This work presents a comprehensive understanding of the estimation of a planted low-rank signal from a general spiked tensor model near the computational threshold. Relying on standard tools from the theory of large random matrices, we characterize the large-dimensional spectral behavior of the unfoldings of the data tensor and exhibit relevant signal-to-noise ratios governing the detectability of the principal directions of the signal. These results allow to accurately predict the reconstruction performance of truncated multilinear SVD (MLSVD) in the non-trivial regime. This is particularly important since it serves as an initialization of the higher-order orthogonal iteration (HOOI) scheme, whose convergence to the best low-multilinear-rank approximation depends entirely on its initialization. We give a sufficient condition for the convergence of HOOI and show that the number of iterations before convergence tends to $1$ in the large-dimensional limit.

The performance of spectral clustering relies on the fluctuations of the entries of the eigenvectors of a similarity matrix, which has been left uncharacterized until now. In this letter, it is shown that thesignal$+$noisestructure of a general spike random matrix model is transferred to the eigenvectors of the corresponding Gram kernel matrix and the fluctuations of their entries are Gaussian in the large-dimensional regime. This CLT-like result was the last missing piece to precisely predict the classification performance of spectral clustering. The proposed proof is very general and relies solely on the rotational invariance of the noise. Numerical experiments on synthetic and real data illustrate the universality of this phenomenon.

This paper considers the analysis of bivariate signals using complex Singular Spectrum Analysis (SSA). It introduces a pseudo-correlation based criterion in the grouping step of complex SSA. The advantage of using pseudo-correlation rather than correlation measures when analyzing polarized signals with complex SSA is demonstrated theoretically. This criterion is shown to be effective to extract bivariate signals modeled as particular complex Linear Recurrence Relations (LRR) of order 2. These elementary complex bricks offer a high interpretability in terms of polarization. Illustration of the proposed grouping technique is made through polarized component extraction on a real-world data example.

This article introduces a random matrix framework for the analysis of clustering on high-dimensional data streams, a particularly relevant setting for a more sober processing of large amounts of data with limited memory and energy resources. Assuming data $\mathbf{x}_1, \mathbf{x}_2, \ldots$ arrives as a continuous flow and a small number $L$ of them can be kept in the learning pipeline, one has only access to the diagonal elements of the Gram kernel matrix: $\left[ \mathbf{K}_L \right]_{i, j} = \frac{1}{p} \mathbf{x}_i^\top \mathbf{x}_j \mathbf{1}_{\left\lvert i - j \right\rvert < L}$. Under a large-dimensional data regime, we derive the limiting spectral distribution of the banded kernel matrix $\mathbf{K}_L$ and study its isolated eigenvalues and eigenvectors, which behave in an unfamiliar way. We detail how these results can be used to perform efficient online kernel spectral clustering and provide theoretical performance guarantees. Our findings are empirically confirmed on image clustering tasks. Leveraging on optimality results of spectral methods for clustering, this work offers insights on efficient online clustering techniques for high-dimensional data.

The article introduces an elementary cost and storage reduction method for spectral clustering and principal component analysis. The method consists in randomly “puncturing” both the data matrix $X\in\mathbb{C}^{p\times n}$ (or $\mathbb{R}^{p\times n}$) and its corresponding kernel (Gram) matrix $K$ through Bernoulli masks: $S\in\{0,1\}^{p\times n}$ for $X$ and $B\in\{0,1\}^{n\times n}$ for $K$. The resulting “two-way punctured” kernel is thus given by $K=\frac1p[(X\odot S)^\H (X\odot S)]\odot B$. We demonstrate that, for $X$ composed of independent columns drawn from a Gaussian mixture model, as $n,p\to\infty$ with $p/n\to c_0\in(0,\infty)$, the spectral behavior of $K$ – its limiting eigenvalue distribution, as well as its isolated eigenvalues and eigenvectors – is fully tractable and exhibits a series of counter-intuitive phenomena. We notably prove, and empirically confirm on various image databases, that it is possible to drastically puncture the data, thereby providing possibly huge computational and storage gains, for a virtually constant (clustering or PCA) performance. This preliminary study opens as such the path towards rethinking, from a large dimensional standpoint, computational and storage costs in elementary machine learning models.

