SEARCH · Engineering Papers

Results for “Tensor”

Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

SourceResearch connectionFrom yearThrough year

Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.

At least 73 records · Page 4

All sources NASA NTRS DOE OSTI

DOE OSTI2021

A High Performance Sparse Tensor Algebra Compiler in MLIR

Sparse tensor algebra is widely used in many applications, including scientific computing, machine learning, and data analytics. The performance of sparse tensor algebra kernels strongly depends on the intrinsic characteristics of the input tensors, hence many storage formats are designed for tensors to achieve optimal performance for particular applications/architectures, which makes it challenging to implement and optimize every tensor operation of interest on a given architecture. We propose a tensor algebra domain-specific language (DSL) and compiler framework to automatically generate kernels for mixed sparse-dense tensor algebra operations. The proposed DSL provides high-level programming abstractions that resemble the familiar Einstein notation to represent tensor algebra operations. The compiler introduces a new Sparse Tensor Algebra dialect built on top of LLVM's extensible MLIR compiler infrastructure for efficient code generation while covering a wide range of tensor storage formats. Our compiler also leverages input-dependent code optimization to enhance data locality for better performance. Our results show that the performance of automatically generated kernels outperforms the state-of-the-art sparse tensor algebra compiler, with up to 20.92x, 6.39x, and 13.9x performance improvement over state-of-the-art tensor algebra compilers, for parallel SpMV, SpMM, and TTM, respectively.

Tian, Ruiqin↗

NASA NTRS1994

Databases post-processing in Tensoral

The Center for Turbulent Research (CTR) post-processing effort aims to make turbulence simulations and data more readily and usefully available to the research and industrial communities. The Tensoral language, introduced in this document and currently existing in prototype form, is the foundation of this effort. Tensoral provides a convenient and powerful protocol to connect users who wish to analyze fluids databases with the authors who generate them. In this document we introduce Tensoral and its prototype implementation in the form of a user's guide. This guide focuses on use of Tensoral for post-processing turbulence databases. The corresponding document - the Tensoral 'author's guide' - which focuses on how authors can make databases available to users via the Tensoral system - is currently unwritten. Section 1 of this user's guide defines Tensoral's basic notions: we explain the class of problems at hand and how Tensoral abstracts them. Section 2 defines Tensoral syntax for mathematical expressions. Section 3 shows how these expressions make up Tensoral statements. Section 4 shows how Tensoral statements and expressions are embedded into other computer languages (such as C or Vectoral) to make Tensoral programs. We conclude with a complete example program.

Dresselhaus, Eliot↗

NASA NTRS1997

The Topology of Symmetric Tensor Fields

Combinatorial topology, also known as "rubber sheet geometry", has extensive applications in geometry and analysis, many of which result from connections with the theory of differential equations. A link between topology and differential equations is vector fields. Recent developments in scientific visualization have shown that vector fields also play an important role in the analysis of second-order tensor fields. A second-order tensor field can be transformed into its eigensystem, namely, eigenvalues and their associated eigenvectors without loss of information content. Eigenvectors behave in a similar fashion to ordinary vectors with even simpler topological structures due to their sign indeterminacy. Incorporating information about eigenvectors and eigenvalues in a display technique known as hyperstreamlines reveals the structure of a tensor field. The simplify and often complex tensor field and to capture its important features, the tensor is decomposed into an isotopic tensor and a deviator. A tensor field and its deviator share the same set of eigenvectors, and therefore they have a similar topological structure. A a deviator determines the properties of a tensor field, while the isotopic part provides a uniform bias. Degenerate points are basic constituents of tensor fields. In 2-D tensor fields, there are only two types of degenerate points; while in 3-D, the degenerate points can be characterized in a Q'-R' plane. Compressible and incompressible flows share similar topological feature due to the similarity of their deviators. In the case of the deformation tensor, the singularities of its deviator represent the area of vortex core in the field. In turbulent flows, the similarities and differences of the topology of the deformation and the Reynolds stress tensors reveal that the basic addie-viscosity assuptions have their validity in turbulence modeling under certain conditions.

