Comparing distributions and shapes using the kernel distance

  • Authors:

  • Sarang Joshi;Raj Varma Kommaraji;Jeff M. Phillips;Suresh Venkatasubramanian

  • Affiliations:

  • University of Utah, Salt Lake city, UT, USA;Microsoft, Salt Lake CIty, UT, USA;University of Utah, Salt Lake City, UT, USA;University of Utah, Salt Lake City, UT, USA

  • Venue:
  • Proceedings of the twenty-seventh annual symposium on Computational geometry
  • Year:
  • 2011

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Abstract

Starting with a similarity function between objects, it is possible to define a distance metric (the kernel distance) on pairs of objects, and more generally on probability distributions over them. These distance metrics have a deep basis in functional analysis and geometric measure theory, and have a rich structure that includes an isometric embedding into a Hilbert space. They have recently been applied to numerous problems in machine learning and shape analysis. SIn this paper, we provide the first algorithmic analysis of these distance metrics. Our main contributions are as follows: We present fast approximation algorithms for computing the kernel distance between two point sets P and Q that runs in near-linear time in the size of P ∪ Q (an explicit calculation would take quadratic time). We present polynomial-time algorithms for approximately minimizing the kernel distance under rigid transformation; they run in time O(n + poly(1/ε, log n)). We provide several general techniques for reducing complex objects to convenient sparse representations (specifically to point sets or sets of points sets) which approximately preserve the kernel distance. In particular, this allows us to reduce problems of computing the kernel distance between various types of objects such as curves, surfaces, and distributions to computing the kernel distance between point sets.