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A Unified Framework for Domain Adaptation Using Metric Learning on Manifolds

  • Sridhar MahadevanEmail author
  • Bamdev Mishra
  • Shalini Ghosh
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11052)

Abstract

We present a novel framework for domain adaptation, whereby both geometric and statistical differences between a labeled source domain and unlabeled target domain can be reconciled using a unified mathematical framework that exploits the curved Riemannian geometry of statistical manifolds. We exploit a simple but important observation that as the space of covariance matrices is both a Riemannian space as well as a homogeneous space, the shortest path geodesic between two covariances on the manifold can be computed analytically. Statistics on the SPD matrix manifold, such as the geometric mean of two SPD matries can be reduced to solving the well-known Riccati equation. We show how the Ricatti-based solution can be constrained to not only reduce the statistical differences between the source and target domains, such as aligning second order covariances and minimizing the maximum mean discrepancy, but also the underlying geometry of the source and target domains using diffusions on the underlying source and target manifolds. Our solution also emerges as a consequence of optimal transport theory, which shows that the optimal transport mapping between source and target distributions that are multivariate Gaussians is a function of the geometric mean of the source and target covariances, a quantity that also minimizes the Wasserstein distance. A key strength of our proposed approach is that it enables integrating multiple sources of variation between source and target in a unified way, by reducing the combined objective function to a nested set of Ricatti equations where the solution can be represented by a cascaded series of geometric mean computations. In addition to showing the theoretical optimality of our solution, we present detailed experiments using standard transfer learning testbeds from computer vision comparing our proposed algorithms to past work in domain adaptation, showing improved results over a large variety of previous methods. Code related to this paper is available at: https://github.com/sridharmahadevan/Geodesic-Covariance-Alignment.

Notes

Acknowledgments

Portions of this research were completed when the first and third authors were at SRI International, Menlo Park, CA and when the second author was at Amazon.com, Bangalore, India.

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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Sridhar Mahadevan
    • 1
    Email author
  • Bamdev Mishra
    • 2
  • Shalini Ghosh
    • 3
  1. 1.University of MassachusettsAmherstUSA
  2. 2.MicrosoftHyderabadIndia
  3. 3.Samsung Research AmericaMountain ViewUSA

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