Pierre Comon

Pierre Comon, Gene Golub, Lek-Heng Lim et al.

A symmetric tensor is a higher order generalization of a symmetric matrix. In this paper, we study various properties of symmetric tensors in relation to a decomposition into a sum of symmetric outer product of vectors. A rank-1 order-k tensor is the outer product of k non-zero vectors. Any symmetric tensor can be decomposed into a linear combination of rank-1 tensors, each of them being symmetric or not. The rank of a symmetric tensor is the minimal number of rank-1 tensors that is necessary to reconstruct it. The symmetric rank is obtained when the constituting rank-1 tensors are imposed to be themselves symmetric. It is shown that rank and symmetric rank are equal in a number of cases, and that they always exist in an algebraically closed field. We will discuss the notion of the generic symmetric rank, which, due to the work of Alexander and Hirschowitz, is now known for any values of dimension and order. We will also show that the set of symmetric tensors of symmetric rank at most r is not closed, unless r = 1.

Lek-Heng Lim, Pierre Comon

We discuss how recently discovered techniques and tools from compressed sensing can be used in tensor decompositions, with a view towards modeling signals from multiple arrays of multiple sensors. We show that with appropriate bounds on a measure of separation between radiating sources called coherence, one could always guarantee the existence and uniqueness of a best rank-r approximation of the tensor representing the signal. We also deduce a computationally feasible variant of Kruskal's uniqueness condition, where the coherence appears as a proxy for k-rank. Problems of sparsest recovery with an infinite continuous dictionary, lowest-rank tensor representation, and blind source separation are treated in a uniform fashion. The decomposition of the measurement tensor leads to simultaneous localization and extraction of radiating sources, in an entirely deterministic manner.

Yang Qi, Pierre Comon, Lek-Heng Lim

We show that for a nonnegative tensor, a best nonnegative rank-r approximation is almost always unique, its best rank-one approximation may always be chosen to be a best nonnegative rank-one approximation, and that the set of nonnegative tensors with non-unique best rank-one approximations form an algebraic hypersurface. We show that the last part holds true more generally for real tensors and thereby determine a polynomial equation so that a real or nonnegative tensor which does not satisfy this equation is guaranteed to have a unique best rank-one approximation. We also establish an analogue for real or nonnegative symmetric tensors. In addition, we prove a singular vector variant of the Perron--Frobenius Theorem for positive tensors and apply it to show that a best nonnegative rank-r approximation of a positive tensor can never be obtained by deflation. As an aside, we verify that the Euclidean distance (ED) discriminants of the Segre variety and the Veronese variety are hypersurfaces and give defining equations of these ED discriminants.

Pierre Comon, Lek-Heng Lim, Yang Qi et al.

We study path-connectedness and homotopy groups of sets of tensors defined by tensor rank, border rank, multilinear rank, as well as their symmetric counterparts for symmetric tensors. We show that over $\mathbb{C}$, the set of rank-$r$ tensors and the set of symmetric rank-$r$ symmetric tensors are both path-connected if $r$ is not more than the complex generic rank; these results also extend to border rank and symmetric border rank over $\mathbb{C}$. Over $\mathbb{R}$, the set of rank-$r$ tensors is path-connected if it has the expected dimension but the corresponding result for symmetric rank-$r$ symmetric $d$-tensors depends on the order $d$: connected when $d$ is odd but not when $d$ is even. Border rank and symmetric border rank over $\mathbb{R}$ have essentially the same path-connectedness properties as rank and symmetric rank over $\mathbb{R}$. When $r$ is greater than the complex generic rank, we are unable to discern any general pattern: For example, we show that border-rank-three tensors in $\mathbb{R}^2 \otimes \mathbb{R}^2 \otimes \mathbb{R}^2$ fall into four connected components. For multilinear rank, the manifold of $d$-tensors of multilinear rank $(r_1,\dots,r_d)$ in $\mathbb{C}^{n_1} \otimes \cdots \otimes \mathbb{C}^{n_d}$ is always path-connected, and the same is true in $\mathbb{R}^{n_1} \otimes \cdots \otimes \mathbb{R}^{n_d}$ unless $n_i = r_i = \prod_{j \ne i} r_j$ for some $i\in\{1, \dots, d\}$. Beyond path-connectedness, we determine, over both $\mathbb{R}$ and $\mathbb{C}$, the fundamental and higher homotopy groups of the set of tensors of a fixed small rank, and, taking advantage of Bott periodicity, those of the manifold of tensors of a fixed multilinear rank. We also obtain analogues of these results for symmetric tensors of a fixed symmetric rank or a fixed symmetric multilinear rank.

Pierre Comon

A matrix (and any associated linear system) will be referred to as structured if it has a small displacement rank. It is known that the inverse of a structured matrix is structured, which allows fast inversion (or solution), and reduced storage requirements. According to two definitions of displacement structure of practical interest, it is shown here that several types of inverses are also structured, including the Moore-Penrose inverse of rank-deficient matrices.

