The ability to map brain networks at the macroscale in living individuals is fundamental in effor... more The ability to map brain networks at the macroscale in living individuals is fundamental in efforts to chart the relation between human behavior, health and disease. We present a framework to encode structural brain connectomes and diffusion-weighted magnetic resonance data into multidimensional arrays (tensors). The framework overcomes current limitations in building connectomes; it prevents information loss by integrating the relation between connectome nodes, edges, fascicles and diffusion data. We demonstrate the utility of the framework for in vivo white matter mapping and anatomical computing. The framework reduces dramatically storage requirements for connectome evaluation methods, with up to 40x compression factors. We apply the framework to evaluate 1,980 connectomes, thirteen tractography methods, and three data sets. We describe a general equation to predicts connector resolution (number of fascicles) given data quality and tractography model parameters. Finally, we provi...
This paper proposes a new method for blind inversion of a monotonic nonlinear map applied to a su... more This paper proposes a new method for blind inversion of a monotonic nonlinear map applied to a sum of random variables. Such kinds of mixtures of random variables are found in source separation and Wiener system inversion problems, for example. The importance of our proposed method is based on the fact that it permits to decouple the estimation of the nonlinear part (nonlinear compensation) from the estimation of the linear one (source separation matrix or deconvolution filter), which can be solved by applying any convenient linear algorithm. Our new nonlinear compensation algorithm, the MaxEnt algorithm, generalizes the idea of Gaussianization of the observation by maximizing its entropy instead. We developed two versions of our algorithm based either in a polynomial or a neural network parameterization of the nonlinear function. We provide a sufficient condition on the nonlinear function and the probability distribution that gives a guarantee for the MaxEnt method to succeed compensating the distortion. Through an extensive set of simulations, MaxEnt is compared with existing algorithms for blind approximation of nonlinear maps. Experiments show that MaxEnt is able to successfully compensate monotonic distortions outperforming other methods in terms of the obtained Signal to Noise Ratio in many important cases, for example when the number of variables in a mixture is small. Besides its ability for compensating nonlinearities, MaxEnt is very robust, i.e. showing small variability in the results.
In this paper, a new algorithm for estimating sparse non-negative sources from a set of noisy lin... more In this paper, a new algorithm for estimating sparse non-negative sources from a set of noisy linear mixtures is proposed. In particular, difficult situations with high noise levels and more sources than sensors (underdetermined case) are considered. It is shown that, when sources are very sparse in time and overlapped at some locations, they can be recovered even with very low SNR and by using much fewer sensors than sources. A theoretical analysis based on Bayesian estimation tools is included showing strong connections with algorithms in related areas of research such as ICA, NMF, FOCUSS, and sparse representation of data with overcomplete dictionaries. Our algorithm uses a Bayesian approach by modelling sparse signals through mixed-state random variables. This new model for priors imposes 0 norm based sparsity. We start our analysis for the case of non-overlapped sources (1−sparse), which allows us to simplify the search of the posterior maximum avoiding a combinatorial search. General algorithms for overlapped cases, such as 2−sparse and k−sparse sources, are derived by using the algorithm for 1−sparse signals recursively. Additionally, a combination of our MAP algorithm with the NN-KSVD algorithm is proposed for estimating the mixing matrix and the sources simultaneously in a real blind fashion. A complete set of simulation results is included showing the performance of our algorithm.
We developed two new tools to deal with synthetic aperture radar (SAR) images, based on two diffe... more We developed two new tools to deal with synthetic aperture radar (SAR) images, based on two different and powerful mathematical formalism: wavelets (WT) and Karhunen Loève transforms (KLT). WT are used to remove the undesirable speckle, which comes from the specific measurement process of SAR systems, and KLT for the co-registration of SAR images.
