Computing the Distance between unbalanced Distributions -- The flat Metric

• URL: http://arxiv.org/abs/2308.01039v2
• Date: Mon, 16 Jun 2025 11:43:39 GMT
• Title: Computing the Distance between unbalanced Distributions -- The flat Metric
• Authors: Henri Schmidt, Christian Düll,
• Abstract summary: We provide an implementation to compute the flat metric in any dimension.<n>Our implementation adapts very well to mass differences and uses them to distinguish between different distributions.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We provide an implementation to compute the flat metric in any dimension. The flat metric, also called dual bounded Lipschitz distance, generalizes the well-known Wasserstein distance $W_1$ to the case that the distributions are of unequal total mass. Thus, our implementation adapts very well to mass differences and uses them to distinguish between different distributions. This is of particular interest for unbalanced optimal transport tasks and for the analysis of data distributions where the sample size is important or normalization is not possible. The core of the method is based on a neural network to determine an optimal test function realizing the distance between two given measures. Special focus was put on achieving comparability of pairwise computed distances from independently trained networks. We tested the quality of the output in several experiments where ground truth was available as well as with simulated data.

Related papers

• Kernel Trace Distance: Quantum Statistical Metric between Measures through RKHS Density Operators [11.899035547580201]
  We introduce a novel distance between measures that compares them through a Schatten norm of their kernel covariance operators.<n>We show that this new distance is an integral probability metric that can be framed between a Maximum Mean Discrepancy (MMD) and a Wasserstein distance.
  arXiv  Detail & Related papers  (2025-07-08T14:56:44Z)
• Optimal Transport with Heterogeneously Missing Data [12.896319628045967]
  We consider the problem of solving the optimal transport problem between two empirical distributions with missing values.<n>We show that the Wasserstein distance between empirical distributions and linear Monge maps can be debiased without significantly affecting the sample complexity.
  arXiv  Detail & Related papers  (2025-05-22T21:16:22Z)
• Bias Detection via Maximum Subgroup Discrepancy [2.236957801565796]
  We propose a new notion of distance, the Maximum Subgroup Discrepancy (MSD)<n>In this metric, two distributions are close if, roughly, discrepancies are low for all feature subgroups.<n>We show that the sample complexity is linear in the number of features, thus making it feasible for practical applications.
  arXiv  Detail & Related papers  (2025-02-04T11:01:03Z)
• Distributed Markov Chain Monte Carlo Sampling based on the Alternating Direction Method of Multipliers [143.6249073384419]
  In this paper, we propose a distributed sampling scheme based on the alternating direction method of multipliers. We provide both theoretical guarantees of our algorithm's convergence and experimental evidence of its superiority to the state-of-the-art. In simulation, we deploy our algorithm on linear and logistic regression tasks and illustrate its fast convergence compared to existing gradient-based methods.
  arXiv  Detail & Related papers  (2024-01-29T02:08:40Z)
• Energy-Based Sliced Wasserstein Distance [47.18652387199418]
  A key component of the sliced Wasserstein (SW) distance is the slicing distribution. We propose to design the slicing distribution as an energy-based distribution that is parameter-free. We then derive a novel sliced Wasserstein metric, energy-based sliced Waserstein (EBSW) distance.
  arXiv  Detail & Related papers  (2023-04-26T14:28:45Z)
• Unsupervised Learning of Sampling Distributions for Particle Filters [80.6716888175925]
  We put forward four methods for learning sampling distributions from observed measurements. Experiments demonstrate that learned sampling distributions exhibit better performance than designed, minimum-degeneracy sampling distributions.
  arXiv  Detail & Related papers  (2023-02-02T15:50:21Z)
• Statistical Efficiency of Score Matching: The View from Isoperimetry [96.65637602827942]
  We show a tight connection between statistical efficiency of score matching and the isoperimetric properties of the distribution being estimated. We formalize these results both in the sample regime and in the finite regime.
  arXiv  Detail & Related papers  (2022-10-03T06:09:01Z)
• Nonlinear Sufficient Dimension Reduction for Distribution-on-Distribution Regression [9.086237593805173]
  We introduce a new approach to nonlinear sufficient dimension reduction in cases where both the predictor and the response are distributional data. Our key step is to build universal kernels (cc-universal) on the metric spaces.
  arXiv  Detail & Related papers  (2022-07-11T04:11:36Z)
• Near-optimal estimation of smooth transport maps with kernel sums-of-squares [81.02564078640275]