Improperness testing for complex-valued vectors and processes has been of interest lately due to the potential applications of complex-valued time series analysis in several research areas. This paper provides exact distribution characterization of the GLRT (Generalized Likelihood Ratio Test) statistics for Gaussian complex-valued signals under the null hypothesis of properness. This distribution is a special case of the Wilks's lambda distribution, as are the distributions of the GLRT statistics in multivariate analysis of variance (MANOVA) procedures. In the high dimensional setting, i.e. when the size of the vectors grows at the same rate as the number of samples, a closed form expression is obtained for the asymptotic distribution of the GLRT statistics. This is, to our knowledge, the first exact characterization for the GLRT-based improperness testing.

In this paper, a new robust regression method based on the Least Trimmed Squares (LTS) is proposed. The novelty of this approach consists in a simple adaptive estimation of the number of outliers. This method can be applied to baseline estimation, for example to improve the detection of gas spectral signature in astronomical hyperspectral data such as those produced by the new Multi Unit Spectroscopic Explorer (MUSE) instrument. To do so a method following the general idea of the LOWESS algorithm, a classical robust smoothing method, is developed. It consists in a windowed local linear regression, the local regression being done here by the new adaptive LTS approach. The developed method is compared with state-of-the art baseline estimated algorithms on simulated data closed to the real data produced by the MUSE instrument.

This paper presents several results about isotropic random walks and multiple scattering processes on hyperspheres Sp-1. It allows one to derive the Fourier expansions on Sp-1of these processes. A result of unimodality for the multiconvolution of symmetrical probability density functions on Sp-1is also introduced. Such processes are then studied in the case where the scattering distribution is von Mises-Fisher (vMF). Asymptotic distributions for the multiconvolution of vMFs on Sp-1are obtained. Both Fourier expansion and asymptotic approximation allow us to compute estimation bounds for the parameters of compound cox processes on Sp-1.

This paper considers the problem of parameter estimation for multiple scattering process on the sphere. Using harmonic analysis, a Fourier expansion of the pdf of the process is obtained. Based on the Fourier coefficient statistics, we consider the problem of estimating the parameter of the process using an Approximate Bayesian Computation (ABC) approach. Simulations show the ability of the proposed approach for the density estimation of intensity and concentration parameters for the von Mises Fisher multiple scattering process.

In this study, a method that aims at detecting small and faint objects in noisy hyperspectral astrophysical images is presented. The particularity of the hyperspectral images that we are interested in is the high dynamics between object intensities. Detection of the smallest and faintest objects is challenging, because their signal-to-noise ratio is low, and if the brightest objects are not well reconstructed, their residuals can be more energetic than faint objects. This paper proposes a marked point process within a nonparametric Bayesian framework for the detection of galaxies in hyperspectral data. The efficiency of the method is demonstrated on synthetic images, and it provides good results for very faint objects in quasi-real astrophysical hyperspectral data.

This paper presents a “method of moments” estimation technique for the study of multiple scattering on the hypersphere. The proposed model is similar to a compound Poisson process evolving on a special manifold: the unit hypersphere. The presented work makes use of an approximation result for multiply convolved von Mises-Fisher distributions on hyperspheres. Comparison with other approximations show the accuracy of the proposed model to provide estimators for the mean free path and concentration parameters when studying a multiple scattering process. Such a process is classically used to model the propagation of waves or particles in random media.

This paper addresses the problem of the supervised signal classification, by using a hierarchical Bayesian method. Each signal is characterized by a set of parameters, the features, which are estimated from a set of learning signals. Moreover, these parameters are distributed according to a class-specific posterior distribution which allows one to capture the variability of the features within the same class. Within the hierarchical Bayesian framework, the feature extraction step and the learning step can be performed jointly. Unfortunately, the estimation of the class-specific distribution parameters requires the computation of intractable multi-dimensional integrals. Then a Markov-chain Monte Carlo (MCMC) algorithm is used to sample the posterior distributions of the features over all the training signals of each class. An application to electrical transient classification for non-intrusive load monitoring is introduced. Simulations over real-world electrical transients signals are driven and show the capacity of the proposed methodology to discriminate two classes of transients.

