Tensor-Based Algorithms for Image Classification

Interest in machine learning with tensor networks has been growing rapidly in recent years. We show that tensor-based methods developed for learning the governing equations of dynamical systems from data can, in the same way, be used for supervised learning problems and propose two novel approaches for image classification. One is a kernel-based reformulation of the previously introduced multidimensional approximation of nonlinear dynamics (MANDy), the other an alternating ridge regression in the tensor train format. We apply both methods to the MNIST and fashion MNIST data set and show that the approaches are competitive with state-of-the-art neural network-based classifiers

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

A literature survey of low-rank tensor approximation techniques

During the last years, low-rank tensor approximation has been established as a new tool in scientific computing to address large-scale linear and multilinear algebra problems, which would be intractable by classical techniques. This survey attempts to give a literature overview of current developments in this area, with an emphasis on function-related tensors

A continuous analogue of the tensor-train decomposition

We develop new approximation algorithms and data structures for representing and computing with multivariate functions using the functional tensor-train (FT), a continuous extension of the tensor-train (TT) decomposition. The FT represents functions using a tensor-train ansatz by replacing the three-dimensional TT cores with univariate matrix-valued functions. The main contribution of this paper is a framework to compute the FT that employs adaptive approximations of univariate fibers, and that is not tied to any tensorized discretization. The algorithm can be coupled with any univariate linear or nonlinear approximation procedure. We demonstrate that this approach can generate multivariate function approximations that are several orders of magnitude more accurate, for the same cost, than those based on the conventional approach of compressing the coefficient tensor of a tensor-product basis. Our approach is in the spirit of other continuous computation packages such as Chebfun, and yields an algorithm which requires the computation of "continuous" matrix factorizations such as the LU and QR decompositions of vector-valued functions. To support these developments, we describe continuous versions of an approximate maximum-volume cross approximation algorithm and of a rounding algorithm that re-approximates an FT by one of lower ranks. We demonstrate that our technique improves accuracy and robustness, compared to TT and quantics-TT approaches with fixed parameterizations, of high-dimensional integration, differentiation, and approximation of functions with local features such as discontinuities and other nonlinearities

Tensor completion in hierarchical tensor representations

Compressed sensing extends from the recovery of sparse vectors from undersampled measurements via efficient algorithms to the recovery of matrices of low rank from incomplete information. Here we consider a further extension to the reconstruction of tensors of low multi-linear rank in recently introduced hierarchical tensor formats from a small number of measurements. Hierarchical tensors are a flexible generalization of the well-known Tucker representation, which have the advantage that the number of degrees of freedom of a low rank tensor does not scale exponentially with the order of the tensor. While corresponding tensor decompositions can be computed efficiently via successive applications of (matrix) singular value decompositions, some important properties of the singular value decomposition do not extend from the matrix to the tensor case. This results in major computational and theoretical difficulties in designing and analyzing algorithms for low rank tensor recovery. For instance, a canonical analogue of the tensor nuclear norm is NP-hard to compute in general, which is in stark contrast to the matrix case. In this book chapter we consider versions of iterative hard thresholding schemes adapted to hierarchical tensor formats. A variant builds on methods from Riemannian optimization and uses a retraction mapping from the tangent space of the manifold of low rank tensors back to this manifold. We provide first partial convergence results based on a tensor version of the restricted isometry property (TRIP) of the measurement map. Moreover, an estimate of the number of measurements is provided that ensures the TRIP of a given tensor rank with high probability for Gaussian measurement maps.Comment: revised version, to be published in Compressed Sensing and Its Applications (edited by H. Boche, R. Calderbank, G. Kutyniok, J. Vybiral

From continuous-time formulations to discretization schemes: tensor trains and robust regression for BSDEs and parabolic PDEs

The numerical approximation of partial differential equations (PDEs) poses formidable challenges in high dimensions since classical grid-based methods suffer from the so-called curse of dimensionality. Recent attempts rely on a combination of Monte Carlo methods and variational formulations, using neural networks for function approximation. Extending previous work (Richter et al., 2021), we argue that tensor trains provide an appealing framework for parabolic PDEs: The combination of reformulations in terms of backward stochastic differential equations and regression-type methods holds the promise of leveraging latent low-rank structures, enabling both compression and efficient computation. Emphasizing a continuous-time viewpoint, we develop iterative schemes, which differ in terms of computational efficiency and robustness. We demonstrate both theoretically and numerically that our methods can achieve a favorable trade-off between accuracy and computational efficiency. While previous methods have been either accurate or fast, we have identified a novel numerical strategy that can often combine both of these aspects