Sparsity-Aware Roofline Models for Sparse Matrix-Matrix Multiplication

TL;DR

This paper develops sparsity-aware roofline models for sparse matrix-matrix multiplication, highlighting how matrix structure influences performance and the need for tailored modeling approaches.

Contribution

It introduces and evaluates sparsity-aware roofline models that account for matrix structure, improving performance prediction accuracy for SpMM implementations.

Findings

Single roofline models are insufficient across diverse sparsity patterns.

Structured sparsity and blocking significantly impact arithmetic intensity.

Performance analysis must consider matrix structure and data layout.

Abstract

Sparse matrix-dense matrix multiplication (SpMM) is a critical kernel in scientific computing, graph analytics, and machine learning, whose performance is often constrained by memory bandwidth. In this work, we investigate the applicability and limitations of roofline modeling for SpMM by explicitly accounting for the impact of matrix sparsity structure on arithmetic intensity and attainable performance. We evaluate three SpMM implementations: Compressed Sparse Row (CSR), Compressed Sparse Blocks (CSB), and Intel's Math Kernel Library (MKL). Each implementation was tested using large-scale matrices from the SuiteSparse collection and grouped by sparsity pattern, including block-structured, banded (diagonal), scale-free, and uniformly random matrices. We derive sparsity-aware roofline models that incorporate memory traffic, cache locality, and blocking behavior, and demonstrate that a single model is insufficient to accurately predict performance across diverse structures. Experiments were conducted on an AMD-based Perlmutter compute node with a varying number of columns in the dense matrix. In particular, blocking and structured sparsity significantly alter effective arithmetic intensity. The results show that accurate roofline-based performance analysis of SpMM requires sparsity-aware modeling, and that data layout and blocking strategies must be evaluated in the context of matrix structure rather than through a single unified model.