The idea that one might extend numerical linear algebra, the collection of matrix computational methods that form the workhorse of scientific and engineering computing, to numerical multilinear algebra, an analogous collection of tools involving hypermatrices/tensors, appears very promising and has attracted a lot of attention recently. We examine here the computational tractability of some core problems in numerical multilinear algebra. We show that tensor analogues of several standard problems that are readily computable in the matrix (i.e. 2-tensor) case are NP hard. Our list here includes: determining the feasibility of a system of bilinear equations, determining an eigenvalue, a singular value, or the spectral norm of a 3-tensor, determining a best rank-1 approximation to a 3-tensor, determining the rank of a 3-tensor over R or C. Hence making tensor computations feasible is likely to be a challenge.

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 coherence, one could always guarantee the existence and uniqueness of a best rank-r approximation of a tensor. In particular, we obtain a computationally feasible variant of Kruskal’s uniqueness condition with coherence as a proxy for k-rank. We treat sparsest recovery and lowest-rank recovery problems in a uniform fashion by considering Schatten and nuclear norms of tensors of arbitrary order and dictionaries that comprise a continuum of uncountably many atoms. Résumé Traitement du signal multi-antenne: les décompositions tensorielles rejoignent l’échantillonnage compressé. Nous décrivons comment les techniques et outils d’échantillonnage compressé récemment découverts peuvent être utilisés dans les décompositions tensorielles, avec pour illustration une modélisation des signaux provenant de plusieurs antennes multicapteurs. Nous montrons qu’en posant des bornes appropriées sur la cohérence, on pouvait toujours garantir l’existence et l’unicité d’une meilleure approximation de rang r d’un tenseur. En particulier, nous obtenons une variante calculable de la condition d’unicité de Kruskal, où la cohérence apparaît comme une mesure du rang. Nous

Abstract—Modern applications such as web knowledge base, network traffic monitoring and online social networks have made available an unprecedented amount of network data with rich types of interactions carrying multiple attributes, for instance, port number and time tick in the case of network traffic. The design of algorithms to leverage this structured relationship with the power of computing to assist researchers and practitioners for better understanding, exploration and navigation of this space of information has become a challenging, albeit rewarding, topic in social network analysis and data mining. The constantly growing scale and enriching genres of network data always demand higher levels of efficiency, robustness and generalizability where existing approaches with successes on small, homogeneous network data are likely to fall short. We introduce MultiAspectForensics, a handy tool to automatically detect and visualize novel subgraph patterns within a local community of nodes in a heterogenous network, such as a set of vertices that form a dense bipartite graph whose edges share exactly the same set of attributes. We apply the proposed method on three data sets from distinct application domains, present empirical results and discuss insights derived from these patterns discovered. Our algorithm, built on scalable tensor analysis procedures, captures spectral properties of network data and reveals informative signals for subsequent domain-specific study and investigation, such as suspicious port-scanning activities in the scenario of cybersecurity monitoring. I.

We show that any explicit example for a tensor A: [n] r → F with tensor-rank ≥ nr·(1−o(1)) , (where r = r(n) ≤ log n / log log n), implies an explicit super-polynomial lower bound for the size of general arithmetic formulas over F. This shows that strong enough lower bounds for the size of arithmetic formulas of depth 3 imply superpolynomial lower bounds for the size of general arithmetic formulas. One component of our proof is a new approach for homogenization and multilinearization of arithmetic formulas, that gives the following results: We show that for any n-variate homogenous polynomial f of degree r, if there exists a (fanin-2) ( formula of size s and depth d for f then there exists a homogenous (d+r+1)) formula of size O r · s for f. In particular, for any r ≤ log n, if there exists a polynomial size formula for f then there exists a polynomial size homogenous formula for f. This refutes a conjecture of Nisan and Wigderson [NW95] and shows that superpolynomial lower bounds for homogenous formulas for polynomials of small degree imply super-polynomial lower bounds for general formulas. We show that for any n-variate set-multilinear polynomial f of degree r, if there exists a (fanin-2) formula of size s and depth d for f then there exists a set-multilinear formula of size O ((d + 2) r · s) for f. In particular, for any r ≤ log n / log log n, if there exists a polynomial size formula for f then there exists a polynomial size set-multilinear formula for f. This shows that super-polynomial lower bounds for set-multilinear formulas for polynomials of small degree imply super-polynomial lower bounds for general formulas.

