T-SVD based incomplete multi-view clustering (IMVC) has received wide attention due to its ability to capture high-order correlations. However, t-SVD suffers from rotation sensitivity, failing to fully explore both inter- and intra-view consistencies. Besides, current methods mainly consider inter- or intra-view correlations, ignoring the low-rank information of sample features within views. To address these weaknesses, we first propose a feature space recovery based IMVC (FSR-IMVC) method, where low-rank feature space recovery and low-rank tensor ring based consistency learning are considered into a unified framework. Furthermore, we extend FSR-IMVC by incorporating anchor learning on the latent feature space, resulting in a scalable FSR-IMVC (sFSR-IMVC) approach that is well-suited to large-scale data. In an iterative way, the learned inter- and intra-view correlations will guide the recovery of missing features, while the explored low-rank information from feature spaces will in turn facilitate consistency exploration, eventually achieving outstanding clustering performance. Experimental results show that FSR-IMVC provides a significant improvement over known state-of-the-art algorithms in terms of ACC, NMI and Purity. Compared with FSR-IMVC, sFSR-IMVC performs slightly worse in clustering accuracy, but offers a notable advantage in computational efficiency, particularly for large-scale datasets. The codes of FSR-IMVC and sFSR-IMVC are publicly available athttps://github.com/longzhen520/sFSR-IMVC.

Incomplete multi-view clustering (IMVC), based on imputation and clustering unification, has received wide attention due to its ability to exploit hidden information from missing views. However, current methods mainly consider inter/intra-view correlations, ignoring the structural information of sample features within views. In this paper, we propose a feature space recovery based IMVC method, where low-rank feature space recovery and consensus representation learning of inter/intra-views are considered into a unified framework. Moreover, low-rank tensor ring approximation is used to capture the correlations of self-representation tensor. In an iterative way, the learned inter/intra-view correlations will guide the recovery of missing features, while the explored low-rank information from feature spaces will in turn facilitate self-representation learning, eventually achieving out-standing clustering performance. Experimental results show our method has a very significant improvement over known state-of-the-art algorithms in terms of ACC, NMI and Purity.

Hyperspectral unmixing allows to represent mixed pixels as a set of pure materials weighted by their abundances. Spectral features alone are often insufficient, so it is common to rely on other features of the scene. Matrix models become insufficient when the hyperspectral image is represented as a high-order tensor with additional features in a multimodal, multi-feature framework. Tensor models such as Canonical polyadic decomposition allow for this kind of unmixing, but lack a general framework and interpretability of the results. In this paper, we propose an interpretable methodological framework for low-rank Multi-feature hyperspectral unmixing based on tensor decomposition (MultiHU-TD) which incorporates the abundance sum-to-one constraint in the Alternating optimization ADMM algorithm, and provide in-depth mathematical, physical and graphical interpretation and connections with the extended linear mixing model. As additional features, we propose to incorporate mathematical morphology and reframe a previous work on neighborhood patches within MultiHU-TD. Experiments on real hyperspectral images showcase the interpretability of the model and the analysis of the results. Python and MATLAB implementations are made available on GitHub.

Extensive literature exists on how to test for normality, especially for identically and independently distributed (i.i.d) processes. The case of dependent samples has also been addressed, but only for scalar random processes. For this reason, we have proposed a joint normality test for multivariate time-series, extending Mardia’s Kurtosis test. In the continuity of this work, we provide here an original performance study of the latter test applied to two-dimensional projections. By leveraging copula, we conduct a comparative study between the bivariate tests and their scalar counterparts. This simulation study reveals that one-dimensional random projections lead to notably less powerful tests than two-dimensional ones.

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.

