Nonparametric curve estimation on stiefel manifolds

Lee, Jeffrey M.; Ruymgaart, F.H.

doi:10.1080/10485259608832663

Cited by 10 publications

(6 citation statements)

References 14 publications

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“…We now use the elementary inequality log(1 + x) ≥ x/(1 + x) for x ∈ (−1, +∞), and a lower bound on π O |O| − U 0 2 F ≤ c 2 O (given by Lee and Ruymgaart, 1996). Therefore…”

Section: Upper Bound On the Kullback-leibler Divergencementioning

confidence: 99%

Revisiting Clustering as Matrix Factorisation on the Stiefel Manifold

Chrétien

Guedj

2020

Machine Learning, Optimization, and Data Science

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“…We now use the elementary inequality log(1 + x) ≥ x/(1 + x) for x ∈ (−1, +∞), and a lower bound on π O |O| − U 0 2 F ≤ c 2 O (given by Lee and Ruymgaart, 1996). Therefore…”

Section: Upper Bound On the Kullback-leibler Divergencementioning

confidence: 99%

Revisiting Clustering as Matrix Factorisation on the Stiefel Manifold

Chrétien

Guedj

2020

Machine Learning, Optimization, and Data Science

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“…We now use the elementary inequality log(1 + x) ≥ x/(1 + x) for x ∈ (−1, +∞), and a lower bound on Lee and Ruymgaart, 1996). Therefore…”

Section: Upper Bound On the Kullback-leibler Divergencementioning

confidence: 99%

Revisiting clustering as matrix factorisation on the Stiefel manifold

Chrétien¹,

Guedj²

2019

Preprint

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This paper studies clustering for possibly high dimensional data (e.g. images, time series, gene expression data, and many other settings), and rephrase it as low rank matrix estimation in the PAC-Bayesian framework. Our approach leverages the well known Burer-Monteiro factorisation strategy from large scale optimisation, in the context of low rank estimation. Moreover, our Burer-Monteiro factors are shown to lie on a Stiefel manifold. We propose a new generalized Bayesian estimator for this problem and prove novel prediction bounds for clustering. We also devise a componentwise Langevin sampler on the Stiefel manifold to compute this estimator.

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“…This manifold is defined as the set of orthonormal m-frames in R d , and includes the sphere and the orthogonal group as special cases. Prentice [15] introduces an extension of spherical regression to this setting, and Lee and Ruymgaart [16] consider non-parametric regression estimation using caps, i.e. intersections of closed balls in the ambient space with the manifold, following an earlier work on density estimation on compact submanifolds of a Euclidean space [17].…”

Section: Introductionmentioning

confidence: 99%

Non-parametric regression estimation on closed Riemannian manifolds

Pelletier

2006

Journal of Nonparametric Statistics

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The non-parametric estimation of the regression function of a real-valued random variable Y on a random object X valued in a closed Riemannian manifold M is considered. A regression estimator which generalizes kernel regression estimators on Euclidean sample spaces is introduced. Under classical assumptions on the kernel and the bandwidth sequence, the asymptotic bias and variance are obtained, and the estimator is shown to converge at the same L 2 -rate as kernel regression estimators on Euclidean spaces.

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Nonparametric curve estimation on stiefel manifolds

Cited by 10 publications

References 14 publications

Revisiting Clustering as Matrix Factorisation on the Stiefel Manifold

Revisiting Clustering as Matrix Factorisation on the Stiefel Manifold

Revisiting clustering as matrix factorisation on the Stiefel manifold

Non-parametric regression estimation on closed Riemannian manifolds

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