Levin, Yingmei↗

DOE OSTI2025

A Linear-Complexity Tensor Butterfly Algorithm for Compressing High-Dimensional Oscillatory Integral Operators

This paper presents a multilevel tensor compression algorithm called tensor butterfly algorithm for efficiently representing large-scale and high-dimensional oscillatory integral operators, including Green's functions for wave equations and integral transforms such as Radon transforms and Fourier transforms. The proposed algorithm leverages a tensor extension of the so-called complementary low-rank property of existing matrix butterfly algorithms. The algorithm partitions the discretized integral operator tensor into subtensors of multiple levels and factorizes each subtensor at the middle level as a Tucker-type interpolative decomposition, whose factor matrices are formed in a multilevel fashion. For a d-dimensional (d > 1) integral operator discretized into a 2d-mode tensor with n2d entries, the overall CPU time and memory requirement scale as O(nd), in stark contrast to the O(nd log n) complexity of existing matrix algorithms such as matrix butterfly algorithms and fast Fourier transforms (FFTs), where n is the number of points per direction. When comparing with other tensor algorithms such as quantized tensor train (QTT), the proposed algorithm also shows superior CPU and memory performance for tensor contraction. Remarkably, the tensor butterfly algorithm can efficiently model high-frequency Green's function interactions between two unit cubes, each spanning 512 wavelengths per direction, which represents problems of scale over 512× larger than that existing butterfly algorithms can handle, with the same amount of computation resources. On the other hand, for a problem representing 64 wavelengths per direction, which is the largest size existing algebraic matrix algorithms can handle, our tensor butterfly algorithm exhibits 200x speedups and 30× memory reduction compared with existing ones. Moreover, the tensor butterfly algorithm also permits O(nd)-complexity FFTs and Radon transforms up to d = 6 dimensions.

Kielstra, P Michael↗

NASA NTRS1993

Tensoral for post-processing users and simulation authors

The CTR post-processing effort aims to make turbulence simulations and data more readily and usefully available to the research and industrial communities. The Tensoral language, which provides the foundation for this effort, is introduced here in the form of a user's guide. The Tensoral user's guide is presented in two main sections. Section one acts as a general introduction and guides database users who wish to post-process simulation databases. Section two gives a brief description of how database authors and other advanced users can make simulation codes and/or the databases they generate available to the user community via Tensoral database back ends. The two-part structure of this document conforms to the two-level design structure of the Tensoral language. Tensoral has been designed to be a general computer language for performing tensor calculus and statistics on numerical data. Tensoral's generality allows it to be used for stand-alone native coding of high-level post-processing tasks (as described in section one of this guide). At the same time, Tensoral's specialization to a minute task (namely, to numerical tensor calculus and statistics) allows it to be easily embedded into applications written partly in Tensoral and partly in other computer languages (here, C and Vectoral). Embedded Tensoral, aimed at advanced users for more general coding (e.g. of efficient simulations, for interfacing with pre-existing software, for visualization, etc.), is described in section two of this guide.

Dresselhaus, Eliot↗

DOE OSTI2021

QBTNs - Quantum Boolean Tensor Networks

We develop algorithms and software that uses the D-Wave 2000Q quantum annealer to solve several types of Boolean tensor factorization problems. Boolean tensor factorization refers to the problem of representing a high-dimensional tensor filled with Boolean values as a product of smaller Boolean core tensors and Boolean matrices. We consider different tensor factorization models, including Boolean Tensor Train, Boolean Tucker, and Boolean Hierarchical Tucker. As an exact decomposition of a given type may not exist in the general case, the objective is to minimize the difference between the input high-dimensional tensor and the product of the lower-dimensional tensors of the proposed factorization, using a specified tensor norm. In our approach, we reduce the Boolean tensor factorization problem to a sequence of quadratic unconstrained binary optimization problems suitable for the D-Wave 2000Q quantum annealer. Although current quantum technology is still fairly restricted in the problems it can tackle, we show that complex tensor factorization problems as the ones addressed by us can be solved efficiently and accurately.

Alexandrov, Boian↗

DOE OSTI2021

DNTTD (Distributed Non-Negative Tensor Train Decomposition)

The era of exascale computing opens new venues for innovations and discoveries in many scientific, engineering, and commercial fields. However, with the exa flops also come the extra-large high-dimensional data generated by high performance computing. High-dimensional data is presented as multidimensional arrays, aka tensors. The presence of latent (not directly observable) structures in the tensor allows a unique representation and compression of the data by classical tensor factorization techniques. However, the classical tensor methods are not always stable or they can be exponential in their memory requirements, which makes them not suitable for high-dimensional tensors. Tensor train (TT) is a state-of-the-art tensor network introduced for factorization of high-dimensional tensors. TT transforms the initial high-dimensional tensor in a network of three-dimensional tensors that requires only a linear storage. Many real-world data, such as, density, temperature, population, probability, etc., are non-negative and for an easy interpretation, the algorithms preserving non-negativity are preferred. Here, we introduce a distributed non-negative tensor-train and demonstrate its scalability and the compression on synthetic and real world big datasets.