Stochastic Neighbour Embedding methods (SNE, t-SNE) aim at finding a faithful low-dimensional representation of a high-dimensional dataset. Despite their popularity, being solution to a non-convex optimization, the behavior of these tools is not well understood. This work provides first answers by leveraging a large dimensional statistics approach, where the number n and dimension p of the large-dimensional data are of the same magnitude. We derive and study the canonical equation verified by the critical points of this non-convex optimization problem. The study notably reveals that, in a simple setup, the achievable SNE solutions correspond to a subset of those critical points. In particular, when the clusters composing the dataset are balanced in size, these solutions are symmetrical and assume closed-form expressions.As a major conclusion, the analysis rigorously proves a long-standing heuristic statement on the "proper normalization" of the symmetric SNE: out of two natural normalization choices, only the claimed proper one leads to non-trivial solutions.

Numerous works have shown the versatility of deterministic constrained Cramér-Rao bound for estimation performance analysis and design of a system of measurements. Indeed, most of factors impacting the asymptotic estimation performance of the parameters of interest can be taken into account via equality constraints. In this communication, we introduce a new constrained Cramér-Rao- like bound for observations where the probability density function (p.d.f.) parameterized by unknown deterministic parameters results from the marginalization of a joint p.d.f. depending on random variables as well. In this setting, it is now possible to consider random equality constraints, i.e., equality constraints on the unknown deterministic parameters depending on the random parameters, which can not be addressed with the usual constrained Cramér-Rao bound. The usefulness of the proposed bound is illustrated by way of a coupled canonical polyadic model with linear constraints applied to the hyperspectral super-resolution problem.

Over the last two decades, tensor‐based methods have received growing attention in the signal processing community. In this work, the authors proposed a comprehensive overview of tensor‐based models and methods for multisensor signal processing. They presented for instance the Tucker decomposition, the canonical polyadic decomposition, the tensor‐train decomposition (TTD), the structured TTD, including nested Tucker train, as well as the associated optimisation strategies. More precisely, they gave synthetic descriptions of state‐of‐the‐art estimators as the alternating least square (ALS) algorithm, the high‐order singular value decomposition (HOSVD), and of more advanced algorithms as the rectified ALS, the TT‐SVD/TT‐HSVD and the Joint dImensionally Reduction and Factor retrieval Estimator scheme. They illustrated the efficiency of the introduced methodological and algorithmic concepts in the context of three important and timely signal processing‐based applications: the direction‐of‐arrival estimation based on sensor arrays, multidimensional harmonic retrieval and multiple‐input–multiple‐output wireless communication systems.

Hyperspectral (HS) super-resolution, which aims at enhancing the spatial resolution of hyperspectral images (HSIs), has recently attracted considerable attention. A common way of HS super-resolution is to fuse the HSI with a higher spatial-resolution multispectral image (MSI). Various approaches have been proposed to solve this problem by establishing the degradation model of low spatial-resolution HSIs and MSIs based on matrix factorization methods, e.g., unmixing and sparse representation. However, this category of approaches cannot well construct the relationship between the high-spatial-resolution (HR) HSI and MSI. In fact, since the HSI and the MSI capture the same scene, these two image sources must have common factors. In this paper, a nonlocal tensor decomposition model for hyperspectral and multispectral image fusion (HSI-MSI fusion) is proposed. First, the nonlocal similar patch tensors of the HSI are constructed according to the MSI for the purpose of calculating the smooth order of all the patches for clustering. Then, the relationship between the HR HSI and the MSI is explored through coupled tensor canonical polyadic (CP) decomposition. The fundamental idea of the proposed model is that the factor matrices in the CP decomposition of the HR HSI's nonlocal tensor can be shared with the matrices factorized by the MSI's nonlocal tensor. Alternating direction method of multipliers is used to solve the proposed model. Through this method, the spatial structure of the MSI can be successfully transferred to the HSI. Experimental results on three synthetic data sets and one real data set suggest that the proposed method substantially outperforms the existing state-of-the-art HSI-MSI fusion methods.

We propose a novel approach for hyperspectral super-resolution that is based on low-rank tensor approximation for a coupled low-rank multilinear (Tucker) model. We show that the correct recovery holds for a wide range of multilinear ranks. For coupled tensor approximation, we propose an SVD-based algorithm that is simple and fast, but with a performance comparable to that of the state-of-the-art methods.

