Flows on convex polytopes

• URL: http://arxiv.org/abs/2503.10232v2
• Date: Sat, 15 Mar 2025 11:45:12 GMT
• Title: Flows on convex polytopes
• Authors: Tomek Diederen, Nicola Zamboni,
• Abstract summary: We show that any full-dimensional polytope is homeomorphic to a unit ball, and our approach harnesses flows defined on the ball, mapping them back to the original polytope.<n>Our experiments take inspiration from applications in metabolic flux analysis and demonstrate that our methods achieve competitive density estimation, sampling accuracy, as well as fast training and inference times.
• Score: 0.0
• License: http://creativecommons.org/licenses/by-sa/4.0/
• Abstract: We present a framework for modeling complex, high-dimensional distributions on convex polytopes by leveraging recent advances in discrete and continuous normalizing flows on Riemannian manifolds. We show that any full-dimensional polytope is homeomorphic to a unit ball, and our approach harnesses flows defined on the ball, mapping them back to the original polytope. Furthermore, we introduce a strategy to construct flows when only the vertex representation of a polytope is available, employing maximum entropy barycentric coordinates and Aitchison geometry. Our experiments take inspiration from applications in metabolic flux analysis and demonstrate that our methods achieve competitive density estimation, sampling accuracy, as well as fast training and inference times.

Related papers

• Bayesian Circular Regression with von Mises Quasi-Processes [57.88921637944379]
  In this work we explore a family of expressive and interpretable distributions over circle-valued random functions.<n>For posterior inference, we introduce a new Stratonovich-like augmentation that lends itself to fast Gibbs sampling.<n>We present experiments applying this model to the prediction of wind directions and the percentage of the running gait cycle as a function of joint angles.
  arXiv  Detail & Related papers  (2024-06-19T01:57:21Z)
• Scaling Riemannian Diffusion Models [68.52820280448991]
  We show that our method enables us to scale to high dimensional tasks on nontrivial manifold. We model QCD densities on $SU(n)$ lattices and contrastively learned embeddings on high dimensional hyperspheres.
  arXiv  Detail & Related papers  (2023-10-30T21:27:53Z)
• Generative Modeling on Manifolds Through Mixture of Riemannian Diffusion Processes [57.396578974401734]
  We introduce a principled framework for building a generative diffusion process on general manifold. Instead of following the denoising approach of previous diffusion models, we construct a diffusion process using a mixture of bridge processes. We develop a geometric understanding of the mixture process, deriving the drift as a weighted mean of tangent directions to the data points.
  arXiv  Detail & Related papers  (2023-10-11T06:04:40Z)
• Flow Matching on General Geometries [43.252817099263744]
  We propose a simple yet powerful framework for training continuous normalizing flows on manifold geometries. We show that it is simulation-free on simple geometries, does not require divergence, and computes its target vector field in closed-form. Our method achieves state-of-the-art performance on many real-world non-Euclidean datasets.
  arXiv  Detail & Related papers  (2023-02-07T18:21:24Z)
• Riemannian Diffusion Models [11.306081315276089]
  Diffusion models are recent state-of-the-art methods for image generation and likelihood estimation. In this work, we generalize continuous-time diffusion models to arbitrary Riemannian manifold. Our proposed method achieves new state-of-the-art likelihoods on all benchmarks.
  arXiv  Detail & Related papers  (2022-08-16T21:18:31Z)
• Counting Phases and Faces Using Bayesian Thermodynamic Integration [77.34726150561087]
  We introduce a new approach to reconstruction of the thermodynamic functions and phase boundaries in two-parametric statistical mechanics systems. We use the proposed approach to accurately reconstruct the partition functions and phase diagrams of the Ising model and the exactly solvable non-equilibrium TASEP.
  arXiv  Detail & Related papers  (2022-05-18T17:11:23Z)
• Semi-Discrete Normalizing Flows through Differentiable Tessellation [31.474420819149724]
  We propose a tessellation-based approach that learns quantization boundaries on a continuous space, complete with exact likelihood evaluations. This is done through constructing normalizing flows on convex polytopes parameterized through a differentiable Voronoi tessellation. We show improvements over existing methods across a range of structured data modalities, and find that we can achieve a significant gain from just adding Voronoi mixtures to a baseline model.
  arXiv  Detail & Related papers  (2022-03-14T03:06:31Z)
• Data-driven modelling of nonlinear dynamics by polytope projections and memory [0.0]
  We present a numerical method to model dynamical systems from data. We project points from a Euclidean space to convex polytopes and represent these projected states of a system in new, lower-dimensional coordinates. We then introduce a specific nonlinear transformation to construct a model of the dynamics in the polytope and to transform back into the original state space.
  arXiv  Detail & Related papers  (2021-12-13T15:49:36Z)
• PU-Flow: a Point Cloud Upsampling Networkwith Normalizing Flows [58.96306192736593]
  We present PU-Flow, which incorporates normalizing flows and feature techniques to produce dense points uniformly distributed on the underlying surface. Specifically, we formulate the upsampling process as point in a latent space, where the weights are adaptively learned from local geometric context. We show that our method outperforms state-of-the-art deep learning-based approaches in terms of reconstruction quality, proximity-to-surface accuracy, and computation efficiency.
  arXiv  Detail & Related papers  (2021-07-13T07:45:48Z)
• Improving Metric Dimensionality Reduction with Distributed Topology [68.8204255655161]
  DIPOLE is a dimensionality-reduction post-processing step that corrects an initial embedding by minimizing a loss functional with both a local, metric term and a global, topological term. We observe that DIPOLE outperforms popular methods like UMAP, t-SNE, and Isomap on a number of popular datasets.
  arXiv  Detail & Related papers  (2021-06-14T17:19:44Z)
• Continuous normalizing flows on manifolds [0.342658286826597]
  We describe how the recently introduced Neural ODEs and continuous normalizing flows can be extended to arbitrary smooth manifold. We propose a general methodology for parameterizing vector fields on these spaces and demonstrate how gradient-based learning can be performed.
  arXiv  Detail & Related papers  (2021-03-14T15:35:19Z)

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.