Bhattarai, Manish↗

DOE OSTI2019

Beyond toy models: distilling tensor networks in full AdS/CFT

We present a general procedure for constructing tensor networks that accurately reproduce holographic states in conformal field theories (CFTs). Given a state in a large-N CFT with a static, semiclassical gravitational dual, we build a tensor network by an iterative series of approximations that eliminate redundant degrees of freedom and minimize the bond dimensions of the resulting network. We argue that the bond dimensions of the tensor network will match the areas of the corresponding bulk surfaces. For “tree” tensor networks (i.e., those that are constructed by discretizing spacetime with nonintersecting Ryu-Takayanagi surfaces), our arguments can be made rigorous using a version of one-shot entanglement distillation in the CFT. Using the known quantum error correcting properties of AdS/CFT, we show that bulk legs can be added to the tensor networks to create holographic quantum error correcting codes. These codes behave similarly to previous holographic tensor network toy models, but describe actual bulk excitations in continuum AdS/CFT. By assuming some natural generalizations of the “holographic entanglement of purification” conjecture, we are able to construct tensor networks for more general bulk discretizations, leading to finer-grained networks that partition the information content of a Ryu-Takayanagi surface into tensor-factorized subregions. While the granularity of such a tensor network must be set larger than the string/Planck scales, we expect that it can be chosen to lie well below the AdS scale. However, we also prove a no-go theorem which shows that the bulk-to-boundary maps cannot all be isometries in a tensor network with intersecting Ryu-Takayanagi surfaces.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

DOE OSTI2021

Efficient Parallel Sparse Symmetric Tucker Decomposition for High-Order Tensors

Tensor based methods are receiving renewed attention in recent years due to their prevalence in diverse real-world applications. There is considerable literature on tensor representations and algorithms for tensor decompositions, both for dense and sparse tensors. Many applications in hypergraph analytics, machine learning, psychometry, and signal processing result in tensors that are both sparse and symmetric, making it an important class for further study. Similar to the critical Tensor Times Matrix chain operation (TTMc) in general sparse tensors, the Sparse Symmetric Tensor Times Same Matrix chain (S3TTMc) operation is compute and memory intensive due to high tensor order and the associated factorial explosion in the number of non-zeros. In this work, we present a novel compressed storage format CSS for sparse symmetric tensors, along with an efficient parallel algorithm for the S3TTMc operation. We theoretically establish that S3TTMc on CSS achieves a better memory versus run-time trade-off compared to state-of-the-art implementations. We demonstrate experimental findings that confirm these results and achieve up to 2.9× speedup on synthetic and real datasets.

Shivakumar, Shruti↗

DOE OSTI2020

A Sparse Tensor Benchmark Suite for CPUs and GPUs

Tensor computations present significant performance chal- lenges that impact a wide spectrum of applications ranging from machine learning, healthcare analytics, social network analysis, data mining to quantum chemistry and signal processing. Efforts to improve the perfor- mance of tensor computations include exploring data layout, execution scheduling, and parallelism in common tensor kernels. This work presents a benchmark suite for arbitrary-order sparse tensor kernels using state- of-the-art tensor formats: coordinate (COO) and hierarchical coordinate (HiCOO) on CPUs and GPUs. It presents a set of reference tensor kernel implementations that are compatible with real-world tensors and power law tensors extended from synthetic graph generation techniques. We also propose Roofline performance models for these kernels to provide insights of computer platforms from sparse tensor view. This benchmark suite along with the synthetic tensor generator is publicly available.

