Accelerating Bidiagonalization of Banded Matrices through Memory-Aware Bulge-Chasing on GPUs

Abstract

The reduction of a banded matrix to bidiagonal form is a critical step in the calculation of Singular Values, a cornerstone of scientific computing and AI. Although inherently parallel, this step has traditionally been considered unsuitable for GPUs due to its memory-bound nature. However, recent advances in GPU architectures, such as increased L1 memory per Streaming Multiprocessor or Compute Unit and larger L2 caches, have shifted this paradigm. In this work, we present the first GPU-accelerated algorithm for reducing a banded matrix to bidiagonal form, integrated into an open-source software package. Our algorithm builds on prior multicore CPU cache-efficient bulge-chasing methods, adapted to modern GPU architectures to optimize throughput. Leveraging Julia's high-level array abstractions and KernelAbstractions.jl, we implement a single function that is both hardware-agnostic and data-precision-aware, running efficiently across NVIDIA, AMD, Intel, and Apple Metal GPUs. We develop a hardware-aware performance model to guide tuning and identify key hyperparameters that govern optimal GPU performance for memory-bound workloads. We show that such workloads, when carefully optimized, can achieve substantial speed-ups on modern GPUs: our implementation outperforms multithreaded CPU libraries (PLASMA,SLATE) starting from matrix sizes as small as 1024x1024, and achieves over 100x speed-up on 32k x 32k matrices. Moreover, the algorithm's performance scales linearly with the matrix bandwidth, enabling efficient reduction of matrices with larger bandwidths, previously considered impractical.