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Tensors in signal processing

Higher-order tensors (HOTs) can be viewed as multiway arrays having order (number of ways or modes) greater than two. HOTs are very useful for processing and analyzing large-scale multimodal data sets, i.e. big data. Decomposing tensors into factor matrices or tensors is a powerful tool for storing, representing, visualizing and classifying (possibly noisy and incomplete) large data sets, and also for designing complex systems such as, e.g., MIMO wireless communication systems. In particular, a recently proposed class of tensor decompositions (TD) called tensor train (TT) decomposes a tensor into a train of interconnected tensors of low (second and third) order. In certain signal processing applications, factor matrices of TDs can be Toeplitz, Hankel, Vandermonde or circulant. Efficient algorithms are needed for estimating the parameters of such structured TDs, and also for recovering missing values of an incomplete data tensor. The latter problem, called tensor completion (TC), or more generally tensor recovery when measurements are acquired by means of an arbitrary linear operator, is usually solved under a low-rank assumption. Low-rank tensor recovery (LRTR) constitutes an extension of the low-rank matrix recovery (LRMR) problem, which is itself an extension of compressive sensing (CS), i.e. the estimation of sparse signals from undercomplete linear measurements. In the case of HOTs, sparsity is linked to the low-rank assumption, which implies a reduced number of degrees of freedom.
The aim of our research activity is two-fold. On one hand, we develop new tensor decompositions, studying their uniqueness/identifiability properties, and numerical algorithms for estimating structured tensor decompositions (with performance analysis by means of Cramér-Rao bounds) and for addressing the LRTR problem. On the other hand, we exploit TDs to (i) design new MIMO (point-to-point and cooperative) wireless communication systems, (ii) model and identify nonlinear dynamic systems, (iii) recover incomplete hyperspectral images and road traffic data and (iv) analyze biomedical signals (ECG, EEG, MEG). In a recent past, we have also applied multilinear analysis tools for data compression and target recognition/classification in synthetic aperture radar (SAR) images [hal-00719293].
In the following, we briefly present our main results concerning tensor models, estimation algorithms for structured tensors and LRTR, and tensor-based solutions for the design of wireless communication systems and for the modeling/identification of nonlinear systems.
Tensor models
During the last decade, our research works on wireless communication systems has led to several new tensor models: block PARAFAC [hal-00417636], constrained PARAFAC (CONFAC) [hal-00417627], PARATUCK [hal-01246045], generalized PARATUCK [hal-01245336, hal-01246038], nested PARAFAC [hal-01246044, hal-01246037, hal-01312376], and nested Tucker [hal-01312373], these last two ones being particular cases of TT decompositions. We recently presented an overview of constrained PARAFAC models in [hal-01246040] and an overview of tensor decompositions applied to communications in [hal-01246056].
Parameter estimation algorithms for structured CPD
We have studied the estimation of CP decompositions (CPD) characterized by structured matrix factors. Such TDs arise in signal processing applications like Wiener-Hammerstein system identification and cumulant-based communication channel estimation. A general formulation of the considered structured CPDs (SCPDs) was introduced, specialized parameter estimators were devised, and closed-form expressions for Cramér-Rao bounds (CRBs) were established under the assumption of additive white Gaussian noise [hal-01072538, hal-01118725, hal-01169880, hal-01246855]. In the case of circulant-constrained CPDs (CCPDs), i.e. CPDs with circulant factor matrices, we proposed a novel algebraic approach based on applying the multidimensional Fourier transform to the data tensor, which leads to a set of homogeneous monomial equations whose solutions provide estimates for the parameters of the CCPD [hal-00967263].
Algorithms for low-rank tensor recovery
TC and LRTR find applications in many fields such as computer vision, medical and hyper-spectral imaging, radar and seismic data processing, text mining and recommender systems. Numerous methods have been proposed in the literature for solving these problems, each  one relying on three fundamental choices: a TD for modeling the data tensor, with an underlying notion of rank, a cost function to be minimized, and an optimization algorithm. The formulated optimization problem aims at finding the tensor model which best fits the data and has the lowest possible rank. One feature of HOTs is that, contrarily to matrices, there exist different notions of rank, each one being associated with a different class of TDs, i.e. CP, Tucker and TT ranks.  Choosing an appropriate tensor rank is a key issue for TC/LRTR. Our work has been focused on the iterative hard thresholding (IHT) approach, which is very popular in CS and LRMR, using Tucker decompositions with a low multilinear rank constraint, i.e. assuming that the dimensions of the modal subspaces are low. IHT algorithms are very attractive due to their simplicity and effectiveness. They are composed of two steps: one gradient-type correction followed by a projection onto a set of low-multilinear rank tensors, which consists in computing a best multilinear rank approximation. We proposed two main contributions: (i) a new step size selection heuristic for accelerating the convergence of tensor IHT (TIHT) [hal-01132367]; and (ii) a new TIHT algorithm using sequentially optimal modal projections, named SeMPIHT, which is less costly than TIHT based on the standard truncated high-order SVD (HOSVD). Recovery guarantees relying on restricted isometry constants were established for SeMPIHT. A continuation technique for gradually increasing the model complexity via its multilinear rank was also developed, offering the double advantage of a estimation error stabilization and a convergence acceleration. The performance of SeMPIHT was illustrated with the completion of hyperspectral imaging data [hal-01364008, submitted to SIAM’2016 (adresse hal à compléter)].

