Tree-Sliced Wasserstein Distance with Nonlinear Projection

• URL: http://arxiv.org/abs/2505.00968v2
• Date: Mon, 09 Jun 2025 10:00:36 GMT
• Title: Tree-Sliced Wasserstein Distance with Nonlinear Projection
• Authors: Thanh Tran, Viet-Hoang Tran, Thanh Chu, Trang Pham, Laurent El Ghaoui, Tam Le, Tan M. Nguyen,
• Abstract summary: Tree-Sliced methods have emerged as an alternative to the traditional Sliced Wasserstein (SW) distance.<n>We propose a novel nonlinear projectional framework for the Tree-Sliced Wasserstein (TSW) distance, replacing the linear projections in earlier versions with general projections.<n>We validate our proposed metric through extensive numerical experiments for Euclidean and spherical datasets.
• Score: 8.996793030061324
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Tree-Sliced methods have recently emerged as an alternative to the traditional Sliced Wasserstein (SW) distance, replacing one-dimensional lines with tree-based metric spaces and incorporating a splitting mechanism for projecting measures. This approach enhances the ability to capture the topological structures of integration domains in Sliced Optimal Transport while maintaining low computational costs. Building on this foundation, we propose a novel nonlinear projectional framework for the Tree-Sliced Wasserstein (TSW) distance, substituting the linear projections in earlier versions with general projections, while ensuring the injectivity of the associated Radon Transform and preserving the well-definedness of the resulting metric. By designing appropriate projections, we construct efficient metrics for measures on both Euclidean spaces and spheres. Finally, we validate our proposed metric through extensive numerical experiments for Euclidean and spherical datasets. Applications include gradient flows, self-supervised learning, and generative models, where our methods demonstrate significant improvements over recent SW and TSW variants.

Related papers

• Enforcing Latent Euclidean Geometry in Single-Cell VAEs for Manifold Interpolation [79.27003481818413]
  We introduce FlatVI, a training framework that regularises the latent manifold of discrete-likelihood variational autoencoders towards Euclidean geometry.<n>By encouraging straight lines in the latent space to approximate geodesics on the decoded single-cell manifold, FlatVI enhances compatibility with downstream approaches.
  arXiv  Detail & Related papers  (2025-07-15T23:08:14Z)
• Spherical Tree-Sliced Wasserstein Distance [6.967581304933985]
  We present an adaptation of tree systems on OT problems for measures supported on a sphere.<n>We obtain an efficient metric for measures on the sphere, named Spherical Tree-Sliced Wasserstein (STSW) distance.
  arXiv  Detail & Related papers  (2025-03-14T10:00:13Z)
• Distance-Based Tree-Sliced Wasserstein Distance [6.967581304933985]
  We propose a novel class of splitting maps that generalizes the existing one studied in Tree-Sliced Wasserstein distance on Systems of Lines.<n>We show that Db-TSW significantly improves accuracy compared to recent SW variants while maintaining low computational cost.
  arXiv  Detail & Related papers  (2025-03-14T03:36:44Z)
• Bridging Geometric States via Geometric Diffusion Bridge [79.60212414973002]
  We introduce the Geometric Diffusion Bridge (GDB), a novel generative modeling framework that accurately bridges initial and target geometric states. GDB employs an equivariant diffusion bridge derived by a modified version of Doob's $h$-transform for connecting geometric states. We show that GDB surpasses existing state-of-the-art approaches, opening up a new pathway for accurately bridging geometric states.
  arXiv  Detail & Related papers  (2024-10-31T17:59:53Z)
• Verification of Geometric Robustness of Neural Networks via Piecewise Linear Approximation and Lipschitz Optimisation [57.10353686244835]
  We address the problem of verifying neural networks against geometric transformations of the input image, including rotation, scaling, shearing, and translation. The proposed method computes provably sound piecewise linear constraints for the pixel values by using sampling and linear approximations in combination with branch-and-bound Lipschitz. We show that our proposed implementation resolves up to 32% more verification cases than present approaches.
  arXiv  Detail & Related papers  (2024-08-23T15:02:09Z)
• Tree-Sliced Wasserstein Distance: A Geometric Perspective [6.63487092120036]
  We propose to replace one-dimensional lines with a more intricate structure, called tree systems.<n>This structure is metrizable by a tree metric, which yields a closed-form expression for OT problems on tree systems.
  arXiv  Detail & Related papers  (2024-06-19T17:40:11Z)
• 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)
• A Unifying and Canonical Description of Measure-Preserving Diffusions [60.59592461429012]
  A complete recipe of measure-preserving diffusions in Euclidean space was recently derived unifying several MCMC algorithms into a single framework. We develop a geometric theory that improves and generalises this construction to any manifold.
  arXiv  Detail & Related papers  (2021-05-06T17:36:55Z)
• GELATO: Geometrically Enriched Latent Model for Offline Reinforcement Learning [54.291331971813364]
  offline reinforcement learning approaches can be divided into proximal and uncertainty-aware methods. In this work, we demonstrate the benefit of combining the two in a latent variational model. Our proposed metrics measure both the quality of out of distribution samples as well as the discrepancy of examples in the data.
  arXiv  Detail & Related papers  (2021-02-22T19:42:40Z)
• Augmented Sliced Wasserstein Distances [55.028065567756066]
  We propose a new family of distance metrics, called augmented sliced Wasserstein distances (ASWDs) ASWDs are constructed by first mapping samples to higher-dimensional hypersurfaces parameterized by neural networks. Numerical results demonstrate that the ASWD significantly outperforms other Wasserstein variants for both synthetic and real-world problems.
  arXiv  Detail & Related papers  (2020-06-15T23:00:08Z)

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.