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In mathematics, a submanifold of a manifold M is a subset S which itself has the structure of a manifold, and for which the inclusion map S → M satisfies certain properties. There are different types of submanifolds depending on exactly which properties are required. Different authors often have different definitions. Formal definition In the following we assume all manifolds are differentiable manifolds of class Cr for a fixed r ≥ 1, and all morphisms are differentiable of class Cr. Immersed submanifolds An immersed submanifold of a manifold M is the image S of an immersion map f : N → M; in general this image will not be a submanifold as a subset, and an immersion map need not even be injective (one-to-one) – it can have self-intersections. More narrowly, one can require that the map f : N → M be an injection (one-to-one), in which we call it an inj

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Riemannian Optimization For High-Dimensional Tensor Completion

Michael Maximilian Steinlechner

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.

Siam Publications2016

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MATHICSE Technical Report : Riemannian optimization for high-dimensional tensor completion

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.

MATHICSE2015

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Low-Rank Matrix Completion By Riemannian Optimization

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.

Siam Publications2013

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MATH-512: Optimization on manifolds

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This course is an introduction to the theory of Riemann surfaces. Riemann surfaces naturally appear is mathematics in many different ways: as a result of analytic continuation, as quotients of complex domains under discontinuous group actions, as algebraic curves.