Cyber-physical-social big data generated from ubiquitous devices and diverse spaces generally are multi-source, heterogeneous, and deeply intertwined. To efficiently analyze and handle the ubiquitous cyber-physical-social big data, tensor is considered as an effective tool, but the curse of dimensionality is still the main bottleneck of tensor-based big data analysis. Tensor networks can considerably alleviate or overcome it through the tensor approximate theory. Therefore, this paper focuses on developing an efficient big data processing framework based on tensor networks and providing an incremental tensor train decomposition approach for the streaming big data. Concretely, this paper first presents a hierarchical cyber-physical-social big data processing framework composed of three planes, namely, data representation and decomposition, data storage and processing, and data analysis and service, in which tensor train (TT) and quantized TT decompositions are particularly introduced to remarkably overcome the curse of dimensionality. Besides, to efficiently handle the continuous streaming big data and avoid the repeated decomposition for the history data, an incremental tensor train decomposition (ITTD) approach is proposed and the complexities are further analyzed in detail. Experimental results demonstrate that ITTD demonstrably outperforms the nonincremental TT decomposition in execution time on the precise of guaranteeing the nearly equal approximation error.
Cyber-physical-social big data generated from ubiquitous devices and diverse spaces generally are multi-source, heterogeneous, and deeply intertwined. To efficiently analyze and handle the ubiquitous cyber-physical-social big data, tensor is considered as an effective tool, but the curse of dimensionality is still the main bottleneck of tensor-based big data analysis. Tensor networks can considerably alleviate or overcome it through the tensor approximate theory. Therefore, this paper focuses on developing an efficient big data processing framework based on tensor networks and providing an incremental tensor train decomposition approach for the streaming big data. Concretely, this paper first presents a hierarchical cyber-physical-social big data processing framework composed of three planes, namely, data representation and decomposition, data storage and processing, and data analysis and service, in which tensor train (TT) and quantized TT decompositions are particularly introduced to remarkably overcome the curse of dimensionality. Besides, to efficiently handle the continuous streaming big data and avoid the repeated decomposition for the history data, an incremental tensor train decomposition (ITTD) approach is proposed and the complexities are further analyzed in detail. Experimental results demonstrate that ITTD demonstrably outperforms the nonincremental TT decomposition in execution time on the precise of guaranteeing the nearly equal approximation error.
C yber- P hysical- S ocial S ystems (CPSS) integrating the cyber, physical, and social worlds is a key technology to provide proactive and personalized services for humans. In this paper, we studied CPSS by taking h uman- i nteraction-aware b ig d ata (HIBD) as the starting point. However, the HIBD collected from all aspects of our daily lives are of high-order and large-scale, which bring ever-increasing challenges for their cleaning, integration, processing, and interpretation. Therefore, new strategies for representing and processing of HIBD become increasingly important in the provision of CPSS services. As an emerging technique, tensor is proving to be a suitable and promising representation and processing tool of HIBD. In particular, tensor networks, as a significant tensor decomposition technique, bring advantages of computing, storage, and applications of HIBD. Furthermore, T ensor- T rain (TT), a type of tensor network, is particularly well suited for representing and processing high-order data by decomposing a high-order tensor into a series of low-order tensors. However, at present, there is still need for an efficient Tensor-Train decomposition method for massive data. Therefore, for larger-scale HIBD, a highly-efficient computational method of Tensor-Train is required. In this paper, a d istributed T ensor- T rain (DTT) decomposition method is proposed to process the high-order and large-scale HIBD. The high performance of the proposed DTT such as the execution time is demonstrated with a case study on a typical form of CPSS data, C omputed T omography (CT) image data.
In the context of big data, high-order tensor decompositions have to face a new challenge in terms of storage and computational costs. The tensor train (TT) decomposition provides a very useful graph-based model reduction, whose storage cost grows linearly with the tensor order D. The computation of the TT-core tensors and TT-ranks can be done in a stable sequential (i.e., noniterative) way thanks to the popular TT-SVD algorithm. In this paper, we exploit the ideas developed for the hierarchical/tree Tucker decomposition in the context of the TT decomposition. Specifically, a new efficient estimation scheme, called TT-HSVD (Tensor-Train Hierarchical SVD), is proposed as a solution to compute the TT decomposition of a high-order tensor. The new algorithm simultaneously delivers the TT-core tensors and their TT-ranks in a hierarchical way. It is a stable (i.e., noniterative) and computationally more efficient algorithm than TT-SVD, which is very important when dealing with large-scale data. The TT-HSVD algorithm uses a new reshaping strategy and a tailored partial SVD, which allows us to deal with smaller matrices compared to those of the TT-SVD. In addition, TT-HSVD is well suited for a parallel processing architecture. An algebraic analysis of the two algorithms is carried out, showing that TT-SVD and TT-HSVD compute the same TT-ranks and TT-core tensors up to specific bases. Simulation results for different tensor orders and dimensions corroborate the effectiveness of the proposed algorithm.