Li, Jiajia↗

NASA NTRS1994

Tensor-GMRES method for large sparse systems of nonlinear equations

This paper introduces a tensor-Krylov method, the tensor-GMRES method, for large sparse systems of nonlinear equations. This method is a coupling of tensor model formation and solution techniques for nonlinear equations with Krylov subspace projection techniques for unsymmetric systems of linear equations. Traditional tensor methods for nonlinear equations are based on a quadratic model of the nonlinear function, a standard linear model augmented by a simple second order term. These methods are shown to be significantly more efficient than standard methods both on nonsingular problems and on problems where the Jacobian matrix at the solution is singular. A major disadvantage of the traditional tensor methods is that the solution of the tensor model requires the factorization of the Jacobian matrix, which may not be suitable for problems where the Jacobian matrix is large and has a 'bad' sparsity structure for an efficient factorization. We overcome this difficulty by forming and solving the tensor model using an extension of a Newton-GMRES scheme. Like traditional tensor methods, we show that the new tensor method has significant computational advantages over the analogous Newton counterpart. Consistent with Krylov subspace based methods, the new tensor method does not depend on the factorization of the Jacobian matrix. As a matter of fact, the Jacobian matrix is never needed explicitly.

Feng, Dan↗

DOE OSTI2022

Towards Compact Neural Networks via End-to-End Training: A Bayesian Tensor Approach with Automatic Rank Determination

Post-training model compression can reduce the inference costs of deep neural networks, but uncompressed training still consumes enormous hardware resources and energy. To enable low-energy training on edge devices, it is highly desirable to directly train a compact neural network from scratch with a low memory cost. Low-rank tensor decomposition is an effective approach to reduce the memory and computing costs of large neural networks. However, directly training low-rank tensorized neural networks is a very challenging task because it is hard to determine a proper tensor rank a priori, and the tensor rank controls both model complexity and accuracy. Here, this paper presents a novel end-to-end framework for low-rank tensorized training. We first develop a Bayesian model that supports various low-rank tensor formats (e.g., CANDECOMP/PARAFAC, Tucker, tensor-train, and tensor-train matrix) and reduces neural network parameters with automatic rank determination during training. Then we develop a customized Bayesian solver to train large-scale tensorized neural networks. Our training methods shows orders-of-magnitude parameter reduction and little accuracy loss (or even better accuracy) in the experiments. On a very large deep learning recommendation system with over 4.2 ×10 9 model parameters, our method can reduce the parameter number to 1.6 ×10 5 automatically in the training process (i.e., by 2.6 ×10 4 times) while achieving almost the same accuracy. Code is available at https://github.com/colehawkins/bayesian-tensor-rank-determination.

compact neural networks↗

DOE OSTI2022

The ITensor Software Library for Tensor Network Calculations

ITensor is a system for programming tensor network calculations with an interface modeled on tensor diagrams, allowing users to focus on the connectivity of a tensor network without manually bookkeeping tensor indices. The ITensor interface rules out common programming errors and enables rapid prototyping of algorithms. After discussing the philosophy behind the ITensor approach, we show examples of each part of the interface including Index objects, the ITensor product operator, tensor factorizations, tensor storage types, algorithms for matrix product state (MPS) and matrix product operator (MPO) tensor networks, quantum number conserving block sparse tensors, and the NDTensors library. We also review publications that have used ITensor for quantum many-body physics and for other areas where tensor networks are increasingly applied. To conclude we discuss promising features and optimizations to be added in the future.

Fishman, Matthew↗

DOE OSTI2020

Estimating Higher-Order Moments Using Symmetric Tensor Decomposition

In this paper, we consider the problem of decomposing higher-order moment tensors, i.e., the sum of symmetric outer products of data vectors. Such a decomposition can be used to estimate the means in a Gaussian mixture model and for other applications in machine learning. The dth-order empirical moment tensor of a set of p observations of n variables is a symmetric d-way tensor. Our goal is to nd a low-rank tensor approximation comprising r $\ll$ p symmetric outer products. The challenge is that forming the empirical moment tensor costs O(pn d ) operations and O(n d ) storage, which may be prohibitively expensive; additionally, the algorithm to compute the low-rank approximation costs O(n d ) per iteration. Our contribution is avoiding formation of the moment tensor, computing the low-rank tensor approximation of the moment tensor implicitly using O(pnr) operations per iteration and no extra memory. This advance opens the door to more applications of higher-order moments since they can now be efficiently computed. We present numerical evidence of the computational savings and show an example of estimating the means for higher-order moments.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2020

Enhanced Tensor Completion Based Approaches for State Estimation in Distribution Systems

