arXiv subjects

Julian Bellavita

Publications and source records attributed to Julian Bellavita.

7 recordsLinked to original sources

arXiv2026

SpSYRK: Half the Work in Distributed Sparse Matrix Multiplication

The symmetric rank-$k$ update (SYRK), $\C = \A\A^\top$, computes the dot product between each pair of rows of $\A$, producing the Gram matrix $\C$. Its sparse variant underpins similarity search in machine learning, graph analytics, and genomics, including Jaccard similarity on datasets too large for a single node. Yet, despite the symmetry in its inputs and outputs, existing distributed sparse matrix multiplication algorithms, such as Sparse SUMMA, treat sparse SYRK as generic multiplication, leaving performance untapped. In this paper, we present distributed sparse SYRK approaches that leverage symmetry. The approach partitions the off-diagonal blocks of the output between the upper and lower triangular portions of the process grid and computes only the lower-triangular part of each diagonal block, reducing per-process communication and computation compared with state-of-the-art distributed SpGEMM. A second variant reorders communication to avoid materializing $\A^\top$. On 32 nodes of the Perlmutter supercomputer, the algorithm achieves a $2\times$ speedup over an optimized Sparse SUMMA on matrices where local multiplication dominates the runtime; the advantage narrows on communication-bound inputs, a dependence the cost model predicts from the arithmetic intensity. Our variant rectifies this and consistently achieves superior scaling at high process counts. The approach is a drop-in replacement for any application computing $\C = \A \A^\top$ via a distributed SpGEMM routine, and its triangular output can be consumed directly by subsequent operations, reducing both compute and memory footprint.

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arXiv2026

Communication-Avoiding SpGEMM via Trident Partitioning on Hierarchical GPU Interconnects

The multiplication of two sparse matrices, known as SpGEMM, is a key kernel in scientific computing and large-scale data analytics, underpinning graph algorithms, machine learning, simulations, and computational biology, where sparsity is often highly unstructured. The unstructured sparsity makes achieving high performance challenging because it limits both memory efficiency and scalability. In distributed memory, the cost of exchanging and merging partial products across nodes further constrains performance. These issues are exacerbated on modern heterogeneous supercomputers with deep, hierarchical GPU interconnects. Current SpGEMM implementations overlook the gap between intra-node and inter-node bandwidth, resulting in unnecessary data movement and synchronization not fully exploiting the fast intra-node interconnect. To address these challenges, we introduce Trident, a hierarchy-aware 2D distributed SpGEMM algorithm that uses communication-avoiding techniques and asynchronous communication to exploit the hierarchical and heterogeneous architecture of modern supercomputing interconnect. Central to Trident is the novel trident partitioning scheme, which enables hierarchy-aware decomposition and reduces internode communication by leveraging the higher bandwidth between GPUs within a node compared to across nodes. Here, we evaluate Trident on unstructured matrices, achieving up to $2.38\times$ speedup over a 2D SpGEMM with a corresponding geometric mean speedup of $1.54\times$. Trident reduces internode communication volume by up to $2\times$ on NERSC's Perlmutter supercomputer. Furthermore, we demonstrate the effectiveness of Trident in speeding up Markov Clustering, achieving up to $2\times$ speedup compared to competing strategies.

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arXiv2026

Communication-Avoiding Linear Algebraic Kernel K-Means on GPUs

Clustering is an important tool in data analysis, with K-means being popular for its simplicity and versatility. However, it cannot handle non-linearly separable clusters. Kernel K-means addresses this limitation but requires a large kernel matrix, making it computationally and memory intensive. Prior work has accelerated Kernel K-means by formulating it using sparse linear algebra primitives and implementing it on a single GPU. However, that approach cannot run on datasets with more than approximately 80,000 samples due to limited GPU memory. In this work, we address this issue by presenting a suite of distributed-memory parallel algorithms for large-scale Kernel K-means clustering on multi-GPU systems. Our approach maps the most computationally expensive components of Kernel K-means onto communication-efficient distributed linear algebra primitives uniquely tailored for Kernel K-means, enabling highly scalable implementations that efficiently cluster million-scale datasets. Central to our work is the design of partitioning schemes that enable communication-efficient composition of the linear algebra primitives that appear in Kernel K-means. Our 1.5D algorithm consistently achieves the highest performance, enabling Kernel K-means to scale to data one to two orders of magnitude larger than previously practical. On 256 GPUs, it achieves a geometric mean weak scaling efficiency of $79.7\%$ and a geometric mean strong scaling speedup of $4.2\times$. Compared to our 1D algorithm, the 1.5D approach achieves up to a $3.6\times$ speedup on 256 GPUs and reduces clustering time from over an hour to under two seconds relative to a single-GPU sliding window implementation. Our results show that distributed algorithms designed with application-specific linear algebraic formulations can achieve substantial performance improvement.

