The average condition number of most tensor rank decomposition problems is infinite

The tensor rank decomposition, or canonical polyadic decomposition, is the decomposition of a tensor into a sum of rank-1 tensors. The condition number of the tensor rank decomposition measures the sensitivity of the rank-1 summands with respect to structured perturbations. Those are perturbations preserving the rank of the tensor that is decomposed. On the other hand, the angular condition number measures the perturbations of the rank-1 summands up to scaling. We show for random rank-2 tensors with Gaussian density that the expected value of the condition number is infinite. Under some mild additional assumption, we show that the same is true for most higher ranks $r\geq 3$ as well. In fact, as the dimensions of the tensor tend to infinity, asymptotically all ranks are covered by our analysis. On the contrary, we show that rank-2 Gaussian tensors have finite expected angular condition number. Our results underline the high computational complexity of computing tensor rank decompositions. We discuss consequences of our results for algorithm design and for testing algorithms that compute the CPD. Finally, we supply numerical experiments

On the average condition number of tensor rank decompositions

We compute the expected value of powers of the geometric condition number of random tensor rank decompositions. It is shown in particular that the expected value of the condition number of $n_1\times n_2 \times 2$ tensors with a random rank-$r$ decomposition, given by factor matrices with independent and identically distributed standard normal entries, is infinite. This entails that it is expected and probable that such a rank-$r$ decomposition is sensitive to perturbations of the tensor. Moreover, it provides concrete further evidence that tensor decomposition can be a challenging problem, also from the numerical point of view. On the other hand, we provide strong theoretical and empirical evidence that tensors of size $n_1~\times~n_2~\times~n_3$ with all $n_1,n_2,n_3 \ge 3$ have a finite average condition number. This suggests there exists a gap in the expected sensitivity of tensors between those of format $n_1\times n_2 \times 2$ and other order-3 tensors. For establishing these results, we show that a natural weighted distance from a tensor rank decomposition to the locus of ill-posed decompositions with an infinite geometric condition number is bounded from below by the inverse of this condition number. That is, we prove one inequality towards a so-called condition number theorem for the tensor rank decomposition

Coulomb interaction and ferroelectric instability of BaTiO3

Using first-principles calculations, the phonon frequencies at the $\Gamma$ point and the dielectric tensor are determined and analysed for the cubic and rhombohedral phases of BaTiO$_{3}$. The dipole-dipole interaction is then separated \`a la Cochran from the remaining short-range forces, in order to investigate their respective influence on lattice dynamics. This analysis highlights the delicate balance of forces leading to an unstable phonon in the cubic phase and demonstrates the extreme sensitivity of this close compensation to minute effective charge changes. Within our decomposition, the stabilization of the unstable mode in the rhombohedral phase or under isotropic pressure has a different origin.Comment: 9 pages, 4 tables, 1 figur

On ANOVA decompositions of kernels and Gaussian random field paths

The FANOVA (or "Sobol'-Hoeffding") decomposition of multivariate functions has been used for high-dimensional model representation and global sensitivity analysis. When the objective function f has no simple analytic form and is costly to evaluate, a practical limitation is that computing FANOVA terms may be unaffordable due to numerical integration costs. Several approximate approaches relying on random field models have been proposed to alleviate these costs, where f is substituted by a (kriging) predictor or by conditional simulations. In the present work, we focus on FANOVA decompositions of Gaussian random field sample paths, and we notably introduce an associated kernel decomposition (into 2^{2d} terms) called KANOVA. An interpretation in terms of tensor product projections is obtained, and it is shown that projected kernels control both the sparsity of Gaussian random field sample paths and the dependence structure between FANOVA effects. Applications on simulated data show the relevance of the approach for designing new classes of covariance kernels dedicated to high-dimensional kriging

The Average Condition Number of Most Tensor Rank Decomposition Problems is Infinite

The tensor rank decomposition, or canonical polyadic decomposition, is the decomposition of a tensor into a sum of rank-1 tensors. The condition number of the tensor rank decomposition measures the sensitivity of the rank-1 summands with respect to structured perturbations. Those are perturbations preserving the rank of the tensor that is decomposed. On the other hand, the angular condition number measures the perturbations of the rank-1 summands up to scaling. We show for random rank-2 tensors that the expected value of the condition number is infinite for a wide range of choices of the density. Under a mild additional assumption, we show that the same is true for most higher ranks r?3r?3 as well. In fact, as the dimensions of the tensor tend to infinity, asymptotically all ranks are covered by our analysis. On the contrary, we show that rank-2 tensors have finite expected angular condition number. Based on numerical experiments, we conjecture that this could also be true for higher ranks. Our results underline the high computational complexity of computing tensor rank decompositions. We discuss consequences of our results for algorithm design and for testing algorithms computing tensor rank decompositions.We thank the reviewers for helpful suggestions. Part of this work was made while the second and third author were visiting the Universidad de Cantabria, supported by the funds of Grant 21.SI01.64658 (Banco Santander and Universidad de Cantabria), Grant MTM2017-83816-P from the Spanish Ministry of Science. The third author was additionally supported by the FWO Grant for a long stay abroad V401518N. We thank these institutions for their support