Tatsuya Yokota

Smooth PARAFAC Decomposition for Tensor Completion

In recent years, low-rank based tensor completion, which is a higher order extension of matrix completion, has received considerable attention. However, the low-rank assumption is not sufficient for the recovery of visual data, such as color and 3D images, when the ratio of missing data is extremely high. In this paper, we consider “smoothness” constraints as well as low-rank approximations and propose an efficient algorithm for performing tensor completion that is particularly powerful regarding visual data. The proposed method admits significant advantages, owing to the integration of smooth PARAFAC decomposition for incomplete tensors and the efficient selection of models in order to minimize the tensor rank. Thus, our proposed method is termed as “smooth PARAFAC tensor completion (SPC).” In order to impose the smoothness constraints, we employ two strategies, total variation (SPC-TV) and quadratic variation (SPC-QV), and invoke the corresponding algorithms for model learning. Extensive experimental evaluations on both synthetic and real-world visual data illustrate the significant improvements of our method, in terms of both prediction performance and efficiency, compared with many state-of-the-art tensor completion methods.

Linked PARAFAC/CP tensor decomposition and its fast implementation for multi-block tensor analysis

In this paper we propose a new flexible group tensor analysis model called the linked CP tensor decomposition (LCPTD). The LCPTD method can decompose given multiple tensors into common factor matrices, individual factor matrices, and core tensors, simultaneously. We applied the Hierarchical Alternating Least Squares (HALS) algorithm to the LCPTD model; besides we impose additional constraints to obtain sparse and nonnegative factors. Furthermore, we conducted some experiments of this model to demonstrate its advantages over existing models.

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. HighlightsWe proposed a new efficient algorithm for smooth NMF.Moreover, we extended it to nonnegative tensor factorization (NTF) models.Proposed methods were applied to blind source separation and parts analysis.Proposed smooth NMF/NTF was more robust to noise than existing NMF/NTF.

Robust Multilinear Tensor Rank Estimation Using Higher Order Singular Value Decomposition and Information Criteria

Model selection in tensor decomposition is important for real applications if the rank of the original data tensor is unknown and the observed tensor is noisy. In the Tucker model, the minimum description length (MDL) or Bayesian information criteria have been applied to tensors via matrix unfolding, but these methods are sensitive to noise when the tensors have a multilinear low rank structure given by the Tucker model. In this study, we propose new methods for improving the MDL so it is more robust to noise. The proposed methods are justified theoretically by analyzing the “multilinear low-rank structure” of tensors. Extensive experiments including numerical simulations and a real application to image denoising are provided to illustrate the advantages of the proposed methods.

B-Spline Smoothing of Feature Vectors in Nonnegative Matrix Factorization

Nonnegative Matrix Factorization (NMF) captures nonnegative, sparse and parts-based feature vectors from the set of observed nonnegative vectors. In many applications, the features are also expected to be locally smooth. To incorporate the information on the local smoothness to the optimization process, we assume that the features vectors are conical combinations of higher degree B-splines with a given number of knots. Due to this approach the computational complexity of the optimization process does not increase considerably with respect to the standard NMF model. The numerical experiments, which were carried out for the blind spectral unmixing problem, demonstrate the robustness of the proposed method.

Tensor completion via functional smooth component deflation

For the matrix/tensor completion problem with very high missing ratio, the standard local (e.g., patch, probabilistic, and smoothness) and global (e.g., low-rank) structure-based methods do not work well. To address this issue, we proposed to use local and global data structures at the same time by applying a novel functional smooth PARAFAC decomposition model for the tensor completion. This decomposition model is constructed as a sum of the outer product of functional smooth component vectors, which are represented by linear combinations of smooth basis functions. A new algorithm was developed by applying greedy deflation and smooth rank-one tensor decomposition. Our extensive experiments demonstrated the high performance and advantages of our algorithm in comparison to existing state-of-the-art methods.