We present two new algorithms for the estimation of the temperature from a realization of a 2-D I... more We present two new algorithms for the estimation of the temperature from a realization of a 2-D Ising model. The methods here introduced are based in the maximization of pseudo likelihood and on the minimum mean squared error (MSE) fit of the conditional probability function. We derive the analytical expressions of these estimators and also we include computational results comparing these new techniques with the traditional method. A very good performance in terms of the average absolute error and the average standard deviation is demonstrated through simulations in a 100 × 100 lattice in the ferromagnetic and antiferromagnetic cases. Summarizing, we have provided two new useful computational tools that allow us to measure the "virtual" temperature of an Ising like system.
A multilinear subspace regression model based on so called latent variable decomposition is intro... more A multilinear subspace regression model based on so called latent variable decomposition is introduced. Unlike standard regression methods which typically employ matrix (2D) data representations followed by vector subspace transformations, the proposed approach uses tensor subspace transformations to model common latent variables across both the independent and dependent data. The proposed approach aims to maximize the correlation between the so derived latent variables and is shown to be suitable for the prediction of multidimensional dependent data from multidimensional independent data, where for the estimation of the latent variables we introduce an algorithm based on Multilinear Singular Value Decomposition (MSVD) on a specially defined cross-covariance tensor. It is next shown that in this way we are also able to unify the existing Partial Least Squares (PLS) and N-way PLS regression algorithms within the same framework. Simulations on benchmark synthetic data confirm the advantages of the proposed approach, in terms of its predictive ability and robustness, especially for small sample sizes. The potential of the proposed technique is further illustrated on a real world task of the decoding of human intracranial electrocorticogram (ECoG) from a simultaneously recorded scalp electroencephalograph (EEG).
... Not. R. Astron. Soc., 318, 769-780, (2000) 7. EE Kuruoglu, L. Bedini, MT Paratore, E. Salerno... more ... Not. R. Astron. Soc., 318, 769-780, (2000) 7. EE Kuruoglu, L. Bedini, MT Paratore, E. Salerno, A. Tonazzini, "Source separation in astrophysical maps using independent factor analysis", Neural Networks, 6, 479-491, (2003). ...
Tractography uses diffusion MRI to estimate the trajectory and cortical projection zones of white... more Tractography uses diffusion MRI to estimate the trajectory and cortical projection zones of white matter fascicles in the living human brain. There are many different tractography algorithms and each requires the user to set several parameters, such as curvature threshold. Choosing a single algorithm with specific parameters poses two challenges. First, different algorithms and parameter values produce different results. Second, the optimal choice of algorithm and parameter value may differ between different white matter regions or different fascicles, subjects, and acquisition parameters. We propose using ensemble methods to reduce algorithm and parameter dependencies. To do so we separate the processes of fascicle generation and evaluation. Specifically, we analyze the value of creating optimized connectomes by systematically combining candidate streamlines from an ensemble of algorithms (deterministic and probabilistic) and systematically varying parameters (curvature and stoppin...
Abstract When dealing with nonlinear blind processing algorithms (deconvolution or post-nonlinear... more Abstract When dealing with nonlinear blind processing algorithms (deconvolution or post-nonlinear source separation), complex mathematical estimations must be done giving as a result very slow algorithms. This is the case, for example, in speech processing, spike signals deconvolution or microarray data analysis. In this paper, we propose a simple method to reduce computational time for the inversion of Wiener systems or the separation of post-nonlinear mixtures, by using a linear approximation in a minimum mutual ...
