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

TL;DR

This paper extends private distributed matrix multiplication codes from specific partitioning schemes to a more general grid partitioning, improving performance and introducing a new scheme that outperforms existing methods.

Contribution

The authors develop extension operations for existing codes to support grid partitioning, and introduce a new GP scheme that surpasses current state-of-the-art performance.

Findings

Extension operations enable existing codes to support grid partitioning.

The new GP scheme outperforms previous schemes for various parameters.

Some extended schemes are limited by additional combinatorial constraints.

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.