Abstract
In this chapter we present numerical methods for low-rank matrix and tensor problems that explicitly make use of the geometry of rank constrained matrix and tensor spaces. We focus on two types of problems: The first are optimization problems, like matrix and tensor completion, solving linear systems and eigenvalue problems. Such problems can be solved by numerical optimization for manifolds, called Riemannian optimization methods. We will explain the basic elements of differential geometry in order to apply such methods efficiently to rank constrained matrix and tensor spaces. The second type of problem is ordinary differential equations, defined on matrix and tensor spaces. We show how their solution can be approximated by the dynamical low-rank principle, and discuss several numerical integrators that rely in an essential way on geometric properties that are characteristic to sets of low rank matrices and tensors.
Highlights
The following chapter is an outline of Riemannian optimization and integration methods on manifolds of low-rank matrices and tensors
It may appear at this point that it is difficult to deal with the tensor train (TT) tensor format computationally, but this is not the case
This is due to the structure (9.35) of tangent vectors as sums of TT decompositions that vary in a single core each [50]
Summary
The following chapter is an outline of Riemannian optimization and integration methods on manifolds of low-rank matrices and tensors. 9 Geometric Methods on Low-Rank Matrix and Tensor Manifolds geometry of non-symmetric fixed rank matrices was quite explicitly exploited in numerical algorithms is [59]. It introduced the dynamical low-rank approximation method for calculating low-rank approximations when integrating a matrix that satisfies a set of ordinary differential equations (ODEs), as we will explain in Sect. For optimization problems with rank constraints, several Riemannian optimization methods were first presented in [79, 98, 113] that each use slightly different geometries of the sets fixed rank matrices. Some examples and references for successful application of such methods will be presented in some details later
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