Image classification has been at the core of remote sensing applications. Optical remote sensing imaging systems naturally acquire images with spectral features corresponding to pixels. Spectral classification ignores the spatial distribution of the data which is becoming more relevant with the development of spatial resolution sensors, and many works aim to incorporate spatial features based on neighborhood through for example, Mathematical Morphology (MM). Additionally, one could stack multiple morphological transformations of the image resulting in a highly complex block of data. Since classification is a tool that requires a matrix of samples and features, and simply stacking the different sets of features can lead to the problem of high dimensionality, we propose a way to create a matrix of low dimensional feature space by modeling the data as tensors and thanks to Canonical Polyadic (CP) decomposition. Experiments on real image show the effectiveness of the proposed method.

In this letter, we investigated the connection between information and estimation measures for mismatched Gaussian models. In addition to the input prior mismatch, we take into account the noise mismatch and establish a new relation between relative entropy and excess mean square error. The derived formula shows that the input prior mismatch may be canceled by the noise mismatch. Finally, an example illustrates the impact of model mismatches on estimation accuracy.

Directivity gain patterns are treated as a physical diversity for tensor array processing, replacing space diversity, in addition to time and space shift diversities. We show that the tensor formulation allows to estimate directions of arrival under the assumption of unknown gain patterns, improving the performance of the omnidirectional case. We propose a trilinear model where one dimension of the multiway data array is fully provided by gain patterns, allowing tensor approaches even when space diversity is missing due to sensor overlap.

As a noninvasive technique, electroencephalography (EEG) is commonly used to monitor the brain signals of patients with epilepsy such as the interictal epileptic spikes. However, the recorded data are often corrupted by artifacts originating, for example, from muscle activities, which may have much higher amplitudes than the interictal epileptic signals of interest. To remove these artifacts, a number of independent component analysis (ICA) techniques were successfully applied. In this paper, we propose a new deflation ICA algorithm, called penalized semialgebraic unitary deflation (P-SAUD) algorithm, that improves upon classical ICA methods by leading to a considerably reduced computational complexity at equivalent performance. This is achieved by employing a penalized semialgebraic extraction scheme, which permits us to identify the epileptic components of interest (interictal spikes) first and obviates the need of extracting subsequent components. The proposed method is evaluated on physiologically plausible simulated EEG data and actual measurements of three patients. The results are compared to those of several popular ICA algorithms as well as second-order blind source separation methods, demonstrating that P-SAUD extracts the epileptic spikes with the same accuracy as the best ICA methods, but reduces the computational complexity by a factor of 10 for 32-channel recordings. This superior computational efficiency is of particular interest considering the increasing use of high-resolution EEG recordings, whose analysis requires algorithms with low computational cost.

Our goal is to devise a wideband High-Resolution technique that does not require a priori knowledge of DoA rough estimates, and that is able to exploit multiple spatial invariances. Existing tensor array processing techniques are limited to the narrowband case. On the other hand, wideband Esprit has only been proposed with focusing matrices, requiring a priori DoA knowledge. We resort to the decomposition of tensors built on space, space translation and frequency diversities, and demonstrate the good behavior of the algorithm proposed.

We propose a noniterative algorithm, called SeROAP,1 to estimate a rank-1 approximation of a tensor in the real or complex field. Our algorithm is based on a sequence of singular value decompositions followed by a sequence of projections onto Kronecker vectors. For three-way tensors, we show that our algorithm is always at least as good as the state-of-the-art truncation algorithm, ST-HOSVD,2in terms of approximation error. Thus, it gives a good starting point to iterative rank-1 tensor approximation algorithms. By means of computational experiments, it also turns out that for fourth order tensors, SeROAP yields a better approximation with high probability when compared to the standard THOSVD3algorithm.

