An extrinsic look at the Riemannian Hessian

P. A. Absil; Robert Mahony; Jochen Trumpf

doi:10.1007/978-3-642-40020-9_39

An extrinsic look at the Riemannian Hessian

P. A. Absil, Robert Mahony, Jochen Trumpf

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

77 Citations (Scopus)

Abstract

Let f be a real-valued function on a Riemannian submanifold of a Euclidean space, and let f̄ be a local extension of f. We show that the Riemannian Hessian of f can be conveniently obtained from the Euclidean gradient and Hessian of f̄ by means of two manifold-specific objects: the orthogonal projector onto the tangent space and the Weingarten map. Expressions for the Weingarten map are provided on various specific submanifolds.

Original language	English
Title of host publication	Geometric Science of Information - First International Conference, GSI 2013, Proceedings
Pages	361-368
Number of pages	8
DOIs	https://doi.org/10.1007/978-3-642-40020-9_39
Publication status	Published - 2013
Event	1st International SEE Conference on Geometric Science of Information, GSI 2013 - Paris, France Duration: 28 Aug 2013 → 30 Aug 2013

Publication series

Name	Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume	8085 LNCS
ISSN (Print)	0302-9743
ISSN (Electronic)	1611-3349

Conference

Conference	1st International SEE Conference on Geometric Science of Information, GSI 2013
Country/Territory	France
City	Paris
Period	28/08/13 → 30/08/13

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10.1007/978-3-642-40020-9_39

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Absil, P. A., Mahony, R., & Trumpf, J. (2013). An extrinsic look at the Riemannian Hessian. In Geometric Science of Information - First International Conference, GSI 2013, Proceedings (pp. 361-368). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 8085 LNCS). https://doi.org/10.1007/978-3-642-40020-9_39

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title = "An extrinsic look at the Riemannian Hessian",

abstract = "Let f be a real-valued function on a Riemannian submanifold of a Euclidean space, and let {\=f} be a local extension of f. We show that the Riemannian Hessian of f can be conveniently obtained from the Euclidean gradient and Hessian of {\=f} by means of two manifold-specific objects: the orthogonal projector onto the tangent space and the Weingarten map. Expressions for the Weingarten map are provided on various specific submanifolds.",

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Absil, PA, Mahony, R & Trumpf, J 2013, An extrinsic look at the Riemannian Hessian. in Geometric Science of Information - First International Conference, GSI 2013, Proceedings. Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 8085 LNCS, pp. 361-368, 1st International SEE Conference on Geometric Science of Information, GSI 2013, Paris, France, 28/08/13. https://doi.org/10.1007/978-3-642-40020-9_39

An extrinsic look at the Riemannian Hessian. / Absil, P. A.; Mahony, Robert ; Trumpf, Jochen.
Geometric Science of Information - First International Conference, GSI 2013, Proceedings. 2013. p. 361-368 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 8085 LNCS).

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

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T1 - An extrinsic look at the Riemannian Hessian

AU - Absil, P. A.

AU - Mahony, Robert

AU - Trumpf, Jochen

PY - 2013

Y1 - 2013

N2 - Let f be a real-valued function on a Riemannian submanifold of a Euclidean space, and let f̄ be a local extension of f. We show that the Riemannian Hessian of f can be conveniently obtained from the Euclidean gradient and Hessian of f̄ by means of two manifold-specific objects: the orthogonal projector onto the tangent space and the Weingarten map. Expressions for the Weingarten map are provided on various specific submanifolds.

AB - Let f be a real-valued function on a Riemannian submanifold of a Euclidean space, and let f̄ be a local extension of f. We show that the Riemannian Hessian of f can be conveniently obtained from the Euclidean gradient and Hessian of f̄ by means of two manifold-specific objects: the orthogonal projector onto the tangent space and the Weingarten map. Expressions for the Weingarten map are provided on various specific submanifolds.

KW - Euclidean Hessian

KW - Riemannian Hessian

KW - Weingarten map

KW - shape operator

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DO - 10.1007/978-3-642-40020-9_39

M3 - Conference contribution

SN - 9783642400193

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

SP - 361

EP - 368

BT - Geometric Science of Information - First International Conference, GSI 2013, Proceedings

T2 - 1st International SEE Conference on Geometric Science of Information, GSI 2013

Y2 - 28 August 2013 through 30 August 2013

ER -

An extrinsic look at the Riemannian Hessian

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