Vertex structure of fiber products of probability polytopes

• URL: http://arxiv.org/abs/2603.19479v1
• Date: Thu, 19 Mar 2026 21:21:08 GMT
• Title: Vertex structure of fiber products of probability polytopes
• Abstract summary: We develop tools for characterizing vertices of fiber products of polytopes.<n>In the theory of simplicial distributions, a pair of simplicial sets encoding measurement and outcome spaces determines a convex polytope.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We develop tools for characterizing vertices of fiber products of polytopes and apply them to simplicial distribution polytopes, a class of probability polytopes arising in quantum foundations and quantum information. In the theory of simplicial distributions, a pair of simplicial sets encoding measurement and outcome spaces determines a convex polytope of compatible probability assignments. Our first results give geometric criteria for detecting vertices of fiber products in terms of support data. These results are obtained in the more general framework of inverse limits of diagrams of polytopes in standard form, and they translate to corresponding criteria for simplicial distributions on arbitrary colimits of measurement spaces. We then focus on one-dimensional measurement spaces, where simplicial distributions recover and generalize local marginal polytopes in graphical models. In this setting, our sharpest results concern dipole graphs, for which we obtain a complete characterization of vertices and refine it to a graph-theoretic criterion. These characterizations are reminiscent of the classical support-graph criteria for transportation polytopes, but they arise in a richer class of polytopes in which vertex structure depends not only on support acyclicity but also on additional geometric compatibility data. Using the collapsing method from simplicial topology, we transfer the dipole characterization to rose graphs and obtain analogous results there. Finally, we apply collapsing to complete bipartite graphs, which encode physically relevant bipartite Bell scenarios, and more generally to arbitrary connected graphs, yielding lower bounds on the number of vertices.

Related papers

• Tensor network representations of discrete maximum entropy distributions via mean polytopes [0.8021197489470758]
  We present tensor network representations for discrete maximum entropy distributions under expectation constraints.<n>By leveraging the geometry of the convex polytope of realizable expectation vectors, we represent any maximum entropy distribution in the same architecture.<n>We derive explicit representations for the support within the CompActNet architecture.
  arXiv  Detail & Related papers  (2026-09-07T08:14:54Z)
• Approximation Depth of Convex Polytopes [1.6180992915701702]
  We study the ability to approximate a target polytope by polytopes of a given depth.<n>Our main results imply that simplices can only be additive sums.
  arXiv  Detail & Related papers  (2025-07-10T13:58:55Z)
• Linear Convergence of the Frank-Wolfe Algorithm over Product Polytopes [18.454334675619936]
  We study the linear convergence of Frank-Wolfe algorithms over product polytopes.<n>For convex objectives that are $mu$-Polyak-Lojasiewicz, we show linear convergence rates quantified in terms of the resulting condition numbers.
  arXiv  Detail & Related papers  (2025-05-16T13:50:55Z)
• The geometry of simplicial distributions on suspension scenarios [0.0]
  The geometrical structure of simplicial distributions can be seen as a resource for applications in quantum information theory.<n>We show that applying the cone construction to the measurement space makes the corresponding non-signaling polytope equal to the join of $m$ copies of the original polytope.<n>The decomposition done for simplicial distributions on a cone measurement space provides deeper insights into the geometry of simplicial distributions on a suspension measurement space.
  arXiv  Detail & Related papers  (2024-12-14T20:47:53Z)
• SpaceMesh: A Continuous Representation for Learning Manifold Surface Meshes [61.110517195874074]
  We present a scheme to directly generate manifold, polygonal meshes of complex connectivity as the output of a neural network.<n>Our key innovation is to define a continuous latent connectivity space at each mesh, which implies the discrete mesh.<n>In applications, this approach not only yields high-quality outputs from generative models, but also enables directly learning challenging geometry processing tasks such as mesh repair.
  arXiv  Detail & Related papers  (2024-09-30T17:59:03Z)
• Generalization of Geometric Graph Neural Networks with Lipschitz Loss Functions [84.01980526069075]
  We study the generalization capabilities of geometric graph neural networks (GNNs)<n>We prove a generalization gap between the optimal empirical risk and the optimal statistical risk of this GNN.<n>We verify this theoretical result with experiments on multiple real-world datasets.
  arXiv  Detail & Related papers  (2024-09-08T18:55:57Z)
• Gaussian Entanglement Measure: Applications to Multipartite Entanglement of Graph States and Bosonic Field Theory [50.24983453990065]
  An entanglement measure based on the Fubini-Study metric has been recently introduced by Cocchiarella and co-workers. We present the Gaussian Entanglement Measure (GEM), a generalization of geometric entanglement measure for multimode Gaussian states. By providing a computable multipartite entanglement measure for systems with a large number of degrees of freedom, we show that our definition can be used to obtain insights into a free bosonic field theory.
  arXiv  Detail & Related papers  (2024-01-31T15:50:50Z)
• Improving embedding of graphs with missing data by soft manifolds [51.425411400683565]
  The reliability of graph embeddings depends on how much the geometry of the continuous space matches the graph structure. We introduce a new class of manifold, named soft manifold, that can solve this situation. Using soft manifold for graph embedding, we can provide continuous spaces to pursue any task in data analysis over complex datasets.
  arXiv  Detail & Related papers  (2023-11-29T12:48:33Z)
• Data Topology-Dependent Upper Bounds of Neural Network Widths [52.58441144171022]
  We first show that a three-layer neural network can be designed to approximate an indicator function over a compact set. This is then extended to a simplicial complex, deriving width upper bounds based on its topological structure. We prove the universal approximation property of three-layer ReLU networks using our topological approach.
  arXiv  Detail & Related papers  (2023-05-25T14:17:15Z)
• One-step Bipartite Graph Cut: A Normalized Formulation and Its Application to Scalable Subspace Clustering [56.81492360414741]
  We show how to enforce a one-step normalized cut for bipartite graphs, especially with linear-time complexity. In this paper, we first characterize a novel one-step bipartite graph cut criterion with normalized constraints, and theoretically prove its equivalence to a trace problem. We extend this cut criterion to a scalable subspace clustering approach, where adaptive anchor learning, bipartite graph learning, and one-step normalized bipartite graph partitioning are simultaneously modeled.
  arXiv  Detail & Related papers  (2023-05-12T11:27:20Z)
• Curvature Graph Generative Adversarial Networks [31.763904668737304]
  Generative adversarial network (GAN) is widely used for generalized and robust learning on graph data. Existing GAN-based graph representation methods generate negative samples by random walk or traverse in discrete space. CurvGAN consistently and significantly outperforms the state-of-the-art methods across multiple tasks.
  arXiv  Detail & Related papers  (2022-03-03T10:00:32Z)
• Heterogeneous manifolds for curvature-aware graph embedding [6.3351090376024155]
  Graph embeddings are used in a broad range of Graph ML applications. The quality of such embeddings crucially depends on whether the geometry of the space matches that of the graph.
  arXiv  Detail & Related papers  (2022-02-02T18:18:35Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.