Pierre Comon

CNRS, University of Grenoble · Gipsa-Lab

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 mu...

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...

Multidimensional signal analysis has become an important part of many signal processing problems. This type of analysis allows to take advantage of different diversities of a signal in order to extract useful information. This paper focuses on the design and development of multidimensional data decomposition algorithms called Canonical Polyadic (CP...

Most normality tests in the literature are performed for scalar and independent samples. Thus, they become unreliable when applied to colored processes, hampering their use in realistic scenarios. We focus on Mardia’s multivariate kurtosis, derive closed-form expressions of its asymptotic distribution for statistically dependent samples, under the...

Performances of the Multivariate Kurtosis are investigated when applied to colored data, with or without Auto-Regressive pre-whitening, and with or without projection onto a lower-dimensional random subspace. Computer experiments demonstrate the importance of taking into account the possible color of the process in calculating the power of the norm...

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...

Gas sensors lack repeatability over time. They are affected by drift, the result of changes at the sensor level and in the environment. A solution is to design software methods that compensate for the drift. Existing methods are often based on calibration samples acquired at the start of each new measurement session. However, finding a good referen...

We propose a theoretical performance analysis for a class of reconstruction problems, formulated as coupled canonical polyadic decompositions of two low-resolution tensor observations. We study a particular case when all the modes of the tensors are coupled. Unlike the case of a single coupling constraint, a fully-coupled model requires nonlinear c...

Permutation and scaling ambiguities are relevant issues in tensor decomposition and source separation algorithms. Although these ambiguities are inevitable when working on real data sets, it is preferred to eliminate these uncertainties for evaluating algorithms on synthetic data sets. As shown in the paper, the existing performance indices for thi...

With the help of auxiliary data, tensor completion may better recover a low rank multidimensional array from limited observation entries. Most existing methods, including coupled matrix-tensor factorization and coupled tensor rank minimization, mainly focus on how to extract and incorporate subspace or directly use auxiliary data for tensor complet...

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...

Most normality tests in the literature are performed for scalar and independent samples. Thus, they become unreliable when applied to colored processes, hampering their use in realistic scenarios. We focus on Mardia's multivariate kurtosis, derive closed-form expressions of its asymptotic distribution for statistically dependent samples, under the...

Tensor models play an increasingly prominent role in many fields, notably in machine learning. In several applications of such models, such as community detection, topic modeling and Gaussian mixture learning, one must estimate a low-rank signal from a noisy tensor. Hence, understanding the fundamental limits and the attainable performance of estim...

The Canonical Polyadic (CP) tensor decomposition has become an attractive mathematical tool in several fields during the last ten years. This decomposition is very powerful for representing and analyzing multidimensional data. The most attractive feature of the CP decomposition is its uniqueness, contrary to rank-revealing matrix decompositions, wh...

Over the last two decades, tensor-based methods have received a growing attention in the signal processing community. In this work, we propose a comprehensive overview of tensor-based models and methods for multisensor signal processing. We present for instance the Tucker decomposition, the Canonical Polyadic Decomposition (CPD), the Tensor-Train D...

We propose a theoretical performance analysis for the hyperspectral super-resolution task, formulated as a coupled canonical polyadic decomposition. We introduce two probabilistic scenarios along with different parameterizations, then derive constrained Cramér-Rao lower bounds (CCRB) for the proposed scenarios. We then illustrate the versatility of...

Robot olfaction takes inspiration from animals for locating a gas source in the environment, such as a gas leak to fix or an explosive to neutralize. end In these cases, gas sources end emit Volatile Organic Compounds (VOCs) which can be measured with an electronic nose. This instrument can detect a broad variety of VOCs, so the same device can the...

In this paper, we propose a gradient based block coordinate descent (BCD-G) framework to solve the joint approximate diagonalization of matrices defined on the product of the complex Stiefel manifold and the special linear group. Instead of the cyclic fashion, we choose the block for optimization in a way based on the Riemannian gradient. To update...

