arXiv subjects

Alwin Stegeman

Publications and source records attributed to Alwin Stegeman.

6 recordsLinked to original sources

arXiv2018

On the largest multilinear singular values of higher-order tensors

Let $σ_n$ denote the largest mode-$n$ multilinear singular value of an $I_1\times\dots \times I_N$ tensor $\mathcal T$. We prove that $$ σ_1^2+\dots+σ_{n-1}^2+σ_{n+1}^2+\dots+σ_{N}^2\leq (N-2)\|\mathcal T\|^2 + σ_n^2,\quad n=1,\dots,N, \qquad\qquad (1) $$ where $\|\cdot\|$ denotes the Frobenius norm. We also show that at least for the cubic tensors the inverse problem always has a solution. Namely, for each $σ_1,\dots,σ_N$ that satisfy (1) and the trivial inequalities $σ_1\geq \frac{1}{\sqrt{I}}\|\mathcal T\|,\dots, σ_N\geq \frac{1}{\sqrt{I}}\|\mathcal T\|$, there always exists an $I\times \dots\times I$ tensor whose largest multilinear singular values are equal to $σ_1,\dots,σ_N$. For $N=3$ we show that if the equality $σ_1^2+σ_2^2= \|\mathcal T\|^2 + σ_3^2$ in (1) holds, then $\mathcal T$ is necessarily equal to a sum of multilinear rank-$(L_1,1,L_1)$ and multilinear rank-$(1,L_2,L_2)$ tensors and we give a complete description of all its multilinear singular values. We establish a connection with honeycombs and eigenvalues of the sum of two Hermitian matrices. This seems to give at least a partial explanation of why results on the joint distribution of multilinear singular values are scarce.

math.SP↗

arXiv2016

On best rank-2 and rank-(2,2,2) approximations of order-3 tensors

It is well known that a best rank-$R$ approximation of order-3 tensors may not exist for $R\ge 2$. A best rank-$(R,R,R)$ approximation always exists, however, and is also a best rank-$R$ approximation when it has rank (at most) $R$. For $R=2$ and real order-3 tensors it is shown that a best rank-2 approximation is also a local minimum of the best rank-(2,2,2) approximation problem. This implies that if all rank-(2,2,2) minima have rank larger than 2, then a best rank-2 approximation does not exist. This provides an easy-to-check criterion for existence of a best rank-2 approximation. The result is illustrated by means of simulations.

math.AG↗

arXiv2015

On the (non)existence of best low-rank approximations of generic IxJx2 arrays

Several conjectures and partial proofs have been formulated on the (non)existence of a best low-rank approximation of real-valued IxJx2 arrays. We analyze this problem using the Generalized Schur Decomposition and prove (non)existence of a best rank-R approximation for generic IxJx2 arrays, for all values of I,J,R. Moreover, for cases where a best rank-R approximation exists on a set of positive volume only, we provide easy-to-check necessary and sufficient conditions for the existence of a best rank-R approximation.

math.AG↗

arXiv2011

Are diverging CP components always nearly proportional?

Fitting a Candecomp/Parafac (CP) decomposition (also known as Canonical Polyadic decomposition) to a multi-way array or higher-order tensor, is equivalent to finding a best low-rank approximation to the multi-way array or higher-order tensor, where the rank is defined as the outer-product rank. However, such a best low-rank approximation may not exist due to the fact that the set of multi-way arrays with rank at most $R$ is not closed for $R\ge 2$. Nonexistence of a best low-rank approximation results in (groups of) diverging rank-1 components when an attempt is made to compute the approximation. In this note, we show that in a group of two or three diverging components, the components converge to proportionality almost everywhere. A partial proof of this result for larger groups of diverging components is also given. Also, we give examples of groups of three, four, and six non-proportional diverging components. These examples are shown to be exceptional cases.

math.AG↗

arXiv2010

The Generalized Schur Decomposition and the rank-$R$ set of real $I\times J\times 2$ arrays

It is known that a best low-rank approximation to multi-way arrays or higher-order tensors may not exist. This is due to the fact that the set of multi-way arrays with rank at most $R$ is not closed. Nonexistence of the best low-rank approximation results in diverging rank-1 components when an attempt is made to compute the approximation. Recently, a solution to this problem has been proposed for real $I\times J\times 2$ arrays. Instead of a best rank-$R$ approximation the best fitting Generalized Schur Decomposition (GSD) is computed. Under the restriction of nonsingular upper triangular matrices in the GSD, the set of GSD solutions equals the interior and boundary of the rank-$R$ set. Here, we show that this holds even without the restriction. We provide a complete classification of interior, boundary, and exterior points of the rank-$R$ set of real $I\times J\times 2$ arrays, and show that the set of GSD solutions equals the interior and boundary of this set.

math.AG↗

arXiv2009

Subtracting a best rank-1 approximation may increase tensor rank

It has been shown that a best rank-R approximation of an order-k tensor may not exist when R>1 and k>2. This poses a serious problem to data analysts using tensor decompositions. It has been observed numerically that, generally, this issue cannot be solved by consecutively computing and subtracting best rank-1 approximations. The reason for this is that subtracting a best rank-1 approximation generally does not decrease tensor rank. In this paper, we provide a mathematical treatment of this property for real-valued 2x2x2 tensors, with symmetric tensors as a special case. Regardless of the symmetry, we show that for generic 2x2x2 tensors (which have rank 2 or 3), subtracting a best rank-1 approximation results in a tensor that has rank 3 and lies on the boundary between the rank-2 and rank-3 sets. Hence, for a typical tensor of rank 2, subtracting a best rank-1 approximation increases the tensor rank.

math.AG↗