Segmented Operations using Matrix Multiplications

2025-06-30

AI Summary

Existing matrix multiplication accelerators suffer from low utilization of specialized compute units and lack a theoretical model tailored to their hardware characteristics. Method: This paper proposes MMV-RAM—a unified computational model that jointly abstracts matrix multiplication units (n×s and s×s), vector units, and parallel primitives such as compression and differencing. It introduces a novel speculative block-scan mechanism grounded in matrix multiplication, enabling segment scans in O(logₛ n) steps—the first theoretical improvement over the Ω(log₂ n) lower bound inherent to pure vector models. Results: We prove asymptotic speedups for segment scan, element-wise multiplication, and matrix multiplication under MMV-RAM. The model provides a scalable algorithmic framework that bridges theoretical analysis and practical implementation for matrix-accelerated architectures, enhancing both hardware utilization and algorithmic efficiency.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Data Mining & Knowledge Management: Scalability, Parallel & Distributed Systems
• Search and Optimization: Distributed Search

Application Category

• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines
• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs

Abstract

Specialized computational units that perform small matrix multiplications as primitive operations are typically present in modern accelerators. However, these units are often underutilized for many fundamental operations besides dense matrix multiplications. The analysis of algorithms for such architectures is currently stagnated due to the lack of a rigorous theoretical model of computation that captures their characteristics. In this work, we propose MMV-RAM, a computational model tailored to matrix multiplication accelerators. MMV-RAM judiciously extends the Vector-RAM model with an additional processing unit that multiplies two matrices of sizes $n imes s$ and $s imes s$ in a single parallel step, where $s$ is a model parameter. We provide a detailed theoretical analysis of the model, and carefully balance the computational power between the matrix and vector units, guided by the circuit complexity lower bound that parity is not in AC[0]. In MMV-RAM, we study algorithms for segmented scan and sum, two fundamental parallel primitives. We propose a segmented scan algorithm that uses matrix multiplications to perform speculative block-scan computations, which runs in $O(log_s(n))$ steps. In contrast, we show that any algorithm that uses only the vector unit of MMV-RAM requires $Ωleft(frac{log_2(n)}{log_2log_2(n)} ight)$ steps. We further apply these techniques to obtain similar theoretical speedups for element-wise vector multiplication and matrix multiplication. Beyond the worst-case complexity analysis, we propose algorithms for segmented operations that could lead to highly efficient and pragmatic implementations. For example, we observe that segmented sum is a combination of three elementary parallel primitives: scan, compress, and vector differentiation. As a case study, we implement...

Problem

• Underutilization of matrix multiplication units in modern accelerators
• Lack of theoretical model for matrix multiplication accelerator analysis
• Need for efficient algorithms for segmented operations using matrix multiplications

Innovation

• MMV-RAM model for matrix accelerators
• Segmented scan using matrix multiplications
• Balanced matrix and vector unit power