Canonical Polyadic (CP) tensor decomposition is useful in many real-world applications due to its uniqueness, and the ease of interpretation of its factor matrices. This work addresses the problem of calculating the CP decomposition of tensors in difficult cases where the factor matrices in one or all modes are almost collinear – i.e. bottleneck or...

In some applications, blind source separation can be performed by computing an approximate block-term tensor decomposition (BTD), under much milder constraints than matrix-based techniques. However, choosing the BTD model structure (i.e., the number of blocks and their ranks) is a difficult problem, and the standard least-squares formulation can be...

Chiral discrimination is a key problem in analytical chemistry. It is generally performed using expensive instruments or highly-specific miniaturized sensors. An artificial nose is a bio-inspired instrument capable after training of discriminating a wide variety of analytes. However, generality is achieved at the cost of specificity which makes chi...

Hyperspectral Image (HSI) classification refers to classifying hyperspectral data into features, where labels are given to pixels sharing the same features, distinguishing the present materials of the scene from one another. Naturally a HSI acquires spectral features of pixels, but spatial features based on neighborhood information are also importa...

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 introd...

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 two SVD-based algorithms that are simple and fast, but with a pe...

Jacobi-type algorithms for simultaneous approximate diagonalization of symmetric real tensors (or partially symmetric complex tensors) have been widely used in independent component analysis (ICA) because of its high performance. One natural way of choosing the index pairs in Jacobi-type algorithms is the classical cyclic ordering, while the other...

In this paper, we propose a Jacobi-type algorithm to solve the low rank orthogonal approximation problem of symmetric tensors. This algorithm includes as a special case the well-known Jacobi CoM2 algorithm for the approximate orthogonal diagonalization problem of symmetric tensors. We first prove the weak convergence of this algorithm, \textit{i.e....

We derive the constrained Cramér-Rao bounds for a coupled CP model with linear constraints applied to the hyperspectral super-resolution problem. For this problem, we consider two tensors representing low-resolution hyperspectral and mul-tispectral images. In a practical measurement setup, white Gaussian noise sequences are added to each tensor wit...

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 establish...

Text mining, as a special case of data mining, refers to the estimation of knowledge or parameters necessary for certain purposes, such as unsupervised clustering by observing various documents. In this context, the topic of a document can be seen as a hidden variable, and words are multi-view variables related to each other by a topic. The main go...

Text mining is addressed by identifying multiview models. Following the recent works of A.Anandkumar, we show that the algorithms previously proposed exhibit important drawbacks. The algorithm subsequently proposed permits to decompose a nonnegative tensor built with sample joint probabilities. It compares favorably to the former on synthetically g...

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 wo...

Nous proposons une nouvelle approche pour la super-résolution hyperspectrale, fondée sur l'approximation tensorielle de rang faible d'un modèle de Tucker couplé. Ce papier montre que la reconstruction de l'image de super-résolution est possible pour un panel de rangs multilinéaires. De plus, nous proposons un algorithme fondé sur la SVD, simple et...

A Hyperspectral Image (HSI) is an image that is acquired by means of spatial and spectral acquisitions, over an almost continuous spectrum. Pixelwise classification is an important application in HSI due to the natural spectral diversity that the latter brings. There are many works where spatial information (e.g., contextual relations in a spatial...

We propose a gradient-based Jacobi algorithm for a class of maximization problems on the unitary group, with a focus on approximate diagonalization of complex matrices and tensors by unitary transformations.We provide weak convergence results, and prove local linear convergence of this algorithm.The convergence results also apply to the case of rea...

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 two SVD-based algorithms that are simple and fast, but with a pe...

Tensor decompositions are still in the process of study and development. In this paper, we point out a problem existing in nonnegative tensor decompositions, stemming from the representation of decomposable tensors by outer products of vectors, and propose approaches to solve it. In fact, a scaling indeterminacy appears whereas it is not inherent i...

We study path-connectedness and homotopy groups of sets of tensors defined by tensor rank, border rank, multilinear rank, as well as their symmetric counterparts for symmetric tensors. We show that over $\mathbb{C}$, the set of rank-$r$ tensors and the set of symmetric rank-$r$ symmetric tensors are both path-connected if $r$ is not more than the c...