We provide a solution to the BSS problem for the special case of statistically dependent sources.... more We provide a solution to the BSS problem for the special case of statistically dependent sources. We propose the MaxNG algorithm based on the maximization of a non-Gaussianity (NG) measure which is equivalent to minimizing the Shannon entropy of source estimates. We compare our algorithm against a strategy commonly used which is based on the minimization of mutual information (MinMI). It is shown that, for uncorrelated sources, both strategies arrive at similar solutions but when sources are dependent (correlated), better results are obtained using MaxNG. In order to measure NG, we use a non-parametric density estimation technique, namely Parzen windows, and L 2 -Euclidean distance in the space of density functions. A wide set of simulations based on real world data with complex dependence structures is presented, showing that our MaxNG algorithm successfully separates the sources, even when the original sources are strongly dependent for which traditional MinMI algorithms, such as ICA, usually fail. Many experimental results are provided to evaluate the performance of our algorithm for two and six sources. Comparisons of MaxNG with some popular BSS algorithms are provided. The main conclusion of the present work is that, our NG measure provides a useful tool for separating dependent signals since original sources usually represent local maxima of this measure. r
The number of neuroimaging data sets publicly available is growing at fast rate. The increase in ... more The number of neuroimaging data sets publicly available is growing at fast rate. The increase in availability and resolution of neuroimaging data requires modern approaches to signal processing for data analysis and results validation. We introduce the application of sparse multiway decomposition methods to linearized neuroimaging models. We show that decomposed models are more compact but as accurate as full models and can be successfully used for fast data analysis. We focus as example on a recent model for the evaluation of white matter connectomes . We show that the multiway decomposed model achieves accuracy comparable to the full model, while requiring only a small fraction of the memory and compute time. The approach has implications for a majority of neuroimaging methods using linear approximations to measured signals.
Blind Source Separation (BSS) problem has been extensively studied for signals composed by statis... more Blind Source Separation (BSS) problem has been extensively studied for signals composed by statistical indepen dent components. As it is well known, the applied methods usually fail when sources exhibit some degree of dependence. Our work points to solve the BBS problem removing the condition of statistical independence looking for sources estimates that maximize a measure of nonGaussianity. We present a measure of nongaussianity based on the L 2 Euclidean distance using a non parametric technique for the estimation of probability densities. These mathematical tools have allowed us to build new algorithms for BSS which may have good performance for dependent as well as independent nonGaussian realworld sources.
A binary additive communication channel with different noise processes is analyzed. Several noise... more A binary additive communication channel with different noise processes is analyzed. Several noise processes are generated according to Bernoulli, Markov, Polya, and Logarithmic distribu- tions. A noise process based on the two dimensional Ising model (Markov 2D) is also studied. In all cases, a maximum likelihood decoding algorithm is derived. We obtain interesting results since in many cases, the most probable code-word is either the closest to the input, or that farthest away, depending on the model parameters. x = (1;0;0;1;1;0;1;1;1;0;1;0;0) z = (0;0;1;1;0;0;0;0;1;1;0;0;1) y = (1;0;1;0;1;0;1;1;0;1;1;0;1)
The widespread use of multi-sensor technology and the emergence of big datasets has highlighted t... more The widespread use of multi-sensor technology and the emergence of big datasets has highlighted the limitations of standard flat-view matrix models and the necessity to move towards more versatile data analysis tools. We show that higher-order tensors (i.e., multiway arrays) enable such a fundamental paradigm shift towards models that are essentially polynomial and whose uniqueness, unlike the matrix methods, is guaranteed under very mild and natural conditions. Benefiting from the power of multilinear algebra as their mathematical backbone, data analysis techniques using tensor decompositions are shown to have great flexibility in the choice of constraints that match data properties, and to find more general latent components in the data than matrix-based methods. A comprehensive introduction to tensor decompositions is provided from a signal processing perspective, starting from the algebraic foundations, via basic Canonical Polyadic and Tucker models, through to advanced cause-effect and multi-view data analysis schemes. We show that tensor decompositions enable natural generalizations of some commonly used signal processing paradigms, such as canonical correlation and subspace techniques, signal separation, linear regression, feature extraction and classification. We also cover computational aspects, and point out how ideas from compressed sensing and scientific computing may be used for addressing the otherwise unmanageable storage and manipulation problems associated with big datasets. The concepts are supported by illustrative real world case studies illuminating the benefits of the tensor framework, as efficient and promising tools for modern signal processing, data analysis and machine learning applications; these benefits also extend to vector/matrix data through tensorization.
