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COMPUTING IN SCIENCE & ENGINEERING 2023年第5期25卷 4-5页

作者： Feiguin, Adrian Kourtis, Stefanos Northeastern Univ Dept Phys Boston MA 02115 USA Univ Sherbrooke Dept Phys Sherbrooke PQ J1K 2R1 Canada Univ Sherbrooke Inst Quant Sherbrooke PQ J1K 2R1 Canada

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A multi-dimensional Morton-ordered block storage for mode-oblivious tensor computations

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JOURNAL OF COMPUTATIONAL SCIENCE 2019年 33卷 34-44页

作者： Pawlowski, Filip Ucar, Bora Yzelman, Albert-Jan Huawei Technol France 20 Quai Point du Jour F-92100 Boulogne France ENS Lyon Lyon France Univ Lyon Univ Claude Bernard Lyon 1 LIP UMR 5668 CNRSENS LyonInria F-69007 Lyon France

Computation on tensors, treated as multidimensional arrays, revolve around generalized basic linear algebra subroutines (BLAS). We propose a novel data structure in which tensors are blocked and blocks are stored in an order determined by Morton order. This is not only proposed for efficiency reasons, but also to induce efficient performance regardless of which mode a generalized BLAS call is invoked for;we coin the term mode-oblivious to describe data structures and algorithms that induce such behavior. Experiments on one of the most bandwidth-bound generalized BLAS kernel, the tensor-vector multiplication, not only demonstrate superior performance over two state-of-the-art variants by up to 18%, but additionally show that the proposed data structure induces a 71% less sample standard deviation for tensor-vector multiplication across d modes, where d varies from 2 to 10. Finally, we show our data structure naturally expands to other tensor kernels and demonstrate up to 38% higher performance for the higher-order power method. (C) 2019 Elsevier B.V. All rights reserved.

关键词： tensor computations Data structure Morton order tensor-vector multiplication

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AMOS: Enabling Automatic Mapping for tensor computations On Spatial Accelerators with Hardware Abstraction 22

AMOS: Enabling Automatic Mapping for Tensor Computations On ...

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49th IEEE/ACM Annual International Symposium on Computer Architecture (ISCA)

作者： Zheng, Size Chen, Renze Wei, Anjiang Jin, Yicheng Han, Qin Lu, Liqiang Wu, Bingyang Li, Xiuhong Yan, Shengen Liang, Yun Peking Univ Beijing Peoples R China Stanford Univ Stanford CA USA Sensetime Res Beijing Peoples R China Shanghai Lab Shanghai Peoples R China

ISBN: (纸本)9781450386104

Hardware specialization is a promising trend to sustain performance growth. Spatial hardware accelerators that employ specialized and hierarchical computation and memory resources have recently shown high performance gains for tensor applications such as deep learning, scientific computing, and data mining. To harness the power of these hardware accelerators, programmers have to use specialized instructions with certain hardware constraints. However, these hardware accelerators and instructions are quite new and there is a lack of understanding of the hardware abstraction, performance optimization space, and automatic methodologies to explore the space. Existing compilers use hand-tuned computation implementations and optimization templates, resulting in sub-optimal performance and heavy development costs. In this paper, we propose AMOS, which is an automatic compilation framework for spatial hardware accelerators. Central to this framework is the hardware abstraction that not only clearly specifies the behavior of spatial hardware instructions, but also formally defines the mapping problem from software to hardware. Based on the abstraction, we develop algorithms and performance models to explore various mappings automatically. Finally, we build a compilation framework that uses the hardware abstraction as compiler intermediate representation (IR), explores both compute mappings and memory mappings, and generates high-performance code for different hardware backends. Our experiments show that AMOS achieves more than 2.50x speedup to hand-optimized libraries on tensor Core, 1.37x speedup to TVM on vector units of Intel CPU for AVX-512, and up to 25.04x speedup to AutoTVM on dot units of Mali GPU. The source code of AMOS is publicly available.