Semi-Automated Biomarker Discovery from Pharmacodynamic Effects on EEG in ADHD Rodent Models

We propose a novel semi-automatic approach to design biomarkers for capturing pharmacodynamic effects induced by pharmacological agents on the spectral power of electroencephalography (EEG) recordings. We apply this methodology to investigate the pharmacodynamic effects of methylphenidate (MPH) and atomoxetine (ATX) on attention deficit/hyperactivity disorder (ADHD), using rodent models. We inject the two agents into the spontaneously hypertensive rat (SHR) model of ADHD, the Wistar-Kyoto rat (WKY), and the Wistar rat (WIS), and record their EEG patterns. To assess individual EEG patterns quantitatively, we use an integrated methodological approach, which consists of calculating the mean, slope and intercept parameters of temporal records of EEG spectral power using a smoothing filter, outlier truncation, and linear regression. We apply Fisher discriminant analysis (FDA) to identify dominant discriminants to be heuristically consolidated into several new composite biomarkers. Results of the analysis of variance (ANOVA) and t-test show benefits in pharmacodynamic parameters, especially the slope parameter. Composite biomarker evaluation confirms their validity for genetic model stratification and the effects of the pharmacological agents used. The methodology proposed is of generic use as an approach to investigating thoroughly the dynamics of the EEG spectral power.

Simultaneous Visual Data Completion and Denoising Based on Tensor Rank and Total Variation Minimization and Its Primal-Dual Splitting Algorithm

Tensor completion has attracted attention because of its promising ability and generality. However, there are few studies on noisy scenarios which directly solve an optimization problem consisting of a noise inequality constraint. In this paper, we propose a new tensor completion and denoising model including tensor total variation and tensor nuclear norm minimization with a range of values and noise inequalities. Furthermore, we developed its solution algorithm based on a primal-dual splitting method, which is computationally efficient as compared to tensor decomposition based non-convex optimization. Lastly, extensive experiments demonstrated the advantages of the proposed method for visual data retrieval such as for color images, movies, and 3D-volumetric data.

Landmark-based reconstruction of 3D smooth structures from serial histological sections

Given microscope images, one can observe 2D cross-sections of 3D micro anatomical structures with high spatial resolutions. Each of the 2D microscope images alone is, though, not suitable for studying the 3D anatomical structures and hence many works have been done on a 3D image reconstruction from a given series of microscope images of histological sections obtained from a single target tissue. For the 3D image reconstruction, an image registration technique is necessary because there exists the independent translation, rotation, and non-rigid deformation of the histological sections. In this paper, a landmark-based method of fully non-rigid image registration for the 3D image reconstruction is proposed. The proposed method first detects landmarks corresponded between given images by using a template matching and then non-rigidly deforms the images so that the corresponding landmarks detected in different images are located along a single smooth curve in the reconstructed 3D image. Most of all conventional methods for the reconstruction of 3D microscope image registers two consecutive images at a time and many micro anatomical structures often have unnatural straight shape along the vertical (z) direction in the resultant 3D image because, roughly speaking, the conventional methods registers two given images so that pixels with the same coordinates in the two images have the same pixel value. The proposed method, on the other hand, determine the deformations of all given images by referring to the all images and deforms them simultaneously. In the experiments, a 3D microscope image of the pancreas of a KPC mouse was reconstructed from a series of microscope images of the histological sections.

Super-Resolution of Magnetic Resonance Images via Convex Optimization with Local and Global Prior Regularization and Spectrum Fitting

Given a low-resolution image, there are many challenges to obtain a super-resolved, high-resolution image. Many of those approaches try to simultaneously upsample and deblur an image in signal domain. However, the nature of the super-resolution is to restore high-frequency components in frequency domain rather than upsampling in signal domain. In that sense, there is a close relationship between super-resolution of an image and extrapolation of the spectrum. In this study, we propose a novel framework for super-resolution, where the high-frequency components are theoretically restored with respect to the frequency fidelities. This framework helps to introduce multiple simultaneous regularizers in both signal and frequency domains. Furthermore, we propose a new super-resolution model where frequency fidelity, low-rank (LR) prior, low total variation (TV) prior, and boundary prior are considered at once. The proposed method is formulated as a convex optimization problem which can be solved by the alternating direction method of multipliers. The proposed method is the generalized form of the multiple super-resolution methods such as TV super-resolution, LR and TV super-resolution, and the Gerchberg method. Experimental results show the utility of the proposed method comparing with some existing methods using both simulational and practical images.