2014 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), 2014
ABSTRACT We introduce a reconstruction formula that allows one to recover an N-order tensor X ϵ R... more ABSTRACT We introduce a reconstruction formula that allows one to recover an N-order tensor X ϵ RI1×...×In from a reduced set of multi-way compressive measurements by exploiting its low multilinear rank structure. It is proved that, in the matrix case (N = 2), the proposed reconstruction is stable in the sense that the approximation error is proportional to the one provided by the best low-rank approximation, i.e ||X - X||2 ≤ K||X - X0||2, where K is a constant and X0 is the corresponding truncated SVD of X. We also present simulation results indicating that the same stable behavior is observed with higher order tensors (N > 2). In addition, it is shown that, an interesting property of multi-way measurements allows us to build the reconstruction based on compressive linear measurements of fibers taken only in two selected modes, independently of the tensor order N. Simulation results using real-world 2D and 3D signals are presented illustrating our results and comparing the reconstructions against the best low multilinear rank approximations and the reconstructions obtained by using the Kronecker-CS approach.
We report some of our results of a particular blind source separation technique applied to spectr... more We report some of our results of a particular blind source separation technique applied to spectral unmixing of remote-sensed hyperspectral images. Different nongaussianity measures are introduced in the learning procedure, and the results are compared to assess their relative efficiencies, with respect to both the output signal-to-interference ratio and the overall computational complexity. This study has been conducted on both simulated and real data sets, and the first results show that skewness is a powerful and unexpensive tool to extract the typical sources that characterize remote-sensed images.
In this paper we apply a novel tensor decomposition model of SOD (slice oriented decomposition) t... more In this paper we apply a novel tensor decomposition model of SOD (slice oriented decomposition) to extract slice features from the multichannel time-frequency representation of EEG signals measured for MI (motor imagery) tasks in application to BCI (brain computer interface). The advantages of the SOD based feature extraction approach lie in its capability to obtain slice matrix components across the space, time and frequency domains and the discriminative features across different classes without any prior knowledge of the discriminative frequency bands. Furthermore, the combination of horizontal, lateral and frontal slice features makes our method more robust for the outlier problem. The experiment results demonstrate the effectiveness and robustness of our method.
In the framework of multidimensional Compressed Sensing (CS), we introduce an analytical reconstr... more In the framework of multidimensional Compressed Sensing (CS), we introduce an analytical reconstruction formula that allows one to recover an N th-order data tensor X ∈ R I1×I2×···×IN from a reduced set of multi-way compressive measurements by exploiting its low multilinear-rank structure. Moreover, we show that, an interesting property of multi-way measurements allows us to build the reconstruction based on compressive linear measurements taken only in two selected modes, independently of the tensor order N . In addition, it is proved that, in the matrix case and in a particular case with 3rd-order tensors where the same 2D sensor operator is applied to all mode-3 slices, the proposed reconstruction X τ is stable in the sense that the approximation error is comparable to the one provided by the best low-multilinear-rank approximation, where τ is a threshold parameter that controls the approximation error. Through the analysis of the upper bound of the approximation error we show that, in the 2D case, an optimal value for the threshold parameter τ = τ 0 > 0 exists, which is confirmed by our simulation results. On the other hand, our experiments on 3D datasets show that very good reconstructions are obtained using τ = 0, which means that this parameter does not need to be tuned. Our extensive simulation results demonstrate the stability and robustness of the method when it is applied to real-world 2D and 3D signals. A comparison with state-of-the-arts sparsity based CS methods specialized for multidimensional signals is also included. A very attractive characteristic of the proposed method is that it provides a direct computation, i.e. it is non-iterative in contrast to all existing sparsity based CS algorithms, thus providing super fast computations, even for large datasets.