Marked point processes have received a great attention in the recent years, for their ability to extract objects in large data sets as those obtained in biological studies or hyperspectral remote sensing frameworks. This paper focuses on an original Bayesian point process estimation for the detection of galaxies from the hyperspectral data ‘cubes’ provided by the Multi Unit Spectroscopic Explorer (MUSE) instrument. It is shown that this approach allows to obtain a synthetic representation of the detection problem and circumvent the computational complexity inherent to high dimensional pixel based approaches. The reversible jump Monte Carlo Markov Chain implemented to sample the parameters is detailed, and the results obtained on benchmark data mimicking the real instrument are provided.

This paper proposes a Bayesian algorithm to estimate the parameters of a smooth transition regression model. With in this model, time series are divided into segments and a linear regression analysis is performed on each segment. Unlike a piecewise regression model, smooth transition functions are introduced to model smooth transitions between the sub-models. Appropriate prior distributions are associated with each parameter to penalize a data-driven criterion, leading to a fully Bayesian model. Then, a reversible jump Markov Chain Monte Carlo algorithm is derived to sample the parameter posterior distributions. It allows one to compute standard Bayesian estimators, providing a sparse representation of the data. Results are obtained for real-world electrical transients with a view to non-intrusive load monitoring applications.

Information contained in polarimetric images can be characterized by a scalar parameter called the polarization degree. This parameter is usually estimated using four polarimetric images. However, acquisition and registration of four images are complex and costly. Thus, reducing the number of required images can be of great interest for practical applications. In coherent illumination, these images are degraded by speckle noise. This noise can be reduced by transforming the original single-look images into multi-look images. We provide maximum likelihood estimators of the polarization degree in the general case of multi-look intensity images. The estimators are derived based on only two measurements, under coherent illumination and fully developed speckle. We evaluate our method on synthetic data and compare its performance with that of moment-based methods.

A new family of distributions, constructed by summing two correlated gamma random variables, is studied. First, a simple closed form expression for their density is derived. Second, the three parameters characterizing such a density are estimated by using the maximum likelihood (ML) principle. Numerical simulations are conducted to compare the performance of the ML estimator against those of the conventional estimator of moments. Finally, a multiresolution multivariate gamma based modeling of Internet traffic illustrates the potential interest of the proposed distributions for the detection of anomalies. Aggregated times series of IP packet counts are split into adjacent non overlapping time blocks. The distribution of the resulting time series are modeled by the proposed multivariate gamma based distributions, over a collection of different aggregation levels. The anomaly detection strategy is based on tracking changes along time of the corresponding multiresolution parameters.

This paper studies a family of distributions constructed from multivariate gamma distributions to model the statistical properties of multisensor synthetic aperture radar (SAR) images. These distributions referred to as multisensor multivariate gamma distributions (MuMGDs) are potentially interesting for detecting changes in SAR images acquired by different sensors having different numbers of looks. The first part of this paper compares different estimators for the parameters of MuMGDs. These estimators are based on the maximum likelihood principle, the method of inference function for margins, and the method of moments. The second part of the paper studies change detection algorithms based on the estimated correlation coefficient of MuMGDs. Simulation results conducted on synthetic and real data illustrate the performance of these change detectors.

This paper addresses the problem of estimating the parameters of a family of bivariate gamma distributions whose margins have different shape parameters. These distributions are interesting to detect changes in two synthetic radar aperture (SAR) images acquired by different sensors and having different numbers of looks. The estimators based on the maximum likelihood method and the method of moments are studied for these distributions. An application to change detection is finally discussed.

This paper evaluates the potential interest of using bivariate gamma distributions for image registration and change detection. The first part of this paper studies estimators for the parameters of bivariate gamma distributions based on the maximum likelihood principle and the method of moments. The performance of both methods are compared in terms of estimated mean square errors and theoretical asymptotic variances. The mutual information is a classical similarity measure which can be used for image registration or change detection. The second part of the paper studies some properties of the mutual information for bivariate Gamma distributions. Image registration and change detection techniques based on bivariate gamma distributions are finally investigated. Simulation results conducted on synthetic and real data are very encouraging. Bivariate gamma distributions are good candidates allowing us to develop new image registration algorithms and new change detectors.

Estimating the parameters of multivariate distributions whose densities or masses cannot be expressed in tractable closed-form is a challenging problem. This paper concentrates on a family of such discrete distributions referred to as multivariate mixed Poisson distributions (MMPDs). These distributions are interesting for modeling correlations between adjacent pixels of active and astronomical images. Several estimators of MMPD parameters are investigated. These estimators include a composite likelihood estimator and a non-linear least squares estimator