From manifold to manifold: geometry-aware dimensionality reduction for SPD matrices

Mehrtash T. Harandi, Mathieu Salzmann, Richard Hartley

Research output: Chapter in Book/Report/Conference proceedingConference PaperResearchpeer-review

79 Citations (Scopus)

Abstract

Representing images and videos with Symmetric Positive Definite (SPD) matrices and considering the Riemannian geometry of the resulting space has proven beneficial for many recognition tasks. Unfortunately, computation on the Riemannian manifold of SPD matrices -especially of high-dimensional ones- comes at a high cost that limits the applicability of existing techniques. In this paper we introduce an approach that lets us handle high-dimensional SPD matrices by constructing a lower-dimensional, more discriminative SPD manifold. To this end, we model the mapping from the high-dimensional SPD manifold to the low-dimensional one with an orthonormal projection. In particular, we search for a projection that yields a low-dimensional manifold with maximum discriminative power encoded via an affinity-weighted similarity measure based on metrics on the manifold. Learning can then be expressed as an optimization problem on a Grassmann manifold. Our evaluation on several classification tasks shows that our approach leads to a significant accuracy gain over state-of-the-art methods.

Original languageEnglish
Title of host publicationComputer Vision - ECCV 2014
Subtitle of host publication13th European Conference Zurich, Switzerland, September 6-12, 2014 Proceedings, Part II
EditorsDavid Fleet, Tomas Pajdla, Bernt Schiele, Tinne Tuytelaars
Place of PublicationCham Switzerland
PublisherSpringer
Pages17-32
Number of pages16
ISBN (Electronic)9783319105840
ISBN (Print)9783319106045
DOIs
Publication statusPublished - 2014
Externally publishedYes
EventEuropean Conference on Computer Vision 2014 - Zurich, Switzerland
Duration: 6 Sep 201412 Sep 2014
Conference number: 13th
http://eccv2014.org/

Publication series

NameLecture Notes in Computer Science
PublisherSpringer
Volume8690
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349

Conference

ConferenceEuropean Conference on Computer Vision 2014
Abbreviated titleECCV 2014
CountrySwitzerland
CityZurich
Period6/09/1412/09/14
Internet address

Keywords

  • dimensionality reduction
  • Grassmann manifold
  • Riemannian geometry
  • SPD manifold
  • visual recognition

Cite this

Harandi, M. T., Salzmann, M., & Hartley, R. (2014). From manifold to manifold: geometry-aware dimensionality reduction for SPD matrices. In D. Fleet, T. Pajdla, B. Schiele, & T. Tuytelaars (Eds.), Computer Vision - ECCV 2014: 13th European Conference Zurich, Switzerland, September 6-12, 2014 Proceedings, Part II (pp. 17-32). (Lecture Notes in Computer Science; Vol. 8690). Cham Switzerland: Springer. https://doi.org/10.1007/978-3-319-10605-2_2
Harandi, Mehrtash T. ; Salzmann, Mathieu ; Hartley, Richard. / From manifold to manifold : geometry-aware dimensionality reduction for SPD matrices. Computer Vision - ECCV 2014: 13th European Conference Zurich, Switzerland, September 6-12, 2014 Proceedings, Part II. editor / David Fleet ; Tomas Pajdla ; Bernt Schiele ; Tinne Tuytelaars. Cham Switzerland : Springer, 2014. pp. 17-32 (Lecture Notes in Computer Science).
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Harandi, MT, Salzmann, M & Hartley, R 2014, From manifold to manifold: geometry-aware dimensionality reduction for SPD matrices. in D Fleet, T Pajdla, B Schiele & T Tuytelaars (eds), Computer Vision - ECCV 2014: 13th European Conference Zurich, Switzerland, September 6-12, 2014 Proceedings, Part II. Lecture Notes in Computer Science, vol. 8690, Springer, Cham Switzerland, pp. 17-32, European Conference on Computer Vision 2014, Zurich, Switzerland, 6/09/14. https://doi.org/10.1007/978-3-319-10605-2_2

From manifold to manifold : geometry-aware dimensionality reduction for SPD matrices. / Harandi, Mehrtash T.; Salzmann, Mathieu; Hartley, Richard.

Computer Vision - ECCV 2014: 13th European Conference Zurich, Switzerland, September 6-12, 2014 Proceedings, Part II. ed. / David Fleet; Tomas Pajdla; Bernt Schiele; Tinne Tuytelaars. Cham Switzerland : Springer, 2014. p. 17-32 (Lecture Notes in Computer Science; Vol. 8690).

Research output: Chapter in Book/Report/Conference proceedingConference PaperResearchpeer-review

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Harandi MT, Salzmann M, Hartley R. From manifold to manifold: geometry-aware dimensionality reduction for SPD matrices. In Fleet D, Pajdla T, Schiele B, Tuytelaars T, editors, Computer Vision - ECCV 2014: 13th European Conference Zurich, Switzerland, September 6-12, 2014 Proceedings, Part II. Cham Switzerland: Springer. 2014. p. 17-32. (Lecture Notes in Computer Science). https://doi.org/10.1007/978-3-319-10605-2_2