José Henrique de Morais Goulart, Romain Couillet, Pierre Comon

Tensor models play an increasingly prominent role in many fields, notably in machine learning. In several applications, such as community detection, topic modeling and Gaussian mixture learning, one must estimate a low-rank signal from a noisy tensor. Hence, understanding the fundamental limits of estimators of that signal inevitably calls for the study of random tensors. Substantial progress has been recently achieved on this subject in the large-dimensional limit. Yet, some of the most significant among these results--in particular, a precise characterization of the abrupt phase transition (with respect to signal-to-noise ratio) that governs the performance of the maximum likelihood (ML) estimator of a symmetric rank-one model with Gaussian noise--were derived based of mean-field spin glass theory, which is not easily accessible to non-experts. In this work, we develop a sharply distinct and more elementary approach, relying on standard but powerful tools brought by years of advances in random matrix theory. The key idea is to study the spectra of random matrices arising from contractions of a given random tensor. We show how this gives access to spectral properties of the random tensor itself. For the aforementioned rank-one model, our technique yields a hitherto unknown fixed-point equation whose solution precisely matches the asymptotic performance of the ML estimator above the phase transition threshold in the third-order case. A numerical verification provides evidence that the same holds for orders 4 and 5, leading us to conjecture that, for any order, our fixed-point equation is equivalent to the known characterization of the ML estimation performance that had been obtained by relying on spin glasses. Moreover, our approach sheds light on certain properties of the ML problem landscape in large dimensions and can be extended to other models, such as asymmetric and non-Gaussian.

Jianze Li, Konstantin Usevich, Pierre Comon

In this paper, we study the approximate orthogonal diagonalization problem of third order symmetric tensors. We define several classes of approximately diagonal tensors, including the ones corresponding to the stationary points of this problem. We study the relationships between these classes, and other well-known objects, such as tensor Z-eigenvalue and Z-eigenvector. We also prove results on convergence of the cyclic Jacobi (or Jacobi CoM2) algorithm.

Jianze Li, Konstantin Usevich, Pierre Comon

In this paper, we consider a family of Jacobi-type algorithms for simultaneous orthogonal diagonalization problem of symmetric tensors. For the Jacobi-based algorithm of [SIAM J. Matrix Anal. Appl., 2(34):651--672, 2013], we prove its global convergence for simultaneous orthogonal diagonalization of symmetric matrices and 3rd-order tensors. We also propose a new Jacobi-based algorithm in the general setting and prove its global convergence for sufficiently smooth functions.

Pierre Comon, Yang Qi, Konstantin Usevich

In this paper, we study a polynomial decomposition model that arises in problems of system identification, signal processing and machine learning. We show that this decomposition is a special case of the X-rank decomposition --- a powerful novel concept in algebraic geometry that generalizes the tensor CP decomposition. We prove new results on generic/maximal rank and on identifiability of a particular polynomial decomposition model. In the paper, we try to make results and basic tools accessible for general audience (assuming no knowledge of algebraic geometry or its prerequisites).

Alex Pereira da Silva, Pierre Comon, Andre Lima Ferrer de Almeida

Because of the attractiveness of the canonical polyadic (CP) tensor decomposition in various applications, several algorithms have been designed to compute it, but efficient ones are still lacking. Iterative deflation algorithms based on successive rank-1 approximations can be used to perform this task, since the latter are rather easy to compute. We first present an algebraic rank-1 approximation method that performs better than the standard higher-order singular value decomposition (HOSVD) for three-way tensors. Second, we propose a new iterative rank-1 approximation algorithm that improves any other rank-1 approximation method. Third, we describe a probabilistic framework allowing to study the convergence of deflation CP decomposition (DCPD) algorithms based on successive rank-1 approximations. A set of computer experiments then validates theoretical results and demonstrates the efficiency of DCPD algorithms compared to other ones.

José Henrique De Morais Goulart, Maxime Boizard, Rémy Boyer et al.

The computation of a structured canonical polyadic decomposition (CPD) is useful to address several important modeling problems in real-world applications. In this paper, we consider the identification of a nonlinear system by means of a Wiener-Hammerstein model, assuming a high-order Volterra kernel of that system has been previously estimated. Such a kernel, viewed as a tensor, admits a CPD with banded circulant factors which comprise the model parameters. To estimate them, we formulate specialized estimators based on recently proposed algorithms for the computation of structured CPDs. Then, considering the presence of additive white Gaussian noise, we derive a closed-form expression for the Cramer-Rao bound (CRB) associated with this estimation problem. Finally, we assess the statistical performance of the proposed estimators via Monte Carlo simulations, by comparing their mean-square error with the CRB.

Konstantin Usevich, Pierre Comon

The completion of matrices with missing values under the rank constraint is a non-convex optimization problem. A popular convex relaxation is based on minimization of the nuclear norm (sum of singular values) of the matrix. For this relaxation, an important question is whether the two optimization problems lead to the same solution. This question was addressed in the literature mostly in the case of random positions of missing elements and random known elements. In this contribution, we analyze the case of structured matrices with fixed pattern of missing values, in particular, the case of Hankel and quasi-Hankel matrix completion, which appears as a subproblem in the computation of symmetric tensor canonical polyadic decomposition. We extend existing results on completion of rank-one real Hankel matrices to completion of rank-r complex Hankel and quasi-Hankel matrices.

Lek-Heng Lim, Pierre Comon

We study the decomposition of a nonnegative tensor into a minimal sum of outer product of nonnegative vectors and the associated parsimonious naive Bayes probabilistic model. We show that the corresponding approximation problem, which is central to nonnegative PARAFAC, will always have optimal solutions. The result holds for any choice of norms and, under a mild assumption, even Bregman divergences.