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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

© 1991-2012 IEEE. Multiway datasets are widespread in signal processing and play an important role in blind signal separation, array processing, and biomedical signal processing, among others. One key strength of tensors is that their decompositions are unique under mild conditions, which allows the recovery of features or source signals. In several applications, such as classification, we wish to compare factor matrices of the decompositions. Though this is possible by first computing the tensor decompositions and subsequently comparing the factors, these decompositions are often computationally expensive. In this paper, we present a similarity method that indicates whether the factors in two modes are essentially equal without explicitly computing them. Essential equality conditions, which ensure the theoretical validity of our approach, are provided for various underlying tensor decompositions. The developed algorithm provides a computationally efficient way to compare factors. The method is illustrated in a context of emitter movement detection and fluorescence data analysis. ispartof: IEEE Transactions on Signal Processing vol:66 issue:5 pages:1273-1285 status: published

© 2017 IEEE. Nonnegative matrix factorizationisakey toolinmany data analysis applications such as feature extraction, compression, and noise filtering. Many existing algorithms impose additional constraintstotake into account prior knowledgeandtoimprove the physical interpretation. This letter proposes a novel algorithm for nonnegative matrix factorization, in which the factors are modeled by nonnegative polynomials. Using a parametric representation of finite-interval nonnegative polynomials, we obtain an optimization problem without external nonnegativity constraints, which can be solved using conventional quasi-Newton or nonlinear least-squares methods. The polynomial model guarantees smooth solutions and may realize a noise reduction. A dedicated orthogonal compression enables a significant reduction of the matrix dimensions, without sacrificing accuracy. The overall approach scales well to large matrices. The approach is illustrated with applications in hyperspectral imaging and chemical shift brain imaging. ispartof: IEEE Signal Processing Letters vol:24 issue:7 pages:948-952 status: published

© 1991-2012 IEEE. Many real-life signals are compressible, meaning that they depend on much fewer parameters than their sample size. In this paper, we use low-rank matrix or tensor representations for signal compression. We propose a new deterministic method for blind source separation that exploits the low-rank structure, enabling a unique separation of the source signals and providing a way to cope with large-scale data. We explain that our method reformulates the blind source separation problem as the computation of a tensor decomposition, after reshaping the observed data matrix into a tensor. This deterministic tensorization technique is called segmentation and is closely related to Hankel-based tensorization. We apply the same strategy to the mixing coefficients of the blind source separation problem, as in many large-scale applications, the mixture is also compressible because of many closely located sensors. Moreover, we combine both strategies, resulting in a general technique that allows us to exploit the underlying compactness of the sources and the mixture simultaneously. We illustrate the techniques for fetal electrocardiogram extraction and direction-of-arrival estimation in large-scale antenna arrays. ispartof: IEEE Transactions on Signal Processing vol:65 issue:2 pages:346-358 status: published

© 1991-2012 IEEE. A new blind signal separation (BSS) technique is proposed, enabling a deterministic separation of signals into rational functions. Rational functions can take on a wide range of forms, such as the well-known pole-like shape. The approach is a possible alternative for the well-known independent component analysis when the theoretical sources are not independent, such as for frequency spectra, or when only a small number of samples is available. The technique uses a low-rank decomposition on the tensorized version of the observed data matrix. The deterministic tensorization with Löwner matrices is comprehensively analyzed in this paper. Uniqueness properties are investigated, and a connection with the separation into exponential polynomials is made. Finally, the technique is illustrated for fetal electrocardiogram extraction and with an application in the domain of fluorescence spectroscopy, enabling the identification of chemical analytes using only a single excitation-emission matrix. ispartof: IEEE Transactions on Signal Processing vol:64 issue:8 pages:1909-1918 status: published

© 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

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.