A new convergence proof for the higher-order power method and generalizations

A proof for the point-wise convergence of the factors in the higher-order power method for tensors towards a critical point is given. It is obtained by applying established results from the theory of \L{}ojasiewicz inequalities to the equivalent, unconstrained alternating least squares algorithm for best rank-one tensor approximation

Convergence results for projected line-search methods on varieties of low-rank matrices via \L{}ojasiewicz inequality

The aim of this paper is to derive convergence results for projected line-search methods on the real-algebraic variety $\mathcal{M}_{\le k}$ of real $m \times n$ matrices of rank at most $k$. Such methods extend Riemannian optimization methods, which are successfully used on the smooth manifold $\mathcal{M}_k$ of rank-$k$ matrices, to its closure by taking steps along gradient-related directions in the tangent cone, and afterwards projecting back to $\mathcal{M}_{\le k}$. Considering such a method circumvents the difficulties which arise from the nonclosedness and the unbounded curvature of $\mathcal{M}_k$. The pointwise convergence is obtained for real-analytic functions on the basis of a \L{}ojasiewicz inequality for the projection of the antigradient to the tangent cone. If the derived limit point lies on the smooth part of $\mathcal{M}_{\le k}$, i.e. in $\mathcal{M}_k$, this boils down to more or less known results, but with the benefit that asymptotic convergence rate estimates (for specific step-sizes) can be obtained without an a priori curvature bound, simply from the fact that the limit lies on a smooth manifold. At the same time, one can give a convincing justification for assuming critical points to lie in $\mathcal{M}_k$: if $X$ is a critical point of $f$ on $\mathcal{M}_{\le k}$, then either $X$ has rank $k$, or $\nabla f(X) = 0$

On convergence of the maximum block improvement method

Abstract. The MBI (maximum block improvement) method is a greedy approach to solving optimization problems where the decision variables can be grouped into a finite number of blocks. Assuming that optimizing over one block of variables while fixing all others is relatively easy, the MBI method updates the block of variables corresponding to the maximally improving block at each iteration, which is arguably a most natural and simple process to tackle block-structured problems with great potentials for engineering applications. In this paper we establish global and local linear convergence results for this method. The global convergence is established under the Lojasiewicz inequality assumption, while the local analysis invokes second-order assumptions. We study in particular the tensor optimization model with spherical constraints. Conditions for linear convergence of the famous power method for computing the maximum eigenvalue of a matrix follow in this framework as a special case. The condition is interpreted in various other forms for the rank-one tensor optimization model under spherical constraints. Numerical experiments are shown to support the convergence property of the MBI method

Alternating least squares as moving subspace correction

In this note we take a new look at the local convergence of alternating optimization methods for low-rank matrices and tensors. Our abstract interpretation as sequential optimization on moving subspaces yields insightful reformulations of some known convergence conditions that focus on the interplay between the contractivity of classical multiplicative Schwarz methods with overlapping subspaces and the curvature of low-rank matrix and tensor manifolds. While the verification of the abstract conditions in concrete scenarios remains open in most cases, we are able to provide an alternative and conceptually simple derivation of the asymptotic convergence rate of the two-sided block power method of numerical algebra for computing the dominant singular subspaces of a rectangular matrix. This method is equivalent to an alternating least squares method applied to a distance function. The theoretical results are illustrated and validated by numerical experiments.Comment: 20 pages, 4 figure

Finding a low-rank basis in a matrix subspace

For a given matrix subspace, how can we find a basis that consists of low-rank matrices? This is a generalization of the sparse vector problem. It turns out that when the subspace is spanned by rank-1 matrices, the matrices can be obtained by the tensor CP decomposition. For the higher rank case, the situation is not as straightforward. In this work we present an algorithm based on a greedy process applicable to higher rank problems. Our algorithm first estimates the minimum rank by applying soft singular value thresholding to a nuclear norm relaxation, and then computes a matrix with that rank using the method of alternating projections. We provide local convergence results, and compare our algorithm with several alternative approaches. Applications include data compression beyond the classical truncated SVD, computing accurate eigenvectors of a near-multiple eigenvalue, image separation and graph Laplacian eigenproblems