  Under smoothness conditions, the squared Wasserstein distance between two distributions could be efficiently computed with appealing statistical error upper bounds. The object of interest for applications such as generative modeling is the underlying optimal transport map. We propose the first tractable algorithm for which the statistical $L2$ error on the maps nearly matches the existing minimax lower-bounds for smooth map estimation.
  arXiv  Detail & Related papers  (2021-12-03T13:45:36Z)
• Density Ratio Estimation via Infinitesimal Classification [85.08255198145304]
  We propose DRE-infty, a divide-and-conquer approach to reduce Density ratio estimation (DRE) to a series of easier subproblems. Inspired by Monte Carlo methods, we smoothly interpolate between the two distributions via an infinite continuum of intermediate bridge distributions. We show that our approach performs well on downstream tasks such as mutual information estimation and energy-based modeling on complex, high-dimensional datasets.
  arXiv  Detail & Related papers  (2021-11-22T06:26:29Z)
• Meta Learning Low Rank Covariance Factors for Energy-Based Deterministic Uncertainty [58.144520501201995]
  Bi-Lipschitz regularization of neural network layers preserve relative distances between data instances in the feature spaces of each layer. With the use of an attentive set encoder, we propose to meta learn either diagonal or diagonal plus low-rank factors to efficiently construct task specific covariance matrices. We also propose an inference procedure which utilizes scaled energy to achieve a final predictive distribution.
  arXiv  Detail & Related papers  (2021-10-12T22:04:19Z)
• Tangent Space and Dimension Estimation with the Wasserstein Distance [10.118241139691952]
  Consider a set of points sampled independently near a smooth compact submanifold of Euclidean space. We provide mathematically rigorous bounds on the number of sample points required to estimate both the dimension and the tangent spaces of that manifold.
  arXiv  Detail & Related papers  (2021-10-12T21:02:06Z)
• Kernel distance measures for time series, random fields and other structured data [71.61147615789537]
  kdiff is a novel kernel-based measure for estimating distances between instances of structured data. It accounts for both self and cross similarities across the instances and is defined using a lower quantile of the distance distribution. Some theoretical results are provided for separability conditions using kdiff as a distance measure for clustering and classification problems.
  arXiv  Detail & Related papers  (2021-09-29T22:54:17Z)
• The Exploitation of Distance Distributions for Clustering [3.42658286826597]
  In cluster analysis, different properties for distance distributions are judged to be relevant for appropriate distance selection. By systematically investigating this specification using distribution analysis through a mirrored-density plot, it is shown that multimodal distance distributions are preferable in cluster analysis. Experiments are performed on several artificial datasets and natural datasets for the task of clustering.
  arXiv  Detail & Related papers  (2021-08-22T06:22:08Z)
• Depth-based pseudo-metrics between probability distributions [1.1470070927586016]
  We propose two new pseudo-metrics between continuous probability measures based on data depth and its associated central regions. In contrast to the Wasserstein distance, the proposed pseudo-metrics do not suffer from the curse of dimensionality. The regions-based pseudo-metric appears to be robust w.r.t. both outliers and heavy tails.
  arXiv  Detail & Related papers  (2021-03-23T17:33:18Z)
• Two-sample Test using Projected Wasserstein Distance [18.46110328123008]
  We develop a projected Wasserstein distance for the two-sample test, a fundamental problem in statistics and machine learning. A key contribution is to couple optimal projection to find the low dimensional linear mapping to maximize the Wasserstein distance between projected probability distributions.
  arXiv  Detail & Related papers  (2020-10-22T18:08:58Z)
• Learning to Match Distributions for Domain Adaptation [116.14838935146004]
  This paper proposes Learning to Match (L2M) to automatically learn the cross-domain distribution matching. L2M reduces the inductive bias by using a meta-network to learn the distribution matching loss in a data-driven way. Experiments on public datasets substantiate the superiority of L2M over SOTA methods.
  arXiv  Detail & Related papers  (2020-07-17T03:26:13Z)
• On Projection Robust Optimal Transport: Sample Complexity and Model Misspecification [101.0377583883137]
  Projection robust (PR) OT seeks to maximize the OT cost between two measures by choosing a $k$-dimensional subspace onto which they can be projected. Our first contribution is to establish several fundamental statistical properties of PR Wasserstein distances. Next, we propose the integral PR Wasserstein (IPRW) distance as an alternative to the PRW distance, by averaging rather than optimizing on subspaces.
  arXiv  Detail & Related papers  (2020-06-22T14:35:33Z)

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.