Maxime Boizard

Tensor CP Decomposition With Structured Factor Matrices: Algorithms and Performance

The canonical polyadic decomposition (CPD) of high-order tensors, also known as Candecomp/Parafac, is very useful for representing and analyzing multidimensional data. This paper considers a CPD model having structured matrix factors, as e.g. Toeplitz, Hankel or circulant matrices, and studies its associated estimation problem. This model arises in signal processing applications such as Wiener-Hammerstein system identification and cumulant-based wireless communication channel estimation. After introducing a general formulation of the considered structured CPD (SCPD), we derive closed-form expressions for the Cramér-Rao bound (CRB) of its parameters under the presence of additive white Gaussian noise. Formulas for special cases of interest, as when the CPD contains identical factors, are also provided. Aiming at a more relevant statistical evaluation from a practical standpoint, we discuss the application of our formulas in a Bayesian context, where prior distributions are assigned to the model parameters. Three existing algorithms for computing SCPDs are then described: a constrained alternating least squares (CALS) algorithm, a subspace-based solution and an algebraic solution for SCPDs with circulant factors. Subsequently, we present three numerical simulation scenarios, in which several specialized estimators based on these algorithms are proposed for concrete examples of SCPD involving circulant factors. In particular, the third scenario concerns the identification of a Wiener-Hammerstein system via the SCPD of an associated Volterra kernel. The statistical performance of the proposed estimators is assessed via Monte Carlo simulations, by comparing their Bayesian mean-square error with the expected CRB.

Low-rank filter and detector for multidimensional data based on an alternative unfolding HOSVD: application to polarimetric STAP

This paper proposes an extension of the higher order singular value decomposition (HOSVD), namely the alternative unfolding HOSVD (AU-HOSVD), in order to exploit the correlated information in multidimensional data. We show that the properties of the AU-HOSVD are proven to be the same as those for HOSVD: the orthogonality and the low-rank (LR) decomposition. We next derive LR filters and LR detectors based on AU-HOSVD for multidimensional data composed of one LR structure contribution. Finally, we apply our new LR filters and LR detectors in polarimetric space-time adaptive processing (STAP). In STAP, it is well known that the response of the background is correlated in time and space and has a LR structure in space-time. Therefore, our approach based on AU-HOSVD seems to be appropriate when a dimension (like polarimetry in this paper) is added. Simulations based on signal-to-interferenceplus-noise ratio (SINR) losses, probability of detection (Pd), and probability of false alarm (Pfa) show the interest of our approach: LR filters and LR detectors which can be obtained only from AU-HOSVD outperform the vectorial approach and those obtained from a single HOSVD.

Performance estimation for tensor CP decomposition with structured factors

The Canonical Polyadic tensor decomposition (CPD), also known as Candecomp/Parafac, is very useful in numerous scientific disciplines. Structured CPDs, i.e. with Toeplitz, circulant, or Hankel factor matrices, are often encountered in signal processing applications. As subsequently pointed out, specialized algorithms were recently proposed for estimating the deterministic parameters of structured CP decompositions. A closed-form expression of the Cramér-Rao bound (CRB) is derived, related to the problem of estimating CPD parameters, when the observed tensor is corrupted with an additive circular i.i.d. Gaussian noise. This CRB is provided for arbitrary tensor rank and sizes. Finally, the proposed CRB expression is used to asses the statistical efficiency of the existing algorithms by means of simulation results in the cases of third-order tensors having three circulant factors on one hand, and an Hankel factor on the other hand.

