Joint blind source separation (J-BSS) is an emerging data-driven technique for multi-set data-fusion. In this paper, J-BSS is addressed from a tensorial perspective. We show how, by using second-order multi-set statistics in J-BSS, a specific double coupled canonical polyadic decomposition (DC-CPD) problem can be formulated. We propose an algebraic DC-CPD algorithm based on a coupled rank-1 detection mapping. This algorithm converts a possibly underdetermined DC-CPD to a set of overdetermined CPDs. The latter can be solved algebraically via a generalized eigenvalue decomposition based scheme. Therefore, this algorithm is deterministic and returns the exact solution in the noiseless case. In the noisy case, it can be used to effectively initialize optimization based DC-CPD algorithms. In addition, we obtain the deterministic and generic uniqueness conditions for DC-CPD, which are shown to be more relaxed than their CPD counterpart. We also introduce optimization based DC-CPD methods, including alternating least squares, and structured data fusion based methods. Experiment results are given to illustrate the superiority of DC-CPD over standard CPD based BSS methods and several existing J-BSS methods, with regards to uniqueness and accuracy.

Tensor decompositions play an important role in a variety of applications, such as signal processing and machine learning. In practice, the tensor can be incomplete or very large, making it difficult to analyze and process using conventional tensor techniques. In this paper we focus on the basic canonical polyadic decomposition (CPD). We propose an algebraic framework for finding the CPD of tensors that have missing fibers. This includes extensions of multilinear algebraic as well as generic uniqueness conditions originally developed for the CPD of fully observed tensors. Computationally, we reduce the CPD of a tensor with missing fibers to relatively simple matrix completion problems via a matrix eigenvalue decomposition (EVD). Under the given conditions, the EVD-based algorithm is guaranteed to return the exact CPD. The derivation establishes connections with so-called coupled CPDs, an emerging concept that has proven to be of great interest in a range of array processing and signal processing applications. It will become clear that relatively few fibers are needed in order to compute the CPD. This makes fiber sampling interesting for large scale tensor decompositions. Numerical experiments show that the algebraic framework may significantly speed up more common optimization-based computation schemes for the estimation of the CPD of incomplete noisy data tensors.

During the past 20 years, low-rank tensor and matrix decomposition models (LRDMs) have become indispensable tools for signal processing, machine learning, and data science. LRDMs represent high-dimensional, multiaspect, and multimodal data using low-dimensional latent factors in a succinct and parsimonious way. LRDMs can serve a variety of purposes, e.g., data embedding (dimensionality reduction), denoising, latent variable analysis, model parameter estimation, and big data compression; see [1]-[5] for surveys of applications.

The Canonical Polyadic Decomposition (CPD) plays an important role for signal separation in array processing. The CPD model requires arrays composed of several displaced but identical subarrays. Consequently, it is less appropriate for more complex array geometries. In this paper we explain that coupled CPD allows a much more flexible modeling that can handle multiple shift-invariance structures, i.e., arrays that can be decomposed into multiple but not identical displaced subarrays. Both deterministic and generic identifiability conditions are presented. We also point out that, under mild conditions, the signal separation problem can in the exact case be solved by means of an eigenvalue decomposition. This is similar to ESPRIT, although the working conditions are much more relaxed. Borrowing tools from algebraic geometry, we derive generic uniqueness bounds for L-shaped, frame-shaped and triangular-shaped arrays that come close to bounds that are necessary for uniqueness. Recognizing multiple shift-invariance can be a bit of an art by itself. We explain that any centro-symmetric array processing problem can be interpreted in terms of a coupled CPD. In addition, we demonstrate that the coupled CPD model allows us to significantly relax the far-field assumption commonly used in CPD-based array processing.

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. Wh...