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Exploiting Symmetry in Tensors for High Performance: Multiplication with Symmetric Tensors

Martin D. Schatz, Tze Meng Low, Robert A. van de Geijn, Tamara G. Kolda
2014 SIAM Journal on Scientific Computing  
However, the benefits of exploiting symmetry in order to reduce storage and computation is in conflict with a desire to simplify memory access patterns.  ...  We propose an algorithm-by-blocks, already shown of benefit for matrix computations, that exploits this storage format by utilizing a series of temporary tensors to avoid redundant computation.  ...  We would like to thank Grey Ballard for his insights into restructuring many parts of this paper.  ... 
doi:10.1137/130907215 fatcat:kxrbt5i6ibh4tlewp3f73qnt2y

Accelerating iterative CT reconstruction algorithms using Tensor Cores

Mohsen Nourazar, Bart Goossens
2021 Journal of Real-Time Image Processing  
We then presented an approach for real-time and memory-limited applications by exploiting the symmetry of the system (i.e., the acquisition geometry).  ...  In this paper, we demonstrate the feasibility of using NVIDIA Tensor Cores for the acceleration of a non-machine learning application: iterative Computed Tomography (CT) reconstruction.  ...  Acknowledgements The authors would like to thank Asli Kumcu for proofreading the manuscript.  ... 
doi:10.1007/s11554-020-01069-5 fatcat:7o3nt5ojtfcpdpenlbilqboqpm

Effective Utilization of Tensor Symmetry in Operation Optimization of Tensor Contraction Expressions

Pai-Wei Lai, Huaijian Zhang, Samyam Rajbhandari, Edward Valeev, Karol Kowalski, P. Sadayappan
2012 Procedia Computer Science  
In this paper, we address the effective exploitation of symmetry properties of tensors in performing algebraic transformations for minimizing operation count of tensor expressions.  ...  Furthermore, we show that tensor expressions optimized for a few input parameter combinations can be used to achieve operation counts within 3% of the optimal, for the entire parameter space of interest  ...  We thank NSF for the support of this work at The Ohio State University via award OCI-0926688. We are also grateful to Dr. Marcel Nooijen for his guidance and support.  ... 
doi:10.1016/j.procs.2012.04.044 fatcat:n7znbmaizbac3k5do3abzq6ckm

Analysis and tuning of libtensor framework on multicore architectures

Khaled Z. Ibrahim, Samuel W. Williams, Evgeny Epifanovsky, Anna I. Krylov
2014 2014 21st International Conference on High Performance Computing (HiPC)  
It has been optimized for symmetry and sparsity to be memory efficient. This allows it to run efficiently on the ubiquitous and cost-effective SMP architectures.  ...  In the end, our optimizations can improve the performance of contractions implemented in Libtensor by up to 2× on representative Ivy Bridge, Nehalem, and Opteron SMPs.  ...  Handling Symmetry and Sparsity Symmetries in the physical model in many-body theories lead to structure in the tensors used for model description.  ... 
doi:10.1109/hipc.2014.7116881 dblp:conf/hipc/IbrahimWEK14 fatcat:fi63t7ifjne57bzfqor2rhvkua

High Performance Tensor Decomposition Computations

Grey Ballard, Tamara Kolda
2017 Figshare  
-Can computations can be optimized for common sparsity patterns, coming from a particular set of applications? • How do we exploit (partial and full) symmetry?  ...  Tensor decompositions produce better representations of multidimensional data in terms of interpretability and number of parameters required. 3 Why do we need high performance?  ... 
doi:10.6084/m9.figshare.5317813.v1 fatcat:do5hkufvkzefln2vmqchshv7sy

Tensor-Based Methods for Blind Spatial Signature Estimation in Multidimensional Sensor Arrays

Paulo R. B. Gomes, André L. F. de Almeida, João Paulo C. L. da Costa, João C. M. Mota, Daniel Valle de Lima, Giovanni Del Galdo
2017 International Journal of Antennas and Propagation  
The second tensor-based algorithm is formulated for the case in which the sources are uncorrelated and exploits the dual-symmetry of the covariance tensor.  ...  Additionally, a new tensor-based formulation is proposed for anL-shaped array configuration.  ...  For high SNR values, the performance of the MUSIC method comes close to that of our proposal.  ... 
doi:10.1155/2017/1615962 fatcat:pv7nnw7hanahflamusliaadi7i

Cyclops Tensor Framework: Reducing Communication and Eliminating Load Imbalance in Massively Parallel Contractions

Edgar Solomonik, Devin Matthews, Jeff Hammond, James Demmel
2013 2013 IEEE 27th International Symposium on Parallel and Distributed Processing  
CTF aims to scale high-dimensional tensor contractions such as those required in the Coupled Cluster (CC) electronic structure method to massively-parallel supercomputers.  ...  Cyclops (cyclic-operations) Tensor Framework (CTF) 1 is a distributed library for tensor contractions.  ...  The challenge in exploiting high-dimensional symmetry is that the contractions can no longer be trivially reduced to dense matrix multiplication.  ... 
doi:10.1109/ipdps.2013.112 dblp:conf/ipps/SolomonikMHD13 fatcat:ql44ablwgvgotea3zl6cuknnda

A beginner's guide to non-abelian iPEPS for correlated fermions [article]