We study path-connectedness and homotopy groups of sets of tensors defined by tensor rank, border rank, multilinear rank, as well as their symmetric counterparts for symmetric tensors. We show that over $\mathbb{C}$, the set of rank-$r$ tensors and the set of symmetric rank-$r$ symmetric tensors are both path-connected if $r$ is not more than the c...

In this paper, we study the orthogonal diagonalization problem of third order symmetric tensors. We define several classes of approximately diagonal tensors, including the ones corresponding to stationary points of the problem. We study relationships between these classes, and other well-known objects, such as tensor eigenvalues and eigenvectors. W...

In this paper, we study the approximate orthogonal diagonalization problem of third order symmetric tensors. We define several classes of approximately diagonal tensors, including the ones corresponding to the stationary points of this problem. We study the relationships between these classes, and other well-known objects, such as tensor Z-eigenval...

Under the action of the general linear group, the ranks of matrices $A$ and $B$ forming a $m \times n$ pencil $A + \lambda B$ can change, but in a restricted manner. Specifically, to every pencil one can associate a pair of minimal ranks, which is unique up to a permutation. This notion can be defined for matrix pencils and, more generally, also fo...

Over the past decades, a multitude of different brain source imaging algorithms have been developed to identify the neural generators underlying the surface electroencephalography measurements. While most of these techniques focus on determining the source positions, only a small number of recently developed algorithms provides an indication of the...

In this paper we present and analyze the performance of multidimensional ESPRIT ( $N$ -D ESPRIT) method for estimating parameters of $N$ -D superimposed damped and/or undamped exponentials. $N$ -D ESPRIT algorithm is based on low-rank decomposition of multilevel Hankel matrices formed by the $N$ -D data. In order to reduce the computational complex...

This paper establishes a tensor model for wideband coherent array processing including multiple physical diversities. A separable coherent focusing operation is proposed as a pre-processing step in order to ensure the multilinearity of the interpolated data. We propose an ALS algorithm to process tensor data, taking into account the noise correlati...

In this paper, 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 c...

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 (DoAs) under the assumption of unknown gain patterns, improving the performance of the omnidirectional cas...

In this paper, a new method for the estimation of the parameters of multidimensional (R-D) harmonic and damped complex signals in noise is presented. The problem is formulated as R simultaneous sparse approximations of multiple 1-D signals. To get a method able to handle large size signals while maintaining a sufficient resolution, a multigrid dict...

In subspace-based methods for mulditimensional harmonic retrieval, the modes can be estimated either from eigenvalues or eigenvectors. The purpose of this study is to find out which way is the best. We compare the state-of-the art methods N-D ESPRIT and IMDF, propose a modification of IMDF based on least-squares criterion, and derive expressions of...

In this paper, we consider a family of Jacobi-type algorithms for simultaneous orthogonal diagonalization problem of symmetric tensors. For the Jacobi-based algorithm of [SIAM J. Matrix Anal. Appl., 2(34):651--672, 2013], we prove its global convergence for simultaneous orthogonal diagonalization of symmetric matrices and 3rd-order tensors. We also...

In this paper, we consider a family of Jacobi-type algorithms for simultaneous orthogonal diagonalization problem of symmetric tensors. For the Jacobi-based algorithm of [SIAM J. Matrix Anal. Appl., 2(34):651--672, 2013], we prove its global convergence for simultaneous orthogonal diagonalization of symmetric matrices and 3rd-order tensors. We also...

Some theoretical difficulties that arise from dimensionality reduction for tensors with non-negative coefficients is discussed in this paper. A necessary and sufficient condition is derived for a low non-negative rank tensor to admit a non-negative Tucker decomposition with a core of the same non-negative rank. Moreover, we provide evidence that th...

This paper introduces directivity gain pattern as a physical diversity for tensor array processing, in addition to time and space shift diversities. We show that tensor formulation allows to estimate Directions of Arrival (DoAs) under the assumption of unknown gain pattern, improving the performance of the omnidirectional case. The proposed approac...