We present two new algorithms for the estimation of the temperature from a realization of a 2-D I... more We present two new algorithms for the estimation of the temperature from a realization of a 2-D Ising model. The methods here introduced are based in the maximization of pseudo likelihood and on the minimum mean squared error (MSE) fit of the conditional probability function. We derive the analytical expressions of these estimators and also we include computational results comparing
The ability to map brain networks at the macroscale in living individuals is fundamental in effor... more The ability to map brain networks at the macroscale in living individuals is fundamental in efforts to chart the relation between human behavior, health and disease. We present a framework to encode structural brain connectomes and diffusion-weighted magnetic resonance data into multidimensional arrays (tensors). The framework overcomes current limitations in building connectomes; it prevents information loss by integrating the relation between connectome nodes, edges, fascicles and diffusion data. We demonstrate the utility of the framework for in vivo white matter mapping and anatomical computing. The framework reduces dramatically storage requirements for connectome evaluation methods, with up to 40x compression factors. We apply the framework to evaluate 1,980 connectomes, thirteen tractography methods, and three data sets. We describe a general equation to predicts connector resolution (number of fascicles) given data quality and tractography model parameters. Finally, we provi...
This paper proposes a new method for blind inversion of a monotonic nonlinear map applied to a su... more This paper proposes a new method for blind inversion of a monotonic nonlinear map applied to a sum of random variables. Such kinds of mixtures of random variables are found in source separation and Wiener system inversion problems, for example. The importance of our proposed method is based on the fact that it permits to decouple the estimation of the nonlinear part (nonlinear compensation) from the estimation of the linear one (source separation matrix or deconvolution filter), which can be solved by applying any convenient linear algorithm. Our new nonlinear compensation algorithm, the MaxEnt algorithm, generalizes the idea of Gaussianization of the observation by maximizing its entropy instead. We developed two versions of our algorithm based either in a polynomial or a neural network parameterization of the nonlinear function. We provide a sufficient condition on the nonlinear function and the probability distribution that gives a guarantee for the MaxEnt method to succeed compensating the distortion. Through an extensive set of simulations, MaxEnt is compared with existing algorithms for blind approximation of nonlinear maps. Experiments show that MaxEnt is able to successfully compensate monotonic distortions outperforming other methods in terms of the obtained Signal to Noise Ratio in many important cases, for example when the number of variables in a mixture is small. Besides its ability for compensating nonlinearities, MaxEnt is very robust, i.e. showing small variability in the results.
In this paper, a new algorithm for estimating sparse non-negative sources from a set of noisy lin... more In this paper, a new algorithm for estimating sparse non-negative sources from a set of noisy linear mixtures is proposed. In particular, difficult situations with high noise levels and more sources than sensors (underdetermined case) are considered. It is shown that, when sources are very sparse in time and overlapped at some locations, they can be recovered even with very low SNR and by using much fewer sensors than sources. A theoretical analysis based on Bayesian estimation tools is included showing strong connections with algorithms in related areas of research such as ICA, NMF, FOCUSS, and sparse representation of data with overcomplete dictionaries. Our algorithm uses a Bayesian approach by modelling sparse signals through mixed-state random variables. This new model for priors imposes 0 norm based sparsity. We start our analysis for the case of non-overlapped sources (1−sparse), which allows us to simplify the search of the posterior maximum avoiding a combinatorial search. General algorithms for overlapped cases, such as 2−sparse and k−sparse sources, are derived by using the algorithm for 1−sparse signals recursively. Additionally, a combination of our MAP algorithm with the NN-KSVD algorithm is proposed for estimating the mixing matrix and the sources simultaneously in a real blind fashion. A complete set of simulation results is included showing the performance of our algorithm.
We developed two new tools to deal with synthetic aperture radar (SAR) images, based on two diffe... more We developed two new tools to deal with synthetic aperture radar (SAR) images, based on two different and powerful mathematical formalism: wavelets (WT) and Karhunen Loève transforms (KLT). WT are used to remove the undesirable speckle, which comes from the specific measurement process of SAR systems, and KLT for the co-registration of SAR images.
