Nonnegative Tucker Decomposition Research Articles

Muscle Synergy during Wrist Movements Based on Non-Negative Tucker Decomposition.

Modular control of the muscle, which is called muscle synergy, simplifies control of the movement by the central nervous system. The purpose of this study was to explore the synergy in both the frequency and movement domains based on the non-negative Tucker decomposition (NTD) method. Surface electromyography (sEMG) data of 8 upper limb muscles in 10 healthy subjects under wrist flexion (WF) and wrist extension (WE) were recorded. NTD was selected for exploring the multi-domain muscle synergy from the sEMG data. The results showed two synergistic flexor pairs, Palmaris longus-Flexor Digitorum Superficialis (PL-FDS) and Extensor Carpi Radialis-Flexor Carpi Radialis (ECR-FCR), in the WF stage. Their spectral components are mainly in the respective bands 0-20 Hz and 25-50 Hz. And the spectral components of two extensor pairs, Extensor Digitorum-Extensor Carpi Ulnar (ED-ECU) and Extensor Carpi Radialis-Brachioradialis (ECR-B), are mainly in the respective bands 0-20 Hz and 7-45 Hz in the WE stage. Additionally, further analysis showed that the Biceps Brachii (BB) muscle was a shared muscle synergy module of the WE and WF stage, while the flexor muscles FCR, PL and FDS were the specific synergy modules of the WF stage, and the extensor muscles ED, ECU, ECR and B were the specific synergy modules of the WE stage. This study showed that NTD is a meaningful method to explore the multi-domain synergistic characteristics of multi-channel sEMG signals. The results can help us to better understand the frequency features of muscle synergy and shared and specific synergies, and expand the study perspective related to motor control in the nervous system.

Non-negative Tucker decomposition with graph regularization and smooth constraint for clustering

Non-negative Tucker decomposition (NTD) and its graph regularized extensions are the most popular techniques for representing high-dimensional non-negative data, which are typically found in a low-dimensional sub-manifold of ambient space, from a geometric perspective. Therefore, the performance of the graph-based NTD methods relies heavily on the low-dimensional representation of the original data. However, most existing approaches treat the last factor matrix in NTD as a low-dimensional representation of the original data. This treatment leads to the loss of the original data’s multi-linear structure in the low-dimensional subspace. To remedy this defect, we propose a novel graph regularized Lp smooth NTD (GSNTD) method for high-dimensional data representation by incorporating graph regularization and an Lp smoothing constraint into NTD. The new graph regularization term constructed by the product of the core tensor and the last factor matrix in NTD, and it is used to uncover hidden semantics while maintaining the intrinsic multi-linear geometric structure of the data. The addition of the Lp smoothing constraint to NTD may produce a more accurate and smoother solution to the optimization problem. The update rules and the convergence of the GSNTD method are proposed. In addition, a randomized variant of the GSNTD algorithm based on fiber sampling is proposed. Finally, the experimental results on four standard image databases show that the proposed method and its randomized variant have better performance than some other state-of-the-art graph-based regularization methods for image clustering.

Robust Feature Extraction via ℓ∞-Norm Based Nonnegative Tucker Decomposition

Feature extraction plays an indispensable role in image and video technology. However, it is difficult for traditional matrix based feature extraction methods to handle massive multi-dimensional data. This, alongside with the ubiquitous uncertainty (noise) in real-world data, resulted in many robust tensor based feature extraction models. However, these existing models did not consider the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">worst-case</i> model performance (i.e., the largest fitting error among all samples), which is critically important from a robust optimization perspective. In this paper, we propose a novel robust feature extraction model via ℓ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> -norm based nonnegative Tucker decomposition. The model is to minimize the maximum sample fitting error so as to overcome the influence of data uncertainty. Although the new model is nonconvex and nonsmooth, we design an effective iterative optimization algorithm with theoretical guarantee on its convergence for it. The performance of the new model on five real-world benchmark object classification and face recognition datasets under various corruption scenarios are evaluated, and the experimental results show the excellence of the new model by comparing to many existing models.