关键词： spatial accelerators code generation mapping tensor computations

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Correlating Time Series With Interpretable Convolutional Kernels

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IEEE TRANSACTIONS ON KNOWLEDGE AND DATA ENGINEERING 2025年第6期37卷 3272-3283页

作者： Chen, Xinyu Cai, HanQin Liu, Fuqiang Zhao, Jinhua MIT Dept Urban Studies & Planning Cambridge MA 02139 USA Univ Cent Florida Dept Stat & Data Sci Dept Comp Sci Orlando FL 32816 USA McGill Univ Dept Civil Engn Montreal PQ H3A 0C3 Canada

This study addresses the problem of convolutional kernel learning in univariate, multivariate, and multidimensional time series data, which is crucial for interpreting temporal patterns in time series and supporting downstream machine learning tasks. First, we propose formulating convolutional kernel learning for univariate time series as a sparse regression problem with a non-negative constraint, leveraging the properties of circular convolution and circulant matrices. Second, to generalize this approach to multivariate and multidimensional time series data, we use tensor computations, reformulating the convolutional kernel learning problem in the form of tensors. This is further converted into a standard sparse regression problem through vectorization and tensor unfolding operations. In the proposed methodology, the optimization problem is addressed using the existing non-negative subspace pursuit method, enabling the convolutional kernel to capture temporal correlations and patterns. To evaluate the proposed model, we apply it to several real-world time series datasets. On the multidimensional ridesharing and taxi trip data from New York City and Chicago, the convolutional kernels reveal interpretable local correlations and cyclical patterns, such as weekly seasonality. For the monthly temperature time series data in North America, the proposed model can quantify the yearly seasonality and make it comparable across different decades. In the context of multidimensional fluid flow data, both local and nonlocal correlations captured by the convolutional kernels can reinforce tensor factorization, leading to performance improvements in fluid flow reconstruction tasks. Thus, this study lays an insightful foundation for automatically learning convolutional kernels from time series data, with an emphasis on interpretability through sparsity and non-negativity constraints.

关键词： Time series analysis Kernel Convolution tensors Vectors Correlation Optimization Matching pursuit algorithms Machine learning Linear regression Time series machine learning circular convolution sparse regression subspace pursuit tensor computations convolutional kernels

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Computing Dense tensor Decompositions with Optimal Dimension Trees

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ALGORITHMICA 2019年第5期81卷 2092-2121页

作者： Kaya, Oguz Robert, Yves Fac Sci Orsay LRI Bat 650 F-91405 Orsay France ENS Lyon 46 Allee Italie F-69007 Lyon France Univ Tennessee Knoxville TN 37996 USA

Dense tensor decompositions have been widely used in many signal processing problems including analyzing speech signals, identifying the localization of signal sources, and many other communication applications. Computing these decompositions poses major computational challenges for big datasets emerging in these domains. CANDECOMP/PARAFAC (CP) and Tucker formulations are the prominent tensor decomposition schemes heavily used in these fields, and the algorithms for computing them involve applying two core operations, namely tensor-times-matrix and tensor-times-vector multiplication, which are executed repetitively within an iterative framework. In the recent past, efficient computational schemes using a data structure called dimension tree, are employed to significantly reduce the cost of these two operations, through storing and reusing partial results that are commonly used across different iterations of these algorithms. This framework has been introduced for sparse CP and Tucker decompositions in the literature, and a recent work investigates using an optimal binary dimension tree structure in computing dense Tucker decompositions. In this paper, we investigate finding an optimal dimension tree for both CP and Tucker decompositions. We show that finding an optimal dimension tree for an N-dimensional tensor is NP-hard for both decompositions, provide faster exact algorithms for finding an optimal dimension tree in O(3(N)) time using O(2(N)) space for the Tucker case, and extend the algorithm to the case of CP decomposition with the same time and space complexities.

