem Detecting financial crises by analyzing portfolio dependencies.
method Exact and approximate algorithms for volume computation of specific convex bodies.
result Practical algorithms can accurately predict financial crises.
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.
Let n>2 and let M be an orientable complete finite volume hyperbolic n-manifold with (possibly empty) geodesic boundary having Riemannian volume vol(M) and simplicial volume ||M||. A celebrated result by Gromov and Thurston states that if M has empty boundary then the ratio between vol(M) and ||M|| is equal to v_n, whe…
A correspondence between three-dimensional flat connections and constant curvature four-dimensional simplices is used to give a novel quantization of geometry via complex SL(2,C) Chern-Simons theory. The resulting quantum geometrical states are hence represented by the 3d blocks of analytically continued Chern-Simons t…
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.
Study on Funk geometry volume growth and polytope flags, verifying conjectures.
problem Volume minimization in Funk geometry and polytopes.
method Analyzing Holmes--Thompson volume, studying asymptotics, computing coefficients.
result Second highest volume coefficient minimized by unique center point, maximized by regular polygons.
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.
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…
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…
Robust Volume Minimization-Based Matrix Factorization for Remote Sensing and Document Clusteringstat.ML
This paper considers \emph{volume minimization} (VolMin)-based structured matrix factorization (SMF). VolMin is a factorization criterion that decomposes a given data matrix into a basis matrix times a structured coefficient matrix via finding the minimum-volume simplex that encloses all the columns of the data matrix.…
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.
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…
Paper proposes a neural network for sparse recovery using Laplace techniques.
problem Sparse recovery from compressed measurements.
method Designing a neural network to compute the centroid of a polytope.
result Analytical computation of volume and centroid enables efficient sparse recovery.
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.