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arXiv2025

Parallel GPU-Enabled Algorithms for SpGEMM on Arbitrary Semirings with Hybrid Communication

Sparse General Matrix Multiply (SpGEMM) is key for various High-Performance Computing (HPC) applications such as genomics and graph analytics. Using the semiring abstraction, many algorithms can be formulated as SpGEMM, allowing redefinition of addition, multiplication, and numeric types. Today large input matrices require distributed memory parallelism to avoid disk I/O, and modern HPC machines with GPUs can greatly accelerate linear algebra computation. In this paper, we implement a GPU-based distributed-memory SpGEMM routine on top of the CombBLAS library. Our implementation achieves a speedup of over 2x compared to the CPU-only CombBLAS implementation and up to 3x compared to PETSc for large input matrices. Furthermore, we note that communication between processes can be optimized by either direct host-to-host or device-to-device communication, depending on the message size. To exploit this, we introduce a hybrid communication scheme that dynamically switches data paths depending on the message size, thus improving runtimes in communication-bound scenarios.

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arXiv2025

Popcorn: Accelerating Kernel K-means on GPUs through Sparse Linear Algebra

K-means is a popular clustering algorithm with significant applications in numerous scientific and engineering areas. One drawback of K-means is its inability to identify non-linearly separable clusters, which may lead to inaccurate solutions in certain cases. Kernel K-means is a variant of classical K-means that can find non-linearly separable clusters. However, it scales quadratically with respect to the size of the dataset, taking several minutes to cluster even medium-sized datasets on traditional CPU-based machines. In this paper, we present a formulation of Kernel K-means using sparse-dense matrix multiplication (SpMM) and sparse matrix-vector multiplication (SpMV), and we show that our formulation enables the rapid implementation of a fast GPU-based version of Kernel K-means with little programming effort. Our implementation, named Popcorn, is the first open-source GPU-based implementation of Kernel K-means. Popcorn achieves a speedup of up to 123.8x over a CPU implementation of Kernel K-means and a speedup of up to 2.6x over a GPU implementation of Kernel K-means that does not use sparse matrix computations. Our results support the effectiveness of sparse matrices as tools for efficient parallel programming.

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arXiv2023

Tackling the Matrix Multiplication Micro-kernel Generation with Exo

The optimization of the matrix multiplication (or GEMM) has been a need during the last decades. This operation is considered the flagship of current linear algebra libraries such as BLIS, OpenBLAS, or Intel OneAPI because of its widespread use in a large variety of scientific applications. The GEMM is usually implemented following the GotoBLAS philosophy, which tiles the GEMM operands and uses a series of nested loops for performance improvement. These approaches extract the maximum computational power of the architectures through small pieces of hardware-oriented, high-performance code called micro-kernel. However, this approach forces developers to generate, with a non-negligible effort, a dedicated micro-kernel for each new hardware. In this work, we present a step-by-step procedure for generating micro-kernels with the Exo compiler that performs close to (or even better than) manually developed microkernels written with intrinsic functions or assembly language. Our solution also improves the portability of the generated code, since a hardware target is fully specified by a concise library-based description of its instructions.

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arXiv2022

Studying Scientific Data Lifecycle in On-demand Distributed Storage Caches

The XRootD system is used to transfer, store, and cache large datasets from high-energy physics (HEP). In this study we focus on its capability as distributed on-demand storage cache. Through exploring a large set of daily log files between 2020 and 2021, we seek to understand the data access patterns that might inform future cache design. Our study begins with a set of summary statistics regarding file read operations, file lifetimes, and file transfers. We observe that the number of read operations on each file remains nearly constant, while the average size of a read operation grows over time. Furthermore, files tend to have a consistent length of time during which they remain open and are in use. Based on this comprehensive study of the cache access statistics, we developed a cache simulator to explore the behavior of caches of different sizes. Within a certain size range, we find that increasing the XRootD cache size improves the cache hit rate, yielding faster overall file access. In particular, we find that increase the cache size from 40TB to 56TB could increase the hit rate from 0.62 to 0.89, which is a significant increase in cache effectiveness for modest cost.

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