New hyperspectral missions will collect huge amounts of hyperspectral data. In addition, it is possible now to acquire time series and multiangular hyperspectral images. The process and analysis of these big data collections will require common hyperspectral techniques to be adapted or reformulated. The tensor decomposition, which is also known as multiway analysis, is a technique to decompose multiway arrays, i.e., hypermatrices with more than two dimensions (ways). Hyperspectral time series and multiangular acquisitions can be represented as a three-way tensor. Here, we apply canonical polyadic (CP) tensor decomposition techniques to the blind analysis ohyperspectral big data. In order to do so, we use a novel compression-based nonnegative CP decomposition. We show that the proposed methodology can be interpreted as multilinear blind spectral unmixing, i.e., a higher order extension of the widely known spectral unmixing. In the proposed approach, the big hyperspectral tensor is decomposed in three sets of factors, which can be interpreted as spectral signatures, their spatial distribution, and temporal/angular changes. We provide experimental validation using a study case of the snow coverage of the French Alps during the snow season.

We show that for a nonnegative tensor, a best nonnegative rank-r approximation is almost always unique, its best rank-one approximation may always be chosen to be a best nonnegative rank-one approximation, and the set of nonnegative tensors with nonunique best rank-one approximations forms an algebraic hypersurface. We show that the last part holds true more generally for real tensors and, thereby, determine a polynomial equation, so that a real or nonnegative tensor that does not satisfy this equation is guaranteed to have a unique best rank-one approximation. We also establish an analogue for real or nonnegative symmetric tensors. In addition, we prove a singular vector variant of the Perron-Frobenius theorem for positive tensors and apply it to show that a best nonnegative rank-r approximation of a positive tensor can never be obtained by deflation. As an aside, we verify that the Euclidean distance (ED) discriminants of the Segre variety and the Veronese variety are hypersurfaces and give defining equations of these ED discriminants.

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.

The Canonical Polyadic (CP) tensor decomposition has become an attractive mathematical tool these last ten years in various fields. Yet, efficient algorithms are still lacking to compute the full CP decomposition, whereas rank-one approximations are rather easy to compute. We propose a new deflation-based iterative algorithm allowing to compute the full CP decomposition, by resorting only to rank-one approximations. An analysis of convergence issues is included, as well as computer experiments. Our theoretical and experimental results show that the algorithm converges almost surely.

In signal processing, tensor decompositions have gained in popularity this last decade. In the meantime, the volume of data to be processed has drastically increased. This calls for novel methods to handle Big Data tensors. Since most of these huge data are issued from physical measurements, which are intrinsically real nonnegative, being able to compress nonnegative tensors has become mandatory. Following recent works on HOSVD compression for Big Data, we detail solutions to decompose a nonnegative tensor into decomposable terms in a compressed domain.

The objective of brain source imaging consists in reconstructing the cerebral activity everywhere within the brain based on EEG or MEG measurements recorded on the scalp. This requires solving an ill-posed linear inverse problem. In order to restore identifiability, additional hypotheses need to be imposed on the source distribution, giving rise to an impressive number of brain source imaging algorithms. However, a thorough comparison of different methodologies is still missing in the literature. In this paper, we provide an overview of priors that have been used for brain source imaging and conduct a comparative simulation study with seven representative algorithms corresponding to the classes of minimum norm, sparse, tensor-based, subspace-based, and Bayesian approaches. This permits us to identify new benchmark algorithms and promising directions for future research.

In tensor factorization approach to blind separation of multidimensional sources two formulas for calculating the source tensor have emerged. In practice, it is observed that these two schemes exhibit different levels of robustness against perturbations of the factors involved in the tensor model. Motivated by both practical reasons and the will to better figure this out, we present error analyses in source tensor estimation performed by low-rank factorization of three-way tensors. To that aim, computer simulations as well as the analytical calculation of the theoretical error are carried out. The conclusions drawn from these numerical and analytical error analyses are supported by the results obtained thanks to the tensor factorization based blind decomposition of an experimental multispectral image of a skin tumor.

