arXiv research

We introduce a globally-convergent algorithm for optimizing the tree-reweighted (TRW) variational objective over the marginal polytope. The algorithm is based on the conditional gradient method (Frank-Wolfe) and moves pseudomarginals within the marginal polytope through repeated maximum a posteriori (MAP) calls. This m…

We consider three different approaches to define natural Riemannian metrics on polytopes of stochastic matrices. First, we define a natural class of stochastic maps between these polytopes and give a metric characterization of Chentsov type in terms of invariance with respect to these maps. Second, we consider the Fish…

New method bounds causal effects using local consistency of marginals.

To construct flexible nonlinear predictive distributions, the paper introduces a family of softplus function based regression models that convolve, stack, or combine both operations by convolving countably infinite stacked gamma distributions, whose scales depend on the covariates. Generalizing logistic regression that…

We propose in this paper a general framework for deriving loss functions for structured prediction. In our framework, the user chooses a convex set including the output space and provides an oracle for projecting onto that set. Given that oracle, our framework automatically generates a corresponding convex and smooth l…

We study the problem of identifying the causal relationship between two discrete random variables from observational data. We recently proposed a novel framework called entropic causality that works in a very general functional model but makes the assumption that the unobserved exogenous variable has small entropy in t…

The Frank-Wolfe (FW) optimization algorithm has lately re-gained popularity thanks in particular to its ability to nicely handle the structured constraints appearing in machine learning applications. However, its convergence rate is known to be slow (sublinear) when the solution lies at the boundary. A simple less-know…

The study broadens the concept of cyclic polytopes to Veronese polytopes.

The paper studies deformation spaces of Coxeter truncation polytopes.

Neural networks approximate unit spheres as polytopes.

The study classifies all compact hyperbolic polytopes with eight facets.

Proves stability in Weyl polytopes using optimal transport.

Faces of quasi-arithmetic Coxeter polytopes are also quasi-arithmetic.

Given a finite collection P of convex n-polytopes in RP^n (n>1), we consider a real projective manifold M which is obtained by gluing together the polytopes in P along their facets in such a way that the union of any two adjacent polytopes sharing a common facet is convex. We prove that the real projective structure on…

We analyze variational inference for highly symmetric graphical models such as those arising from first-order probabilistic models. We first show that for these graphical models, the tree-reweighted variational objective lends itself to a compact lifted formulation which can be solved much more efficiently than the sta…

This article announces the completion of the classification of rank 4 locally projective polytopes and their quotients. There are seventeen universal locally projective polytopes (nine nondegenerate). Amongst their 441 quotients are a further four (nonuniversal) regular polytopes, and 152 nonregular but section regular…

The study classifies all compact 5D polytopes with 9 facets.

Smooth deformation space of Coxeter polytopes proven for orderable orbifolds.

We consider the problem of jointly estimating the parameters as well as the structure of binary valued Markov Random Fields, in contrast to earlier work that focus on one of the two problems. We formulate the problem as a maximization of $\ell_1$-regularized surrogate likelihood that allows us to find a sparse solution…

The study classifies 331 specific 4D polytopes with 7 facets.

Contact manifolds' momentum polytopes are convex.

Given a lattice L of R^n, a polytope D is called a Delaunay polytope in L if the set of its vertices is S\cap L where S is a sphere having no lattice points in its interior. D is called perfect if the only ellipsoid in R^n that contains S\cap L is exactly S. For a vector v of the Leech lattice Λ_{24} we define Λ_{24}(v…

Smooth approximations bound dihedral angles of convex polytopes.

Polytopic Matrix Factorization models data as latent vectors from a polytope, maximizing determinant for identifiability.

Examines nonrational polytopes and fans in toric geometry.

The article studies factorization structures in geometry and their applications to cones and polytopes.

New methods classify hyperbolic polytopes with up to 40 facets.

The paper explains how to parameterize facets of moment polytopes in real symplectic geometry.

We introduce two operations named biflip and puzzle-move on simple polytopes producing polytopes with diffeomorphic moment-angle manifolds.

Proves necessity of at least log2(n) layers to compute maximum of n numbers.

Unified cosmological and Einstein polytope theories.

The study of symmetries in manifolds derived from colored polytopes.

Diffeomorphisms of convex polytopes form a Lie group.

Efficiently projects points onto polytopes, especially useful in web-scale applications.

We study the Newton polytopes of determinants of square matrices defined over rings of twisted Laurent polynomials. We prove that such Newton polytopes are single polytopes (rather than formal differences of two polytopes); this result can be seen as analogous to the fact that determinants of matrices over commutative …

Geometric constraints help classify hyperbolic polytopes.

New noncompact Coxeter polytopes found in various dimensions.

Proves weight polytope matches with energy vectors in toric varieties.

Maximum entropy distributions with discrete support in $m$ dimensions arise in machine learning, statistics, information theory, and theoretical computer science. While structural and computational properties of max-entropy distributions have been extensively studied, basic questions such as: Do max-entropy distributio…

Unique floating and buoyancy surfaces identify convex polytopes.

The study proves curvature rigidity for convex polytopes.

The paper classifies compact hyperbolic Coxeter polytopes and improves upper bounds.

We study unbounded 2-dimensional metric polytopes such as those arising as Kähler quotients of complete Kähler 4-manifolds with two commuting symmetries and zero scalar curvature. Under a mild closedness condition, we obtain a complete classification of metrics on such polytopes, and as a result classify all possible m…

We investigate polyhedral $2k$-manifolds as subcomplexes of the boundary complex of a regular polytope. We call such a subcomplex {\it $k$-Hamiltonian} if it contains the full $k$-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 study the space of generalized translation invariant valuations on a finite-dimensional vector space and construct a partial convolution which extends the convolution of smooth translation invariant valuations. Our main theorem is that McMullen's polytope algebra is a subalgebra of the (partial) convolution algebra …

Real moment-angle manifolds of combinatorially equivalent simple polytopes are equivariantly diffeomorphic.

Proved inequality for anti-self-polar polytopes.

We show that the Grothendieck group associated to integral polytopes in $\mathbb{R}^n$ is free-abelian by providing an explicit basis. Moreover, we identify the involution on this polytope group given by reflection about the origin as a sum of Euler characteristic type. We also compute the kernel of the norm map sendin…