Many problems in computational neuroscience, neuroinformatics, pattern/image recognition, signal processing and machine learning generate massive amounts of multidimensional data with multiple aspects and high dimensionality. Tensors (i.e., multi-way arrays) provide often a natural and compact representation for such massive multidimensional data via suitable low-rank approximations. Big data analytics require novel technologies to efficiently process huge datasets within tolerable elapsed times. Such a new emerging technology for multidimensional big data is a multiway analysis via tensor networks (TNs) and tensor decompositions (TDs) which represent tensors by sets of factor (component) matrices and lower-order (core) tensors. Dynamic tensor analysis allows us to discover meaningful hidden structures of complex data and to perform generalizations by capturing multi-linear and multi-aspect relationships. We will discuss some fundamental TN models, their mathematical and graphical descriptions and associated learning algorithms for large-scale TDs and TNs, with many potential applications including: Anomaly detection, feature extraction, classification, cluster analysis, data fusion and integration, pattern recognition, predictive modeling, regression, time series analysis and multiway component analysis. Keywords: Large-scale HOSVD, Tensor decompositions, CPD, Tucker models, Hierarchical Tucker (HT) decomposition, low-rank tensor approximations (LRA), Tensorization/Quantization, tensor train (TT/QTT) - Matrix Product States (MPS), Matrix Product Operator (MPO), DMRG, Strong Kronecker Product (SKP).
In this paper we review basic and emerging models and associated algorithms for large-scale tensor networks, especially Tensor Train (TT) decompositions using novel mathematical and graphical representations. We discus the concept of tensorization (i.e., creating very high-order tensors from lower-order original data) and super compression of data achieved via quantized tensor train (QTT) networks. The purpose of a tensorization and quantization is to achieve, via low-rank tensor approximations "super" compression, and meaningful, compact representation of structured data. The main objective of this paper is to show how tensor networks can be used to solve a wide class of big data optimization problems (that are far from tractable by classical numerical methods) by applying tensorization and performing all operations using relatively small size matrices and tensors and applying iteratively optimized and approximative tensor contractions. Keywords: Tensor networks, tensor train (TT) decompositions, matrix product states (MPS), matrix product operators (MPO), basic tensor operations, tensorization, distributed representation od data optimization problems for very large-scale problems: generalized eigenvalue decomposition (GEVD), PCA/SVD, canonical correlation analysis (CCA).
Relational tensor decomposition aims to uncover the underlying relationships between entities represented in a multi-dimensional tensor. Traditional methods often struggle with scalability and interpretability, particularly when dealing with complex relational structures. This paper presents a novel approach to relational tensor decomposition that leverages the efficiency of Tensor Trains (TTs) combined with a sparsity-inducing constraint based on relational dependencies. We decompose relational tensors using TT decomposition, and then introduce a sparsity pattern that prioritizes connections between related entities. This process effectively builds a sparse TT representation that explicitly encodes the relational structure of the data. The resulting method demonstrates improved efficiency and enhanced interpretability compared to standard TT decomposition, offering a promising solution for analyzing complex relational data. The core claim is improved efficiency and interpretability of relational tensor decomposition by sparsifying the Tensor Train representation based on relational dependencies. The core mechanism is the application of Tensor Train decomposition to relational tensors, coupled with a sparsity constraint prioritizing connections between related entities.
We present two novel algorithms for approximating and incrementally updating the hierarchical Tucker (HT) decomposition of tensor streams. The first algorithm, Batch Hierarchical Tucker - leaf to root ( BHT-l2r ), introduces a restructured HT topology that anchors the batch dimension at the root node, enabling efficient compression and downstream latent space extraction for batches of similar tensors. The second algorithm, Hierarchical Tucker - Rapid Incremental Subspace Expansion ( HT-RISE ), provides a deterministic subspace tracking approach to continuously update this representation as new data batches arrive. Designed for the online setting, HT-RISE never requires full storage or reconstruction of the historical data. Unlike randomized sketching methods that defer network construction, HT-RISE explicitly maintains a fully orthogonal tensor network state at every time step, dynamically adapting tensor ranks to satisfy certified, user-defined error bounds. Empirical evaluations on several datasets demonstrate the superior performance of both methods. BHT-l2r achieves up to 6.2 × greater compression and a 3.7 × speedup over standard HT formats, while HT-RISE delivers up to 3.1 × higher compression and a 3.2 × speedup compared to state-of-the-art incremental Tensor Train algorithms.