Optimal Transport with Heterogeneously Missing Data

• URL: http://arxiv.org/abs/2505.17291v1
• Date: Thu, 22 May 2025 21:16:22 GMT
• Title: Optimal Transport with Heterogeneously Missing Data
• Authors: Linus Bleistein, Aurélien Bellet, Julie Josse,
• Abstract summary: 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.
• Score: 12.896319628045967
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We consider the problem of solving the optimal transport problem between two empirical distributions with missing values. Our main assumption is that the data is missing completely at random (MCAR), but we allow for heterogeneous missingness probabilities across features and across the two distributions. As a first contribution, we show that the Wasserstein distance between empirical Gaussian distributions and linear Monge maps between arbitrary distributions can be debiased without significantly affecting the sample complexity. Secondly, we show that entropic regularized optimal transport can be estimated efficiently and consistently using iterative singular value thresholding (ISVT). We propose a validation set-free hyperparameter selection strategy for ISVT that leverages our estimator of the Bures-Wasserstein distance, which could be of independent interest in general matrix completion problems. Finally, we validate our findings on a wide range of numerical applications.

Related papers

• Sublinear Algorithms for Wasserstein and Total Variation Distances: Applications to Fairness and Privacy Auditing [7.81603404636933]
  We propose a generic framework to learn the probability and cumulative distribution functions (PDFs and CDFs) of a sub-Weibull.<n>We compute mergeable summaries of distributions from the stream of samples while requiring only sublinear space.<n>Our algorithms significantly improves on the existing methods for distance estimation incurring super-linear time and linear space complexities.
  arXiv  Detail & Related papers  (2025-03-10T18:57:48Z)
• Towards Self-Supervised Covariance Estimation in Deep Heteroscedastic Regression [102.24287051757469]
  We study self-supervised covariance estimation in deep heteroscedastic regression.<n>We derive an upper bound on the 2-Wasserstein distance between normal distributions.<n>Experiments over a wide range of synthetic and real datasets demonstrate that the proposed 2-Wasserstein bound coupled with pseudo label annotations results in a computationally cheaper yet accurate deep heteroscedastic regression.
  arXiv  Detail & Related papers  (2025-02-14T22:37:11Z)
• On the Wasserstein Convergence and Straightness of Rectified Flow [54.580605276017096]
  Rectified Flow (RF) is a generative model that aims to learn straight flow trajectories from noise to data.<n>We provide a theoretical analysis of the Wasserstein distance between the sampling distribution of RF and the target distribution.<n>We present general conditions guaranteeing uniqueness and straightness of 1-RF, which is in line with previous empirical findings.
  arXiv  Detail & Related papers  (2024-10-19T02:36:11Z)
• Distributional Matrix Completion via Nearest Neighbors in the Wasserstein Space [8.971989179518216]
  Given a sparsely observed matrix of empirical distributions, we seek to impute the true distributions associated with both observed and unobserved matrix entries. We utilize tools from optimal transport to generalize the nearest neighbors method to the distributional setting.
  arXiv  Detail & Related papers  (2024-10-17T00:50:17Z)
• Computing the Distance between unbalanced Distributions -- The flat Metric [0.0]
  The flat metric generalizes the well-known Wasserstein distance W1 to the case that the distributions are of unequal total mass. The core of the method is based on a neural network to determine on optimal test function realizing the distance between two measures.
  arXiv  Detail & Related papers  (2023-08-02T09:30:22Z)
• Approximating a RUM from Distributions on k-Slates [88.32814292632675]
  We find a generalization-time algorithm that finds the RUM that best approximates the given distribution on average. Our theoretical result can also be made practical: we obtain a that is effective and scales to real-world datasets.
  arXiv  Detail & Related papers  (2023-05-22T17:43:34Z)
• Minimax estimation of discontinuous optimal transport maps: The semi-discrete case [14.333765302506658]
  We consider the problem of estimating the optimal transport map between two probability distributions, $P$ and $Q$ in $mathbb Rd$. We show that an estimator based on entropic optimal transport converges at the minimax-optimal rate $n-1/2$, independent of dimension.
  arXiv  Detail & Related papers  (2023-01-26T18:41:38Z)
• Optimal Efficiency-Envy Trade-Off via Optimal Transport [33.85971515753188]
  We consider the problem of allocating a distribution of items to $n$ recipients where each recipient has to be allocated a fixed, prespecified fraction of all items. We show that this problem can be formulated as a variant of the semi-discrete optimal transport (OT) problem, whose solution structure in this case has a concise representation and a simple geometric interpretation.
  arXiv  Detail & Related papers  (2022-09-25T00:39:43Z)
• A Robust and Flexible EM Algorithm for Mixtures of Elliptical Distributions with Missing Data [71.9573352891936]
  This paper tackles the problem of missing data imputation for noisy and non-Gaussian data. A new EM algorithm is investigated for mixtures of elliptical distributions with the property of handling potential missing data. Experimental results on synthetic data demonstrate that the proposed algorithm is robust to outliers and can be used with non-Gaussian data.
  arXiv  Detail & Related papers  (2022-01-28T10:01:37Z)
• 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)
• Semi-Discrete Optimal Transport: Hardness, Regularization and Numerical Solution [8.465228064780748]
  We prove that computing the Wasserstein distance between a discrete probability measure supported on two points is already #P-hard. We introduce a distributionally robust dual optimal transport problem whose objective function is smoothed with the most adverse disturbance distributions. We show that smoothing the dual objective function is equivalent to regularizing the primal objective function.
  arXiv  Detail & Related papers  (2021-03-10T18:53:59Z)
• Variational Transport: A Convergent Particle-BasedAlgorithm for Distributional Optimization [106.70006655990176]
  A distributional optimization problem arises widely in machine learning and statistics. We propose a novel particle-based algorithm, dubbed as variational transport, which approximately performs Wasserstein gradient descent. We prove that when the objective function satisfies a functional version of the Polyak-Lojasiewicz (PL) (Polyak, 1963) and smoothness conditions, variational transport converges linearly.
  arXiv  Detail & Related papers  (2020-12-21T18:33:13Z)

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.