Graph regularized discriminative nonnegative tucker decomposition for tensor data representation

Graph regularized discriminative nonnegative tucker decomposition for tensor data representation

ETucker: a constrained tensor decomposition for single trial ERP extraction

Objective. In this paper, we propose a new tensor decomposition to extract event-related potentials (ERP) by adding a physiologically meaningful constraint to the Tucker decomposition. Approach. We analyze the performance of the proposed model and compare it with Tucker decomposition by synthesizing a dataset. The simulated dataset is generated using a 12th-order autoregressive model in combination with independent component analysis (ICA) on real no-task electroencephalogram (EEG) recordings. The dataset is manipulated to contain the P300 ERP component and to cover different SNR conditions, ranging from 0 to −30 dB, to simulate the presence of the P300 component in extremely noisy recordings. Furthermore, in order to assess the practicality of the proposed methodology in real-world scenarios, we utilized the brain-computer interface (BCI) competition III-dataset II. Main results. Our primary results demonstrate the superior performance of our approach compared to conventional methods commonly employed for single-trial estimation. Additionally, our method outperformed both Tucker decomposition and non-negative Tucker decomposition in the synthesized dataset. Furthermore, the results obtained from real-world data exhibited meaningful performance and provided insightful interpretations for the extracted P300 component. Significance. The findings suggest that the proposed decomposition is eminently capable of extracting the target P300 component’s waveform, including latency and amplitude as well as its spatial location, using single-trial EEG recordings.

Accurate regularized Tucker decomposition for image restoration

We propose a new accurate regularized Tucker decomposition (ARTD) method for image restoration (IR), which considers global low-rankness and local similarity of intrinsic image characteristics. Specifically, global low-rankness is represented by a sparse Tucker core tensor, whereas the local similarity is captured using nonnegative factor matrices and manifold regularization terms. Sparse nonnegative Tucker decomposition (SNTD) and graph nonnegative Tucker decomposition (GNTD) can be considered a special case of ARTD. We propose and implement an effective Alternating Proximal Gradient (APG) based algorithm to solve the ARTD model and deduce the closed-form updating rules. Notably, ARTD does not need to tune the Tucker rank and provides an initialization strategy and a first-order feedback control rule to accelerate its convergence. Experiments on real IR problems show that our method outperforms some existing state-of-the-art methods.

Adaptive graph regularized non-negative Tucker decomposition for multiway dimensionality reduction

Adaptive graph regularized non-negative Tucker decomposition for multiway dimensionality reduction

VOLUME-REGULARIZED NONNEGATIVE TUCKER DECOMPOSITION WITH IDENTIFIABILITY GUARANTEES.

It is well-known that the Tucker decomposition of a multi-dimensional tensor is not unique, because its factors are subject to rotation ambiguities similar to matrix factorization models. Inspired by the recent success in the identifiability of nonnegative matrix factorization, the goal of this work is to achieve similar results for nonnegative Tucker decomposition (NTD). We propose to add a matrix volume regularization as the identifiability criterion, and show that NTD is indeed identifiable if all of the Tucker factors satisfy the sufficiently scattered condition. We then derive an algorithm to solve the modified formulation of NTD that minimizes the generalized Kullback-Leibler divergence of the approximation plus the proposed matrix volume regularization. Numerical experiments show the effectiveness of the proposed method.

Discriminative Nonnegative Tucker Decomposition for Tensor Data Representation

Nonnegative Tucker decomposition (NTD) is an unsupervised method and has been extended in many applied fields. However, NTD does not make use of the label information of sample data, even though such label information is available. To remedy the defect, in this paper, we propose a label constraint NTD method, namely Discriminative NTD (DNTD), which considers a fraction of the label information of the sample data as a discriminative constraint. Differing from other label-based methods, the proposed method enforces the sample data, with the same label to be aligned on the same axis or line. Combining the NTD and the label-discriminative constraint term, DNTD can not only extract the part-based representation of the data tensor but also boost the discriminative ability of the NTD. An iterative updating algorithm is provided to solve the objective function of DNTD. Finally, the proposed DNTD method is applied to image clustering. Experimental results on ORL, COIL20, Yale datasets show the clustering accuracy of DNTD is improved by 8.47–32.17% and the normalized mutual information is improved by 10.43–29.64% compared with the state-of-the-art approaches.

Local homeomorphism regularized non-negative Tucker decomposition of tensor data

Non-negative Tucker decomposition of tensor data has a wide range of applications in machine learning. With the non-negative constraints, tensor data can be decomposed into the mode product of a core tensor and a series of projection matrices. The core tensor usually is be regarded as the low-dimensional representation of the original tensor. The process of dimensionality reduction preserves the global properties of tensor data. But many applications in machine learning expect the continuous dependencies of data local feature to remain unchanged during dimensionality reduction. To this end, this paper proposes a local homeomorphism regularized non-negative Tucker decomposition algorithm for tensor data. The proposed method introduce a local homeomorphism regularized term to tensor non-negative Tucker decomposition constrain for effectively preserving the global and local characters of tensor. Experiments on four commonly used real data sets and six compared algorithms have demonstrate the well performance of the proposed algorithms.

