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Front Matter |
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Abstract
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Abstract
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Robust Multilinear Decomposition of Low Rank Tensors |
Xu Han,Laurent Albera,Amar Kachenoura,Huazhong Shu,Lotfi Senhadji |
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Abstract
Although several methods are available to compute the multilinear rank of multi-way arrays, they are not sufficiently robust with respect to the noise. In this paper, we propose a novel Multilinear Tensor Decomposition (MTD) method, namely R-MTD (Robust MTD), which is able to identify the multilinear rank even in the presence of noise. The latter is based on sparsity and group sparsity of the core tensor imposed by means of the . norm and the mixed-norm, respectively. After several iterations of R-MTD, the underlying core tensor is sufficiently well estimated, which allows for an efficient use of the minimum description length approach and an accurate estimation of the multilinear rank. Computer results show the good behavior of R-MTD.
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Multichannel Audio Modeling with Elliptically Stable Tensor Decomposition |
Mathieu Fontaine,Fabian-Robert Stöter,Antoine Liutkus,Umut Şimşekli,Romain Serizel,Roland Badeau |
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Abstract
This paper introduces a new method for multichannel speech enhancement based on a versatile modeling of the residual noise spectrogram. Such a model has already been presented before in the single channel case where the noise component is assumed to follow an alpha-stable distribution for each time-frequency bin, whereas the speech spectrogram, supposed to be more regular, is modeled as Gaussian. In this paper, we describe a multichannel extension of this model, as well as a Monte Carlo Expectation - Maximisation algorithm for parameter estimation. In particular, a multichannel extension of the Itakura-Saito nonnegative matrix factorization is exploited to estimate the spectral parameters for speech, and a Metropolis-Hastings algorithm is proposed to estimate the noise contribution. We evaluate the proposed method in a challenging multichannel denoising application and compare it to other state-of-the-art algorithms.
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Sum Conditioned Poisson Factorization |
Gökhan Çapan,Semih Akbayrak,Taha Yusuf Ceritli,Ali Taylan Cemgil |
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Abstract
We develop an extension to Poisson factorization, to model Multinomial data using a moment parametrization. Our construction is an alternative to the canonical construction of generalized linear models. This is achieved by defining . component Poisson Factorization models and constraining the sum of observation tensors across components. A family of fully conjugate tensor decomposition models for binary, ordinal or multinomial data is devised as a result, which can be used as a generic building block in hierarchical models for arrays of such data. We give parameter estimation and approximate inference procedures based on Expectation Maximization and variational inference. The flexibility of the resulting model on binary and ordinal matrix factorizations is illustrated. Empirical evaluation is performed for movie recommendation on ordinal ratings matrix, and for knowledge graph completion on binary tensors. The model is tested for both prediction and producing ranked lists.
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Curve Registered Coupled Low Rank Factorization |
Jeremy Emile Cohen,Rodrigo Cabral Farias,Bertrand Rivet |
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Abstract
We propose an extension of the canonical polyadic (CP) tensor model where one of the latent factors is allowed to vary through data slices in a constrained way. The components of the latent factors, which we want to retrieve from data, can vary from one slice to another up to a diffeomorphism. We suppose that the diffeomorphisms are also unknown, thus merging curve registration and tensor decomposition in one model, which we call registered CP. We present an algorithm to retrieve both the latent factors and the diffeomorphism, which is assumed to be in a parametrized form. At the end of the paper, we show simulation results comparing registered CP with other models from the literature.
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Source Analysis and Selection Using Block Term Decomposition in Atrial Fibrillation |
Pedro Marinho R. de Oliveira,Vicente Zarzoso |
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Atrial fibrillation (AF) is the most common sustained cardiac arrhythmia in clinical practice, and is becoming a major public health concern. To better understand the mechanisms of this arrhythmia an accurate analysis of the atrial activity (AA) signal in electrocardiogram (ECG) recordings is necessary. The block term decomposition (BTD), a tensor factorization technique, has been recently proposed as a tool to extract the AA in ECG signals using a blind source separation (BSS) approach. This paper makes a deep analysis of the sources estimated by BTD, showing that the classical method to select the atrial source among the other sources may not work in some cases, even for the matrix-based methods. In this context, we propose two new automated methods to select the atrial source by considering another novel parameter. Experimental results on ten patients show the validity of the proposed methods.
