Nonnegative Canonical Polyadic Decomposition Research Articles

Zero-truncated Poisson regression for sparse multiway count data corrupted by false zeros

Abstract We propose a novel statistical inference methodology for multiway count data that is corrupted by false zeros that are indistinguishable from true zero counts. Our approach consists of zero-truncating the Poisson distribution to neglect all zero values. This simple truncated approach dispenses with the need to distinguish between true and false zero counts and reduces the amount of data to be processed. Inference is accomplished via tensor completion that imposes low-rank tensor structure on the Poisson parameter space. Our main result shows that an $N$-way rank-$R$ parametric tensor $\boldsymbol{\mathscr{M}}\in (0,\infty )^{I\times \cdots \times I}$ generating Poisson observations can be accurately estimated by zero-truncated Poisson regression from approximately $IR^2\log _2^2(I)$ non-zero counts under the nonnegative canonical polyadic decomposition. Our result also quantifies the error made by zero-truncating the Poisson distribution when the parameter is uniformly bounded from below. Therefore, under a low-rank multiparameter model, we propose an implementable approach guaranteed to achieve accurate regression in under-determined scenarios with substantial corruption by false zeros. Several numerical experiments are presented to explore the theoretical results.

Online Nonnegative and Sparse Canonical Polyadic Decomposition of Fluorescence Tensors

The NonNegative Canonical Polyadic Decomposition (NN-CPD) is used in many fields such as in chemistry, biology and medicine. The data coming from these fields can be dynamic which lead to use real-time or “online” decomposition. Even though there are a variety of online tensor decomposition algorithms, the main assumption of all these algorithms is that the rank of the decomposition is known and/or does not vary over time. However this should not be the case in experimental conditions. In this work, we propose three algorithms to compute the online NN-CPD based on sparse dictionary learning for tracking chemical components in water by using a set of Emission and Excitation Matrices (EEMs) of fluorescence. The methods developed in this work is not limited to this application field and it addresses the major challenges posed by the variation of the CPD rank in real-time. First, the algorithms take into account the unknown factors and the variation of tensor rank. Second, previous extracted information are used to decompose upcoming new tensors. In addition to the development of these algorithms, one of the contributions of this paper is the real-time acquisition of fluorescence data in a semi-controlled environment. These algorithms were applied on these real datasets and compared to state of the art algorithms.

Nonnegative Canonical Polyadic Decomposition for Tissue-Type Differentiation in Gliomas.

Magnetic resonance spectroscopic imaging (MRSI) reveals chemical information that characterizes different tissue types in brain tumors. Blind source separation techniques are used to extract the tissue-specific profiles and their corresponding distribution from the MRSI data. We focus on automatic detection of the tumor, necrotic and normal brain tissue types by constructing a 3D MRSI tensor from in vivo 2D-MRSI data of individual glioma patients. Nonnegative canonical polyadic decomposition (NCPD) is applied to the MRSI tensor to differentiate various tissue types. An in vivo study shows that NCPD has better performance in identifying tumor and necrotic tissue type in glioma patients compared to previous matrix-based decompositions, such as nonnegative matrix factorization and hierarchical nonnegative matrix factorization.

Nonnegative Tensor CP Decomposition of Hyperspectral Data

New hyperspectral missions will collect huge amounts of hyperspectral data. In addition, 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, which is also known as multiway analysis, is a technique to decompose multiway arrays, i.e., hypermatrices with more than two dimensions (ways). Hyperspectral time series and multiangular acquisitions can be represented as a three-way tensor. Here, we apply canonical polyadic (CP) tensor decomposition techniques to the blind analysis ohyperspectral big data. In order to do so, we use a novel compression-based nonnegative CP decomposition. We show that the proposed methodology can be interpreted as multilinear blind spectral unmixing, i.e., a higher order extension of the widely known spectral unmixing. In the proposed approach, the big hyperspectral tensor is decomposed in three sets of factors, which can be interpreted as spectral signatures, their spatial distribution, and temporal/angular changes. We provide experimental validation using a study case of the snow coverage of the French Alps during the snow season.

Tensor based tumor tissue type differentiation using magnetic resonance spectroscopic imaging.

