Canonical correlation analysis on Riemannian manifolds and its applications

Hyun Woo Kim, Nagesh Adluru, Barbara B. Bendlin, Sterling C. Johnson, Baba C. Vemuri, Vikas Singh

Research output: Chapter in Book/Report/Conference proceedingConference contribution

13 Citations (Scopus)

Abstract

Canonical correlation analysis (CCA) is a widely used statistical technique to capture correlations between two sets of multi-variate random variables and has found a multitude of applications in computer vision, medical imaging and machine learning. The classical formulation assumes that the data live in a pair of vector spaces which makes its use in certain important scientific domains problematic. For instance, the set of symmetric positive definite matrices (SPD), rotations and probability distributions, all belong to certain curved Riemannian manifolds where vector-space operations are in general not applicable. Analyzing the space of such data via the classical versions of inference models is rather sub-optimal. But perhaps more importantly, since the algorithms do not respect the underlying geometry of the data space, it is hard to provide statistical guarantees (if any) on the results. Using the space of SPD matrices as a concrete example, this paper gives a principled generalization of the well known CCA to the Riemannian setting. Our CCA algorithm operates on the product Riemannian manifold representing SPD matrix-valued fields to identify meaningful statistical relationships on the product Riemannian manifold. As a proof of principle, we present results on an Alzheimer's disease (AD) study where the analysis task involves identifying correlations across diffusion tensor images (DTI) and Cauchy deformation tensor fields derived from T1-weighted magnetic resonance (MR) images.

Original languageEnglish
Title of host publicationComputer Vision, ECCV 2014 - 13th European Conference, Proceedings
PublisherSpringer Verlag
Pages251-267
Number of pages17
EditionPART 2
ISBN (Print)9783319106045
DOIs
Publication statusPublished - 2014 Jan 1
Externally publishedYes
Event13th European Conference on Computer Vision, ECCV 2014 - Zurich, Switzerland
Duration: 2014 Sep 62014 Sep 12

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
NumberPART 2
Volume8690 LNCS
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349

Conference

Conference13th European Conference on Computer Vision, ECCV 2014
CountrySwitzerland
CityZurich
Period14/9/614/9/12

Fingerprint

Canonical Correlation Analysis
Symmetric Positive Definite Matrix
Riemannian Manifold
Vector space
Tensor
Vector spaces
Valued Fields
Magnetic Resonance Image
Alzheimer's Disease
Medical Imaging
Tensors
Computer Vision
Cauchy
Machine Learning
Probability Distribution
Random variable
Medical imaging
Magnetic resonance
Random variables
Formulation

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Computer Science(all)

Cite this

Kim, H. W., Adluru, N., Bendlin, B. B., Johnson, S. C., Vemuri, B. C., & Singh, V. (2014). Canonical correlation analysis on Riemannian manifolds and its applications. In Computer Vision, ECCV 2014 - 13th European Conference, Proceedings (PART 2 ed., pp. 251-267). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 8690 LNCS, No. PART 2). Springer Verlag. https://doi.org/10.1007/978-3-319-10605-2_17

Canonical correlation analysis on Riemannian manifolds and its applications. / Kim, Hyun Woo; Adluru, Nagesh; Bendlin, Barbara B.; Johnson, Sterling C.; Vemuri, Baba C.; Singh, Vikas.

Computer Vision, ECCV 2014 - 13th European Conference, Proceedings. PART 2. ed. Springer Verlag, 2014. p. 251-267 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 8690 LNCS, No. PART 2).

