Linear Classifiers in Mixed Constant Curvature Spaces

02/19/2021 ∙ by Puoya Tabaghi, et al. ∙ 8

Embedding methods for mixed-curvature spaces are powerful techniques for low-distortion and low-dimensional representation of complex data structures. Nevertheless, little is known regarding downstream learning and optimization in the embedding space. Here, we address for the first time the problem of linear classification in a product space form – a mix of Euclidean, spherical, and hyperbolic spaces with different dimensions. First, we revisit the definition of a linear classifier on a Riemannian manifold by using geodesics and Riemannian metrics which generalize the notions of straight lines and inner products in vector spaces, respectively. Second, we prove that linear classifiers in d-dimensional constant curvature spaces can shatter exactly d+1 points: Hence, Euclidean, hyperbolic and spherical classifiers have the same expressive power. Third, we formalize linear classifiers in product space forms, describe a novel perceptron classification algorithm, and establish rigorous convergence results. We support our theoretical findings with simulation results on several datasets, including synthetic data, MNIST and Omniglot. Our results reveal that learning methods applied to small-dimensional embeddings in product space forms significantly outperform their algorithmic counterparts in Euclidean spaces.

READ FULL TEXT
POST COMMENT

Comments

There are no comments yet.

Authors

page 3

page 4

page 6

page 7

page 9

page 15

page 18

page 19

This week in AI

Get the week's most popular data science and artificial intelligence research sent straight to your inbox every Saturday.