Error Analysis of Matrix Multiplication Emulation Using Ozaki-II Scheme

2026-01-30

AI Summary

This work addresses the limitations of the Ozaki-II scheme, which suffers from degraded accuracy when applied to input matrices with widely distributed eigenvalues and lacks a reliable means to predict the number of low-precision matrix multiplications required to achieve a target accuracy. We present the first rigorous deterministic error analysis framework for this scheme, integrating floating-point error modeling, numerical analysis, and a high-precision simulation technique based on the Chinese Remainder Theorem. This framework elucidates the underlying accuracy behavior of Ozaki-II and enables precise estimation of the requisite number of low-precision operations to meet a specified accuracy threshold. Consequently, our approach significantly enhances the predictability, practicality, and computational efficiency of the Ozaki-II method on AI hardware platforms.

Technology Category

• Machine Learning: Calibration & Uncertainty Quantification
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications
• Security and Privacy: Large-scale security measurements

Abstract

The Ozaki-II scheme is an emulation method that leverages the Chinese Remainder Theorem to compute high-precision matrix multiplication via a sequence of low-precision matrix multiplications. In this scheme, the attainable numerical accuracy improves as the number of low-precision matrix multiplications increases. Previous numerical studies have shown that single- and double-precision matrix multiplication using the Ozaki-II scheme achieves higher throughput than that of standard BLAS routines on modern AI hardware equipped with fast INT8 matrix multiply-accumulate units with INT8 inputs and INT32 accumulation. However, the accuracy of the Ozaki-II scheme can degrade when the exponent distribution of the input matrices is wide, in which case a large number of low-precision matrix multiplications is required to obtain high-precision results. In this paper, we present a rigorous deterministic error analysis of the Ozaki-II scheme. The proposed analysis not only clarifies the accuracy behavior of the method but also enables the estimation of the number of low-precision matrix multiplications required to achieve a desired level of numerical accuracy.

Problem

• Ozaki-II scheme
• error analysis
• matrix multiplication
• numerical accuracy
• precision emulation

Innovation

• Ozaki-II scheme
• error analysis
• matrix multiplication emulation
• Chinese Remainder Theorem
• low-precision computing