We present two new algorithms for the estimation of the temperature from a realization of a 2-D I... more We present two new algorithms for the estimation of the temperature from a realization of a 2-D Ising model. The methods here introduced are based in the maximization of pseudo likelihood and on the minimum mean squared error (MSE) fit of the conditional probability function. We derive the analytical expressions of these estimators and also we include computational results comparing these new techniques with the traditional method. A very good performance in terms of the average absolute error and the average standard deviation is demonstrated through simulations in a 100 × 100 lattice in the ferromagnetic and antiferromagnetic cases. Summarizing, we have provided two new useful computational tools that allow us to measure the "virtual" temperature of an Ising like system.
A multilinear subspace regression model based on so called latent variable decomposition is intro... more A multilinear subspace regression model based on so called latent variable decomposition is introduced. Unlike standard regression methods which typically employ matrix (2D) data representations followed by vector subspace transformations, the proposed approach uses tensor subspace transformations to model common latent variables across both the independent and dependent data. The proposed approach aims to maximize the correlation between the so derived latent variables and is shown to be suitable for the prediction of multidimensional dependent data from multidimensional independent data, where for the estimation of the latent variables we introduce an algorithm based on Multilinear Singular Value Decomposition (MSVD) on a specially defined cross-covariance tensor. It is next shown that in this way we are also able to unify the existing Partial Least Squares (PLS) and N-way PLS regression algorithms within the same framework. Simulations on benchmark synthetic data confirm the advantages of the proposed approach, in terms of its predictive ability and robustness, especially for small sample sizes. The potential of the proposed technique is further illustrated on a real world task of the decoding of human intracranial electrocorticogram (ECoG) from a simultaneously recorded scalp electroencephalograph (EEG).
... Not. R. Astron. Soc., 318, 769-780, (2000) 7. EE Kuruoglu, L. Bedini, MT Paratore, E. Salerno... more ... Not. R. Astron. Soc., 318, 769-780, (2000) 7. EE Kuruoglu, L. Bedini, MT Paratore, E. Salerno, A. Tonazzini, "Source separation in astrophysical maps using independent factor analysis", Neural Networks, 6, 479-491, (2003). ...
Tractography uses diffusion MRI to estimate the trajectory and cortical projection zones of white... more Tractography uses diffusion MRI to estimate the trajectory and cortical projection zones of white matter fascicles in the living human brain. There are many different tractography algorithms and each requires the user to set several parameters, such as curvature threshold. Choosing a single algorithm with specific parameters poses two challenges. First, different algorithms and parameter values produce different results. Second, the optimal choice of algorithm and parameter value may differ between different white matter regions or different fascicles, subjects, and acquisition parameters. We propose using ensemble methods to reduce algorithm and parameter dependencies. To do so we separate the processes of fascicle generation and evaluation. Specifically, we analyze the value of creating optimized connectomes by systematically combining candidate streamlines from an ensemble of algorithms (deterministic and probabilistic) and systematically varying parameters (curvature and stoppin...
Abstract When dealing with nonlinear blind processing algorithms (deconvolution or post-nonlinear... more Abstract When dealing with nonlinear blind processing algorithms (deconvolution or post-nonlinear source separation), complex mathematical estimations must be done giving as a result very slow algorithms. This is the case, for example, in speech processing, spike signals deconvolution or microarray data analysis. In this paper, we propose a simple method to reduce computational time for the inversion of Wiener systems or the separation of post-nonlinear mixtures, by using a linear approximation in a minimum mutual ...