Grid state estimation is essential for effective control and management of distribution systems. While weighted least squares has been the conventional method for state estimation, sparsity-aware methods have become popular due to their superior performance with limited data. Matrix completion and compressed sensing-based state estimation approaches exploit the underlying smoothness in the state variables. However, classic matrix completion methods do not take into account the temporal correlation of system states. Compressed sensing methods, on the other hand, require an appropriate choice of sparsifying basis that may not be easy to identify. This paper proposes a blocktensor completion based framework which uses an alternative approach to estimate voltage phasor, power injections and branch currents. This approach utilizes the temporal correlation of the system states in a tensor trace-norm minimization formulation with power flow equations as constraints. Herein, feature scaling is introduced in the problem formulation to benefit from the improved sensitivity of the tensor trace norm to the matrix columns in the scaled unfoldings of the tensor. Weighted tensor norm is utilized to exploit the structures of the different unfoldings of the state measurement tensor to improve the voltage estimation. The estimation accuracy is further improved by alternatively estimating the tensor columns and increasing the available data at each stage in the tensor completion process. The proposed methods are evaluated on the IEEE-33, 37 test systems and a 100- node test system. The proposed methods are shown to provide significant performance gains relative to the classic matrix and tensor completion based approaches.

24 POWER TRANSMISSION AND DISTRIBUTION↗

DOE OSTI2020

Large Scale Tensor Factorization via Parallel Sketches

Tensor factorization methods have recently gained increased popularity. A key feature that renders tensors attractive is the ability to directly model multi-relational data. In this work, we propose ParaSketch, a parallel tensor factorization algorithm that enables massive parallelism, to deal with large tensors. The idea is to compress the large tensor into multiple small tensors, decompose each small tensor in parallel, and combine the results to reconstruct the desired latent factors. Prior art in this direction entails potentially very high complexity in the (Gaussian) compression and final combining stages. Adopting sketching matrices for compression, the proposed method enjoys a dramatic reduction in compression complexity, and features a much lighter combining step. Moreover, theoretical analysis shows that the compressed tensors inherit latent identifiability under mild conditions, hence establishing correctness of the overall approach. Numerical experiments corroborate the theory and demonstrate the effectiveness of the proposed algorithm.

block term decomposition↗

DOE OSTI2022

Reverse-mode differentiation in arbitrary tensor network format: with application to supervised learning.

This paper describes an efficient reverse-mode differentiation algorithm for contraction operations of tensor networks that may have arbitrary and unconventional network topologies. The approach leverages the tensor contraction tree of Evenbly and Pfeifer (2014), which provides an instruction set for the contraction sequence of a network. We show that this tree can be efficiently leveraged for differentiation of a full tensor network contraction using a recursive scheme that exploits (1) the bilinear property of contraction and (2) the property that trees have single path from root to leaves. While differentiation of tensor-tensor contraction is already possible in most automatic differentiation packages, we show that exploiting these two additional properties in the specific context of contraction sequences can improve efficiency. Following a description of the algorithm and computational complexity analysis, we investigate its utility for gradient-based supervised learning for low-rank function recovery and for fitting real-world unstructured datasets. We demonstrate improved performance over alternating least-squares optimization approaches and the capability to handle heterogeneous and arbitrary tensor network formats. When compared to alternating minimization algorithms, we find that the gradient-based approach requires a smaller oversampling ratio (number of samples compared to number model parameters) for recovery. This increased efficiency extends to fitting unstructured data of varying dimensionality and when employing a variety of tensor network formats. Here, we show improved learning using the hierarchical Tucker method over the tensor-train in high-dimensional settings on a number of benchmark problems.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2024

Scalable Tensor Methods for Nonuniform Hypergraphs

While multilinear algebra appears natural for studying the multiway interactions modeled by hypergraphs, tensor methods for general hypergraphs have been stymied by theoretical and practical barriers. A recently proposed adjacency tensor is applicable to nonuniform hypergraphs, but is prohibitively costly to form and analyze in practice. We develop tensor times same vector (TTSV) algorithms for this tensor which improve complexity from $O(n^r)$ to a low-degree polynomial in $r$, where $n$ is the number of vertices and $r$ is the maximum hyperedge size. Our algorithms are implicit, avoiding formation of the order $r$ adjacency tensor. Here, we demonstrate the flexibility and utility of our approach in practice by developing tensor-based hypergraph centrality and clustering algorithms. We also show these tensor measures offer complementary information to analogous graph-reduction approaches on data, and are also able to detect higher-order structure that many existing matrix-based approaches provably cannot.

97 MATHEMATICS AND COMPUTING↗