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A Hierarchical Parallel Tensor Infrastructure (HiParTI), is to support fast essential sparse tensor operations and tensor decompositions on multicore CPU and GPU architectures. It consists of sparse tensor decompositions, CANDECOMP/PARAFAC (CP) and Tucker decompositions, fundamental tensor operations, and tensor transformations
We propose an algorithm to efficiently perform latency-bound communication scenarios that consist of many small messages. In these parallel scenarios, processes typically pass around a lot of small-sized messages of a few KBs of size. Performing communication operations with P2P MPI routines or collective MPI routines (including neighborhood collectives) in such scenarios may not always yield the optimal results and may not resolve the latency bottleneck. To this end, we develop a regular structure called virtual process topology (VPT) on which the messages can be communicated in a structured and controlled manner. Using parameters of this topology, one can tune the rate of aggression in tackling the latency costs. We demonstrate that our communication algorithm is preferable to MPI P2P and collective routines for latency-bound communication and it can easily be adapted only by replacing calls to MPI routines in a parallel application. We show how to adapt existing topology-aware mapping heuristics to address the volume overhead due to communicating messages on the VPT. Moreover, we propose a novel swap-based mapping heuristic to address this overhead by optimizing the maximum volume handled by a process. Experiments on synthetic communication graphs as well as real-world applications such as parallel Canonical Polyadic sparse tensor decomposition and parallel sparse matrix-dense matrix multiplication show that our approach is a powerful way of overcoming the bottlenecks posed by sparse and latency-bound irregular communication.
Abstract not provided.
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.
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.
This project developed the LAPIS compiler framework, built on the Multilevel Intermediate Representation (MLIR), to optimize sparse linear algebra operations and support performance portability across diverse architectures. The main innovation of LAPIS is the Kokkos dialect, which allows for lowering codes from a high productivity language to different architectures in an elegant way. The dialect also allows the conversion of lower-level MLIR code to C++ Kokkos code, facilitating the integration of scientific machine learning (SciML) models into applications. To extend LAPIS for distributed memory architectures, a new partition dialect was created to manage the distribution of sparse tensors and express communication patterns for sparse linear algebra operations. This dialect also supports the distributed execution of operators and includes algorithmic optimizations to minimize communication to improve performance. The project also demonstrates that MLIR can enable effective linear algebra-level optimizations, improving performance on different GPUs for both sparse and dense linear algebra kernels. Key applications of LAPIS include sparse linear algebra and graph kernels, TenSQL, a relational database management solution built on GraphBLAS, and the development of subgraph isomorphism and monomorphism kernels, showcasing performance portability. In summary, the LAPIS framework supports productivity, performance, portability, and distributed memory execution, while also enabling linear algebra-level optimizations that are challenging in traditional programming languages, with successful applications ranging from simple sparse linear algebra to complex graph kernels.
Tensor algebra is a powerful tool for computing, but writing optimized codes that operate on sparse tensors can be very complex. This project enables a Tensor Algebra Compiler (TACO) that simplifies this task from man-years to man-days and extends TACO to support complex and large distributed systems. This report details the hypotheses, approaches used, and findings in this project.
Modern data scientists are tasked to analyze ever-growing data sets with increasingly complex relationships. Tensor decompositions have come to play a central role in identifying underlying latent structures in higher-order data. The problem of fitting tensor models to different distributions is complicated by the combinations of size, dimensionality, and sparsity present in real world data. The situation demands efficient algorithms designed for shared-memory and distributed systems. This work will present new research that tackles these challenges on several different fronts, leveraging optimizations in numerical algorithms and sparse tensor representations in heterogeneous high performance computing environments.
We introduce new methods for integrating nonlinear differential equations on low-rank manifolds. These methods rely on interpolatory projections onto the tangent space, enabling low-rank time integration of vector fields that can be evaluated entry-wise. A key advantage of our approach is that it does not require the vector field to exhibit low-rank structure, thereby overcoming significant limitations of traditional dynamical low-rank methods based on orthogonal projection. To construct the interpolatory projectors, we develop a sparse tensor sampling algorithm based on the discrete empirical interpolation method (DEIM) that parameterizes tensor train manifolds and their tangent spaces with cross interpolation. Using these projectors, we propose two time integration schemes on low-rank tensor train manifolds. The first scheme integrates the solution at selected interpolation indices and constructs the solution with cross interpolation. The second scheme generalizes the well-known orthogonal projector-splitting integrator to interpolatory projectors. We demonstrate the proposed methods with applications to several tensor differential equations arising from the discretization of partial differential equations.
