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© 2019 Society for Industrial and Applied Mathematics Decomposing tensors into simple terms is often an essential step toward discovering and understanding underlying processes or toward compressing data. However, storing the tensor and computing its decomposition is challenging in a large-scale setting. Though in many cases a tensor is structured, it can be represented using few parameters: a sparse tensor is determined by the positions and values of its nonzeros, a polyadic decomposition by its factor matrices, a Tensor Train by its core tensors, a Hankel tensor by its generating vector, etc. The complexity of tensor decomposition algorithms can be reduced significantly in terms of time and memory if these efficient representations are exploited directly. Only a few core operations such as norms and inner products need to be specialized to achieve this, thereby avoiding the explicit construction of multiway arrays. To improve the interpretability of tensor models, constraints are often imposed or multiple datasets are fused through joint factorizations. While imposing these constraints prohibits the use of traditional compression techniques, our framework allows constraints and compression, as well as other efficient representations, to be handled trivially as the underlying optimization variables do not change. To illustrate this, large-scale nonnegative tensor factorization is performed using MLSVD and Tensor Train compression. We also show how vector and matrix data can be analyzed using tensorization while keeping a vector or matrix complexity through the concept of implicit tensorization, as illustrated for Hankelization and Löwnerization. The concepts and numerical properties are extensively investigated through experiments. ispartof: SIAM Journal of Scientific Computing vol:41 issue:2 pages:A789-A815 status: published

Many real-life signals can be described in terms of much fewer parameters than the actual number of samples. Such compressible signals can often be represented very compactly with low-rank matrix and tensor models. The authors have adopted this strategy to enable large-scale instantaneous blind source separation. In this paper, we generalize the approach to the blind identification of large-scale convolutive systems. In particular, we apply the same idea to the system coefficients of finite impulse response systems. This allows us to reformulate blind system identification as a structured tensor decomposition. The tensor is obtained by applying a deterministic tensorization technique called segmentation on the observed output data. Exploiting the low-rank structure of the system coefficients enables a unique identification of the system and estimation of the inputs. We obtain a new type of deterministic uniqueness conditions. Moreover, the compactness of the low-rank models allows one to solve large-scale problems. We illustrate our method for direction-of-arrival estimation in large-scale antenna arrays and neural spike sorting in high-density microelectrode arrays.

Fitting a signal to a sum-of-exponentials model is a basic problem in signal processing. It can be posed and solved as a Hankel structured low-rank matrix approximation problem. Subsequently, local optimization, subspace, and convex relaxation methods can be used for the numerical solution. In this paper, we show another approach, based on the recently developed concept of structured data fusion. Structured data fusion problems are solved in the Tensorlab toolbox by local optimization methods. The approach allows fitting of signals with missing samples and adding constraints on the model, such as fixed exponents and common dynamics in multi-channel estimation problems. These problems are non-trivial to solve by other existing methods. Tensorlab is publicly available and the results presented are reproducible.

© 1991-2012 IEEE. Blind system identification (BSI) is an important problem in signal processing, arising in applications such as wireless telecommunications, biomedical signal processing, and seismic signal processing. In the past decades, tensors have proven to be useful tools for these blind identification and separation problems. Most often, tensor-based methods based on fourth-order statistics are used, which have been studied extensively for independent component analysis and its convolutive extensions. However, these tensor-based methods have two main drawbacks: the accuracy is often limited by the estimation error of the statistics and the computation of these fourth-order statistics is time intensive. In this paper, we propose to counter these drawbacks for BSI by coupling the fourth-order statistics with second-order statistics and by using incomplete tensors. By doing so, we can obtain more accurate results or obtain results in a much faster way. ispartof: IEEE TRANSACTIONS ON SIGNAL PROCESSING vol:66 issue:23 pages:6137-6147 status: published

Various parameters influence face recognition such as expression, pose, and illumination. In contrast to matrices, tensors can be used to naturally accommodate for the different modes of variation. The multilinear singular value decomposition (MLSVD) then allows one to describe each mode with a factor matrix and the interaction between the modes with a coefficient tensor. In this paper, we show that each image in the tensor satisfying an MLSVD model can be expressed as a structured linear system called a Kronecker Product Equation (KPE). By solving a similar KPE for a new image, we can extract a feature vector that allows us to recognize the person with high performance. Additionally, more robust results can be obtained by using multiple images of the same person under different conditions, leading to a coupled KPE. Finally, our method can be used to update the database with an unknown person using only a few images instead of an image for each combination of conditions. We illustrate our method for the extended Yale Face Database B, achieving better performance than conventional methods such as Eigenfaces and other tensor-based techniques.

