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<p>Optimal transport (OT) has recently become very popular in machine learning, with application to several sub-fields such as clustering, dictionary learning and domain adaptation. However, OT is known to face challenges when dealing with high-dimensional data, such as images, texts or omics data. Most current OT approaches for high-dimensional situations rely on projections of the data or measures onto low-dimensional spaces, which inevitably results in information loss. In this work, we consider the case of high-dimensional Gaussian distributions with parsimonious covariance structures and lower intrinsic dimension. We exhibit a simplified closed-form expression of the 2-Wasserstein distance with an efficient and robust calculation procedure based on a low-dimensional decomposition of empirical covariance matrices, without relying on data projections. Furthermore, we provide a closed-form expression for the Monge map, which involves the exact calculation of the square-root and inverse square-root of the source distribution covariance matrix. This approach offers analytical and computational advantages, as demonstrated by our numerical experiments, which quantitatively evaluate these benefits in comparison to existing methods. In addition to being able to compute both the <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(W^2_2\)</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mi>W</mi> <mn>2</mn> <mn>2</mn> </msubsup> </math></EquationSource> </InlineEquation>-distance and the transport map, our method can compete with model-free methods, in high dimension, even in the case of non-Gaussian distributions. Moreover, it reveals to be of particular interest in the context of unsupervised domain adaptation for supervised classification.</p>

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Scaling optimal transport to high-dimensional Gaussian distributions with application to domain adaptation

  • 期刊论文

  • 发表日期：

    2026年2月19日

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  • Charles Bouveyron,
  • Marco Corneli

摘要

Optimal transport (OT) has recently become very popular in machine learning, with application to several sub-fields such as clustering, dictionary learning and domain adaptation. However, OT is known to face challenges when dealing with high-dimensional data, such as images, texts or omics data. Most current OT approaches for high-dimensional situations rely on projections of the data or measures onto low-dimensional spaces, which inevitably results in information loss. In this work, we consider the case of high-dimensional Gaussian distributions with parsimonious covariance structures and lower intrinsic dimension. We exhibit a simplified closed-form expression of the 2-Wasserstein distance with an efficient and robust calculation procedure based on a low-dimensional decomposition of empirical covariance matrices, without relying on data projections. Furthermore, we provide a closed-form expression for the Monge map, which involves the exact calculation of the square-root and inverse square-root of the source distribution covariance matrix. This approach offers analytical and computational advantages, as demonstrated by our numerical experiments, which quantitatively evaluate these benefits in comparison to existing methods. In addition to being able to compute both the \(W^2_2\) W 2 2 -distance and the transport map, our method can compete with model-free methods, in high dimension, even in the case of non-Gaussian distributions. Moreover, it reveals to be of particular interest in the context of unsupervised domain adaptation for supervised classification.