Learning Distances from Data with Normalizing Flows and Score Matching

• URL: http://arxiv.org/abs/2407.09297v1
• Date: Fri, 12 Jul 2024 14:30:41 GMT
• Title: Learning Distances from Data with Normalizing Flows and Score Matching
• Authors: Peter Sorrenson, Daniel Behrend-Uriarte, Christoph Schnörr, Ullrich Köthe,
• Abstract summary: Density-based distances offer an elegant solution to the problem of metric learning. We show that existing methods to estimate Fermat distances suffer from poor convergence in both low and high dimensions. Our work paves the way for practical use of density-based distances, especially in high-dimensional spaces.
• Score: 9.605001452209867
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Density-based distances (DBDs) offer an elegant solution to the problem of metric learning. By defining a Riemannian metric which increases with decreasing probability density, shortest paths naturally follow the data manifold and points are clustered according to the modes of the data. We show that existing methods to estimate Fermat distances, a particular choice of DBD, suffer from poor convergence in both low and high dimensions due to i) inaccurate density estimates and ii) reliance on graph-based paths which are increasingly rough in high dimensions. To address these issues, we propose learning the densities using a normalizing flow, a generative model with tractable density estimation, and employing a smooth relaxation method using a score model initialized from a graph-based proposal. Additionally, we introduce a dimension-adapted Fermat distance that exhibits more intuitive behavior when scaled to high dimensions and offers better numerical properties. Our work paves the way for practical use of density-based distances, especially in high-dimensional spaces.

Related papers

• Dimension reduction and the gradient flow of relative entropy [0.0]
  Dimension reduction, widely used in science, maps high-dimensional data into low-dimensional space. We investigate a basic mathematical model underlying the techniques of neighborhood embedding (SNE) and its popular variant t-SNE. The aim is to map these points to low dimensions in an optimal way so that similar points are closer together.
  arXiv  Detail & Related papers  (2024-09-25T14:23:04Z)
• Graph Laplacian-based Bayesian Multi-fidelity Modeling [1.383698759122035]
  A graph Laplacian constructed from the low-fidelity data is used to define a multivariate Gaussian prior density. Few high-fidelity data points are used to construct a conjugate likelihood term. The results demonstrate that by utilizing a small fraction of high-fidelity data, the multi-fidelity approach can significantly improve the accuracy of a large collection of low-fidelity data points.
  arXiv  Detail & Related papers  (2024-09-12T16:51:55Z)
• A Bayesian Approach Toward Robust Multidimensional Ellipsoid-Specific Fitting [0.0]
  This work presents a novel and effective method for fitting multidimensional ellipsoids to scattered data in the contamination of noise and outliers. We incorporate a uniform prior distribution to constrain the search for primitive parameters within an ellipsoidal domain. We apply it to a wide range of practical applications such as microscopy cell counting, 3D reconstruction, geometric shape approximation, and magnetometer calibration tasks.
  arXiv  Detail & Related papers  (2024-07-27T14:31:51Z)
• Density Estimation via Binless Multidimensional Integration [45.21975243399607]
  We introduce the Binless Multidimensional Thermodynamic Integration (BMTI) method for nonparametric, robust, and data-efficient density estimation. BMTI estimates the logarithm of the density by initially computing log-density differences between neighbouring data points. The method is tested on a variety of complex synthetic high-dimensional datasets, and is benchmarked on realistic datasets from the chemical physics literature.
  arXiv  Detail & Related papers  (2024-07-10T23:45:20Z)
• SPARE: Symmetrized Point-to-Plane Distance for Robust Non-Rigid Registration [76.40993825836222]
  We propose SPARE, a novel formulation that utilizes a symmetrized point-to-plane distance for robust non-rigid registration. The proposed method greatly improves the accuracy of non-rigid registration problems and maintains relatively high solution efficiency.
  arXiv  Detail & Related papers  (2024-05-30T15:55:04Z)
• 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)
• LIDL: Local Intrinsic Dimension Estimation Using Approximate Likelihood [10.35315334180936]
  We propose a novel approach to the problem: Local Intrinsic Dimension estimation using approximate Likelihood (LIDL) Our method relies on an arbitrary density estimation method as its subroutine and hence tries to sidestep the dimensionality challenge. We show that LIDL yields competitive results on the standard benchmarks for this problem and that it scales to thousands of dimensions.
  arXiv  Detail & Related papers  (2022-06-29T19:47:46Z)
• Meta-Learning for Relative Density-Ratio Estimation [59.75321498170363]
  Existing methods for (relative) density-ratio estimation (DRE) require many instances from both densities. We propose a meta-learning method for relative DRE, which estimates the relative density-ratio from a few instances by using knowledge in related datasets. We empirically demonstrate the effectiveness of the proposed method by using three problems: relative DRE, dataset comparison, and outlier detection.
  arXiv  Detail & Related papers  (2021-07-02T02:13:45Z)
• Learning Optical Flow from a Few Matches [67.83633948984954]
  We show that the dense correlation volume representation is redundant and accurate flow estimation can be achieved with only a fraction of elements in it. Experiments show that our method can reduce computational cost and memory use significantly, while maintaining high accuracy.
  arXiv  Detail & Related papers  (2021-04-05T21:44:00Z)
• Nonparametric Density Estimation from Markov Chains [68.8204255655161]
  We introduce a new nonparametric density estimator inspired by Markov Chains, and generalizing the well-known Kernel Density Estor. Our estimator presents several benefits with respect to the usual ones and can be used straightforwardly as a foundation in all density-based algorithms.
  arXiv  Detail & Related papers  (2020-09-08T18:33:42Z)
• Variable Skipping for Autoregressive Range Density Estimation [84.60428050170687]
  We show a technique, variable skipping, for accelerating range density estimation over deep autoregressive models. We show that variable skipping provides 10-100$times$ efficiency improvements when targeting challenging high-quantile error metrics.
  arXiv  Detail & Related papers  (2020-07-10T19:01:40Z)

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.