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

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

Tucker Tensor analysis of Matern functions in spatial statistics

In this work, we describe advanced numerical tools for working with multivariate functions and for the analysis of large data sets. These tools will drastically reduce the required computing time and the storage cost, and, therefore, will allow us to consider much larger data sets or finer meshes. Covariance matrices are crucial in spatio-temporal statistical tasks, but are often very expensive to compute and store, especially in 3D. Therefore, we approximate covariance functions by cheap surrogates in a low-rank tensor format. We apply the Tucker and canonical tensor decompositions to a family of Matern- and Slater-type functions with varying parameters and demonstrate numerically that their approximations exhibit exponentially fast convergence. We prove the exponential convergence of the Tucker and canonical approximations in tensor rank parameters. Several statistical operations are performed in this low-rank tensor format, including evaluating the conditional covariance matrix, spatially averaged estimation variance, computing a quadratic form, determinant, trace, loglikelihood, inverse, and Cholesky decomposition of a large covariance matrix. Low-rank tensor approximations reduce the computing and storage costs essentially. For example, the storage cost is reduced from an exponential $\mathcal{O}(n^d)$ to a linear scaling $\mathcal{O}(drn)$, where $d$ is the spatial dimension, $n$ is the number of mesh points in one direction, and $r$ is the tensor rank. Prerequisites for applicability of the proposed techniques are the assumptions that the data, locations, and measurements lie on a tensor (axes-parallel) grid and that the covariance function depends on a distance, $\Vert x-y \Vert$.Comment: 23 pages, 2 diagrams, 2 tables, 9 figure

Learning Relevant Features of Data with Multi-scale Tensor Networks

Inspired by coarse-graining approaches used in physics, we show how similar algorithms can be adapted for data. The resulting algorithms are based on layered tree tensor networks and scale linearly with both the dimension of the input and the training set size. Computing most of the layers with an unsupervised algorithm, then optimizing just the top layer for supervised classification of the MNIST and fashion-MNIST data sets gives very good results. We also discuss mixing a prior guess for supervised weights together with an unsupervised representation of the data, yielding a smaller number of features nevertheless able to give good performance.Comment: 12 pages, 13 figure

The connectome mapper: an open-source processing pipeline to map connectomes with MRI.

Researchers working in the field of global connectivity analysis using diffusion magnetic resonance imaging (MRI) can count on a wide selection of software packages for processing their data, with methods ranging from the reconstruction of the local intra-voxel axonal structure to the estimation of the trajectories of the underlying fibre tracts. However, each package is generally task-specific and uses its own conventions and file formats. In this article we present the Connectome Mapper, a software pipeline aimed at helping researchers through the tedious process of organising, processing and analysing diffusion MRI data to perform global brain connectivity analyses. Our pipeline is written in Python and is freely available as open-source at www.cmtk.org

Deep Positron: A Deep Neural Network Using the Posit Number System

The recent surge of interest in Deep Neural Networks (DNNs) has led to increasingly complex networks that tax computational and memory resources. Many DNNs presently use 16-bit or 32-bit floating point operations. Significant performance and power gains can be obtained when DNN accelerators support low-precision numerical formats. Despite considerable research, there is still a knowledge gap on how low-precision operations can be realized for both DNN training and inference. In this work, we propose a DNN architecture, Deep Positron, with posit numerical format operating successfully at $\leq$8 bits for inference. We propose a precision-adaptable FPGA soft core for exact multiply-and-accumulate for uniform comparison across three numerical formats, fixed, floating-point and posit. Preliminary results demonstrate that 8-bit posit has better accuracy than 8-bit fixed or floating-point for three different low-dimensional datasets. Moreover, the accuracy is comparable to 32-bit floating-point on a Xilinx Virtex-7 FPGA device. The trade-offs between DNN performance and hardware resources, i.e. latency, power, and resource utilization, show that posit outperforms in accuracy and latency at 8-bit and below.Comment: 6 pages, Design, Automation and Test in Europe 201

A Microbenchmark Characterization of the Emu Chick

The Emu Chick is a prototype system designed around the concept of migratory memory-side processing. Rather than transferring large amounts of data across power-hungry, high-latency interconnects, the Emu Chick moves lightweight thread contexts to near-memory cores before the beginning of each memory read. The current prototype hardware uses FPGAs to implement cache-less "Gossamer cores for doing computational work and a stationary core to run basic operating system functions and migrate threads between nodes. In this multi-node characterization of the Emu Chick, we extend an earlier single-node investigation (Hein, et al. AsHES 2018) of the the memory bandwidth characteristics of the system through benchmarks like STREAM, pointer chasing, and sparse matrix-vector multiplication. We compare the Emu Chick hardware to architectural simulation and an Intel Xeon-based platform. Our results demonstrate that for many basic operations the Emu Chick can use available memory bandwidth more efficiently than a more traditional, cache-based architecture although bandwidth usage suffers for computationally intensive workloads like SpMV. Moreover, the Emu Chick provides stable, predictable performance with up to 65% of the peak bandwidth utilization on a random-access pointer chasing benchmark with weak locality