Derivation of the theoretical performance of a Tensor MUSIC algorithm

In this paper, we derive the theoretical performance of a Tensor MUSIC algorithm based on the Higher Order Singular Value Decomposition (HOSVD) that was previously developed to estimate the Direction Of Arrival (DOA) and the polarization parameters of polarized sources. The derivation of this result is done via a perturbation analysis and allows to obtain the theoretical Mean Square Error (MSE) on the DOA and the polarization parameters. The proposed result is also shown to be valid for the Long Vector MUSIC algorithm, i.e. when the multidimensional samples are unfolded into a long vector. The agreement between theoretical and empirical MSEs is illustrated through Monte Carlo simulations. HighlightsConsidered model: polarized source.2 MUSIC algorithms: LV-MUSIC (all data are unfolded into a single vector) and T-MUSIC (based on multilinear algebra).Derivation of theoretical Mean Square Error of DOAs and polarization parameters.Based on a perturbation analysis.

Low Rank Tensor STAP filter based on multilinear SVD

Space Time Adaptive Processing (STAP) is a two-dimensional adaptive filtering technique which uses jointly temporal and spatial dimensions to suppress disturbance and to improve target detection. Disturbance contains both the clutter arriving from signal backscattering of the ground and the thermal noise resulting from the sensors noise. In practical cases, the STAP clutter can be considered to have a low rank structure. Using this assumption, a low rank vector STAP filter is derived based on the projector onto the clutter subspace. With new STAP applications like MIMO STAP or polarimetric STAP, the generalization of the classic filters to multidimensional configurations arises. A possible solution consists in keeping the multidimensional structure and in extending the classic filters with multilinear algebra. Using the low-rank structure of the clutter, we propose in this paper a new low-rank tensor STAP filter based on a generalization of the Higher Order Singular Value Decomposition (HOSVD) in order to use at the same time the simple (for example time, spatial, polarimetric, ...) and the combined information (for example spatio-temporal). Results are shown for two cases: classic 2D STAP and 3D polarimetric STAP. In the classic case, vector and tensor filters are equivalent. In the polarimetric case, we show the enhancement of the tensor filter.

Multidimensional low-rank filter based on the AU-HOSVD for MIMO STAP

We propose in this paper a new low rank filter for MIMO STAP (Multiple Input Multiple Output Space Time Adaptive Processing) based on the AU-HOSVD (Alternative Unfolding Higher Order Singular Value Decomposition). This decomposition called the AU-HOSVD is able to process data in correlated dimensions which is desirable for STAP methods. We apply the new filter to MIMO STAP simulated data. The results are encouraging and outperforms the conventional STAP 2D filter in terms of number of secondary data.

Fast multilinear Singular Value Decomposition for higher-order Hankel tensors

The Higher-Order Singular Value Decomposition (HOSVD) is a possible generalization of the Singular Value Decomposition (SVD) to tensors, which have been successfully applied in various domains. Unfortunately, this decomposition is computationally demanding. Indeed, the HOSVD of a Nth-order tensor involves the computation of the SVD of N matrices. Previous works have shown that it is possible to reduce the complexity of HOSVD for third-order structured tensors. These methods exploit the columns redundancy, which is present in the mode of structured tensors, especially in Hankel tensors. In this paper, we propose to extend these results to fourth order Hankel tensor. We propose two ways to extend Hankel structure to fourth order tensors. For these two types of tensors, a method to build a reordered mode is proposed, which highlights the column redundancy and we derive a fast algorithm to compute their HOSVD. Finally we show the benefit of our algorithms in terms of complexity.

New Low-Rank Filters for MIMO-STAP Based on an Orthogonal Tensorial Decomposition

We develop in this paper a new adaptive low-rank (LR) filter for MIMO-space time adaptive processing (STAP) application based on a tensorial modeling of the data. This filter is based on an extension of the higher order singular value decomposition (HOSVD) (which is also one possible extension of singular value decomposition to the tensor case), called alternative unfolding HOSVD (AU-HOSVD), which allows us to consider the combinations of dimensions. This property is necessary to keep the advantages of the STAP and the MIMO characteristics of the data. We show that the choice of a good partition (as well as the tensorial modeling) is not heuristic but have to follow several features. Thanks to the derivation of the theoretical formulation of multimode ranks for all partitions, the tensorial LR filters are easy to compute. Results on simulated data show the good performance of the AU-HOSVD LR filters in terms of secondary data and clutter notch.