Error Analysis of Low-rank Three-Way Tensor Factorization Approach to Blind Source Separation

International audienceIn tensor factorization approaches 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 tensor-based blind decomposition of an experimental multispectral image of a skin tumor

Robust Manifold Nonnegative Tucker Factorization for Tensor Data Representation

Nonnegative Tucker Factorization (NTF) minimizes the euclidean distance or Kullback-Leibler divergence between the original data and its low-rank approximation which often suffers from grossly corruptions or outliers and the neglect of manifold structures of data. In particular, NTF suffers from rotational ambiguity, whose solutions with and without rotation transformations are equally in the sense of yielding the maximum likelihood. In this paper, we propose three Robust Manifold NTF algorithms to handle outliers by incorporating structural knowledge about the outliers. They first applies a half-quadratic optimization algorithm to transform the problem into a general weighted NTF where the weights are influenced by the outliers. Then, we introduce the correntropy induced metric, Huber function and Cauchy function for weights respectively, to handle the outliers. Finally, we introduce a manifold regularization to overcome the rotational ambiguity of NTF. We have compared the proposed method with a number of representative references covering major branches of NTF on a variety of real-world image databases. Experimental results illustrate the effectiveness of the proposed method under two evaluation metrics (accuracy and nmi)

Tensor dictionary learning with sparse TUCKER decomposition

Dictionary learning algorithms are typically derived for dealing with one or two dimensional signals using vector-matrix operations. Little attention has been paid to the problem of dictionary learning over high dimensional tensor data. We propose a new algorithm for dictionary learning based on tensor factorization using a TUCKER model. In this algorithm, sparseness constraints are applied to the core tensor, of which the n-mode factors are learned from the input data in an alternate minimization manner using gradient descent. Simulations are provided to show the convergence and the reconstruction performance of the proposed algorithm. We also apply our algorithm to the speaker identification problem and compare the discriminative ability of the dictionaries learned with those of TUCKER and K-SVD algorithms. The results show that the classification performance of the dictionaries learned by our proposed algorithm is considerably better as compared to the two state of the art algorithms. © 2013 IEEE

Multiway Array Decomposition Analysis of EEGs in Alzheimer’s Disease

Methods for the extraction of features from physiological datasets are growing needs as clinical investigations of Alzheimer’s disease (AD) in large and heterogeneous population increase. General tools allowing diagnostic regardless of recording sites, such as different hospitals, are essential and if combined to inexpensive non-invasive methods could critically improve mass screening of subjects with AD. In this study, we applied three state of the art multiway array decomposition (MAD) methods to extract features from electroencephalograms (EEGs) of AD patients obtained from multiple sites. In comparison to MAD, spectral-spatial average filter (SSFs) of control and AD subjects were used as well as a common blind source separation method, algorithm for multiple unknown signal extraction (AMUSE). We trained a feed-forward multilayer perceptron (MLP) to validate and optimize AD classification from two independent databases. Using a third EEG dataset, we demonstrated that features extracted from MAD outperformed features obtained from SSFs AMUSE in terms of root mean squared error (RMSE) and reaching up to 100% of accuracy in test condition. We propose that MAD maybe a useful tool to extract features for AD diagnosis offering great generalization across multi-site databases and opening doors to the discovery of new characterization of the disease

Manifold Integration: Data Integration on Multiple Manifolds

In data analysis, data points are usually analyzed based on their relations to other points (e.g., distance or inner product). This kind of relation can be analyzed on the manifold of the data set. Manifold learning is an approach to understand such relations. Various manifold learning methods have been developed and their effectiveness has been demonstrated in many real-world problems in pattern recognition and signal processing. However, most existing manifold learning algorithms only consider one manifold based on one dissimilarity matrix. In practice, multiple measurements may be available, and could be utilized. In pattern recognition systems, data integration has been an important consideration for improved accuracy given multiple measurements. Some data integration algorithms have been proposed to address this issue. These integration algorithms mostly use statistical information from the data set such as uncertainty of each data source, but they do not use the structural information (i.e., the geometric relations between data points). Such a structure is naturally described by a manifold. Even though manifold learning and data integration have been successfully used for data analysis, they have not been considered in a single integrated framework. When we have multiple measurements generated from the same data set and mapped onto different manifolds, those measurements can be integrated using the structural information on these multiple manifolds. Furthermore, we can better understand the structure of the data set by combining multiple measurements in each manifold using data integration techniques. In this dissertation, I present a new concept, manifold integration, a data integration method using the structure of data expressed in multiple manifolds. In order to achieve manifold integration, I formulated the manifold integration concept, and derived three manifold integration algorithms. Experimental results showed the algorithms' effectiveness in classification and dimension reduction. Moreover, for manifold integration, I showed that there are good theoretical and neuroscientific applications. I expect the manifold integration approach to serve as an effective framework for analyzing multimodal data sets on multiple manifolds. Also, I expect that my research on manifold integration will catalyze both manifold learning and data integration research

Nonnegative Tucker decomposition with alpha-divergence

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