Magnetic resonance spectroscopic imaging (MRSI) has the potential to characterise different tissue types in brain tumors. Blind source separation techniques are used to extract the specific tissue profiles and their corresponding distribution from the MRSI data. A 3-dimensional MRSI tensor is constructed from in vivo 2D-MRSI data of individual tumor patients. Non-negative canonical polyadic decomposition (NCPD) with common factor in mode-1 and mode-2 and l(1) regularization on mode-3 is applied on the MRSI tensor to differentiate various tissue types. Initial in vivo study shows that NCPD has better performance in identifying tumor and necrotic tissue type in high grade glioma patients compared to previous matrix-based decompositions, such as non-negative matrix factorization and hierarchical non-negative matrix factorization.

A regularized nonnegative canonical polyadic decomposition algorithm with preprocessing for 3D fluorescence spectroscopy

We consider blind source separation in chemical analysis focussing on the 3D fluorescence spectroscopy framework. We present an alternative method to process the Fluorescence Excitation‐Emission Matrices (FEEM): first, a preprocessing is applied to eliminate the Raman and Rayleigh scattering peaks that clutter the FEEM. To improve its robustness versus possible improper settings, we suggest to associate the classical Zepp's method with a morphological image filtering technique. Then, in the second stage, the Canonical Polyadic (CP or Candecomp/Parafac) decomposition of a nonnegative three‐way array has to be computed. In the fluorescence spectroscopy context, the constituent vectors of the loading matrices should be nonnegative (since standing for spectra and concentrations). Thus, we suggest a new nonnegative third order CP decomposition algorithm (NNCP) based on a nonlinear conjugate gradient optimization algorithm with regularization terms and periodic restarts. Computer simulations performed on real experimental data are provided to enlighten the effectiveness and robustness of the whole processing chain and to validate the approach. Copyright © 2015 John Wiley & Sons, Ltd.

Smooth nonnegative matrix and tensor factorizations for robust multi-way data analysis

In this paper, we discuss new efficient algorithms for nonnegative matrix factorization (NMF) with smoothness constraints imposed on nonnegative components or factors. Such constraints allow us to alleviate certain ambiguity problems, which facilitates better physical interpretation or meaning. In our approach, various basis functions are exploited to flexibly and efficiently represent the smooth nonnegative components. For noisy input data, the proposed algorithms are more robust than the existing smooth and sparse NMF algorithms. Moreover, we extend the proposed approach to the smooth nonnegative Tucker decomposition and smooth nonnegative canonical polyadic decomposition (also called smooth nonnegative tensor factorization). Finally, we conduct extensive experiments on synthetic and real-world multi-way array data to demonstrate the advantages of the proposed algorithms.

Low-rank approximation based non-negative multi-way array decomposition on event-related potentials.

Non-negative tensor factorization (NTF) has been successfully applied to analyze event-related potentials (ERPs), and shown superiority in terms of capturing multi-domain features. However, the time-frequency representation of ERPs by higher-order tensors are usually large-scale, which prevents the popularity of most tensor factorization algorithms. To overcome this issue, we introduce a non-negative canonical polyadic decomposition (NCPD) based on low-rank approximation (LRA) and hierarchical alternating least square (HALS) techniques. We applied NCPD (LRAHALS and benchmark HALS) and CPD to extract multi-domain features of a visual ERP. The features and components extracted by LRAHALS NCPD and HALS NCPD were very similar, but LRAHALS NCPD was 70 times faster than HALS NCPD. Moreover, the desired multi-domain feature of the ERP by NCPD showed a significant group difference (control versus depressed participants) and a difference in emotion processing (fearful versus happy faces). This was more satisfactory than that by CPD, which revealed only a group difference.

MULTI-DOMAIN FEATURE EXTRACTION FOR SMALL EVENT-RELATED POTENTIALS THROUGH NONNEGATIVE MULTI-WAY ARRAY DECOMPOSITION FROM LOW DENSE ARRAY EEG

Non-negative Canonical Polyadic decomposition (NCPD) and non-negative Tucker decomposition (NTD) were compared for extracting the multi-domain feature of visual mismatch negativity (vMMN), a small event-related potential (ERP), for the cognitive research. Since signal-to-noise ratio in vMMN is low, NTD outperformed NCPD. Moreover, we proposed an approach to select the multi-domain feature of an ERP among all extracted features and discussed determination of numbers of extracted components in NCPD and NTD regarding the ERP context.