Non-negative matrix and tensor factorisations with a smoothed Wasserstein loss

• URL: http://arxiv.org/abs/2104.01708v1
• Date: Sun, 4 Apr 2021 22:42:21 GMT
• Title: Non-negative matrix and tensor factorisations with a smoothed Wasserstein loss
• Authors: Stephen Y. Zhang
• Abstract summary: We introduce a general mathematical framework for computing non-negative factorisations of matrices and tensors with respect to an optimal transport loss. We demonstrate the applicability of this approach with several numerical examples.
• Score: 0.0
• License: http://creativecommons.org/licenses/by-nc-nd/4.0/
• Abstract: Non-negative matrix and tensor factorisations are a classical tool in machine learning and data science for finding low-dimensional representations of high-dimensional datasets. In applications such as imaging, datasets can often be regarded as distributions in a space with metric structure. In such a setting, a Wasserstein loss function based on optimal transportation theory is a natural choice since it incorporates knowledge about the geometry of the underlying space. We introduce a general mathematical framework for computing non-negative factorisations of matrices and tensors with respect to an optimal transport loss, and derive an efficient method for its solution using a convex dual formulation. We demonstrate the applicability of this approach with several numerical examples.

Related papers

• Subspace-Constrained Quadratic Matrix Factorization: Algorithm and Applications [1.689629482101155]
  We present a subspace-constrained quadratic matrix factorization model to address challenges in manifold learning. The model is designed to jointly learn key low-dimensional structures, including the tangent space, the normal subspace, and the quadratic form. Results demonstrate that our model outperforms existing methods, highlighting its robustness and efficacy in capturing core low-dimensional structures.
  arXiv  Detail & Related papers  (2024-11-07T13:57:53Z)
• Symmetry Discovery for Different Data Types [52.2614860099811]
  Equivariant neural networks incorporate symmetries into their architecture, achieving higher generalization performance. We propose LieSD, a method for discovering symmetries via trained neural networks which approximate the input-output mappings of the tasks. We validate the performance of LieSD on tasks with symmetries such as the two-body problem, the moment of inertia matrix prediction, and top quark tagging.
  arXiv  Detail & Related papers  (2024-10-13T13:39:39Z)
• Projected Tensor-Tensor Products for Efficient Computation of Optimal Multiway Data Representations [0.0]
  We propose a new projected tensor-tensor product that relaxes the invertibility restriction to reduce computational overhead. We provide theory to prove the matrix mimeticity and the optimality of compressed representations within the projected product framework.
  arXiv  Detail & Related papers  (2024-09-28T16:29:54Z)
• Manifold Learning with Sparse Regularised Optimal Transport [0.17205106391379024]
  Real-world datasets are subject to noisy observations and sampling, so that distilling information about the underlying manifold is a major challenge. We propose a method for manifold learning that utilises a symmetric version of optimal transport with a quadratic regularisation. We prove that the resulting kernel is consistent with a Laplace-type operator in the continuous limit, establish robustness to heteroskedastic noise and exhibit these results in simulations.
  arXiv  Detail & Related papers  (2023-07-19T08:05:46Z)
• Towards a mathematical understanding of learning from few examples with nonlinear feature maps [68.8204255655161]
  We consider the problem of data classification where the training set consists of just a few data points. We reveal key relationships between the geometry of an AI model's feature space, the structure of the underlying data distributions, and the model's generalisation capabilities.
  arXiv  Detail & Related papers  (2022-11-07T14:52:58Z)
• Learning Graphical Factor Models with Riemannian Optimization [70.13748170371889]
  This paper proposes a flexible algorithmic framework for graph learning under low-rank structural constraints. The problem is expressed as penalized maximum likelihood estimation of an elliptical distribution. We leverage geometries of positive definite matrices and positive semi-definite matrices of fixed rank that are well suited to elliptical models.
  arXiv  Detail & Related papers  (2022-10-21T13:19:45Z)
• Efficient Multidimensional Functional Data Analysis Using Marginal Product Basis Systems [2.4554686192257424]
  We propose a framework for learning continuous representations from a sample of multidimensional functional data. We show that the resulting estimation problem can be solved efficiently by the tensor decomposition. We conclude with a real data application in neuroimaging.
  arXiv  Detail & Related papers  (2021-07-30T16:02:15Z)
• Asymptotic Analysis of an Ensemble of Randomly Projected Linear Discriminants [94.46276668068327]
  In [1], an ensemble of randomly projected linear discriminants is used to classify datasets. We develop a consistent estimator of the misclassification probability as an alternative to the computationally-costly cross-validation estimator. We also demonstrate the use of our estimator for tuning the projection dimension on both real and synthetic data.
  arXiv  Detail & Related papers  (2020-04-17T12:47:04Z)
• Eigendecomposition-Free Training of Deep Networks for Linear Least-Square Problems [107.3868459697569]
  We introduce an eigendecomposition-free approach to training a deep network. We show that our approach is much more robust than explicit differentiation of the eigendecomposition. Our method has better convergence properties and yields state-of-the-art results.
  arXiv  Detail & Related papers  (2020-04-15T04:29:34Z)
• Semiparametric Nonlinear Bipartite Graph Representation Learning with Provable Guarantees [106.91654068632882]
  We consider the bipartite graph and formalize its representation learning problem as a statistical estimation problem of parameters in a semiparametric exponential family distribution. We show that the proposed objective is strongly convex in a neighborhood around the ground truth, so that a gradient descent-based method achieves linear convergence rate. Our estimator is robust to any model misspecification within the exponential family, which is validated in extensive experiments.
  arXiv  Detail & Related papers  (2020-03-02T16:40:36Z)

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.