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Some Issues in Computing the CP Decomposition of NonNegative Tensors |
Mohamad Jouni,Mauro Dalla Mura,Pierre Comon |
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Abstract
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 in the decomposition, and the choice of scaling factors has an impact during the execution of iterative algorithms and should not be overlooked. Computer experiments support the interest in the greedy algorithm proposed, in the case of the CP decomposition.
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Abstract
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A Grassmannian Minimum Enclosing Ball Approach for Common Subspace Extraction |
Emilie Renard,Kyle A. Gallivan,P.-A. Absil |
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Abstract
We study the problem of finding a subspace representative of multiple datasets by minimizing the maximal dissimilarity between this subspace and all the subspaces generated by those datasets. After arguing for the choice of the dissimilarity function, we derive some properties of the corresponding formulation. We propose an adaptation of an algorithm used for a similar problem on Riemannian manifolds. Experiments on synthetic data show that the subspace recovered by our algorithm is closer to the true common subspace than the solution obtained using an SVD.
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Decoupling Multivariate Functions Using Second-Order Information and Tensors |
Philippe Dreesen,Jeroen De Geeter,Mariya Ishteva |
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Abstract
The power of multivariate functions is their ability to model a wide variety of phenomena, but have the disadvantages that they lack an intuitive or interpretable representation, and often require a (very) large number of parameters. We study decoupled representations of multivariate vector functions, which are linear combinations of univariate functions in linear combinations of the input variables. This model structure provides a description with fewer parameters, and reveals the internal workings in a simpler way, as the nonlinearities are one-to-one functions. In earlier work, a tensor-based method was developed for performing this decomposition by using first-order derivative information. In this article, we generalize this method and study how the use of second-order derivative information can be incorporated. By doing this, we are able to push the method towards more involved configurations, while preserving uniqueness of the underlying tensor decompositions. Furthermore, even for some non-identifiable structures, the method seems to return a valid decoupled representation. These results are a step towards more general data-driven and noise-robust tensor-based framework for
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Nonnegative PARAFAC2: A Flexible Coupling Approach |
Jeremy E. Cohen,Rasmus Bro |
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Abstract
Modeling variability in tensor decomposition methods is one of the challenges of source separation. One possible solution to account for variations from one data set to another, jointly analysed, is to resort to the PARAFAC2 model. However, so far imposing constraints on the mode with variability has not been possible. In the following manuscript, a relaxation of the PARAFAC2 model is introduced, that allows for imposing nonnegativity constraints on the varying mode. An algorithm to compute the proposed flexible PARAFAC2 model is derived, and its performance is studied on both synthetic and chemometrics data.
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Applications of Polynomial Common Factor Computation in Signal Processing |
Ivan Markovsky,Antonio Fazzi,Nicola Guglielmi |
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Abstract
We consider the problem of computing the greatest common divisor of a set of univariate polynomials and present applications of this problem in system theory and signal processing. One application is blind system identification: given the responses of a system to unknown inputs, find the system. Assuming that the unknown system is finite impulse response and at least two experiments are done with inputs that have finite support and their Z-transforms have no common factors, the impulse response of the system can be computed up to a scaling factor as the greatest common divisor of the Z-transforms of the outputs. Other applications of greatest common divisor problem in system theory and signal processing are finding the distance of a system to the set of uncontrollable systems and common dynamics estimation in a multi-channel sum-of-exponentials model.