SAR Image Reconstruction of Vehicle Targets Based on Tensor Decomposition

Due to the imaging mechanism of Synthetic Aperture Radars (SARs), the target shape on an SAR image is sensitive to the radar incidence angle and target azimuth, but there is strong correlation and redundancy between adjacent azimuth images of SAR targets. This paper studies multi-angle SAR image reconstruction based on non-negative Tucker decomposition using adjacent azimuth images reconstructed to form a sparse tensor. Sparse tensors are used to perform non-negative Tucker decomposition, resulting in non-negative core tensors and factor matrices. The reconstruction tensor is obtained by calculating the n-mode product of the core tensor and the factor matrix, and then image reconstruction is realized. The similarity between the original image and the reconstructed image is calculated by using the structural similarity index and the cosine of the angle between the feature vectors. The reconstruction results of three target images of MSTAR show that the reconstructed image has a similarity higher than 95% with the original image in most cases, which can support target recognition under sparse observation to a certain extent.

Adaptive Band Extraction Based on Low Rank Approximated Nonnegative Tucker Decomposition for Anti-Friction Bearing Faults Diagnosis Using Measured Vibration Data

Condition monitoring and fault diagnosis are topics of growing interest for improving the reliability of modern industrial systems. As critical structural components, anti-friction bearings often operate under harsh conditions and are contributing factors of system failures. Efforts have been cast on bearing diagnostics under the sensor fusion and machine learning framework, whilst challenges remain open on the identification of incipient faults. In this paper, exploiting multi-way representations and decompositions of measured vibration data, a novel band separation method based on the factorization of spectrogram tensors using the low rank approximated nonnegative Tucker decomposition (LRANTD) is proposed and applied to identify detailed fault signatures from the spectral, temporal, and spatial dimensions, flexible for extracting multi-sensor features and multi-dimensional correlations. With the proposed method, informative frequency bands of the latent vibrational components can be automatically extracted, in accordance with the inherent temporal patterns that can be conveniently fed for spectral analysis and fault discrimination. Furthermore, an improved cross-spectrum can be calculated from multi-channel vibrations via LRANTD with enhanced fault features. Based on the real-world vibration data of the accelerated bearing life tests, detailed experimental studies and thorough comparisons to the conventional benchmarks have verified the effectiveness of the reported diagnostic methodology. The proposed method significantly improves the presence of the bearing frequency peaks distinctly over the background noises in the spectrum and hence improves the bearing defect detection process.

Impacts of the COVID-19 pandemic on the spatio-temporal characteristics of a bicycle-sharing system: A case study of Pun Pun, Bangkok, Thailand.

The COVID-19 pandemic is found to be one of the external stimuli that greatly affects mobility of people, leading to a shift of transportation modes towards private individual ones. To properly explain the change in people's transport behavior, especially in pre- and post- pandemic periods, a tensor-based framework is herein proposed and applied to Pun Pun-the only public bicycle-sharing system in Bangkok, Thailand-where multidimensional trip data of Pun Pun are decomposed into four different modes related to their spatial and temporal dimensions by a non-negative Tucker decomposition approach. According to our computational results, the first pandemic wave has a sizable influence not only on Pun Pun but also on other modes of transportation. Nonetheless, Pun Pun is relatively more resilient, as it recovers more quickly than other public transportation modes. In terms of trip patterns, we find that, prior to the pandemic, trips made during weekdays are dominated by business trips with two peak periods (morning and evening peaks), while those made during weekends are more related to leisure activities as they involve stations nearby a public park. However, after the first pandemic wave ends, the patterns of weekday trips have been drastically changed, as the number of business trips sharply drops, while that of educational trips connecting metro/subway stations with a major educational institute in the region significantly rises. These findings may be regarded as a reflection of the ever-changing transport behavior of people seeking a sustainable mode of private transport, with a more positive outlook on the use of bicycle-sharing system in Bangkok, Thailand.

Nonnegative canonical tensor decomposition with linear constraints: nnCANDELINC

AbstractThere is an emerging interest for tensor factorization applications in big‐data analytics and machine learning. To speed up the factorization of extra‐large datasets, organized in multidimensional arrays (also known as tensors), easy to compute compression‐based tensor representations, such as, Tucker and tensor train formats, are used to approximate the initial large‐tensor. Further, tensor factorization is used to extract latent features that can facilitate discoveries of new mechanisms and signatures hidden in the data, where the explainability of the latent features is of principal importance. Nonnegative tensor factorization extracts latent features that are naturally sparse and parts of the data, which makes them easily interpretable. However, to take into account available domain knowledge and subject matter expertise, often additional constraints need to be imposed, which lead us to canonical decomposition with linear constraints (CANDELINC), a canonical polyadic decomposition with rank deficient factors. In CANDELINC, Tucker compression is used as a preprocessing step, which lead to a larger residual error but to more explainable latent features. Here, we propose a nonnegative CANDELINC (nnCANDELINC) accomplished via a specific nonnegative Tucker decomposition; we refer to as minimal or canonical nonnegative Tucker. We derive several results required to understand the specificity of nnCANDELINC, focusing on the difficulties of preserving the nonnegative rank of a tensor to its Tucker core and comparing the real valued to nonnegative case. Finally, we demonstrate nnCANDELINC performance on synthetic and real‐world examples.

