Simplex-structured matrix factorization (SSMF) is a common task encountered in signal processing and machine learning. Minimum-volume constrained unmixing (MVCU) algorithms are among the most widely used methods to perform this task. While MVCU algorithms generally perform well in an offline setting, their direct application to online scenarios suffers from scalability limitations due to memory and computational demands. To overcome these limitations, this paper proposes an approach which can build upon any off-the-shelf MVCU algorithm to operate sequentially, i.e., to handle one observation at a time. The key idea of the proposed method consists in updating the solution of MVCU only when necessary, guided by an online check of the corresponding optimization problem constraints. It only stores and processes observations identified as informative with respect to the geometrical constraints underlying SSMF. We demonstrate the effectiveness of the approach when analyzing synthetic and real datasets, showing that it achieves estimation accuracy comparable to the offline MVCU method upon which it relies, while significantly reducing the computational cost.

We study the estimation of a planted signal hidden in a recently introduced nested matrix-tensor model, which is an extension of the classical spiked rank-one tensor model, motivated by multi-view clustering. Prior work has theoretically examined the performance of a tensor-based approach, which relies on finding a best rank-one approximation, a problem known to be computationally hard. A tractable alternative approach consists in computing instead the best rank-one (matrix) approximation of an unfolding of the observed tensor data, but its performance was hitherto unknown. We quantify here the performance gap between these two approaches, in particular by deriving the precise algorithmic threshold of the unfolding approach and demonstrating that it exhibits a BBP-type transition behavior. This work is therefore in line with recent contributions which deepen our understanding of why tensor-based methods surpass matrix-based methods in handling structured tensor data.

Several machine learning problems such as latent variable model learning and community detection can be addressed by estimating a low-rank signal from a noisy tensor. Despite recent substantial progress on the fundamental limits of the corresponding estimators in the large-dimensional setting, some of the most significant results are based on spin glass theory, which is not easily accessible to non-experts. We propose a sharply distinct and more elementary approach, relying on tools from random matrix theory. The key idea is to study random matrices arising from contractions of a random tensor, which give access to its spectral properties. In particular, for a symmetric $d$th-order rank-one model with Gaussian noise, our approach yields a novel characterization of maximum likelihood (ML) estimation performance in terms of a fixed-point equation valid in the regime where weak recovery is possible. For $d=3$, the solution to this equation matches the existing results. We conjecture that the same holds for any order $d$, based on numerical evidence for $d \in \{4,5\}$. Moreover, our analysis illuminates certain properties of the large-dimensional ML landscape. Our approach can be extended to other models, including asymmetric and non-Gaussian ones.

Volterra models can accurately model numerous nonlinear systems of practical interest, but often at an unacceptable computational cost. If the Volterra kernels of a system have low-rank structure (like, e.g., kernels of bilinear systems), this major drawback can in principle be mitigated. Yet, when one seeks an exact discrete-time model of a mixed-signal chain involving that system, the existing formula that generalizes the impulse invariance principle to Volterra kernels yields discrete-time kernels that do not share the same low rank. At first sight this would seem to seriously complicate the otherwise simple discrete-time realization of low-rank kernels. We show here that this not the case. By defining a cascade operator, the structure of generalized impulse invariance can be unveiled, leading to a realization without an inordinate increase in computational complexity. Finally, we give a numerical example involving a physical system that shows the relevance of our proposal.

The estimation of the atrial activity (AA) signal from electrocardiogram (ECG) recordings is an important step in the noninvasive analysis of atrial fibrillation (AF), the most common sustained arrhythmia encountered in clinical practice. This problem admits a blind source separation (BSS) formulation that has been recently posed as a tensor factorization, using the Hankel-based block term decomposition (BTD), which is particularly well-suited to the estimation of exponential models like AA during AF. However, persistent forms of AF are characterized by short R-R intervals and very disorganized (or weak) AA, making it difficult to model AA directly and perform its successful extraction through Hankel-BTD. To overcome this drawback, the present work proposes a tensor approach to estimate QRS complexes and subtract them from the ECG, resulting in a signal that, ideally, only contains the AA component. Such an approach tackles the problem of blind separation of rational functions, which models QRS complexes explicitly. The data tensor admitting a BTD is built from Löwner matrices generated from each lead of the observed ECG. To this end, this paper formulates a variant of the recently proposed constrained alternating group lasso (CAGL) algorithm that imposes Löwner structure on the decomposition blocks. This is done by performing an orthogonal projection, which we explicitly derive, at each iteration of CAGL. Results from experiments with synthetic data show the consistency of the proposed Löwner-constrained AGL (LCAGL) in extracting the desired sources. Experimental results obtained on a population of 20 patients suffering from persistent AF show that the proposed variant outperforms other tensor-based methods in terms of atrial signal estimation quality from ECG records as short as a single heartbeat.

The Volterra-PARAFAC (VP) nonlinear system model, which consists of a FIR filterbank followed by a memoryless nonlinearity, aims at offering a good compromise between accuracy and parametric complexity. Here, for an even better compromise, we propose a generalization with IIR filters (VPI model) and evaluate both models. For the evaluation, we consider the concrete case of two audio loudspeakers and initially compute reference Volterra kernels from their known physical state-space models, using an efficient procedure. Then, VP and VPI models are derived and their accuracy is tested. As shown, the VPI models have in this case only 15 to 26 % of the parametric complexity of VP models with the same accuracy, which points to a great potential for accurate and efficient nonlinear system modeling.

Despite their generality, conventional Volterra filters are inadequate for some applications, due to the huge number of parameters that may be needed for accurate modelling. When a state-space model of the target system is known, this can be assessed by computing its kernels, which also provides valuable information for choosing an adequate alternate Volterra filter structure, if necessary, and is useful for validating parameter estimation procedures. In this letter, we derive expressions for the kernels by using the Carleman bilinearization method, for which an efficient algorithm is given. Simulation results are presented, which confirm the usefulness of the proposed approach.