pôle SIS / High Order Signal Processing

Pôle SIS (Signal, Images et Systèmes)

Tensor decompositions

Two families of tensor decompositions are studied. The first one concerns decompositions into a sum of low-rank tensors, the most common being the Canonical Polyadic (CP) decomposition, also called Parafac or Canonical decomposition (CanD), that corresponds to a sum of rank-1 tensors [Como02]. See the Tensor Package page for an introduction. Some conjectures are still partially unproved, including the fact that a tensor of rank strictly smaller than generic almost surely admits a unique CP decomposition [ComoLA09], or the fact that the rank and the symmetric rank coincide [ComoGLM08], or that subtracting from a tensor its best rank-1 approximate increases almost sureley its rank [StegmC09].
The second one concerns Tucker type decompositions, the most popular being the high order singular value decomposition (HOSVD), very useful for data compression. Unlike the CP decomposition that is essentially unique under known conditions, the Tucker one is generally not unique unless some constraints are imposed on the core tensor, as in the case of the constrained Tucker model proposed in [FaviB10] for MIMO nonlinear CDMA systems, or the constrained tensor model called CONFAC, in the context of MIMO antenna systems [AlmeFM08].

Identifiability

Uniqueness of tensor decompositions and parameter identifiability are strongly related. Models to identify can be purely multilinear (as in telecommunications, or fluorescent spectrometry at very low concentrations), or multilinear under constraints (e.g. CONFAC, or constrained Tucker). The design of such constrained tensor models along with the study of their uniqueness/identifiability properties will be the subject of our research work in the near future. In these cases, global identifiability is out of reach, since it not proved even in the pure multilinear case. But local identifiability is a relevant problem to address, as subsequently tackled.

Development of numerical algorithms

Tensor users in Engineering often do not use efficient algorithms. That's why a Tensor Package is already under development for several years. It includes various standard optimization techniques, but also less standard (Enhanced Line Search, Hankel structure, positivity constrained preconditioned conjugate gradient...). One or several matrix factors entering the CP tensor model can be structured, e.g. Hankel or block Hankel. Efficient non iterative algorithms can be devised in these circumstances, and do not suffer from the presence of multiple local extrema [ComoST10]. However, there does not exist general algorithms with proved convergence towards the global optimum, for computing the CP of general tensors. The development of algebraic solutions, extending those developed in [BracCMT09] in the complex symmetric case, needs to be pursued. The simplest case is the calculation of the best rank-1 tensor approximation, and is -perhaps surprisingly- still under investigation.

Applications to Telecommunications

The first goal concerns the design of new transceivers with resource allocation, in the context of MIMO antenna systems, using constrained tensor models [hal-00417627]. The second one is to blindly and jointly estimating the channels, the transmitted symbols, and eventually the codes, using data tensor-based approaches. A new constrained Tucker model for MIMO CDMA systems has been recently proposed in [FaviB10] using a nonlinear coding. Several perspectives of this work can be drawn, as for instance the development of adaptive algorithms that take the constrained structure of the input tensors into account, an optimization of the code matrices, and also more general multiantenna/multicode transmission and multipath propagation scenarii. Another research topic concerns the design of blind receivers for nonlinear channels as it is the case with satellite or radio over fiber multiuser communication systems.

Applications to nonlinear system modeling and identification

Two applications have been developed recently. The first one concerns a new family of nonlinear models, the so called Volterra-Parafac models, obtained using a CP decomposition of high-order Volterra kernels viewed as tensors, which can result in a drastic parametric complexity reduction when the rank of the kernels is small with respect to their memory. An extended Kalman filter and LMS type algorithms were proposed to estimate the coefficients of such Volterra-Parafac models [hal-00477178]. Extensions of these results are being considered both from the tensor-based modeling (in particular for MIMO nonlinear communication channels) and algorithmic viewpoints . The second one consists of applying a tensor analysis for determining the structure and estimating the parameters of block-structured nonlinear SISO systems, like Wiener, Hammerstein and Wiener-Hammerstein systems [KibaF10]. These results will be extended to more general block-structured nonlinear MIMO systems. Moreover, the associated Volterra kernels having constrained structures (Toeplitz, band-Toeplitz factors), specific algorithms taking such structures into account will be developed.

Applications to EEG Imaging

We have already demonstrated in [BeckCAHM10] cerebral sources can be localized without solving the Maxwell equations. The idea is to exploit their nonstationarity in space and time. This leads to the decomposition of tensors, which can be supposed to be multilinear in a first approximation.

Applications to Environment

Several tensor decompositions may be studied in the context of environmental applications. For instance, the analysis of water quality by fluorescence spectrometry involves the identification of a tensor model, which is multilinear at low concentrations. The control of bioreactors or the study of microbial ecosystems lead to more complicated tensor decompositions, which can also be studied.

Applications to Radar Imaging

The objective is to use a multilinear analysis for data compression and object recognition/classification in synthetic aperture radar (SAR) images. An high-order tensor can be formed from a SAR image database, each mode corresponding to a variation factor (bearing angle, viewpoint, polarisation ...). A multilinear projection is carried out to project the unknown object into multiple basis that characterize the learned classes from the training set [hal-00489589]. Multilinear projection algorithms will be the subject of future work.

Laboratoire d'Informatique, Signaux et Systèmes de Sophia-Antipolis
I3S - UMR7271 - UNS CNRS
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