There is a significant expansion in both volume and range of applications along with the concomitant increase in the variety of data sources. These ever-expanding trends have highlighted the necessity for more versatile analysis tools that offer greater opportunities for algorithmic developments and computationally faster operations than the standard flat-view matrix approach. Tensors, or multi-way arrays, provide such an algebraic framework which is naturally suited to data of such large volume, diversity, and veracity. Indeed, the associated tensor decompositions have demonstrated their potential in breaking the Curse of Dimensionality associated with traditional matrix methods, where a necessary exponential increase in data volume leads to adverse or even intractable consequences on computational complexity. A key tool underpinning multi-linear manipulation of tensors and tensor networks is the standard Tensor Contraction Product (TCP). However, depending on the dimensionality of the underlying tensors, the TCP also comes at the price of high computational complexity in tensor manipulation. In this work, we resort to diagrammatic tensor network manipulation to calculate such products in an efficient and computationally tractable manner, by making use of Tensor Train decomposition (TTD). This has rendered the underlying concepts easy to perceive, thereby enhancing intuition of the associated underlying operations, while preserving mathematical rigour. In addition to bypassing the cumbersome mathematical multi-linear expressions, the proposed Tensor Train Contraction Product model is shown to accelerate significantly the underlying computational operations, as it is independent of tensor order and linear in the tensor dimension, as opposed to performing the full computations through the standard approach (exponential in tensor order).
There is a significant expansion in both volume and range of applications along with the concomitant increase in the variety of data sources. These ever-expanding trends have highlighted the necessity for more versatile analysis tools that offer greater opportunities for algorithmic developments and computationally faster operations than the standard flat-view matrix approach. Tensors, or multi-way arrays, provide such an algebraic framework which is naturally suited to data of such large volume, diversity, and veracity. Indeed, the associated tensor decompositions have demonstrated their potential in breaking the Curse of Dimensionality associated with traditional matrix methods, where a necessary exponential increase in data volume leads to adverse or even intractable consequences on computational complexity. A key tool underpinning multi-linear manipulation of tensors and tensor networks is the standard Tensor Contraction Product (TCP). However, depending on the dimensionality of the underlying tensors, the TCP also comes at the price of high computational complexity in tensor manipulation. In this work, we resort to diagrammatic tensor network manipulation to calculate such products in an efficient and computationally tractable manner, by making use of Tensor Train decomposition (TTD). This has rendered the underlying concepts easy to perceive, thereby enhancing intuition of the associated underlying operations, while preserving mathematical rigour. In addition to bypassing the cumbersome mathematical multi-linear expressions, the proposed Tensor Train Contraction Product model is shown to accelerate significantly the underlying computational operations, as it is independent of tensor order and linear in the tensor dimension, as opposed to performing the full computations through the standard approach (exponential in tensor order).
Machine learning and data mining algorithms are becoming increasingly important in analyzing large volume, multi-relational and multi--modal datasets, which are often conveniently represented as multiway arrays or tensors. It is therefore timely and valuable for the multidisciplinary research community to review tensor decompositions and tensor networks as emerging tools for large-scale data analysis and data mining. We provide the mathematical and graphical representations and interpretation of tensor networks, with the main focus on the Tucker and Tensor Train (TT) decompositions and their extensions or generalizations. Keywords: Tensor networks, Function-related tensors, CP decomposition, Tucker models, tensor train (TT) decompositions, matrix product states (MPS), matrix product operators (MPO), basic tensor operations, multiway component analysis, multilinear blind source separation, tensor completion, linear/multilinear dimensionality reduction, large-scale optimization problems, symmetric eigenvalue decomposition (EVD), PCA/SVD, huge systems of linear equations, pseudo-inverse of very large matrices, Lasso and Canonical Correlation Analysis (CCA) (This is Part 1)
High-dimensional data arise naturally in many areas of science and engineering, including machine learning, signal processing, computational physics, and statistics. Such data are often represented as tensors, multi-dimensional generalizations of matrices. While tensors provide a natural representation for multi-modal structure, their direct manipulation quickly becomes challenging as the order grows: the number of parameters increases exponentially, and algebraic expressions involving many indices become difficult to interpret and implement. Tensor networks (TNs) provide an effective framework for addressing these challenges. Originally introduced by Penrose and developed extensively in quantum physics, the graphical language of tensor networks encodes contractions as edges in a graph, reducing notational overhead and revealing structural properties obscured by index notation. Despite the central role of high-dimensional tensors in modern machine learning and numerical analysis, tensor network diagrams remain underutilized outside quantum computing, partly due to the lack of a self-contained mathematical reference accessible to a broad technical audience. This manuscript provides a self-contained guide to tensor networks and their use in tensor algebra. We present the main operations on tensors, contractions, products, and reshaping through, graphical notation, and show how classical tensor decompositions and related computations are naturally expressed in this framework. We also illustrate how tensor networks simplify the derivation of gradients and the manipulation of high-dimensional probability distributions. Throughout, we show that the diagrammatic approach yields genuinely shorter and more transparent proofs of classical identities, rank bounds, and gradient formulas that would otherwise require laborious index manipulation.
Tensor-train (TT) decomposition has gained tremendous popularity due to its efficiency in handling high-dimensional data arising in scientific and quantum computing as well as machine learning applications. It provides a compact representation of matrices and vectors with a Kronecker product-like low-rank structure, and hence enables efficient matrix-vector operations in this compressed form. The vector scalar product is among such key operations, comprising a series of tensor contractions in a specific TT network where the order of contractions significantly impacts the computational cost. In this work, we introduce fast algorithms for finding a near-optimal contraction ordering for such networks. We show that our algorithms find such contraction orderings efficiently, outperforming the ordering methods for general tensor networks. Compared to our approaches, the best existing method incurs a higher cost of up to 15% for xTy, twice for xTAy, and more than ten times for xTABy scalar products, where A and B are matrices expressed in TT format.
We discuss extended definitions of linear and multilinear operations such as Kronecker, Hadamard, and contracted products, and establish links between them for tensor calculus. Then we introduce effective low-rank tensor approximation techniques including Candecomp/Parafac (CP), Tucker, and tensor train (TT) decompositions with a number of mathematical and graphical representations. We also provide a brief review of mathematical properties of the TT decomposition as a low-rank approximation technique. With the aim of breaking the curse-of-dimensionality in large-scale numerical analysis, we describe basic operations on large-scale vectors, matrices, and high-order tensors represented by TT decomposition. The proposed representations can be used for describing numerical methods based on TT decomposition for solving large-scale optimization problems such as systems of linear equations and symmetric eigenvalue problems.
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).
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).