A Geometric Take on Metric Learning


Conference Paper


Multi-metric learning techniques learn local metric tensors in different parts of a feature space. With such an approach, even simple classifiers can be competitive with the state-of-the-art because the distance measure locally adapts to the structure of the data. The learned distance measure is, however, non-metric, which has prevented multi-metric learning from generalizing to tasks such as dimensionality reduction and regression in a principled way. We prove that, with appropriate changes, multi-metric learning corresponds to learning the structure of a Riemannian manifold. We then show that this structure gives us a principled way to perform dimensionality reduction and regression according to the learned metrics. Algorithmically, we provide the first practical algorithm for computing geodesics according to the learned metrics, as well as algorithms for computing exponential and logarithmic maps on the Riemannian manifold. Together, these tools let many Euclidean algorithms take advantage of multi-metric learning. We illustrate the approach on regression and dimensionality reduction tasks that involve predicting measurements of the human body from shape data.

Author(s): Soren Hauberg and Oren Freifeld and Michael J. Black
Book Title: Advances in Neural Information Processing Systems (NIPS) 25
Pages: 2033--2041
Year: 2012
Editors: P. Bartlett and F.C.N. Pereira and C.J.C. Burges and L. Bottou and K.Q. Weinberger
Publisher: MIT Press

Department(s): Perceiving Systems
Research Project(s): Smooth Metric Learning
Bibtex Type: Conference Paper (inproceedings)
Paper Type: Conference

Links: PDF
Suppl. material


  title = {A Geometric Take on Metric Learning},
  author = {Hauberg, S{o}ren and Freifeld, Oren and Black, Michael J.},
  booktitle = {Advances in Neural Information Processing Systems (NIPS) 25},
  pages = {2033--2041},
  editors = {P. Bartlett and F.C.N. Pereira and C.J.C. Burges and L. Bottou and K.Q. Weinberger},
  publisher = {MIT Press},
  year = {2012}