We discuss a technique that allows blind recovery of signals or blind identification of mixtures in instances where such recovery or identification were previously thought to be impossible. These instances include: 1) closely located or highly correlated sources in antenna array processing; 2) highly correlated spreading codes in code division multiple access (CDMA) radio communication; and 3) nearly dependent spectra in fluorescence spectroscopy. These have important implications. In the case of antenna array processing, it allows for joint localization and extraction of multiple sources from the measurement of a noisy mixture recorded on multiple sensors in an entirely deterministic manner. In the case of CDMA, it allows the possibility of having a number of users larger than the spreading gain. In the case of fluorescence spectroscopy, it allows for detection of nearly identical chemical constituents. The proposed technique involves the solution of a bounded coherence low-rank multilinear approximation problem. We show that bounded coherence allows us to establish existence and uniqueness of the recovered solution. We will provide some statistical motivation for the approximation problem and discuss greedy approximation bounds. To provide the theoretical underpinnings for this technique, we develop a corresponding theory of sparse separable decompositions of functions, including notions of rank and nuclear norm that can be specialized to the usual ones for matrices and operators and also be applied to hypermatrices and tensors.

This paper deals with the minimum polyadic decomposition of a nonnegative three-way array. The main advantage of the nonnegativity constraint is that the approximation problem becomes well posed. To tackle this problem, we suggest the use of a cost function including penalty terms built with matrix exponentials. Gradient components are then derived, allowing to efficiently implement the decomposition using classical optimization algorithms. In our case ALS and conjugate gradient algorithms are studied and compared with another existing algorithm, thanks to computer simulations performed in the context of data analysis.

This paper deals with the minimal polyadic decomposition (also known as canonical decomposition or Parafac) of a 3-way array, assuming each entry is positive. In this case, the low-rank approximation problem becomes well-posed. The suggested approach consists of taking into account the nonnegative nature of the loading matrices directly in the problem parameterization. Then, the three gradient components are derived allowing to efficiently implement the decomposition using classical optimization algorithms. In our case, we focus on the conjugate gradient algorithm, well matched to large problems. The good behaviour of the proposed approach is illustrated through computer simulations in the context of data analysis and compared to other existing approaches.

In the paper, we address the important problem of tensor decomposition which can be seen as a generalisation of Singular Value Decomposition for matrices. We consider general multilinear and multihomogeneous tensors. We show how to reduce the problem to a truncated moment matrix problem and we give a new criterion for flat extension of Quasi-Hankel matrices. We connect this criterion to the commutation characterisation of border bases. A new algorithm is described: it applies for general multihomogeneous tensors, extending the approach of J.J. Sylvester on binary forms. An example illustrates the algebraic operations involved in this approach and how the decomposition can be recovered from eigenvector computation.

Tensor decompositions permit to estimate in a deterministic way the parameters in a multi-linear model. Applications have been already pointed out in antenna array processing and digital communications, among others, and are extremely attractive provided some diversity at the receiver is available. As opposed to the widely used ALS algorithm, non-iterative algorithms are proposed in this paper to compute the required tensor decomposition into a sum of rank-1 terms, when some factor matrices enjoy some structure, such as block-Hankel, triangular, band, etc.

Independent component analysis (ICA) aims at decomposing an observed random vector into statistically independent variables. Deflation-based implementations, such as the popular one-unit FastICA algorithm and its variants, extract the independent components one after another. A novel method for deflationary ICA, referred to as RobustICA, is put forward in this paper. This simple technique consists of performing exact line search optimization of the kurtosis contrast function. The step size leading to the global maximum of the contrast along the search direction is found among the roots of a fourth-degree polynomial. This polynomial rooting can be performed algebraically, and thus at low cost, at each iteration. Among other practical benefits, RobustICA can avoid prewhitening and deals with real- and complex-valued mixtures of possibly noncircular sources alike. The absence of prewhitening improves asymptotic performance. The algorithm is robust to local extrema and shows a very high convergence speed in terms of the computational cost required to reach a given source extraction quality, particularly for short data records. These features are demonstrated by a comparative numerical analysis on synthetic data. RobustICA's capabilities in processing real-world data involving noncircular complex strongly super-Gaussian sources are illustrated by the biomedical problem of atrial activity (AA) extraction in atrial fibrillation (AF) electrocardiograms (ECGs), where it outperforms an alternative ICA-based technique.