Tensor-based wireless communication systems

Our main contributions concerning tensor-based approaches for wireless communication systems are the following:


Tensor-based modeling and identification of nonlinear systems
Our pioneering work on this topics started ten years ago with the purpose of reducing the parametric complexity of Volterra models using a PARAFAC decomposition of high-order Volterra kernels viewed as tensors, which led to a new class of models called Volterra-PARAFAC [hal-00718858, hal-00477178, hal-00642363]. When the rank of the kernels is small with respect to their memory, such kernels decompositions result in a drastic parametric complexity reduction. The HOSVD was also used for decomposing high-order Volterra kernels [hal-00417549, hal-00576019].
In the context of satellite and radio over fiber multiuser communications, baseband Volterra models are used for representing channel nonlinearities due to high power amplifiers and electro-optic converters. Exploiting the double symmetry of Volterra kernels, baseband Volterra-PARAFAC models were derived, and adaptive algorithms were developed for estimating the parameters of these models [hal-00718858, hal-00718654].
The class of block-structured nonlinear (NL) systems represented by means of Wiener, Hammerstein, or Wiener-Hammerstein models was also considered. Original tensor-based estimation algorithms were developed for such block-structured NL models [hal-00417555, hal-01246201, hal-01246209].  Another work consisted in applying a tensor analysis for determining the structure of block-structured NL SISO systems, like Wiener, Hammerstein and Wiener-Hammerstein systems [hal-00417815].

We have also to mention some research activities related to tensors conducted by Pierre Comon before his departure to GIPSA-Lab (Grenoble), in August 2012: (i) estimation of structured and nonnegative tensor decompositions [hal-00641065, hal-00641052, hal-00618729, hal-00781143, hal-00740572]; (ii) blind source separation of underdetermined mixtures [hal-00537838, hal-00739130, hal-00952039]; (iii) tensor-based processing for source localization and extraction, particularly with EEG and MEG data [hal-00683304, hal-01011856, hal-01190559, hal-00725280, hal-01012083, hal-00990273].

A more recent line of research has looked into the application of tensor decompositions to cardiac signal analysis, in particular in the context of atrial fibrillation, the most common sustained cardiac arrhythmia encountered in clinical practice. We have shown that the noninvasive extraction of atrial activity from electrocardiogram recordings accepts a block term decomposition model [hal-01302686, hal-01302671]. Experimental results demonstrate the superior performance of the tensor approach as compared with alternative blind source separation methods, especially in short data records with limited spatial diversity (reduced number of electrodes). However, the automatic selection of the tensor model parameters remains challenging. To circumvent this difficulty, a preprocessing stage reducing the influence of ventricular artifacts is shown to simplify the tensor model and ease parameter selection while maintaining a satisfactory level of performance [hal-01330512].


Biomedical/Biological signal processing

Researchers working on biomedical signals aim to develop strong local collaborations around the processing, analysis and modeling of signals issued from Medicine and Biology.
Inheriting from the former BIOMED group at I3S, ongoing research partnerships include the Cardiology Departments of Nice University Hospital (CHU) and Monaco Princess Grace Hospital (CHPG), as well as Nice University Sports Department (STAPS).

The Biomedical Signal Processing activity is fully hosted by the Signal team, where we share a common interest for signal processing tools. Based on a solid background in modeling and signal processing, the involved team members have succeeded in developing new signal processing methods adapted to medical demand and clinical goals. The group’s research interests mainly focus on the analysis of heart diseases, with special emphasis on cardiac arrhythmias such as atrial fibrillation, and heart physiology coupled with respiration and blood pressure through exercise records. The modeling of physiological processes is also under the scope of this activity.

An important original goal of this activity is to strengthen and consolidate an "information technology for health" (STIC-Santé) regional cluster of research into heart diseases. This local network is founded on research contracts and open collaborations. Joint researches are not restricted to local collaborations but are also open to international partners such as the IBBE (Warsaw, Poland), Instituto di fisiologica clinica (CNR Roma, Italy), University of Maastricht (DKE, the netherlands), UFSC and UFRJ (Brazil).

The main contributions of this research activity lie in the fields of:
  • Cardiorespiratory coupling and heart rhythm during exercise
  • Electrocardiology
  • Electromyography
  • Biomedical optics 
  • Bio-inspired communications
  • Signal processing theory and methods

Cardiorespiratory coupling and heart rhythm during exercise
Jointly with Dr. S. Bermon (IM2S Monaco) and G. Blain (MCF UNS), we have analyzed the interaction between the autonomous nervous system and the heart rhythm, with emphasis on the influence of the respiratory system. This relation has been accurately quantified during intense exercise where we have shown that the heart rhythm is not only mechanically and neurally modulated by respiration but also by the locomotor activity [IJ-11]. Intense exercise provides a good experimental condition enriching the basal inputs of the cardiac system. This property has been advantageously exploited to develop a new relation linking the PR and the RR intervals [IJ-20], revealing for the first time a hysteresis behavior. This finding has been developed in addition to more theoretical studies in time delay estimation [IJ-19].