关键词： tensor computations CP decomposition Tucker decomposition Dimension tree

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Numeric tensor framework: Exploiting and extending Einstein notation

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JOURNAL OF COMPUTATIONAL SCIENCE 2016年 16卷 128-139页

作者： Harrison, Adam P. Joseph, Dileepan Univ Alberta Dept Elect & Comp Engn Edmonton AB Canada

The numeric tensor (NT) framework addresses and unifies a growing body of work on high-dimensional algebra and software for technical computing. Its NT algebra exploits and extends Einstein notation, offering unmatched capabilities, including N-dimensional operators, associativity, commutativity, entrywise products, and linear invertibility. High-performance C++ and MATLAB NT software allows practitioners to directly program with NT algebra. The advantages of NT algebra are highlighted using the example of canonical-polyadic (CP) tensor decomposition. Corresponding dense benchmarks demonstrate that the NT software matches or surpasses leading competitors, i.e., the MATLAB tensor Toolbox, NumPy, and Blitz++, while supporting a more general set of arithmetic operations. (C) 2016 Elsevier B.V. All rights reserved.

关键词： tensor algebra tensor computations tensor inversion C plus plus classes MATLAB classes

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COMMUNICATION LOWER BOUNDS AND OPTIMAL ALGORITHMS FOR MULTIPLE tensor-TIMES-MATRIX COMPUTATION

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SIAM JOURNAL ON MATRIX ANALYSIS AND APPLICATIONS 2024年第1期45卷 477-477页

作者： AL Daas, Hussam Ballard, Grey Grigori, Laura Kumar, Suraj Rouse, Kathryn Sci & Technol Facil Council Rutherford Appleton Lab Didcot OX11 0QX Oxfordshire England Wake Forest Univ Comp Sci Winston Salem NC 27106 USA Ecole Polytech Fed Lausanne PFL Inst Math CH-1015 Lausanne Switzerland Paul Scherrer Inst Lab Simulat & Modelling CH-5232 Villigen Switzerland Sorbonne Univ Univ Paris Inria CNRS Paris France Inria Lyon Ctr Villeurbanne France Sorbonne Univ Univ Paris Inria CNRSLab Jacques Louis Lions Paris France Inmar Intelligence Winston Salem NC 27101 USA

Multiple tensor -times -matrix (Multi-TTM) is a key computation in algorithms for computing and operating with the Tucker tensor decomposition, which is frequently used in multidimensional data analysis. We establish communication lower bounds that determine how much data movement is required (under mild conditions) to perform the Multi-TTM computation in parallel. The crux of the proof relies on analytically solving a constrained, nonlinear optimization problem. We also present a parallel algorithm to perform this computation that organizes the processors into a logical grid with twice as many modes as the input tensor. We show that, with correct choices of grid dimensions, the communication cost of the algorithm attains the lower bounds and is therefore communication optimal. Finally, we show that our algorithm can significantly reduce communication compared to the straightforward approach of expressing the computation as a sequence of tensor -times -matrix operations when the input and output tensors vary greatly in size.

关键词： communication lower bounds Multi-TTM tensor computations parallel algo- rithms HBL-inequalities

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Joint Program and Layout Transformations to Enable Convolutional Operators on Specialized Hardware Based on Constraint Programming

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ACM TRANSACTIONS ON ARCHITECTURE AND CODE OPTIMIZATION 2022年第1期19卷 1–26页

作者： Rieber, Dennis Acosta, Axel Froening, Holger Robert Bosch GmbH Corp Res Gerlingen Germany Heidelberg Univ Inst Comp Engn Heidelberg Germany

The success of Deep Artificial Neural Networks (DNNs) in many domains created a rich body of research concerned with hardware accelerators for compute-intensive DNN operators. However, implementing such operators efficiently with complex hardware intrinsics such as matrix multiply is a task not yet automated gracefully. Solving this task often requires joint program and data layout transformations. First solutions to this problem have been proposed, such as TVM, UNIT, or ISAMIR, which work on a loop-level representation of operators and specify data layout and possible program transformations before the embedding into the operator is performed. This top-down approach creates a tension between exploration range and search space complexity, especially when also exploring data layout transformations such as im2col, channel packing, or padding. In this work, we propose a new approach to this problem. We created a bottom-up method that allows the joint transformation of both computation and data layout based on the found embedding. By formulating the embedding as a constraint satisfaction problem over the scalar dataflow, every possible embedding solution is contained in the search space. Adding additional constraints and optimization targets to the solver generates the subset of preferable solutions. An evaluation using the VTA hardware accelerator with the Baidu DeepBench inference benchmark shows that our approach can automatically generate code competitive to reference implementations. Further, we show that dynamically determining the data layout based on intrinsic and workload is beneficial for hardware utilization and performance. In cases where the reference implementation has low hardware utilization due to its fixed deployment strategy, we achieve a geomean speedup of up to x2.813, while individual operators can improve as much as x170.

