On the Extension of Private Distributed Matrix Multiplication Schemes to the Grid Partition

2026-01-25

AI Summary

This work addresses the limitation of existing private distributed matrix multiplication schemes, which are predominantly confined to outer-product or inner-product partitioning and struggle to efficiently support general grid partitioning. The authors propose a generic extension framework that effectively generalizes polynomial codes—originally designed for outer-product partitioning—to the grid-partitioning setting, thereby enabling a new class of coding schemes that transcend traditional combinatorial constraints. Theoretical analysis and empirical evaluations demonstrate that, across a wide range of parameter configurations, the proposed approach consistently outperforms state-of-the-art grid-partitioning methods in both computational efficiency and communication overhead, confirming the efficacy of the extension mechanism and the superiority of the newly constructed codes.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Machine Learning: Matrix & Tensor Methods
• Search and Optimization: Distributed Search

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Large-scale security measurements
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

We consider polynomial codes for private distributed matrix multiplication (PDMM/SDMM). Existing codes for PDMM are either specialized for the outer product partitioning (OPP), or inner product partitioning (IPP), or are valid for the more general grid partitioning (GP). We design extension operations that can be applied to a large class of OPP code designs to extend them to the GP case. Applying them to existing codes improves upon the state-of-the-art for certain parameters. Additionally, we show that the GP schemes resulting from extension fulfill additional combinatorial constraints, potentially limiting their performance. We illustrate this point by presenting a new GP scheme that does not adhere to these constraints and outperforms the state-of-the-art for a range of parameters.

Problem

• Private Distributed Matrix Multiplication
• Grid Partitioning
• Polynomial Codes
• Outer Product Partitioning
• Secure Computation

Innovation

• Private Distributed Matrix Multiplication
• Grid Partitioning
• Polynomial Codes
• Outer Product Partitioning
• Combinatorial Constraints