Research into this broad field has focused on the analysis of signals recorded by small (standard 12 leads) or large (Body Surface Potential Maps, BSPM) sets of sensors located on the body surface. Standard 12-lead systems have been used in P-wave shape clustering for obstructive sleep apnoea detection [IJ-16]. This lead system has also been the data generator of very intensive research on atrial fibrillation (AF), the most prevalent cardiac arrhythmia encountered in clinical practice yet still not fully understood. The originality of our approach lies in the exploitation of the multi-lead character of surface recording modalities through the blind source separation (BSS) model. The AF signal has been analyzed after the application of specific BSS techniques for atrial signal extraction developed in the group [IJ-102, IJ-101, IJ-124]. A non-invasive quantitative characterization of the spatio-temporal organization of AF has been successfully achieved using BSPM recordings and data analysis techniques inspired on the BSS model [IJ-14]. The flexibility of BSPM recordings has been also illustrated by applying original shape analysis for the characterization of the ECG spatial variability of patient suffering from myocardial infarction [Th-11, NJ-3].
In close collaboration with Monaco CHPG, the “PERSIST” Young Investigator’s project funded by the ANR (2010-2014) has taken a step forward in the multi-lead characterization of the atrial signal observed on the surface ECG to better predict the outcome of the catheter ablation therapy for persistent AF [hal-00862201hal-00806555, hal-01302694]. The complexity of cardiac arrhythmias calls for multiscale approaches. If therapy outcome can be predicted by processing multidimensional data at organ level (surface ECG), we have also examined lower scale bioelectrical data (cardiomyocytes’ action potentials) in collaboration with Harvard Medical School, Cambridge, USA, where it appears that diabetes increases repolarization variability in agreement with organ-level observations [hal-01276906].

Jointly with the laboratory LAMHESS of Nice University Sports Department (STAPS), work on electromyographic signals (EMG) recorded during vibration has been carried out since 2011. Whole-body vibration has become one of the most popular alternative exercise modalities for high-skilled and recreational athletes, as well as for older and health compromised individuals. Despite the widespread use of whole-body vibration platforms, its scientific investigation lags behind. Even though regular whole-body vibration exercises have been shown to increase sports performance, the underlying neurophysiological mechanisms remain unclear. Another issue to be addressed is the corrupted EMG signal that occurs during vibration at the excitation frequency and its multiple harmonics. Without correction for these motion artifacts, the EMG signal results in overestimation, whereas with the application of common filters, the signal becomes underestimated. Therefore, we aim to develop an innovative approach for the investigation of human neurophysiological signals during vibration exercises.

Biomedical optics
Dealing with Laser-Doppler micro perfusion and oxygenation signals from near infrared spectroscopy, the analysis and decomposition of Laser-Doppler spectra have led us to a new method for estimating the light scattering phase function [hal-00786281, hal-00642672]. Together with a novel acquisition setup this method is now a promising diagnostic technique allowing for non-invasive assessment of tissue microperfusion (collaboration with the Institute of Biocybernetics and Biomedical Engineering, Warsaw) 

Bio-inspired communications
From previous work in the field of audio signal analysis (modulation spectra of speech and animal communications), obvious was to us that many natural communications schemes such as speech and bird songs are great examples of minimal energy communications with good robustness to interference. Bio-inspired schemes (typ. based on continuous phase or frequency modulations), physically achievable with minimal energy requirements and robust to interference/fading with fast decoding at minimal errors were designed with applications in the field of digital communications [inria-00364952]. A famous restriction on the number of usable antennas (Alamouti) was even proved not to hold for these non-linear codes (scattered CPM) [hal-01283021]. Spectral efficiency aspects were studied and bandwidth requirements were improved by introducing variable modulation index (multi-h) CPM [hal-01283004]. Recently, new communication schemes combining multi-h CPM with modified OFDM banks were designed to counteract frequency selective fading [hal-01283018] [hal-01270753]. 

This last scheme proved to be very similar to the way some birds in the canopy do chirping to mitigate frequency selectivity, and consequently motivated us to acquire even deeper understanding on the physiological and ethological aspects of bird communications. Long range aim is to understand how bird vocalizations are produced and perceived (syrinx and cochlea/basilar papilla) and the general schemes used to circumvent interference and fading, as in large flocks of birds such as starling murmurations.

Signal processing theory and methods
The Biomedical Signal Processing activity does not only focus on providing the best solutions to meet the medical or physiological demand but also on designing novel general-purpose signal processing tools. This complementary activity has yielded theoretical results in the areas of: