Stochastic development regression on non-linear manifolds

Line Kühnel; Stefan Horst Sommer

doi:10.1007/978-3-319-59050-9_5

Stochastic development regression on non-linear manifolds

Line Kühnel^*, Stefan Horst Sommer

^*Corresponding author for this work

Department of Computer Science

7 Citations (Scopus)

Abstract

We introduce a regression model for data on non-linear manifolds. The model describes the relation between a set of manifold valued observations, such as shapes of anatomical objects, and Euclidean explanatory variables. The approach is based on stochastic development of Euclidean diffusion processes to the manifold. Defining the data distribution as the transition distribution of the mapped stochastic process, parameters of the model, the non-linear analogue of design matrix and intercept, are found via maximum likelihood. The model is intrinsically related to the geometry encoded in the connection of the manifold. We propose an estimation procedure which applies the Laplace approximation of the likelihood function. A simulation study of the performance of the model is performed and the model is applied to a real dataset of Corpus Callosum shapes.

Original language	English
Title of host publication	Information Processing in Medical Imaging : 25th International Conference, IPMI 2017, Boone, NC, USA, June 25-30, 2017, Proceedings
Number of pages	12
Publisher	Springer
Publication date	2017
Pages	53-64
ISBN (Print)	978-3-319-59049-3
ISBN (Electronic)	978-3-319-59050-9
DOIs	https://doi.org/10.1007/978-3-319-59050-9_5
Publication status	Published - 2017
Event	25th International Conference on Information Processing in Medical Imaging - Boone, United States Duration: 25 Jun 2017 → 30 Jun 2017 Conference number: 25

Conference

Conference	25th International Conference on Information Processing in Medical Imaging
Number	25
Country/Territory	United States
City	Boone
Period	25/06/2017 → 30/06/2017

Series	Lecture notes in computer science
Volume	10265
ISSN	0302-9743

Keywords

Frame bundle
Non-linear statistics
Regression
Statistics on manifolds
Stochastic development

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10.1007/978-3-319-59050-9_5

https://arxiv.org/pdf/1703.00291.pdf

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Stochastic development regression on non-linear manifolds. / Kühnel, Line; Sommer, Stefan Horst.

Information Processing in Medical Imaging: 25th International Conference, IPMI 2017, Boone, NC, USA, June 25-30, 2017, Proceedings. Springer, 2017. p. 53-64 (Lecture notes in computer science, Vol. 10265).

Research output: Chapter in Book/Report/Conference proceeding › Article in proceedings › Research › peer-review

Kühnel, L & Sommer, SH 2017, Stochastic development regression on non-linear manifolds. in Information Processing in Medical Imaging: 25th International Conference, IPMI 2017, Boone, NC, USA, June 25-30, 2017, Proceedings. Springer, Lecture notes in computer science, vol. 10265, pp. 53-64, 25th International Conference on Information Processing in Medical Imaging, Boone, North Carolina, United States, 25/06/2017. https://doi.org/10.1007/978-3-319-59050-9_5

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AB - We introduce a regression model for data on non-linear manifolds. The model describes the relation between a set of manifold valued observations, such as shapes of anatomical objects, and Euclidean explanatory variables. The approach is based on stochastic development of Euclidean diffusion processes to the manifold. Defining the data distribution as the transition distribution of the mapped stochastic process, parameters of the model, the non-linear analogue of design matrix and intercept, are found via maximum likelihood. The model is intrinsically related to the geometry encoded in the connection of the manifold. We propose an estimation procedure which applies the Laplace approximation of the likelihood function. A simulation study of the performance of the model is performed and the model is applied to a real dataset of Corpus Callosum shapes.

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KW - Regression

KW - Statistics on manifolds

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T2 - 25th International Conference on Information Processing in Medical Imaging

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ER -

Stochastic development regression on non-linear manifolds

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