Entropy Tucker model: Mining latent mobility patterns with simultaneous estimation of travel impedance parameters

With the rapid increase in the availability of passive data in the field of transportation, combining machine learning with transportation models has emerged as an important research topic in recent years. This study proposes an entropy Tucker model that integrates (1) a Tucker decomposition technique, i.e., an existing machine learning method, and (2) an entropy maximizing model, i.e., an existing model used for modeling trip distribution in the field of transportation. In addition, an optimization algorithm is presented to empirically identify the proposed model. The proposed model provides a solid theoretical foundation for the machine learning method, substantially improves prediction performance, and provides richer behavioral implications through empirical parameter estimation of travel impedance. We conducted a case study using public transit smart card data. The results showed that the proposed model improves the prediction performance and interpretability of the results compared to the conventional nonnegative Tucker decomposition technique. Further, we empirically confirmed that the travel impedance varies with the origin–destination pair, time of the day, and day of the week. Finally, we discussed how embedding the theoretical foundations of transport modeling into machine learning methods can facilitate the use of various passive data in wider practical contexts of transport policy decision making.

Incremental Nonnegative Tucker Decomposition with Block-Coordinate Descent and Recursive Approaches

Nonnegative Tucker decomposition (NTD) is a robust method used for nonnegative multilinear feature extraction from nonnegative multi-way arrays. The standard version of NTD assumes that all of the observed data are accessible for batch processing. However, the data in many real-world applications are not static or are represented by a large number of multi-way samples that cannot be processing in one batch. To tackle this problem, a dynamic approach to NTD can be explored. In this study, we extend the standard model of NTD to an incremental or online version, assuming volatility of observed multi-way data along one mode. We propose two computational approaches for updating the factors in the incremental model: one is based on the recursive update model, and the other uses the concept of the block Kaczmarz method that belongs to coordinate descent methods. The experimental results performed on various datasets and streaming data demonstrate high efficiently of both algorithmic approaches, with respect to the baseline NTD methods.

Semi-supervised non-negative Tucker decomposition for tensor data representation

Semi-supervised non-negative Tucker decomposition for tensor data representation

Approximately orthogonal nonnegative Tucker decomposition for flexible multiway clustering

Approximately orthogonal nonnegative Tucker decomposition for flexible multiway clustering

Day-to-day dynamic origin–destination flow estimation using connected vehicle trajectories and automatic vehicle identification data

Dynamic vehicular origin–destination (OD) flow is a fundamental component of traffic network modeling and its estimation has long been studied. Although ideal observing conditions and behavioral assumptions are often indispensable for estimation, day-to-day traffic recurrences and variations are seldom utilized to improve the estimation performance. In this paper, we propose a new method to recover day-to-day dynamic OD flows using both connected vehicle (CV) trajectories and automatic vehicle identification (AVI) observations. The method involves two modules: the first module provides reliable prior OD flows given limited observations, while the second module seeks the optimal estimates based on the prior OD flows. In the first module, linear projection is extended to consider temporal and spatial variation of the CV penetration rate, and non-negative Tucker decomposition (NTD) is adopted to address the data sparsity issue caused by the low CV penetration rate. In the second module, a self-supervised learning model called the latency-constrained autoencoder (LCAE) is established to search for the optimal OD flows according to the priors with given robust latent features. To avoid local minima and ensure consistency between estimates, a novel algorithm called adaptive sub-sample correction (ASC) is proposed and integrated into the optimization process of LCAE, which can iteratively correct the most inconsistent samples based on the day-to-day traffic flow characteristics. The proposed method is examined on an empirical urban arterial network, a calibrated simulation network, and a synthetic large-scale grid network. Our results indicated that the proposed method requires very few AVI detectors and CV trajectories to achieve competitive estimation performance against two benchmark models. Furthermore, general robustness to several factors with respect to observing conditions and data quality was investigated, and satisfactory scalability was also demonstrated in terms of both estimation accuracy and computational cost.

Tucker decomposition and applications

Non-negative Tucker decomposition is a well-known higher order tensor decomposition method, where non-negativity is imposed on higher order Tucker decomposition. This paper is associated to the reality behind higher order Tucker decomposition which is computed with the help of HOSVD and its extension HOOI. In this paper, we show that how HOSVD, ALS and HOOI algorithm are useful for Tucker decomposition.