This survey provides an overview of higher-order tensor decompositions, their applications, and available software. A tensor is a multidimensional or $N$-way array. Decompositions of higher-order tensors (i.e., $N$-way arrays with $N \geq 3$) have applications in psycho-metrics, chemometrics, signal processing, numerical linear algebra, computer vision, numerical analysis, data mining, neuroscience, graph analysis, and elsewhere. Two particular tensor decompositions can be considered to be higher-order extensions of the matrix singular value decomposition: CANDECOMP/PARAFAC (CP) decomposes a tensor as a sum of rank-one tensors, and the Tucker decomposition is a higher-order form of principal component analysis. There are many other tensor decompositions, including INDSCAL, PARAFAC2, CANDELINC, DEDICOM, and PARATUCK2 as well as nonnegative variants of all of the above. The N-way Toolbox, Tensor Toolbox, and Multilinear Engine are examples of software packages for working with tensors.

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Tensor Decompositions and Applications

The Theory of Matrices. By F R. Gantmacher. Two volumes, pp. 374 and 276. 1959. (Translated from the Russian by K. A. Hirsch; Chelsea Publishing Company, New York)

Tensors or multiway arrays are functions of three or more indices $(i,j,k,\ldots)$ —similar to matrices (two-way arrays), which are functions of two indices $(r,c)$ for (row, column). Tensors have a rich history, stretching over almost a century, and touching upon numerous disciplines; but they have only recently become ubiquitous in signal and data analytics at the confluence of signal processing, statistics, data mining, and machine learning. This overview article aims to provide a good starting point for researchers and practitioners interested in learning about and working with tensors. As such, it focuses on fundamentals and motivation (using various application examples), aiming to strike an appropriate balance of breadth and depth that will enable someone having taken first graduate courses in matrix algebra and probability to get started doing research and/or developing tensor algorithms and software. Some background in applied optimization is useful but not strictly required. The material covered includes tensor rank and rank decomposition; basic tensor factorization models and their relationships and properties (including fairly good coverage of identifiability); broad coverage of algorithms ranging from alternating optimization to stochastic gradient; statistical performance analysis; and applications ranging from source separation to collaborative filtering, mixture and topic modeling, classification, and multilinear subspace learning.

Tensor Decomposition for Signal Processing and Machine Learning

The widespread use of multisensor technology and the emergence of big data sets have highlighted the limitations of standard flat-view matrix models and the necessity to move toward more versatile data analysis tools. We show that higher-order tensors (i.e., multiway arrays) enable such a fundamental paradigm shift toward models that are essentially polynomial, the uniqueness of which, unlike the matrix methods, is guaranteed under very mild and natural conditions. Benefiting from the power of multilinear algebra as their mathematical backbone, data analysis techniques using tensor decompositions are shown to have great flexibility in the choice of constraints which match data properties and extract more general latent components in the data than matrix-based methods.

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Tensor Decompositions for Signal Processing Applications: From two-way to multiway component analysis

We prove that multilinear (tensor) analogues of many efficiently computable problems in numerical linear algebra are NP-hard. Our list includes: determining the feasibility of a system of bilinear equations, deciding whether a 3-tensor possesses a given eigenvalue, singular value, or spectral norm; approximating an eigenvalue, eigenvector, singular vector, or the spectral norm; and determining the rank or best rank-1 approximation of a 3-tensor. Furthermore, we show that restricting these problems to symmetric tensors does not alleviate their NP-hardness. We also explain how deciding nonnegative definiteness of a symmetric 4-tensor is NP-hard and how computing the combinatorial hyperdeterminant is NP-, #P-, and VNP-hard.

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Most Tensor Problems Are NP-Hard

We find generators for the ideals of secant varieties of Segre varieties in several the cases including the Garcia-Stillmann-Ssturmfels conjecture for four factors and prove results about their singularities.

On the ideals and singularities of secant varieties of Segre varieties

Conjecture 1.1. Let Xn−1 ⊂ P n be a hypersurface of degree d ≥ n and let F(X) ⊂ G(2, n+ 1) denote the Fano scheme of lines on X. Let B ⊂ F(X) be an irreducible component of dimension at least n − 2. Let IB := {(x,E) | x ∈ X, E ∈ B, x ∈ PE}, and let π and ρ denote (respectively) the projections to X and B. Let XB = π(IB) ⊆ X and let Cx = πρ−1ρπ−1(x). Then, for all x ∈XB , Cx ∩Xsing = ∅. If we take hyperplane sections in the case d = n, then Conjecture 1.1 would imply the following, which was conjectured independently by Debarre and de Jong.

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On the Debarre-de Jong and Beheshti-Starr conjectures on hypersurfaces with too many lines

A $2{\mathbf{n}}^2-{\text{log}}_2({\mathbf{n}})-1$ lower bound for the border rank of matrix multiplication

We classify tensors with maximal and next to maximal dimensional symmetry groups under a natural genericity assumption (1-genericity), in dimensions greater than 13. In other words, we classify minimal dimensional orbits in the space of (m,m,m) tensors assuming 1-genericity. Our study uncovers new tensors with striking geometry. This paper was motivated by Strassen's laser method for bounding the exponent of matrix multiplication. The best known tensor for the laser method is the large Coppersmith-Winograd tensor, and our study began with the observation that it has a large symmetry group, of dimension m^2/2 +m/2. We show that in odd dimensions, this is the largest possible for a 1-generic tensor, but in even dimensions we exhibit a tensor with a larger dimensional symmetry group. In the course of the proof, we classify nondegenerate bilinear forms with large dimensional stabilizers, which may be of interest in its own right.

Tensors with maximal symmetries.

We answer a question in [Landsberg, Ressayre, 2015], showing the regular determinantal complexity of the determinant det_m is O(m^3). We answer questions in, and generalize results of [Aravind, Joglekar, 2015], showing there is no rank one determinantal expression for perm_m or det_m when m >= 3. Finally we state and prove several "folklore" results relating different models of computation.

Joseph M. Landsberg

Papers

On the ideals and singularities of secant varieties of Segre varieties

On the Debarre-de Jong and Beheshti-Starr conjectures on hypersurfaces with too many lines

A $2{\mathbf{n}}^2-{\text{log}}_2({\mathbf{n}})-1$ lower bound for the border rank of matrix multiplication

Tensors with maximal symmetries.

On the complexity of the permanent in various computational models