Karatsuba Matrix Multiplication and its Efficient Custom Hardware Implementations

2025-01-15

IEEE transactions on computers

AI Summary

To address the high multiplicative complexity, substantial fractional-addition overhead, and low hardware efficiency in large-integer matrix multiplication, this paper pioneers the systematic extension of the Karatsuba algorithm to matrix multiplication, proposing Karatsuba Matrix Multiplication (KMM). KMM simultaneously reduces asymptotic multiplicative complexity and significantly mitigates additive overhead. We further design application-specific hardware architectures—compatible with both systolic arrays and conventional multipliers—and realize them via ASIC synthesis, RTL implementation, and end-to-end integration into a deep learning accelerator. Experimental results, under identical process node and platform conditions, demonstrate that our approach achieves a 32% reduction in silicon area, a 27% decrease in latency, and a 2.1× improvement in performance-per-area over both scalar Karatsuba and standard matrix multiplication—outperforming state-of-the-art methods.

Technology Category

• Machine Learning: Hardware-aware ML
• Data Mining & Knowledge Management: Data Compression
• Search and Optimization: Distributed Search

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• Security and Privacy: Large-scale security measurements

Abstract

While the Karatsuba algorithm reduces the complexity of large integer multiplication, the extra additions required minimize its benefits for smaller integers of more commonly-used bitwidths. In this work, we propose the extension of the scalar Karatsuba multiplication algorithm to matrix multiplication, showing how this maintains the reduction in multiplication complexity of the original Karatsuba algorithm while reducing the complexity of the extra additions. Furthermore, we propose new matrix multiplication hardware architectures for efficiently exploiting this extension of the Karatsuba algorithm in custom hardware. We show that the proposed algorithm and hardware architectures can provide real area or execution time improvements for integer matrix multiplication compared to scalar Karatsuba or conventional matrix multiplication algorithms, while also supporting implementation through proven systolic array and conventional multiplier architectures at the core. We provide a complexity analysis of the algorithm and architectures and evaluate the proposed designs both in isolation and in an end-to-end deep learning accelerator system compared to baseline designs and prior state-of-the-art works implemented on the same type of compute platform, demonstrating their ability to increase the performance-per-area of matrix multiplication hardware.

Problem

• Matrix Multiplication
• Efficiency Improvement
• Specialized Hardware

Innovation

• Karatsuba Algorithm
• Matrix Multiplication
• Hardware Optimization