Joint analysis of data from multiple sources has the potential to improve our understanding of the underlying structures in complex data sets. For instance, in restaurant recommendation systems, recommendations can be based on rating histories of customers. In addition to rating histories, customers ’ social networks (e.g., Facebook friendships) and restaurant categories information (e.g., Thai or Italian) can also be used to make better recommendations. The task of fusing data, however, is challenging since data sets can be incomplete and heterogeneous, i.e., data consist of both matrices, e.g., the person by person social network matrix or the restaurant by category matrix, and higher-order tensors, e.g., the “ratings ” tensor of the form restaurant by meal by person. In this paper, we are particularly interested in fusing data sets with the goal of capturing their underlying latent structures. We formulate this problem as a coupled matrix and tensor factorization (CMTF) problem where heterogeneous data sets are modeled by fitting outer-product models to higher-order tensors and matrices in a coupled manner. Unlike traditional approaches solving this problem using alternating algorithms, we propose an all-at-once optimization approach called CMTF-OPT (CMTF-OPTimization), which is a gradient-based optimization approach for joint analysis of matrices and higher-order tensors. We also extend the algorithm to handle coupled incomplete data sets. Using numerical experiments, we demonstrate that the proposed all-at-once approach is more accurate than the alternating least squares approach.

rank-k decompositions

When the geometry of 3D space is reconstructed from a pair of views, using the "Fundamental matrix" as the object of analysis, then it is known (as early as the 1940s) that there exists a "critical surface" for which the solution of 3-space is ambiguous. We show that when 3-space is reconstructed from a triplet of views, using the "Trilinear Tensor" as the object of analysis, there are no critical surfaces. In addition to theoretical interest of solving an open problem, this result has profound practical significance. The numerical instability associated with Structure from Motion is largely attributed to the existence of "critical volumes" that arise from the existence of critical surfaces coupled with errors in the image measurements. The lack of critical surfaces in the context of three views (provided that the trilinear tensor is used) suggests that better stability in the presence of errors can be gained. 1 Introduction The geometric relation between three-dimensional (3D) shape...

Introduction Given two n \Theta n matrices, A and B, we wish to examine how fast we can compute the product AB = C. The normal method takes time O(n 3 ) since each entry in C is computed as c i;k = n X j=1 a i;j b j;k ; which means that each of the n 2 entries in C takes time O(n) to compute. The first algorithm to do better than this obvious bound was given by Strassen in 1969 [3]. The idea of this algorithm is to divide each n \Theta n matrix into four parts of size n=2 \Theta n=2. If we view the submatrices of size n=2 \Theta n=2 as single elements, the prob

We present a new method for compact representation of large image datasets. Our method is based on treating small patches from a 2D image as matrices as opposed to the conventional vectorial representation, and encoding these patches as sparse projections onto a set of exemplar orthonormal bases, which are learned a priori from a training set. The end result is a low-error, highly compact image/patch representation that has significant theoretical merits and compares favorably with existing techniques (including JPEG) on experiments involving the compression of ORL and Yale face databases. In the context of learning multiple orthonormal bases, we show the easy tunability of our method to efficiently represent patches of different complexities. Furthermore, we show that our method is extensible in a theoretically sound manner to higher-order matrices (‘tensors’). We demonstrate applications of this theory to compression of well-known color image datasets such as the GaTech face database and show performance competitive with JPEG. Lastly, we also analyze the effect of image noise on the performance of our compression schemes. Index Terms compression, compact representation, sparse projections, singular value decomposition (SVD), higherorder singular value decomposition (HOSVD), greedy algorithm, tensor decompositions. I.

Sandia is a multiprogram laboratory operated by Sandia Corporation,