On orthogonal tensors and best rank-one approximation ratio

As is well known, the smallest possible ratio between the spectral norm and the Frobenius norm of an $m \times n$ matrix with $m \le n$ is $1/\sqrt{m}$ and is (up to scalar scaling) attained only by matrices having pairwise orthonormal rows. In the present paper, the smallest possible ratio between spectral and Frobenius norms of $n_1 \times \dots \times n_d$ tensors of order $d$, also called the best rank-one approximation ratio in the literature, is investigated. The exact value is not known for most configurations of $n_1 \le \dots \le n_d$. Using a natural definition of orthogonal tensors over the real field (resp., unitary tensors over the complex field), it is shown that the obvious lower bound $1/\sqrt{n_1 \cdots n_{d-1}}$ is attained if and only if a tensor is orthogonal (resp., unitary) up to scaling. Whether or not orthogonal or unitary tensors exist depends on the dimensions $n_1,\dots,n_d$ and the field. A connection between the (non)existence of real orthogonal tensors of order three and the classical Hurwitz problem on composition algebras can be established: existence of orthogonal tensors of size $\ell \times m \times n$ is equivalent to the admissibility of the triple $[\ell,m,n]$ to the Hurwitz problem. Some implications for higher-order tensors are then given. For instance, real orthogonal $n \times \dots \times n$ tensors of order $d \ge 3$ do exist, but only when $n = 1,2,4,8$. In the complex case, the situation is more drastic: unitary tensors of size $\ell \times m \times n$ with $\ell \le m \le n$ exist only when $\ell m \le n$. Finally, some numerical illustrations for spectral norm computation are presented

Dynamical low-rank approximation of the Vlasov-Poisson equation with piecewise linear spatial boundary

We consider dynamical low-rank approximation (DLRA) for the numerical simulation of Vlasov--Poisson equations based on separation of space and velocity variables, as proposed in several recent works. The standard approach for the time integration in the DLRA model uses a splitting of the tangent space projector for the low-rank manifold according to the separated variables. It can also be modified to allow for rank-adaptivity. A less studied aspect is the incorporation of boundary conditions in the DLRA model. We propose a variational formulation of the projector splitting which allows to handle inflow boundary conditions on spatial domains with piecewise linear boundary. Numerical experiments demonstrate the principle feasibility of this approach

On the interconnection between the higher-order singular values of real tensors

A higher-order tensor allows several possible matricizations (reshapes into matrices). The simultaneous decay of singular values of such matricizations has crucial implications on the low-rank approximability of the tensor via higher-order singular value decomposition. It is therefore an interesting question which simultaneous properties the singular values of different tensor matricizations actually can have, but it has not received the deserved attention so far. In this paper, preliminary investigations in this direction are conducted. While it is clear that the singular values in different matricizations cannot be prescribed completely independent from each other, numerical experiments suggest that sufficiently small, but otherwise arbitrary perturbations preserve feasibility. An alternating projection heuristic is proposed for constructing tensors with prescribed singular values (assuming their feasibility). Regarding the related problem of characterising sets of tensors having the same singular values in specified matricizations, it is noted that orthogonal equivalence under multilinear matrix multiplication is a sufficient condition for two tensors to have the same singular values in all principal, Tucker-type matricizations, but, in contrast to the matrix case, not necessary. An explicit example of this phenomenon is given

Maximum relative distance between real rank-two and rank-one tensors

It is shown that the relative distance in Frobenius norm of a real symmetric order-$d$ tensor of rank two to its best rank-one approximation is upper bounded by $\sqrt{1-(1-1/d)^{d-1}}$. This is achieved by determining the minimal possible ratio between spectral and Frobenius norm for symmetric tensors of border rank two, which equals $\left(1-{1}/{d}\right)^{(d-1)/{2}}$. These bounds are also verified for arbitrary real rank-two tensors by reducing to the symmetric case.Comment: New resul