Protecting Big Data Privacy Using Randomized Tensor Network Decomposition and Dispersed Tensor Computation

Data privacy is an important issue for organizations and enterprises to securely outsource data storage, sharing, and computation on clouds / fogs. However, data encryption is complicated in terms of the key management and distribution; existing secure computation techniques are expensive in terms of computational / communication cost and therefore do not scale to big data computation. Tensor network decomposition and distributed tensor computation have been widely used in signal processing and machine learning for dimensionality reduction and large-scale optimization. However, the potential of distributed tensor networks for big data privacy preservation have not been considered before, this motivates the current study. Our primary intuition is that tensor network representations are mathematically non-unique, unlinkable, and uninterpretable; tensor network representations naturally support a range of multilinear operations for compressed and distributed / dispersed computation. Therefore, we propose randomized algorithms to decompose big data into randomized tensor network representations and analyze the privacy leakage for 1D to 3D data tensors. The randomness mainly comes from the complex structural information commonly found in big data; randomization is based on controlled perturbation applied to the tensor blocks prior to decomposition. The distributed tensor representations are dispersed on multiple clouds / fogs or servers / devices with metadata privacy, this provides both distributed trust and management to seamlessly secure big data storage, communication, sharing, and computation. Experiments show that the proposed randomization techniques are helpful for big data anonymization and efficient for big data storage and computation

User-transparent Distributed TensorFlow

Deep Learning (DL) algorithms have become the {\em de facto} choice for data analysis. Several DL implementations -- primarily limited to a single compute node -- such as Caffe, TensorFlow, Theano and Torch have become readily available. Distributed DL implementations capable of execution on large scale systems are becoming important to address the computational needs of large data produced by scientific simulations and experiments. Yet, the adoption of distributed DL implementations faces significant impediments: 1) most implementations require DL analysts to modify their code significantly -- which is a show-stopper, 2) several distributed DL implementations are geared towards cloud computing systems -- which is inadequate for execution on massively parallel systems such as supercomputers. This work addresses each of these problems. We provide a distributed memory DL implementation by incorporating required changes in the TensorFlow runtime itself. This dramatically reduces the entry barrier for using a distributed TensorFlow implementation. We use Message Passing Interface (MPI) -- which provides performance portability, especially since MPI specific changes are abstracted from users. Lastly -- and arguably most importantly -- we make our implementation available for broader use, under the umbrella of Machine Learning Toolkit for Extreme Scale (MaTEx) at {\texttt http://hpc.pnl.gov/matex}. We refer to our implementation as MaTEx-TensorFlow.Comment: 9 pages, 8 figure

Neuroimaging study designs, computational analyses and data provenance using the LONI pipeline.

Modern computational neuroscience employs diverse software tools and multidisciplinary expertise to analyze heterogeneous brain data. The classical problems of gathering meaningful data, fitting specific models, and discovering appropriate analysis and visualization tools give way to a new class of computational challenges--management of large and incongruous data, integration and interoperability of computational resources, and data provenance. We designed, implemented and validated a new paradigm for addressing these challenges in the neuroimaging field. Our solution is based on the LONI Pipeline environment [3], [4], a graphical workflow environment for constructing and executing complex data processing protocols. We developed study-design, database and visual language programming functionalities within the LONI Pipeline that enable the construction of complete, elaborate and robust graphical workflows for analyzing neuroimaging and other data. These workflows facilitate open sharing and communication of data and metadata, concrete processing protocols, result validation, and study replication among different investigators and research groups. The LONI Pipeline features include distributed grid-enabled infrastructure, virtualized execution environment, efficient integration, data provenance, validation and distribution of new computational tools, automated data format conversion, and an intuitive graphical user interface. We demonstrate the new LONI Pipeline features using large scale neuroimaging studies based on data from the International Consortium for Brain Mapping [5] and the Alzheimer's Disease Neuroimaging Initiative [6]. User guides, forums, instructions and downloads of the LONI Pipeline environment are available at http://pipeline.loni.ucla.edu