We provide a solution to the BSS problem for the special case of statistically dependent sources.... more We provide a solution to the BSS problem for the special case of statistically dependent sources. We propose the MaxNG algorithm based on the maximization of a non-Gaussianity (NG) measure which is equivalent to minimizing the Shannon entropy of source estimates. We compare our algorithm against a strategy commonly used which is based on the minimization of mutual information (MinMI). It is shown that, for uncorrelated sources, both strategies arrive at similar solutions but when sources are dependent (correlated), better results are obtained using MaxNG. In order to measure NG, we use a non-parametric density estimation technique, namely Parzen windows, and L 2 -Euclidean distance in the space of density functions. A wide set of simulations based on real world data with complex dependence structures is presented, showing that our MaxNG algorithm successfully separates the sources, even when the original sources are strongly dependent for which traditional MinMI algorithms, such as ICA, usually fail. Many experimental results are provided to evaluate the performance of our algorithm for two and six sources. Comparisons of MaxNG with some popular BSS algorithms are provided. The main conclusion of the present work is that, our NG measure provides a useful tool for separating dependent signals since original sources usually represent local maxima of this measure. r
The number of neuroimaging data sets publicly available is growing at fast rate. The increase in ... more The number of neuroimaging data sets publicly available is growing at fast rate. The increase in availability and resolution of neuroimaging data requires modern approaches to signal processing for data analysis and results validation. We introduce the application of sparse multiway decomposition methods to linearized neuroimaging models. We show that decomposed models are more compact but as accurate as full models and can be successfully used for fast data analysis. We focus as example on a recent model for the evaluation of white matter connectomes . We show that the multiway decomposed model achieves accuracy comparable to the full model, while requiring only a small fraction of the memory and compute time. The approach has implications for a majority of neuroimaging methods using linear approximations to measured signals.
Blind Source Separation (BSS) problem has been extensively studied for signals composed by statis... more Blind Source Separation (BSS) problem has been extensively studied for signals composed by statistical indepen dent components. As it is well known, the applied methods usually fail when sources exhibit some degree of dependence. Our work points to solve the BBS problem removing the condition of statistical independence looking for sources estimates that maximize a measure of nonGaussianity. We present a measure of nongaussianity based on the L 2 Euclidean distance using a non parametric technique for the estimation of probability densities. These mathematical tools have allowed us to build new algorithms for BSS which may have good performance for dependent as well as independent nonGaussian realworld sources.
A binary additive communication channel with different noise processes is analyzed. Several noise... more A binary additive communication channel with different noise processes is analyzed. Several noise processes are generated according to Bernoulli, Markov, Polya, and Logarithmic distribu- tions. A noise process based on the two dimensional Ising model (Markov 2D) is also studied. In all cases, a maximum likelihood decoding algorithm is derived. We obtain interesting results since in many cases, the most probable code-word is either the closest to the input, or that farthest away, depending on the model parameters. x = (1;0;0;1;1;0;1;1;1;0;1;0;0) z = (0;0;1;1;0;0;0;0;1;1;0;0;1) y = (1;0;1;0;1;0;1;1;0;1;1;0;1)
The widespread use of multi-sensor technology and the emergence of big datasets has highlighted t... more The widespread use of multi-sensor technology and the emergence of big datasets has highlighted the limitations of standard flat-view matrix models and the necessity to move towards more versatile data analysis tools. We show that higher-order tensors (i.e., multiway arrays) enable such a fundamental paradigm shift towards models that are essentially polynomial and whose uniqueness, unlike the matrix methods, is guaranteed under very mild and natural conditions. Benefiting from the power of multilinear algebra as their mathematical backbone, data analysis techniques using tensor decompositions are shown to have great flexibility in the choice of constraints that match data properties, and to find more general latent components in the data than matrix-based methods. A comprehensive introduction to tensor decompositions is provided from a signal processing perspective, starting from the algebraic foundations, via basic Canonical Polyadic and Tucker models, through to advanced cause-effect and multi-view data analysis schemes. We show that tensor decompositions enable natural generalizations of some commonly used signal processing paradigms, such as canonical correlation and subspace techniques, signal separation, linear regression, feature extraction and classification. We also cover computational aspects, and point out how ideas from compressed sensing and scientific computing may be used for addressing the otherwise unmanageable storage and manipulation problems associated with big datasets. The concepts are supported by illustrative real world case studies illuminating the benefits of the tensor framework, as efficient and promising tools for modern signal processing, data analysis and machine learning applications; these benefits also extend to vector/matrix data through tensorization.