COMET: Domain Specific Compilation in Multi-level IR COMET is a compiler for dense and sparse tensor algebra and a domain-specific-language (DSL) that facilitates development and implementation of high-performance computing, graph analytics, and artificial intelligence (AI) applications.
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.
We propose a novel statistical inference paradigm for zero-inflated multiway count data that dispenses with the need to distinguish between true and false zero counts. Our approach ignores all zero entries and applies zero-truncated Poisson regression on the positive counts. Inference is accomplished via tensor completion that imposes low-rank structure on the Poisson parameter space. Our main result shows that an $\textit{N}$-way rank-R parametric tensor 𝓜 ϵ (0, ∞) $I$Χ∙∙∙Χ$I$ generating Poisson observations can be accurately estimated from approximately $IR^2 \text{log}^2_2(I)$ non-zero counts for a nonnegative canonical polyadic decomposition. Several numerical experiments are presented demonstrating that our zero-truncated paradigm is comparable to the ideal scenario where the locations of false zero counts are known $\textit{a priori}$.
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 them an important class for further study. Similar to the critical Tensor Times Matrix chain operation (TTM c ) in general sparse tensors, the $\underline{S}$ parse $\underline{S}$ ymmetric $\underline{T}$ ensor $\underline{T}$ imes $\underline{S}$ ame $\underline{M}$ atrix $\underline{c}$ hain (S 3 TTM c ) operation is compute and memory intensive due to high tensor order and the associated factorial explosion in the number of non-zeros. We present the novel Compressed Sparse Symmetric (CSS) format for sparse symmetric tensors, along with an efficient parallel algorithm for the S 3 TTM c operation. We theoretically establish that S 3 TTM c on CSS achieves a better memory versus run-time trade-off compared to state-of-the-art implementations, and visualize the variation of the performance gap over the parameter space. We demonstrate experimental findings that confirm these results and achieve up to 2.72× speedup on synthetic and real datasets. The scaling of the algorithm on different test architectures is also showcased to highlight the effect of machine characteristics on algorithm performance.
Sparse symmetric tensors are an important class of tensors, and their decompositions serve as powerful tools for revealing low-rank structures. This paper introduces SymProp, a novel approach for scaling sparse symmetric Tucker decomposition by propagating symmetry through intermediate computations. SymProp optimizes two key computational kernels: Sparse Symmetric Tensor Times Same Matrix chain (S3 TTMc) for Higher-Order Orthogonal Iteration (HOOI) and Sparse Symmetric Tensor Times Same Matrix chain Times Core (S3 TTMcTC) for Higher-Order QR Iteration (HOQRI). Our method employs a metaprogramming-based index iteration approach to efficiently handle the upper triangular parts of intermediate dense symmetric tensors. SymProp achieves up to 50.9× speedup over SPLATT and up to 360.8× over Compressed Sparse Symmetric (CSS) format on the S3 TTMc operation. Moreover, our S3 TTMc and S3 TTMcTC implementations support tensor orders four levels higher than state-of-the-art methods. Our HOQRI demonstrates superior scalability and up to a 33.6× speedup over optimized HOOI. By enabling more scalable Tucker decompositions for higher orders, decomposition ranks, and dimension sizes, SymProp opens new possibilities for analyzing complex hypergraph structures in fields such as network science, data mining, and machine learning.
Tensor decomposition is a well-known tool for multiway data analysis. This work proposes using stochastic gradients for efficient generalized canonical polyadic (GCP) tensor decomposition of large-scale tensors. GCP tensor decomposition is a recently proposed version of tensor decomposition that allows for a variety of loss functions such as Bernoulli loss for binary data or Huber loss for robust estimation. Here, the stochastic gradient is formed from randomly sampled elements of the tensor and is efficient because it can be computed using the sparse matricized-tensor times Khatri--Rao product tensor kernel. For dense tensors, we simply use uniform sampling. For sparse tensors, we propose two types of stratified sampling that give precedence to sampling nonzeros. Numerical results demonstrate the advantages of the proposed approach and its scalability to large-scale problems.