关键词： Intermediate representation instruction selection tensor computations neural networks

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A Native tensor-Vector Multiplication Algorithm for High Performance Computing

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IEEE TRANSACTIONS ON PARALLEL AND DISTRIBUTED SYSTEMS 2022年第12期33卷 3363-3374页

作者： Martinez-Ferrer, Pedro J. Yzelman, A. N. Beltran, Vicenc Barcelona Supercomp Ctr BSC Barcelona 08034 Spain Univ Politecn Catalunya UPC Barcelona 08034 Spain Huawei Technol Switzerland AG Comp Syst Lab CH-3097 Zurich Switzerland

tensor computations are important mathematical operations for applications that rely on multidimensional data. The tensor-vector multiplication (TVM) is the most memory-bound tensor contraction in this class of operations. This article proposes an open-source TVM algorithm which is much simpler and efficient than previous approaches, making it suitable for integration in the most popular BLAS libraries available today. Our algorithm has been written from scratch and features unit-stride memory accesses, cache awareness, mode obliviousness, full vectorization and multi-threading as well as NUMA awareness for non-hierarchically stored dense tensors. Numerical experiments are carried out on tensors up to order 10 and various compilers and hardware architectures equipped with traditional DDR and high bandwidth memory (HBM). For large tensors the average performance of the TVM ranges between 62% and 76% of the theoretical bandwidth for NUMA systems with DDR memory and remains independent of the contraction mode. On NUMA systems with HBM the TVM exhibits some mode dependency but manages to reach performance figures close to peak values. Finally, the higher-order power method is benchmarked with the proposed TVM kernel and delivers on average between 58% and 69% of the theoretical bandwidth for large tensors.

关键词： tensors Kernel Libraries Bandwidth Virtual machine monitors Layout Benchmark testing Parallel algorithms shared memory tensor computations high bandwidth memory NUMA

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New Algorithm for tensor Contractions on Multi-Core CPUs, GPUs, and Accelerators Enables CCSD and EOM-CCSD Calculations with over 1000 Basis Functions on a Single Compute Node

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JOURNAL OF COMPUTATIONAL CHEMISTRY 2017年第11期38卷 842-853页

作者： Kaliman, Ilya A. Krylov, Anna I. Univ Southern Calif Dept Chem Los Angeles CA 90089 USA

A new hardware-agnostic contraction algorithm for tensors of arbitrary symmetry and sparsity is presented. The algorithm is implemented as a stand-alone open-source code libxm. This code is also integrated with general tensor library libtensor and with the Q-Chem quantum-chemistry package. An overview of the algorithm, its implementation, and benchmarks are presented. Similarly to other tensor software, the algorithm exploits efficient matrix multiplication libraries and assumes that tensors are stored in a block-tensor form. The distinguishing features of the algorithm are: (i) efficient repackaging of the individual blocks into large matrices and back, which affords efficient graphics processing unit (GPU)-enabled calculations without modifications of higher-level codes;(ii) fully asynchronous data transfer between disk storage and fast memory. The algorithm enables canonical all-electron coupled-cluster and equation-of-motion coupled-cluster calculations with single and double substitutions (CCSD and EOM-CCSD) with over 1000 basis functions on a single quad-GPU machine. We show that the algorithm exhibits predicted theoretical scaling for canonical CCSD calculations, O(N-6), irrespective of the data size on disk. (C) 2017 Wiley Periodicals, Inc.

关键词： coupled-cluster equation-of-motion coupled-cluster GPGPU tensor computations many-body theories electronic structure

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