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 (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 compared to adaptive sampling techniques such as cross-approximation.
The numerical solution of partial differential equations on high-dimensional domains gives rise to computationally challenging linear systems. When using standard discretization techniques, the size of the linear system grows exponentially with the number of dimensions, making the use of classic iterative solvers infeasible. During the last few years, low-rank tensor approaches have been developed that allow to mitigate this curse of dimensionality by exploiting the underlying structure of the linear operator. In this work, we focus on tensors represented in the Tucker and tensor train formats. We propose two preconditioned gradient methods on the corresponding low-rank tensor manifolds: A Riemannian version of the preconditioned Richardson method as well as an approximate Newton scheme based on the Riemannian Hessian. For the latter, considerable attention is given to the efficient solution of the resulting Newton equation. In numerical experiments, we compare the efficiency of our Riemannian algorithms with other established tensor-based approaches such as a truncated preconditioned Richardson method and the alternating linear scheme. The results show that our approximate Riemannian Newton scheme is significantly faster in cases when the application of the linear operator is expensive.
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