This brief deals with the problem of blind source separation (BSS) via independent component analysis (ICA). We prove that a linear combination of the separator output fourth-order marginal cumulants (kurtoses) is a valid contrast function for ICA under prewhitening if the weights have the same sign as the source kurtoses. If, in addition, the source kurtoses are different and so are the linear combination weights, the contrast eliminates the permutation ambiguity typical to ICA, as the estimated sources are sorted at the separator output according to their kurtosis values in the same order as the weights. If the weights equal the source kurtoses, the contrast is a cumulant matching criterion based on the maximum-likelihood principle. The contrast can be maximized by means of a cost-efficient Jacobi-type pairwise iteration. In the real-valued two-signal case, the asymptotic variance of the resulting Givens angle estimator is determined in closed form, leading to the contrast weights with optimal finite-sample performance. A fully blind solution can be implemented by computing the optimum weights from the initial source estimates obtained by a classical ICA stage. An experimental study validates the features of the proposed technique and shows its superior performance compared to related previous methods.

Space-time block codes can be represented by tensors, like for example the Kathri-Rao space time codes introduced by Sidiropoulos. In this paper, we introduce a parallel factor analysis 2 (PARAFAC2) model for orthogonal space-time block codes (OSTBC). This model, along with a small modification at the transmitter and the use of tensor diagonalisation designed for orthogonal tensors at the receiver, leads us to develop a generic parallel factor analysis (PARAFAC) blind receiver for OSTBC. Unlike most tensor based blind receivers, our channel estimator is based on a data tensor rather than on a cumulant tensor.

The concept of tensor rank, introduced in the twenties, has been popularized at the beginning of the seventies. This has allowed to carry out factor analysis on arrays with more than two indices. The generic rank may be seen as an upper bound to the number of factors that can be extracted from a given tensor, with certain uniqueness conditions. We explain how to obtain numerically the generic rank of tensors of arbitrary dimensions, and compare it with the rare algebraic results already known at order three. In particular, we examine the cases of symmetric tensors, tensors with symmetric matrix slices, or tensors with free entries. Related applications include antenna array processing.

A contrast function for independent component analysis (ICA) is presented incorporating the prior knowledge on the sub-Gaussian or super-Gaussian character of the sources as described by their kurtosis signs. The contrast is related to the maximum likelihood principle, reduces the permutation indeterminacy typical of ICA, and proves particularly useful in the direct extraction of a source signal with distinct kurtosis sign. In addition, its numerical maximization can be performed cost-effectively by a Jacobi-like pairwise iteration. Extensions to standardized cumulants of orders other than four are also given.

The step size leading to the absolute minimum of the constant modulus (CM) criterion along the search direction can be obtained algebraically at each iteration among the roots of a third-degree polynomial. The resulting optimal step-size CMA (OS-CMA) is compared with, other CM-based iterative techniques in terms of performance-versus-complexity trade-off.

In this paper, we investigate the use of output 4th-order cumulants to detect the number of source signals on a multiple-input single-output (MISO) communications channel and blindly identify their respective channel coefficients. More particularly, we are interested in the case of two sources. The proposed cumulant-based order detection principle allows us for recovering the longest channel order and its coefficients. A similar procedure is applied for detecting the presence of a second source as well as estimating its associated channel order and coefficients. When only estimated cumulants are available, we implement two hypothesis tests based on three proposed test-statistics. Computer simulations illustrate the performances obtained with the methods proposed for order detection and channel identification.

Blind identification of under-determined mixtures (UDM) is involved in numerous applications, including multi-way factor analysis (MWA) and signal processing. In the latter case, the use of high-order statistics (HOS) like cumulants leads to the decomposition of symmetric tensors. Yet, little has been published about rank-revealing decompositions of symmetric tensors. Definitions of rank are discussed, and useful results on generic rank are proved, with the help of tools borrowed from algebraic geometry

Algorithm ALESCAF (alternating least squares identification based on the characteristic function) uses the derivatives of the second characteristic function (c.f.) of observations, without any need of sparsity assumption on sources, but assuming their statistical independence. ALESCAF was already proposed by the authors in P. Comon and M. Rajih (2005), where only one derivative order was considered. In this paper, new versions of ALESCAF are proposed, that jointly use derivatives of different orders. We also propose ALESCAS, a new algorithm that uses the knowledge of source c.f.'s. Computer simulations demonstrate that both algorithms accelerate the convergence

