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1. Performance Implication of Tensor Irregularity and Optimization for Distributed Tensor Decomposition

   https://doi.org/10.1145/3580315  

   Miao, Zheng; Calhoun, Jon C; Ge, Rong; Li, Jiajia (June 2023, ACM Transactions on Parallel Computing)

   Tensors are used by a wide variety of applications to represent multi-dimensional data; tensor decompositions are a class of methods for latent data analytics, data compression, and so on. Many of these applications generate large tensors with irregular dimension sizes and nonzero distribution. CANDECOMP/PARAFAC decomposition (Cpd) is a popular low-rank tensor decomposition for discovering latent features. The increasing overhead on memory and execution time ofCpdfor large tensors requires distributed memory implementations as the only feasible solution. The sparsity and irregularity of tensors hinder the improvement of performance and scalability of distributed memory implementations. While previous works have been proved successful inCpdfor tensors with relatively regular dimension sizes and nonzero distribution, they either deliver unsatisfactory performance and scalability for irregular tensors or require significant time overhead in preprocessing. In this work, we focus on medium-grained tensor distribution to address their limitation for irregular tensors. We first thoroughly investigate through theoretical and experimental analysis. We disclose that the main cause of poorCpdperformance and scalability is the imbalance of multiple types of computations and communications and their tradeoffs; and sparsity and irregularity make it challenging to achieve their balances and tradeoffs. Irregularity of a sparse tensor is categorized based on two aspects: very different dimension sizes and a non-uniform nonzero distribution. Typically, focusing on optimizing one type of load imbalance causes other ones more severe for irregular tensors. To address such challenges, we propose irregularity-aware distributedCpdthat leverages the sparsity and irregularity information to identify the best tradeoff between different imbalances with low time overhead. We materialize the idea with two optimization methods: the prediction-based grid configuration and matrix-oriented distribution policy, where the former forms the global balance among computations and communications, and the latter further adjusts the balances among computations. The experimental results show that our proposed irregularity-aware distributedCpdis more scalable and outperforms the medium- and fine-grained distributed implementations by up to 4.4 × and 11.4 × on 1,536 processors, respectively. Our optimizations support different sparse tensor formats, such as compressed sparse fiber (CSF), coordinate (COO), and Hierarchical Coordinate (HiCOO), and gain good scalability for all of them. 

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2. Correlating Time Series With Interpretable Convolutional Kernels

   https://doi.org/10.1109/TKDE.2025.3550877  

   Chen, Xinyu; Cai, HanQin; Liu, Fuqiang; Zhao, Jinhua (June 2025, IEEE Transactions on Knowledge and Data Engineering)

   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. 

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3. Vectorization for digital signal processors via equality saturation

   https://doi.org/10.1145/3445814.3446707  

   VanHattum, Alexa; Nigam, Rachit; Lee, Vincent T.; Bornholt, James; Sampson, Adrian (April 2021, International Conference on Architectural Support for Programming Languages and Operating Systems (ASPLOS 2021))

   null (Ed.)

   Applications targeting digital signal processors (DSPs) benefit from fast implementations of small linear algebra kernels. While existing auto-vectorizing compilers are effective at extracting performance from large kernels, they struggle to invent the complex data movements necessary to optimize small kernels. To get the best performance, DSP engineers must hand-write and tune specialized small kernels for a wide spectrum of applications and architectures. We present Diospyros, a search-based compiler that automatically finds efficient vectorizations and data layouts for small linear algebra kernels. Diospyros combines symbolic evaluation and equality saturation to vectorize computations with irregular structure. We show that a collection of Diospyros-compiled kernels outperform implementations from existing DSP libraries by 3.1× on average, that Diospyros can generate kernels that are competitive with expert-tuned code, and that optimizing these small kernels offers end-to-end speedup for a DSP application. 

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4. Polyhedral Specification and Code Generation of Sparse Tensor Contraction with Co-iteration

   https://doi.org/10.1145/3566054  

   Zhao, Tuowen; Popoola, Tobi; Hall, Mary; Olschanowsky, Catherine; Strout, Michelle (March 2023, ACM Transactions on Architecture and Code Optimization)

   This article presents a code generator for sparse tensor contraction computations. It leverages a mathematical representation of loop nest computations in the sparse polyhedral framework (SPF), which extends the polyhedral model to support non-affine computations, such as those that arise in sparse tensors. SPF is extended to perform layout specification, optimization, and code generation of sparse tensor code: (1) We develop a polyhedral layout specification that decouples iteration spaces for layout and computation; and (2) we develop efficient co-iteration of sparse tensors by combining polyhedra scanning over the layout of one sparse tensor with the synthesis of code to find corresponding elements in other tensors through an SMT solver. We compare the generated code with that produced by a state-of-the-art tensor compiler, TACO. We achieve on average 1.63× faster parallel performance than TACO on sparse-sparse co-iteration and describe how to improve that to 2.72× average speedup by switching the find algorithms. We also demonstrate that decoupling iteration spaces of layout and computation enables additional layout and computation combinations to be supported. 

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5. A Guide for Achieving High Performance with Very Small Matrices on GPU: A Case Study of Batched LU and Cholesky Factorizations

   Haidar, Azzam; Abdelfattah, Ahmad; Zounon, Mawussi; Tomov, Stanimire; Dongarra, Jack (May 2018, IEEE transactions on parallel and distributed systems)

   We present a high-performance GPU kernel with a substantial speedup over vendor libraries for very small matrix computations. In addition, we discuss most of the challenges that hinder the design of efficient GPU kernels for small matrix algorithms. We propose relevant algorithm analysis to harness the full power of a GPU, and strategies for predicting the performance, before introducing a proper implementation. We develop a theoretical analysis and a methodology for high-performance linear solvers for very small matrices. As test cases, we take the Cholesky and LU factorizations and show how the proposed methodology enables us to achieve a performance close to the theoretical upper bound of the hardware. This work investigates and proposes novel algorithms for designing highly optimized GPU kernels for solving batches of hundreds of thousands of small-size Cholesky and LU factorizations. Our focus on efficient batched Cholesky and batched LU kernels is motivated by the increasing need for these kernels in scientific simulations (e.g., astrophysics applications). Techniques for optimal memory traffic, register blocking, and tunable concurrency are incorporated in our proposed design. The proposed GPU kernels achieve performance speedups versus CUBLAS of up to 6× for the factorizations, using double precision arithmetic on an NVIDIA Pascal P100 GPU. 

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