Lifting PCA in the Wasserstein Space

Key points

• Principal Component Analysis (PCA) is a fundamental tool for statistical dimension reduction, yet no canonical extension to distribution-valued data has emerged within the framework of optimal transport.1
• While Wasserstein barycenters and Wasserstein least-squares regression provide canonical liftings of Euclidean averaging and linear regression in the Wasserstein space, an analogous principle for PCA has remained unexplored.1
• Starting from the classical variational formulations of PCA on the Stiefel and Grassmann manifolds, we derive two equivalent minimization problems over probability measures supported on these manifolds that correspond to Wasserstein barycenters with generalized costs.1
• Whereas Euclidean PCA summarizes vectors through projections onto an optimal low-dimensional linear subspace, Lifted PCA summarizes probability distributions through an optimal probability measure supported on multiple low-dimensional subspaces.1

Sources (1)

• [1]Lifting PCA in the Wasserstein Space
  arXiv (AI, ML, NLP, CV, robotics, multi-agent) · Oct 21, 12:57 PM

  In this paper, we introduce Lifted PCA, a canonical extension of Euclidean PCA to Wasserstein space, obtained through the framework of lifting functionals in the space of probability measures by convex extension.

  Principal Component Analysis (PCA) is a fundamental tool for statistical dimension reduction, yet no canonical extension to distribution-valued data has emerged within the framework of optimal transport.

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