Linear mixtures of independent random variables (the so-called sources) are sometimes referred to as under-determined mixtures (UDM) when the number of sources exceeds the dimension of the observation space. The algorithms proposed are able to identify algebraically a UDM using the second characteristic function of the observations. With only two sensors, the first algorithm only needs an SVD. With a larger number of sensors, the second algorithm executes an ALS. The joint use of statistics of different orders is possible, and an LS solution can be computed.

Channel equalization is an important problem in digital communications. The paper studies a hybrid equalization criterion combining the constant modulus (CM) property and the minimum mean square error (MMSE) between the equalizer output and the known pilot sequence. An efficient semi-blind block gradient-descent algorithm is proposed, in which the step size globally minimizing the cost function along the search direction is algebraically computed at each iteration. The use of the optimal step size notably accelerates convergence and can further reduce the impact of local extrema on the semi-blind algorithm's performance. The proposed approach is not restricted to the CM-MMSE principle, but it can benefit other equalization criteria as well.

Static linear mixtures with more sources than sensors are considered. Blind identification (BI) of underdetermined mixtures is addressed by taking advantage of sixth order (SixO) statistics and the virtual array (VA) concept. Surprisingly, identification methods solely based on the hexacovariance matrix succeed well, despite their expected high estimation variance; this is due to the inherently good conditioning of the problem. A computationally simple but efficient algorithm, named BIRTH (Blind Identification of mixtures of sources using Redundancies in the daTa Hexacovariance matrix), is proposed and enables the identification of the steering vectors of up to P=N/sup 2/-N+1 sources for arrays of N sensors with space diversity only, and up to P=N/sup 2/ for those with angular and polarization diversities. Five numerical algorithms are compared.

We describe a family of criteria dedicated to blind SISO equalizers. These criteria are based on alphabet polynomial fitting (APF), and remind us of the well-known constant modulus algorithm (CMA) criterion, and encompass the constant power algorithm (CPA) criterion. Algorithms based on several polynomial criteria have been implemented in block form (including CPA and APF), as well as the CMA and the kurtosis maximization (KMA). Block implementations are indeed more efficient for short data records, and allow the direct computation of the optimal step size in a gradient descent, as shown in the paper. Computational complexities of APF, KMA and CMA are eventually compared, as well as their performance for various digitally modulated inputs.

For about two decades, many second order (SO) and fourth order (FO) blind methods have been developed to separate overdetermined mixtures of statistically independent narrowband (NB) sources. Besides, mainly to overcome some limitations of these methods, sixth order methods have been developed. Nevertheless, despite of this great number of methods, the performance of the latter for arbitrary electromagnetic sources are still almost unknown, which limits their use in operational contexts. The purpose of this paper is to fill the gap previously mentioned by presenting a comparative performance analysis of eight blind source separation (BSS) methods for arbitrary overdetermined mixtures of sources borrowed from the radiocommunications context, and to show off both the advantages and the drawbacks of these methods.

This paper introduces a new blind source separation algorithm for convolutive mixtures. In addition to separate sources, this algorithm respects the paraunitary property of the model considered, obtained after whitening observations. In order to do this, the equalizer is factorized in a novel manner. After a presentation of theoretical results, a numerical algorithm is then derived. This algorithm is based on the solution of a polynomial system, which some values of output cumulant multi-correlations enter. Simulations and performances of the numerical algorithm are presented in the last section.

We consider the problem of convolutive blind signal separation through the optimization of contrast functions. We show that some links between contrasts and joint diagonalization criteria can be exhibited in the convolutive case. This allows us to devise a constructive algorithm performing MIMO blind equalization, with the help of a joint approximate diagonalization of a set of matrices built from the observations. This analytical algorithm can be run block-wise, which is appropriate in the context of short burst communications.

