Manifolds and Differential Geometry

Lee, Jeffrey M.

doi:10.1090/gsm/107

Cited by 147 publications

(124 citation statements)

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“…We note at first that with the same arguments as in the proof of Lemma 4, restriction of to a suitable open neighbourhoodÑ of 0 gives a mapping :Ñ → (Ñ ) ⊆ T. A straightforward computation shows that the tangent map T U belonging to (also see e.g. [30]) is given by the bijection (50). Thus the conditions of the inverse mapping theorem for manifolds are fulfilled, yielding the asserted statement.…”

Section: Lemma 8 Let X Be Defined As In (49) and Letmentioning

confidence: 92%

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On manifolds of tensors of fixed TT-rank

2011

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Recently, the format of TT tensors (Hackbusch and Kühn in J Fourier Anal has turned out to be a promising new format for the approximation of solutions of high dimensional problems. In this paper, we prove some new results for the TT representation of a tensor U ∈ R n 1 ×···×n d and for the manifold of tensors of TT-rank r . As a first result, we prove that the TT (or compression) ranks r i of a tensor U are unique and equal to the respective separation ranks of U if the components of the TT decomposition are required to fulfil a certain maximal rank condition. We then show that the set T of TT tensors of fixed rank r locally forms an embedded manifold in R n 1 ×···×n d , therefore preserving the essential theoretical properties of the Tucker format, but often showing an improved scaling behaviour. Extending a similar approach for matrices (Conte and Lubich in M2AN 44:759, 2010), we introduce certain gauge conditions to obtain a unique representation of the tangent space T U T of T and deduce a local parametrization of the TT manifold. The parametrisation of T U T is often crucial for an algorithmic treatment of high-dimensional time-dependent PDEs and minimisation problems (Lubich in From quantum to classical molecular dynamics: reduced methods and numerical analysis, 2008). We conclude with remarks on those applications and present some numerical examples.

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Section: Lemma 8 Let X Be Defined As In (49) and Letmentioning

confidence: 92%

“…e.g. [30], Theorem 2.25. We note at first that with the same arguments as in the proof of Lemma 4, restriction of to a suitable open neighbourhoodÑ of 0 gives a mapping :Ñ → (Ñ ) ⊆ T. A straightforward computation shows that the tangent map T U belonging to (also see e.g.…”

Section: Lemma 8 Let X Be Defined As In (49) and Letmentioning

confidence: 99%

On manifolds of tensors of fixed TT-rank

2011

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“…For further details, the books I find clearest are Palais (1981) and Bleecker (1981); Nakahara (1990) and Baez and Munian (1994) are also noteworthy. For additional background on the geometry, the classic source on the theory of connections on fiber bundles is Kobayashi and Nomizu (1963), which remains the most comprehensive text available, despite certain pre-modern tendencies; additional helpful sources are Kolář et al (1993), Michor (2009), Lee (2009), andTaubes (2011). One important difference, however, between the presentation here and in the sources just cited is that, as noted in fn.…”

Section: A Primer On the Geometry Of Yang-mills Theory For Philosophmentioning

confidence: 98%

Fiber bundles, Yang–Mills theory, and general relativity

Weatherall

2015

Synthese

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“…In other words, the idea is to linearize the problem. Using the terminology of differential geometry, this means working in the tangent spaces instead of manifolds (see, for instance, [18] for topics in differential geometry).…”

Section: The Surface Unfolding Problemmentioning

confidence: 99%

On a sub‐Stiefel Procrustes problem arising in computer vision

Cardoso

Ziętak

2015

Numerical Linear Algebra App

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A sub-Stiefel matrix is a matrix that results from deleting simultaneously the last row and the last column of an orthogonal matrix. In this paper, we consider a Procrustes problem on the set of sub-Stiefel matrices of order n. For n D 2, this problem has arisen in computer vision to solve the surface unfolding problem considered by R. Fereirra, J. Xavier and J. Costeira. An iterative algorithm for computing the solution of the sub-Stiefel Procrustes problem for an arbitrary n is proposed, and some numerical experiments are carried out to illustrate its performance. For these purposes, we investigate the properties of sub-Stiefel matrices. In particular, we derive two necessary and sufficient conditions for a matrix to be sub-Stiefel. We also relate the sub-Stiefel Procrustes problem with the Stiefel Procrustes problem and compare it with the orthogonal Procrustes problem.

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Manifolds and Differential Geometry

Cited by 147 publications

References 0 publications

On manifolds of tensors of fixed TT-rank

On manifolds of tensors of fixed TT-rank

Fiber bundles, Yang–Mills theory, and general relativity

On a sub‐Stiefel Procrustes problem arising in computer vision

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