A Riemannian trust‐region method for low‐rank tensor completion

Heidel, Gennadij; Schulz, Volker

doi:10.1002/nla.2175

Cited by 22 publications

(22 citation statements)

References 29 publications

(84 reference statements)

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“…Implicit temporal integrators can mitigate this problem, but they require the development of linear solvers on tensor manifolds with constant rank. This can be achieved, e.g., by utilizing Riemannian optimization algorithms [49,44,45,23], or alternating least squares [14,29,43,6]. Let us discretize the spacial derivatives in (29) with second-order centered finite differences on a tensor product evenly-spaced grid in each variable.…”

Section: Stiffness In High-dimensional Pdesmentioning

confidence: 99%

Stability analysis of hierarchical tensor methods for time-dependent PDEs

Rodgers

Venturi

2020

Journal of Computational Physics

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In this paper we address the question of whether it is possible to integrate time-dependent high-dimensional PDEs with hierarchical tensor methods and explicit time stepping schemes. To this end, we develop sufficient conditions for stability and convergence of tensor solutions evolving on tensor manifolds with constant rank. We also argue that the applicability of PDE solvers with explicit time-stepping may be limited by time-step restriction dependent on the dimension of the problem. Numerical applications are presented and discussed for variable coefficients linear hyperbolic and parabolic PDEs.

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Section: Stiffness In High-dimensional Pdesmentioning

confidence: 99%

Stability analysis of hierarchical tensor methods for time-dependent PDEs

Rodgers

Venturi

2020

Journal of Computational Physics

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“…In [16] we called this type of problem a parameter identification problem (PIP). Application areas where PIPs originate are data completion with low-rank matrix and tensor decompositions [15,24,39,45,62,66], geometric modeling [19,49,63], computer vision [33,38,51], and phase retrieval problems in signal processing [9,12].…”

Section: Introductionmentioning

confidence: 99%

The Condition Number of Riemannian Approximation Problems

Breiding¹,

Vannieuwenhoven²

2021

SIAM J. Optim.

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We consider the local sensitivity of least-squares formulations of inverse problems. The sets of inputs and outputs of these problems are assumed to have the structures of Riemannian manifolds. The problems we consider include the approximation problem of finding the nearest point on a Riemannian embedded submanifold from a given point in the ambient space. We characterize the first-order sensitivity, i.e., condition number, of local minimizers and critical points to arbitrary perturbations of the input of the least-squares problem. This condition number involves the Weingarten map of the input manifold, which measures the amount by which the input manifold curves in its ambient space. We validate our main results through experiments with the n-camera triangulation problem in computer vision.

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“…The concept of Riemannian optimization, i.e. optimizing over parameters that live on a smooth manifold is well studied and has gained increasing interest in the domain of Data Science, for example for tensor completion problems (Heidel and Schulz 2018). However, the idea is quite new for Gaussian Mixture Models and Sra (2015, 2020) showed promising results with a Riemannian LBFGS and Riemannian Stochastic Gradient Descent Algorithm.…”

Section: Introductionmentioning

confidence: 99%

A Riemannian Newton trust-region method for fitting Gaussian mixture models

Lena¹,

Burgard²,

Schulz³

2021

Stat Comput

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Gaussian Mixture Models are a powerful tool in Data Science and Statistics that are mainly used for clustering and density approximation. The task of estimating the model parameters is in practice often solved by the expectation maximization (EM) algorithm which has its benefits in its simplicity and low per-iteration costs. However, the EM converges slowly if there is a large share of hidden information or overlapping clusters. Recent advances in Manifold Optimization for Gaussian Mixture Models have gained increasing interest. We introduce an explicit formula for the Riemannian Hessian for Gaussian Mixture Models. On top, we propose a new Riemannian Newton Trust-Region method which outperforms current approaches both in terms of runtime and number of iterations. We apply our method on clustering problems and density approximation tasks. Our method is very powerful for data with a large share of hidden information compared to existing methods.

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A Riemannian trust‐region method for low‐rank tensor completion

Cited by 22 publications

References 29 publications

Stability analysis of hierarchical tensor methods for time-dependent PDEs

Stability analysis of hierarchical tensor methods for time-dependent PDEs

The Condition Number of Riemannian Approximation Problems

A Riemannian Newton trust-region method for fitting Gaussian mixture models

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