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Publication# Riemannian Optimization For High-Dimensional Tensor Completion

Abstract

Tensor completion aims to reconstruct a high-dimensional data set where the vast majority of entries is missing. The assumption of low-rank structure in the underlying original data allows us to cast the completion problem into an optimization problem restricted to the manifold of fixed-rank tensors. Elements of this smooth embedded submanifold can be efficiently represented in the tensor train or matrix product states format with storage complexity scaling linearly with the number of dimensions. We present a nonlinear conjugate gradient scheme within the framework of Riemannian optimization which exploits this favorable scaling. Numerical experiments and comparison to existing methods show the effectiveness of our approach for the approximation of multivariate functions. Finally, we show that our algorithm can obtain competitive reconstructions from uniform random sampling of few entries compared to adaptive sampling techniques such as cross-approximation.

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Optimization problem

In mathematics, computer science and economics, an optimization problem is the problem of finding the best solution from all feasible solutions.
Optimization problems can be divided into two catego

Approximation

An approximation is anything that is intentionally similar but not exactly equal to something else.
Etymology and usage
The word approximation is derived from Latin approximatus, from prox

Tensor

In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such

We extend results on the dynamical low-rank approximation for the treatment of time-dependent matrices and tensors (Koch and Lubich; see [SIAM J. Matrix Anal. Appl., 29 (2007), pp. 434-454], [SIAM J. Matrix Anal. Appl., 31 (2010), pp. 2360-2375]) to the recently proposed hierarchical Tucker (HT) tensor format (Hackbusch and Kuhn; see [J. Fourier Anal. Appl., 15 (2009), pp. 706-722]) and the tensor train (TT) format (Oseledets; see [SIAM J. Sci. Comput., 33 (2011), pp. 2295-2317]), which are closely related to tensor decomposition methods used in quantum physics and chemistry. In this dynamical approximation approach, the time derivative of the tensor to be approximated is projected onto the time-dependent tangent space of the approximation manifold along the solution trajectory. This approach can be used to approximate the solutions to tensor differential equations in the HT or TT format and to compute updates in optimization algorithms within these reduced tensor formats. By deriving and analyzing the tangent space projector for the manifold of HT/TT tensors of fixed rank, we obtain curvature estimates, which allow us to obtain quasi-best approximation properties for the dynamical approximation, showing that the prospects and limitations of the ansatz are similar to those of the dynamical low rank approximation for matrices. Our results are exemplified by numerical experiments.

The matrix completion problem consists of finding or approximating a low-rank matrix based on a few samples of this matrix. We propose a new algorithm for matrix completion that minimizes the least-square distance on the sampling set over the Riemannian manifold of fixed-rank matrices. The algorithm is an adaptation of classical nonlinear conjugate gradients, developed within the framework of retraction-based optimization on manifolds. We describe all the necessary objects from differential geometry necessary to perform optimization over this low-rank matrix manifold, seen as a submanifold embedded in the space of matrices. In particular, we describe how metric projection can be used as retraction and how vector transport lets us obtain the conjugate search directions. Finally, we prove convergence of a regularized version of our algorithm under the assumption that the restricted isometry property holds for incoherent matrices throughout the iterations. The numerical experiments indicate that our approach scales very well for large-scale problems and compares favorably with the state-of-the-art, while outperforming most existing solvers.

Michael Maximilian Steinlechner

Tensor completion aims to reconstruct a high-dimensional data set with a large fraction of missing entries. The assumption of low-rank structure in the underlying original data allows us to cast the completion problem into an optimization problem restricted to the manifold of fixed-rank tensors. Elements of this smooth embedded submanifold can be efficiently represented in the tensor train (TT) or matrix product states (MPS) format with storage complexity scaling linearly with the number of dimensions. We present a nonlinear conjugate gradient scheme within the framework of Riemannian optimization which exploits this favorable scaling. Numerical experiments and comparison to existing methods show the effectiveness of our approach for the approximation of multivariate functions. Finally, we show that our algorithm can obtain competitive reconstructions from uniform random sampling of few entries of compared to adaptive sampling techniques such as cross-approximation.