2014 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), 2014
ABSTRACT We introduce a reconstruction formula that allows one to recover an N-order tensor X ϵ R... more ABSTRACT We introduce a reconstruction formula that allows one to recover an N-order tensor X ϵ RI1×...×In from a reduced set of multi-way compressive measurements by exploiting its low multilinear rank structure. It is proved that, in the matrix case (N = 2), the proposed reconstruction is stable in the sense that the approximation error is proportional to the one provided by the best low-rank approximation, i.e ||X - X||2 ≤ K||X - X0||2, where K is a constant and X0 is the corresponding truncated SVD of X. We also present simulation results indicating that the same stable behavior is observed with higher order tensors (N > 2). In addition, it is shown that, an interesting property of multi-way measurements allows us to build the reconstruction based on compressive linear measurements of fibers taken only in two selected modes, independently of the tensor order N. Simulation results using real-world 2D and 3D signals are presented illustrating our results and comparing the reconstructions against the best low multilinear rank approximations and the reconstructions obtained by using the Kronecker-CS approach.
We report some of our results of a particular blind source separation technique applied to spectr... more We report some of our results of a particular blind source separation technique applied to spectral unmixing of remote-sensed hyperspectral images. Different nongaussianity measures are introduced in the learning procedure, and the results are compared to assess their relative efficiencies, with respect to both the output signal-to-interference ratio and the overall computational complexity. This study has been conducted on both simulated and real data sets, and the first results show that skewness is a powerful and unexpensive tool to extract the typical sources that characterize remote-sensed images.
In this paper we apply a novel tensor decomposition model of SOD (slice oriented decomposition) t... more In this paper we apply a novel tensor decomposition model of SOD (slice oriented decomposition) to extract slice features from the multichannel time-frequency representation of EEG signals measured for MI (motor imagery) tasks in application to BCI (brain computer interface). The advantages of the SOD based feature extraction approach lie in its capability to obtain slice matrix components across the space, time and frequency domains and the discriminative features across different classes without any prior knowledge of the discriminative frequency bands. Furthermore, the combination of horizontal, lateral and frontal slice features makes our method more robust for the outlier problem. The experiment results demonstrate the effectiveness and robustness of our method.
In the framework of multidimensional Compressed Sensing (CS), we introduce an analytical reconstr... more In the framework of multidimensional Compressed Sensing (CS), we introduce an analytical reconstruction formula that allows one to recover an N th-order data tensor X ∈ R I1×I2×···×IN from a reduced set of multi-way compressive measurements by exploiting its low multilinear-rank structure. Moreover, we show that, an interesting property of multi-way measurements allows us to build the reconstruction based on compressive linear measurements taken only in two selected modes, independently of the tensor order N . In addition, it is proved that, in the matrix case and in a particular case with 3rd-order tensors where the same 2D sensor operator is applied to all mode-3 slices, the proposed reconstruction X τ is stable in the sense that the approximation error is comparable to the one provided by the best low-multilinear-rank approximation, where τ is a threshold parameter that controls the approximation error. Through the analysis of the upper bound of the approximation error we show that, in the 2D case, an optimal value for the threshold parameter τ = τ 0 > 0 exists, which is confirmed by our simulation results. On the other hand, our experiments on 3D datasets show that very good reconstructions are obtained using τ = 0, which means that this parameter does not need to be tuned. Our extensive simulation results demonstrate the stability and robustness of the method when it is applied to real-world 2D and 3D signals. A comparison with state-of-the-arts sparsity based CS methods specialized for multidimensional signals is also included. A very attractive characteristic of the proposed method is that it provides a direct computation, i.e. it is non-iterative in contrast to all existing sparsity based CS algorithms, thus providing super fast computations, even for large datasets.
We present two new algorithms for the estimation of the temperature from a realization of a 2-D I... more We present two new algorithms for the estimation of the temperature from a realization of a 2-D Ising model. The methods here introduced are based in the maximization of pseudo likelihood and on the minimum mean squared error (MSE) fit of the conditional probability function. We derive the analytical expressions of these estimators and also we include computational results comparing
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