Cardiac arrhythmia or irregular heartbeats are an important feature to assess the risk on sudden cardiac death and other cardiac disorders. Automatic classification of irregular heartbeats is therefore an important part of ECG analysis. We propose a tensor-based method for single- and multi-channel irregular heartbeat classification. The method tensorizes the ECG data matrix by segmenting each signal beat-by-beat and then stacking the result into a third-order tensor with dimensions channel × time × heartbeat. We use the multilinear singular value decomposition to model the obtained tensor. Next, we formulate the classification task as the computation of a Kronecker Product Equation. We apply our method on the INCART dataset, illustrating promising results. ispartof: pages:438-441 ispartof: Proc. 39th Annual International Conference of the IEEE Engineering in Medicine & Biology Society vol:2017 pages:438-441 ispartof: EMBC 2017 location:JeJu Island, South Korea date:Jul - Jul 2017 status: published

We give an overview of recent developments in numerical optimization-based computation of tensor decompositions that have led to the release of Tensorlab 3.0 in March 2016 (www.tensorlab.net). By careful exploitation of tensor product structure in methods such as quasi-Newton and nonlinear least squares, good convergence is combined with fast computation. A modular approach extends the computation to coupled factorizations and structured factors. Given large datasets, different compact representations (polyadic, Tucker,…) may be obtained by stochastic optimization, randomization, compressed sensing, etc. Exploiting the representation structure allows us to scale the algorithms for constrained/coupled factorizations to large problem sizes.

© 2016 IEEE. Coupled decompositions of multiple tensors are fundamental tools for multi-set data fusion. In this paper, we introduce a coupled version of the rank-(Lm, Ln, •) block term decomposition (BTD), applicable to joint independent subspace analysis. We propose two algorithms for its computation based on a coupled block simultaneous generalized Schur decomposition scheme. Numerical results are given to show the performance of the proposed algorithms. ispartof: pages:2554-2558 ispartof: Proc. 2016 IEEE International Conference on Acoustics, Speech and Signal Processing vol:2016-May pages:2554-2558 ispartof: ICASSP 2016 location:Shanghai, China date:Mar - Mar 2016 status: published

© 2015 EURASIP. A novel deterministic method for blind source separation is presented. In contrast to common methods such as independent component analysis, only mild assumptions are imposed on the sources. On the contrary, the method exploits a hypothesized (approximate) intrinsic low-rank structure of the mixing vectors. This is a very natural assumption for problems with many sensors. As such, the blind source separation problem can be reformulated as the computation of a tensor decomposition by applying a low-rank approximation to the tensorized mixing vectors. This allows the introduction of blind source separation in certain big data applications, where other methods fall short. ispartof: pages:1890-1894 ispartof: Proc. of the 23rd European Signal Processing Conference pages:1890-1894 ispartof: 23rd European Signal Processing Conference (EUSIPCO) location:Nice, France date:31 Aug - 4 Sep 2015 status: published

A novel deterministic blind signal separation technique for separating signals into rational functions is proposed, applicable in various situations. This new technique is based on a tensorization of the observed data matrix into a set of Lowner matrices. The obtained tensor can then be decomposed with a block tensor decomposition, resulting in a unique separation into rational functions under mild conditions. This approach provides a viable alternative to independent component analysis (ICA) in cases where the independence assumption is not valid or where the sources can be modeled well by rational functions, such as frequency spectra. In contrast to ICA, this technique is deterministic and not based on statistics, and therefore works well even with a small number of samples.

© Springer International Publishing Switzerland 2015. Given an instantaneous mixture of some source signals, the blind signal separation (BSS) problem consists of the identification of both the mixing matrix and the original sources. By itself, it is a non-unique matrix factorization problem, while unique solutions can be obtained by imposing additional assumptions such as statistical independence. By mapping the matrix data to a tensor and by using tensor decompositions afterwards, uniqueness is ensured under certain conditions. Tensor decompositions have been studied thoroughly in literature. We discuss the matrix to tensor step and present tensorization as an important concept on itself, illustrated by a number of stochastic and deterministic tensorization techniques. ispartof: pages:3-13 ispartof: Lecture Notes in Computer Science vol:9237 pages:3-13 ispartof: 12th International Conference on Latent Variable Analysis and Signal Separation location:CZECH REPUBLIC, Tech Univ Liberec, Liberec date:25 Aug - 28 Aug 2015 status: published