Optimal Feature Extraction and Classification of Tensors via Matrix Product State Decomposition

© 2015 IEEE. Big data consists of large multidimensional datasets that would often be difficult to analyze if working with the original tensor. There is a rising interest in the use of tensor decompositions for feature extraction due to the ability to extract necessary features from a large dimensional feature space. In this paper the matrix product state (MPS) decomposition is used for feature extraction of large tensors. The novelty of the paper is the introduction of a single core tensor obtained from the MPS that not only contains a significantly reduced feature space, but can perform classification with high accuracy without the need of feature selection methods

Tensor Networks for Dimensionality Reduction and Large-Scale Optimizations. Part 2 Applications and Future Perspectives

Part 2 of this monograph builds on the introduction to tensor networks and their operations presented in Part 1. It focuses on tensor network models for super-compressed higher-order representation of data/parameters and related cost functions, while providing an outline of their applications in machine learning and data analytics. A particular emphasis is on the tensor train (TT) and Hierarchical Tucker (HT) decompositions, and their physically meaningful interpretations which reflect the scalability of the tensor network approach. Through a graphical approach, we also elucidate how, by virtue of the underlying low-rank tensor approximations and sophisticated contractions of core tensors, tensor networks have the ability to perform distributed computations on otherwise prohibitively large volumes of data/parameters, thereby alleviating or even eliminating the curse of dimensionality. The usefulness of this concept is illustrated over a number of applied areas, including generalized regression and classification (support tensor machines, canonical correlation analysis, higher order partial least squares), generalized eigenvalue decomposition, Riemannian optimization, and in the optimization of deep neural networks. Part 1 and Part 2 of this work can be used either as stand-alone separate texts, or indeed as a conjoint comprehensive review of the exciting field of low-rank tensor networks and tensor decompositions.Comment: 232 page

Tensor Networks for Dimensionality Reduction and Large-Scale Optimizations. Part 2 Applications and Future Perspectives

Part 2 of this monograph builds on the introduction to tensor networks and their operations presented in Part 1. It focuses on tensor network models for super-compressed higher-order representation of data/parameters and related cost functions, while providing an outline of their applications in machine learning and data analytics. A particular emphasis is on the tensor train (TT) and Hierarchical Tucker (HT) decompositions, and their physically meaningful interpretations which reflect the scalability of the tensor network approach. Through a graphical approach, we also elucidate how, by virtue of the underlying low-rank tensor approximations and sophisticated contractions of core tensors, tensor networks have the ability to perform distributed computations on otherwise prohibitively large volumes of data/parameters, thereby alleviating or even eliminating the curse of dimensionality. The usefulness of this concept is illustrated over a number of applied areas, including generalized regression and classification (support tensor machines, canonical correlation analysis, higher order partial least squares), generalized eigenvalue decomposition, Riemannian optimization, and in the optimization of deep neural networks. Part 1 and Part 2 of this work can be used either as stand-alone separate texts, or indeed as a conjoint comprehensive review of the exciting field of low-rank tensor networks and tensor decompositions.Comment: 232 page

Learning Models over Relational Data using Sparse Tensors and Functional Dependencies

Integrated solutions for analytics over relational databases are of great practical importance as they avoid the costly repeated loop data scientists have to deal with on a daily basis: select features from data residing in relational databases using feature extraction queries involving joins, projections, and aggregations; export the training dataset defined by such queries; convert this dataset into the format of an external learning tool; and train the desired model using this tool. These integrated solutions are also a fertile ground of theoretically fundamental and challenging problems at the intersection of relational and statistical data models. This article introduces a unified framework for training and evaluating a class of statistical learning models over relational databases. This class includes ridge linear regression, polynomial regression, factorization machines, and principal component analysis. We show that, by synergizing key tools from database theory such as schema information, query structure, functional dependencies, recent advances in query evaluation algorithms, and from linear algebra such as tensor and matrix operations, one can formulate relational analytics problems and design efficient (query and data) structure-aware algorithms to solve them. This theoretical development informed the design and implementation of the AC/DC system for structure-aware learning. We benchmark the performance of AC/DC against R, MADlib, libFM, and TensorFlow. For typical retail forecasting and advertisement planning applications, AC/DC can learn polynomial regression models and factorization machines with at least the same accuracy as its competitors and up to three orders of magnitude faster than its competitors whenever they do not run out of memory, exceed 24-hour timeout, or encounter internal design limitations.Comment: 61 pages, 9 figures, 2 table

Tensor Networks for Big Data Analytics and Large-Scale Optimization 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).Comment: arXiv admin note: text overlap with arXiv:1403.204