Riemannian Geometry and Statistical Machine Learning

  • Lebanon G
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Abstract

Statistical machine learning algorithms deal with the problem of selecting an appropriate statistical model from a model space Θ based on a training set {xi} N i=1 ⊂ X or {(xi, yi)} N i=1 ⊂ X × Y. In doing so they either implicitly or explicitly make assumptions on the geometries of the model space Θ and the data space X. Such assumptions are crucial to the success of the algorithms as different geometries are appropriate for different models and data spaces. By studying these assumptions we are able to develop new theoretical results that enhance our understanding of several popular learning algorithms. Furthermore, using geometrical reasoning we are able to adapt existing algorithms such as radial basis kernels and linear margin classifiers to non-Euclidean geometries. Such adaptation is shown to be useful when the data space does not exhibit Euclidean geometry. In particular, we focus in our experiments on the space of text documents that is naturally associated with the Fisher information metric on corresponding multinomial models.

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Authors

  • Guy Lebanon

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