We study the semialgebraic structure of Dr, the set of nonnegative tensors of nonnegative rank not more than r, and use the results to infer various properties of nonnegative tensor rank. We determine all nonnegative typical ranks for cubical nonnegative tensors and show that the direct sum conjecture is true for nonnegative tensor rank. We show th...

This paper proposes a new perspective on the problem of multidimensional spectral factorization, through helical mapping: $d$-dimensional ($d$D) data arrays are vectorized, processed by $1$D cepstral analysis and then remapped onto the original space. Partial differential equations (PDEs) are the basic framework to describe the evolution of physica...

To deal with large multimodal datasets, coupled canonical polyadic decompositions are used as an approximation model. In this paper, a joint compression scheme is introduced to reduce the dimensions of the dataset. Joint compression allows to solve the approximation problem in a compressed domain using standard coupled decomposition algorithms. Com...

In this paper we analyse the performance of 2-D ESPRIT method for estimating parameters of 2-D superimposed damped exponentials. 2-D ESPRIT algorithm is based on low-rank decomposition of a Hankel-block-Hankel matrix that is formed by the 2-D data. Through a first-order perturbation analysis, we derive closed-form expressions for the variances of t...

Spectral unmixing (SU) is one of the most important and studied topics in hyperspectral image analysis. By means of spectral unmixing it is possible to decompose a hyperspectral image in its spectral components, the so-called endmembers, and their respective fractional spatial distributions, so-called abundance maps. The Canonical Polyadic (CP) ten...

We propose a noniterative algorithm, called SeROAP, 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...

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...

In this paper, we show that a general quadratic multivariate system in the real field can be reduced to a best rank-1 three-way tensor approximation problem. This fact provides a new approach to tackle a system of quadratic polynomials equations. Some experiments using the standard alternating least squares (ALS) algorithm are drawn to evince the u...

Objective Surface EEG recordings are routinely performed for the diagnosis and management of epilepsy. More particularly, they can help to delineate the brain regions involved in interictal epileptic activity. This is achieved by applying distributed source localization algorithms to the EEG data. Over the last two decades, a multitude of different...

This paper proposes a new perspective on the problem of multidimensional spectral factorization, through helical mapping: $d$-dimensional ($d$D) data arrays are vectorized, processed by $1$D cepstral analysis and then remapped onto the original space. Partial differential equations (PDEs) are the basic framework to describe the evolution of physica...

In this paper, we study a polynomial decomposition model that arises in problems of system identification, signal processing and machine learning. We show that this decomposition is a special case of the X-rank decomposition --- a powerful novel concept in algebraic geometry that generalizes the tensor CP decomposition. We prove new results on gene...

The completion of matrices with missing values under the rank constraint is a non-convex optimization problem. A popular convex relaxation is based on minimization of the nuclear norm (sum of singular values) of the matrix. For this relaxation, an important question is whether the two optimization problems lead to the same solution. This question w...

The problem of direction of arrival (DoA) estimation of seismic plane waves impinging on an array of sensors is considered from a new deterministic perspective using tensor decomposition techniques. In addition to temporal and spatial sampling, further information is taken into account, based on the different propagation speed of body waves (P and...

We study the semialgebraic structure of $D_r$, the set of nonnegative tensors of nonnegative rank not more than $r$, and use the results to infer various properties of nonnegative tensor rank. We determine all nonnegative typical ranks for cubical nonnegative tensors and show that the direct sum conjecture is true for nonnegative tensor rank. Under...

As a non-invasive 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 s...

New hyperspectral missions will collect huge amounts of hyperspectral data. Besides, 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, \textit{a.k.a.} multiway an...

A number of application areas such as biomedical engineering require solving an underdetermined linear inverse problem. In such a case, it is necessary to make assumptions on the sources to restore identifiability. This problem is encountered in brain-source imaging when identifying the source signals from noisy electroencephalographic or magnetoen...