Learning Log-Determinant Divergences for Positive Definite Matrices

Anoop Cherian, Panagiotis Stanitsas, Jue Wang, Mehrtash T. Harandi, Vassilios Morellas, Nikos Papanikolopoulos

Research output: Contribution to journalArticlepeer-review

Abstract

Representations in the form of Symmetric Positive Definite (SPD) matrices have been popularized in a variety of visual learning applications due to their demonstrated ability to capture rich second-order statistics of visual data. There exist several similarity measures for comparing SPD matrices with documented benefits. However, selecting an appropriate measure for a given problem remains a challenge and in most cases, is the result of a trial-and-error process. In this paper, we propose to learn similarity measures in a data-driven manner. To this end, we capitalize on the alpha-beta-log-det divergence, which is a meta-divergence parametrized by scalars alpha and beta, subsuming a wide family of popular information divergences on SPD matrices for distinct and discrete values of these parameters. Our key idea is to cast these parameters in a continuum and learn them from data. We systematically extend this idea to learn vector-valued parameters, thereby increasing the expressiveness of the underlying non-linear measure. We conjoin the divergence learning problem with several standard tasks in machine learning, including supervised discriminative dictionary learning and unsupervised SPD matrix clustering. We present Riemannian descent schemes for optimizing our formulations efficiently and show the usefulness of our method on eight standard computer vision tasks.

Original languageEnglish (US)
JournalIEEE Transactions on Pattern Analysis and Machine Intelligence
VolumePP
Early online dateApr 15 2021
DOIs
StateE-pub ahead of print - Apr 15 2021

Bibliographical note

Publisher Copyright:
IEEE

Keywords

  • Covariance matrices
  • Geometry
  • Kernel
  • Measurement
  • Sparse matrices
  • Standards
  • Symmetric matrices
  • action recognition
  • log-det divergence
  • positive definite matrices
  • region covariance matrices
  • texture recognition

PubMed: MeSH publication types

  • Journal Article

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