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Kim, HW, Adluru, N, Bendlin, BB, Johnson, SC, Vemuri, BC & Singh, V 2014, Canonical correlation analysis on Riemannian manifolds and its applications. in Computer Vision, ECCV 2014 - 13th European Conference, Proceedings. PART 2 edn, Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), no. PART 2, vol. 8690 LNCS, Springer Verlag, pp. 251-267, 13th European Conference on Computer Vision, ECCV 2014, Zurich, Switzerland, 14/9/6. https://doi.org/10.1007/978-3-319-10605-2_17
Kim HW, Adluru N, Bendlin BB, Johnson SC, Vemuri BC, Singh V. Canonical correlation analysis on Riemannian manifolds and its applications. In Computer Vision, ECCV 2014 - 13th European Conference, Proceedings. PART 2 ed. Springer Verlag. 2014. p. 251-267. (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); PART 2). https://doi.org/10.1007/978-3-319-10605-2_17
Kim, Hyun Woo ; Adluru, Nagesh ; Bendlin, Barbara B. ; Johnson, Sterling C. ; Vemuri, Baba C. ; Singh, Vikas. / Canonical correlation analysis on Riemannian manifolds and its applications. Computer Vision, ECCV 2014 - 13th European Conference, Proceedings. PART 2. ed. Springer Verlag, 2014. pp. 251-267 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); PART 2).
@inproceedings{7a3fc886efd64f80ab87709b010b293f,
title = "Canonical correlation analysis on Riemannian manifolds and its applications",
abstract = "Canonical correlation analysis (CCA) is a widely used statistical technique to capture correlations between two sets of multi-variate random variables and has found a multitude of applications in computer vision, medical imaging and machine learning. The classical formulation assumes that the data live in a pair of vector spaces which makes its use in certain important scientific domains problematic. For instance, the set of symmetric positive definite matrices (SPD), rotations and probability distributions, all belong to certain curved Riemannian manifolds where vector-space operations are in general not applicable. Analyzing the space of such data via the classical versions of inference models is rather sub-optimal. But perhaps more importantly, since the algorithms do not respect the underlying geometry of the data space, it is hard to provide statistical guarantees (if any) on the results. Using the space of SPD matrices as a concrete example, this paper gives a principled generalization of the well known CCA to the Riemannian setting. Our CCA algorithm operates on the product Riemannian manifold representing SPD matrix-valued fields to identify meaningful statistical relationships on the product Riemannian manifold. As a proof of principle, we present results on an Alzheimer's disease (AD) study where the analysis task involves identifying correlations across diffusion tensor images (DTI) and Cauchy deformation tensor fields derived from T1-weighted magnetic resonance (MR) images.",
author = "Kim, {Hyun Woo} and Nagesh Adluru and Bendlin, {Barbara B.} and Johnson, {Sterling C.} and Vemuri, {Baba C.} and Vikas Singh",
year = "2014",
month = "1",
day = "1",
doi = "10.1007/978-3-319-10605-2_17",
language = "English",
isbn = "9783319106045",
series = "Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)",
publisher = "Springer Verlag",
number = "PART 2",
pages = "251--267",
booktitle = "Computer Vision, ECCV 2014 - 13th European Conference, Proceedings",
edition = "PART 2",

}

TY - GEN

T1 - Canonical correlation analysis on Riemannian manifolds and its applications

AU - Kim, Hyun Woo

AU - Adluru, Nagesh

AU - Bendlin, Barbara B.

AU - Johnson, Sterling C.

AU - Vemuri, Baba C.

AU - Singh, Vikas

PY - 2014/1/1

Y1 - 2014/1/1

N2 - Canonical correlation analysis (CCA) is a widely used statistical technique to capture correlations between two sets of multi-variate random variables and has found a multitude of applications in computer vision, medical imaging and machine learning. The classical formulation assumes that the data live in a pair of vector spaces which makes its use in certain important scientific domains problematic. For instance, the set of symmetric positive definite matrices (SPD), rotations and probability distributions, all belong to certain curved Riemannian manifolds where vector-space operations are in general not applicable. Analyzing the space of such data via the classical versions of inference models is rather sub-optimal. But perhaps more importantly, since the algorithms do not respect the underlying geometry of the data space, it is hard to provide statistical guarantees (if any) on the results. Using the space of SPD matrices as a concrete example, this paper gives a principled generalization of the well known CCA to the Riemannian setting. Our CCA algorithm operates on the product Riemannian manifold representing SPD matrix-valued fields to identify meaningful statistical relationships on the product Riemannian manifold. As a proof of principle, we present results on an Alzheimer's disease (AD) study where the analysis task involves identifying correlations across diffusion tensor images (DTI) and Cauchy deformation tensor fields derived from T1-weighted magnetic resonance (MR) images.

AB - Canonical correlation analysis (CCA) is a widely used statistical technique to capture correlations between two sets of multi-variate random variables and has found a multitude of applications in computer vision, medical imaging and machine learning. The classical formulation assumes that the data live in a pair of vector spaces which makes its use in certain important scientific domains problematic. For instance, the set of symmetric positive definite matrices (SPD), rotations and probability distributions, all belong to certain curved Riemannian manifolds where vector-space operations are in general not applicable. Analyzing the space of such data via the classical versions of inference models is rather sub-optimal. But perhaps more importantly, since the algorithms do not respect the underlying geometry of the data space, it is hard to provide statistical guarantees (if any) on the results. Using the space of SPD matrices as a concrete example, this paper gives a principled generalization of the well known CCA to the Riemannian setting. Our CCA algorithm operates on the product Riemannian manifold representing SPD matrix-valued fields to identify meaningful statistical relationships on the product Riemannian manifold. As a proof of principle, we present results on an Alzheimer's disease (AD) study where the analysis task involves identifying correlations across diffusion tensor images (DTI) and Cauchy deformation tensor fields derived from T1-weighted magnetic resonance (MR) images.

UR - http://www.scopus.com/inward/record.url?scp=84906501984&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84906501984&partnerID=8YFLogxK

U2 - 10.1007/978-3-319-10605-2_17

DO - 10.1007/978-3-319-10605-2_17

M3 - Conference contribution

AN - SCOPUS:84906501984

SN - 9783319106045

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 251

EP - 267

BT - Computer Vision, ECCV 2014 - 13th European Conference, Proceedings

PB - Springer Verlag

ER -