439 research outputs found
The Tensor Networks Anthology: Simulation techniques for many-body quantum lattice systems
We present a compendium of numerical simulation techniques, based on tensor
network methods, aiming to address problems of many-body quantum mechanics on a
classical computer. The core setting of this anthology are lattice problems in
low spatial dimension at finite size, a physical scenario where tensor network
methods, both Density Matrix Renormalization Group and beyond, have long proven
to be winning strategies. Here we explore in detail the numerical frameworks
and methods employed to deal with low-dimension physical setups, from a
computational physics perspective. We focus on symmetries and closed-system
simulations in arbitrary boundary conditions, while discussing the numerical
data structures and linear algebra manipulation routines involved, which form
the core libraries of any tensor network code. At a higher level, we put the
spotlight on loop-free network geometries, discussing their advantages, and
presenting in detail algorithms to simulate low-energy equilibrium states.
Accompanied by discussions of data structures, numerical techniques and
performance, this anthology serves as a programmer's companion, as well as a
self-contained introduction and review of the basic and selected advanced
concepts in tensor networks, including examples of their applications.Comment: 115 pages, 56 figure
Tensor network states and algorithms in the presence of a global U(1) symmetry
Tensor network decompositions offer an efficient description of certain
many-body states of a lattice system and are the basis of a wealth of numerical
simulation algorithms. In a recent paper [arXiv:0907.2994v1] we discussed how
to incorporate a global internal symmetry, given by a compact, completely
reducible group G, into tensor network decompositions and algorithms. Here we
specialize to the case of Abelian groups and, for concreteness, to a U(1)
symmetry, often associated with particle number conservation. We consider
tensor networks made of tensors that are invariant (or covariant) under the
symmetry, and explain how to decompose and manipulate such tensors in order to
exploit their symmetry. In numerical calculations, the use of U(1) symmetric
tensors allows selection of a specific number of particles, ensures the exact
preservation of particle number, and significantly reduces computational costs.
We illustrate all these points in the context of the multi-scale entanglement
renormalization ansatz.Comment: 22 pages, 25 figures, RevTeX
Search-based Model-driven Loop Optimizations for Tensor Contractions
Complex tensor contraction expressions arise in accurate electronic structure models in quantum chemistry, such as the coupled cluster method. The Tensor Contraction Engine (TCE) is a high-level program synthesis system that facilitates the generation of high-performance parallel programs from tensor contraction equations. We are developing a new software infrastructure for the TCE that is designed to allow experimentation with optimization algorithms for modern computing platforms, including for heterogeneous architectures employing general-purpose graphics processing units (GPGPUs). In this dissertation, we present improvements and extensions to the loop fusion optimization algorithm, which can be used with cost models, e.g., for minimizing memory usage or for minimizing data movement costs under a memory constraint. We show that our data structure and pruning improvements to the loop fusion algorithm result in significant performance improvements that enable complex cost models being use for large input equations. We also present an algorithm for optimizing the fused loop structure of handwritten code. It determines the regions in handwritten code that are safe to be optimized and then runs the loop fusion algorithm on the dependency graph of the code. Finally, we develop an optimization framework for generating GPGPU code consisting of loop fusion optimization with a novel cost model, tiling optimization, and layout optimization. Depending on the memory available on the GPGPU and the sizes of the tensors, our framework decides which processor (CPU or GPGPU) should perform an operation and where the result should be moved. We present extensive measurements for tuning the loop fusion algorithm, for validating our optimization framework, and for measuring the performance characteristics of GPGPUs. Our measurements demonstrate that our optimization framework outperforms existing general-purpose optimization approaches both on multi-core CPUs and on GPGPUs
Parameterization of Tensor Network Contraction
We present a conceptually clear and algorithmically useful framework for parameterizing the costs of tensor network contraction. Our framework is completely general, applying to tensor networks with arbitrary bond dimensions, open legs, and hyperedges. The fundamental objects of our framework are rooted and unrooted contraction trees, which represent classes of contraction orders. Properties of a contraction tree correspond directly and precisely to the time and space costs of tensor network contraction. The properties of rooted contraction trees give the costs of parallelized contraction algorithms. We show how contraction trees relate to existing tree-like objects in the graph theory literature, bringing to bear a wide range of graph algorithms and tools to tensor network contraction. Independent of tensor networks, we show that the edge congestion of a graph is almost equal to the branchwidth of its line graph
Application Performance Modeling via Tensor Completion
Performance tuning, software/hardware co-design, and job scheduling are among
the many tasks that rely on models to predict application performance. We
propose and evaluate low-rank tensor decomposition for modeling application
performance. We discretize the input and configuration domains of an
application using regular grids. Application execution times mapped within
grid-cells are averaged and represented by tensor elements. We show that
low-rank canonical-polyadic (CP) tensor decomposition is effective in
approximating these tensors. We further show that this decomposition enables
accurate extrapolation of unobserved regions of an application's parameter
space. We then employ tensor completion to optimize a CP decomposition given a
sparse set of observed execution times. We consider alternative
piecewise/grid-based models and supervised learning models for six applications
and demonstrate that CP decomposition optimized using tensor completion offers
higher prediction accuracy and memory-efficiency for high-dimensional
performance modeling
“An Integer Programming Power Optimization in Storage Systems”
AbstractThis paper presents a linear integer programming framework for effective power management in storage systems. A sample memory system with different data banks is considered for optimal energy consumption during data operations by manipulating the data among banks. The memory bank four-level power state schemes, namely, active, stand-by, nap, and power down states, are included for superior power management of the storage system by formulating a linear integer optimization framework that includes plausible data manipulations, energy consumption levels, data migration, and compression options. The numerical results illustrate the efficiency of the proposed framework in terms of power management of storage systems with respect to available approaches with two-level power state operations
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
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