Benedikt Bruognolo, Jheng-Wei Li, Jan von Delft, Andreas Weichselbaum
2021 arXiv   pre-print
In this review, we discuss the iPEPS construction and some basic properties of this tensor network (TN) ansatz.  ...  The exploitation of non-abelian symmetries substantially increases the performance of the algorithm, enabling the treatment of fermionic systems up to a bond dimension D=24 on a square lattice.  ...  This diagrammatically performed "index bending" exploits the non-uniqueness of the graphical representation for a tensor network [44] .  ... 
arXiv:2006.08289v3 fatcat:c4hn3fx2kbhllgueimbz7trsda

Optimization of symmetric tensor computations

Jonathon Cai, Muthu Baskaran, Benoit Meister, Richard Lethin
2015 2015 IEEE High Performance Extreme Computing Conference (HPEC)  
In this work, we describe novel optimizations that exploit the symmetry in tensors in order to reduce redundancy in computations and storage and effectively parallelize operations involving symmetric tensors  ...  in scaling) for symmetric tensors.  ...  In contrast, in our mttkrp algorithms, we exploit symmetry and traverse distinct elements of the tensor only once.  ... 
doi:10.1109/hpec.2015.7322458 dblp:conf/hpec/CaiBML15 fatcat:sdwsvzxmi5cu7hios6phuleksq

A massively parallel tensor contraction framework for coupled-cluster computations

Edgar Solomonik, Devin Matthews, Jeff R. Hammond, John F. Stanton, James Demmel
2014 Journal of Parallel and Distributed Computing  
CTF serves as the distributed memory contraction engine in Aquarius, a new program designed for high-accuracy and massively-parallel quantum chemical computations.  ...  We present a distributed-memory numerical library (Cyclops Tensor Framework (CTF)) that automatically manages tensor blocking and redistribution to perform any user-specified contractions.  ...  The high-dimensional blocking used in CTF permits the capability to exploit tensor symmetry to lower computational and/or memory costs of contractions whenever possible.  ... 
doi:10.1016/j.jpdc.2014.06.002 fatcat:76at7oi2vfhbxfc6tmbzoe2xyy

Symmetry-protected local minima in infinite DMRG

Robert N. C. Pfeifer
2015 Physical Review B  
Implementations of both finite and infinite DMRG frequently incorporate support for the protection and exploitation of symmetries of the Hamiltonian.  ...  In common with other variational tensor network algorithms, convergence of iDMRG to the ground state is not guaranteed, with the risk that the algorithm may become stuck in a local minimum.  ...  When exploiting symmetries in iDMRG for models with high filling fractions, accessory site iDMRG is consequently the method of choice.  ... 
doi:10.1103/physrevb.92.205127 fatcat:652ynn633nfjjeu7pjc46vztiy

The asymptotic homogenization elasticity tensor properties for composites with material discontinuities

Raimondo Penta, Alf Gerisch
2016 Continuum Mechanics and Thermodynamics  
appearing in the coarse scale elastictype balance equations as a composite material effective elasticity tensor.  ...  The latter are shown for the first time without assuming any equivalence between coarse and fine scale energies (Hill's condition), which, in contrast to the case of representative volume elements, does  ...  We acknowledge Eli Duenisch for programming support. We are indebted to Alfio Grillo and Sara Tiburtius for insightful discussions and hints about the content of this work.  ... 
doi:10.1007/s00161-016-0526-x fatcat:ondbwrcagzdyfibwlhocm7asaa

A beginner's guide to non-abelian iPEPS for correlated fermions

Benedikt Bruognolo, Jheng-Wei Li, Jan von Delft, Andreas Weichselbaum
2021 SciPost Physics Lecture Notes  
In this review, we discuss the iPEPS construction and some basic properties of this tensor network (TN) ansatz.  ...  The exploitation of non-abelian symmetries substantially increases the performance of the algorithm, enabling the treatment of fermionic systems up to a bond dimension D=24D=24 on a square lattice.  ...  This diagrammatically performed "index bending" exploits the non-uniqueness of the graphical representation for a tensor network [45] .  ... 
doi:10.21468/scipostphyslectnotes.25 fatcat:wtbng3gemrg3vix7kw5bkdplam

Inspector-Executor Load Balancing Algorithms for Block-Sparse Tensor Contractions

David Ozog, Jeff R. Hammond, James Dinan, Pavan Balaji, Sameer Shende, Allen Malony
2013 2013 42nd International Conference on Parallel Processing  
This paper explores a set of static and dynamic scheduling algorithms for block-sparse tensor contractions within the NWChem computational chemistry code for different degrees of sparsity (and therefore  ...  Developing effective yet scalable load-balancing methods for irregular computations is critical to the successful application of simulations in a variety of disciplines at petascale and beyond.  ...  Hence, CC calculations cannot exploit more than the 8-fold symmetry of the D 2h group, but this is still a substantial reduction in computational cost.  ... 
doi:10.1109/icpp.2013.12 dblp:conf/icpp/OzogHDBSM13 fatcat:an2u6ksugzgphdrhu6fcdgpooq

Supporting tensor symmetries in EinSum

K. Åhlander
2003 Computers and Mathematics with Applications  
In this paper, the symmetries of tensors are in focus. A convenient notation for describing coordinate-free tensor symmetries is established, based on sets of permutations.  ...  The symmetry information is used to represent tensors with symmetries more economically with respect to memory. In addition, three algorithms that exploit symmetries are presented.  ...  SOME SYMMETRY EXPLOITING ALGORITHMS In the previous section, theory for understanding and representing tensor symmetries was developed.  ... 
doi:10.1016/s0898-1221(03)00041-5 fatcat:bsyoci2zl5cldn5bcn2qrjbn2u
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