Unlike the Schetzen method, this identification method proposed can process inputs with discrete distribution, accepts kernels of any length, tolerates signal-independent zero-mean additive noise irrespective of its color and distribution, and leads to a closed form solution. Identifiability is proved under assumptions on input moments; more accurate conditions are derived for particular discrete inputs (in particular in the complex plane, distributions symmetric about both axes). Several simulations demonstrate its efficiency and its speed for various discrete input distributions.

A polynomial criterion is proposed to perform blind source separation, extending previous works to MIMO systems. The criterion is proved to be asymptotically MAP-equivalent in presence of PSK sources. An efficient minimization algorithm dedicated to polynomial criteria is then developed, improving on the fixed-step stochastic gradient previously utilized in this framework.

Contrast-based separation of sources have a number of advantages. Among others, they are optimal (in a precise sense) in the presence of noise of unknown statistics. Here a new contrast is proposed that allows not only to obtain the optimal solution analytically, but also yields better performances in terms of variance of the estimated mixing matrix. This contrast needs the source kurtosis to have the same sign, and is thus appropriate to multichannel blind equalization in communications.

A class of optimization criteria is proposed whose maximization allows us to carry out blind multichannel deconvolution in the presence of additive noise. Contrasts presented in the paper encompass those related to source separation and independent component analysis problems.

The concept of displacement rank can be defined in a general framework, and turns out to be a useful tool for revealing hidden structures in matrices. When a n*n matrix is structured, it can be described by O(n) parameters. Manipulations of linear systems can then be carried out with much less storage and computational requirements. It is known that the inverse of a structured matrix is structured, but it is proved that this also holds true for pseudo-inverses.>

In various applications it is necessary to keep track of a low-rank approximation of a covariance matrix, R(t), slowly varying with time. It is convenient to track the left singular vectors associated with the largest singular values of the triangular factor, L(t), of its Cholesky factorization. These algorithms are referred to as square-root. The drawback of the eigenvalue decomposition (EVD) or the singular value decompositions (SVD) is usually the volume of the computations. Various numerical methods for carrying out this task are surveyed, and it is shown why this heavy computational burden is questionable in numerous situations and should be revised. Indeed, the complexity per eigenpair is generally a quadratic function of the problem size, but there exist faster algorithms with linear complexity. Finally, in order to make a choice among the large and fuzzy set of available techniques, comparisons based on computer simulations in a relevant signal processing context are made.>

The problem of designing a finite impulse response filter devoted to the joint process problem, and in particular to noise cancellation, is investigated. The goal is restricted to how to choose an optimal order of the FIR filter. The analysis is based on the maximization of a noise reduction criterion which is in fact perfectly matched to the desired performance, namely the minimization of residuals. The criterion includes estimation errors introduced by the substitution of the true covariance matrices for estimates. Since true covariance matrices again enter the noise reduction criterion, they must be replaced by estimates. The criterion is then modified in order to take this fact into account, but the scheme is not iterated any further. Simulations are presented for various simple cases.>

See ibid. vol.75, no.12, p.1692-4 (Dec. 1987). The commenters explain why they disagree with some assertions in the above-titled letter and point out some omissions that they consider important. In particular, they maintain the systolic array of W.M. Gentleman and H.T. Kung cannot be used for subspace estimation in high-resolution methods. The author points out some alternative interpretations that present the original work in a different light.>

Classical optimum adaptive filtering assumes that some second order moments are known, so that the optimality is linked to the accurate knowledge of the letter moments. In most practical cases, this knowledge is uncertain, estimates must be utilized instead of true values, and a considerable decrease in the performance is often deplored. Thus, two related problems arise : first, a single optimization criterion must be clearly defined to provide a joint estimate of the unknown variables and the second order moments; secondly, a measure of the estimated filter performance should be used to enquire into the optimality of the estimated solution.

A simple approach to the adaptation of a linear filter to cope with spectral uncertainties is presented. Based on the calculation of the total output error power, including the errors of spectral estimation, a criterion is proposed for deciding whether the estimated filter is effective at any given frequency. By forcing the filter response to unity at all ineffective frequencies, an improvement in performance is obtained.