MNC: Structure-Exploiting Sparsity Estimation for Matrix Expressions

Efficiently computing linear algebra expressions is central to machine learning (ML) systems. Most systems support sparse formats and operations because sparse matrices are ubiquitous and their dense representation can cause prohibitive overheads. Estimating the sparsity of intermediates, however, remains a key challenge when generating execution plans or performing sparse operations. These sparsity estimates are used for cost and memory estimates, format decisions, and result allocation. Existing estimators tend to focus on matrix products only, and struggle to attain good accuracy with low estimation overhead. However, a key observation is that real-world sparse matrices commonly exhibit structural properties such as a single non-zero per row, or columns with varying sparsity. In this paper, we introduce MNC (Matrix Non-zero Count), a remarkably simple, count-based matrix synopsis that exploits these structural properties for efficient, accurate, and general sparsity estimation. We describe estimators and sketch propagation for realistic linear algebra expressions. Our experiments - on a new estimation benchmark called SparsEst - show that the MNC estimator yields good accuracy with very low overhead. This behavior makes MNC practical and broadly applicable in ML systems.