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Joint Nonnegative Matrix Factorization for Underdetermined Blind Source Separation in Nonlinear Mixt |
Ivica Kopriva |
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Abstract
An approach is proposed for underdetermined blind separation of nonnegative dependent (overlapped) sources from their nonlinear mixtures. The method performs empirical kernel maps based mappings of original data matrix onto reproducible kernel Hilbert spaces (RKHSs). Provided that sources comply with probabilistic model that is sparse in support and amplitude nonlinear underdetermined mixture model in the input space becomes overdetermined linear mixture model in RKHS comprised of original sources and their mostly second-order monomials. It is assumed that linear mixture models in different RKHSs share the same representation, i.e. the matrix of sources. Thus, we propose novel sparseness regularized joint nonnegative matrix factorization method to separate sources shared across different RKHSs. The method is validated comparatively on numerical problem related to extraction of eight overlapped sources from three nonlinear mixtures.
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Image Completion with Nonnegative Matrix Factorization Under Separability Assumption |
Tomasz Sadowski,Rafał Zdunek |
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Abstract
Nonnegative matrix factorization is a well-known unsupervised learning method for part-based feature extraction and dimensionality reduction of a nonnegative matrix with a variety of applications. One of them is a matrix completion problem in which missing entries in an observed matrix is recovered on the basis of partially known entries. In this study, we present a geometric approach to the low-rank image completion problem with separable nonnegative matrix factorization of an incomplete data. The proposed method recursively selects extreme rays of a simplicial cone spanned by an observed image and updates the latent factors with the hierarchical alternating least-squares algorithm. The numerical experiments performed on several images with missing entries demonstrate that the proposed method outperforms other algorithms in terms of computational time and accuracy.
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Feature Selection in Weakly Coherent Matrices |
Stéphane Chrétien,Olivier Ho |
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Abstract
A problem of paramount importance in both pure (Restricted Invertibility problem) and applied mathematics (Feature extraction) is the one of selecting a submatrix of a given matrix, such that this submatrix has its smallest singular value above a specified level. Such problems can be addressed using perturbation analysis. In this paper, we propose a perturbation bound for the smallest singular value of a given matrix after appending a column, under the assumption that its initial coherence is not large, and we use this bound to derive a fast algorithm for feature extraction.
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Variable Projection Applied to Block Term Decomposition of Higher-Order Tensors |
Guillaume Olikier,P.-A. Absil,Lieven De Lathauwer |
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Abstract
Higher-order tensors have become popular in many areas of applied mathematics such as statistics, scientific computing, signal processing or machine learning, notably thanks to the many possible ways of decomposing a tensor. In this paper, we focus on the best approximation in the least-squares sense of a higher-order tensor by a block term decomposition. Using variable projection, we express the tensor approximation problem as a minimization of a cost function on a Cartesian product of Stiefel manifolds. The effect of variable projection on the Riemannian gradient algorithm is studied through numerical experiments.
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Accelerating Likelihood Optimization for ICA on Real Signals |
Pierre Ablin,Jean-François Cardoso,Alexandre Gramfort |
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Abstract
We study optimization methods for solving the maximum likelihood formulation of independent component analysis (ICA). We consider both the problem constrained to white signals and the unconstrained problem. The Hessian of the objective function is costly to compute, which renders Newton’s method impractical for large data sets. Many algorithms proposed in the literature can be rewritten as quasi-Newton methods, for which the Hessian approximation is cheap to compute. These algorithms are very fast on simulated data where the linear mixture assumption really holds. However, on real signals, we observe that their rate of convergence can be severely impaired. In this paper, we investigate the origins of this behavior, and show that the recently proposed Preconditioned ICA for Real Data (Picard) algorithm overcomes this issue on both constrained and unconstrained problems.
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Orthogonally-Constrained Extraction of Independent Non-Gaussian Component from Non-Gaussian Backgrou |
Zbyněk Koldovský,Petr Tichavský,Nobutaka Ono |
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Abstract
We propose a new algorithm for Independent Component Extraction that extracts one non-Gaussian component and is capable to exploit the non-Gaussianity of background signals without decomposing them into independent components. The algorithm is suitable for situations when the signal to be extracted is determined through initialization; it shows an extra stable convergence when the target component is dominant. In simulations, the proposed method is compared with Natural Gradient and One-unit FastICA, and it yields improved results in terms of the Signal-to-Interference ratio and the number of successful extractions.
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发表于 2025-3-21 18:17:46
