Structured Codes for Distributed Matrix Multiplication
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866915994400718848 |
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| author | Malak, Derya |
| author_facet | Malak, Derya |
| contents | Our work addresses the well-known open problem of distributed computing of bilinear functions of two correlated sources ${\bf A}$ and ${\bf B}$. In a setting with two nodes, with the first node having access to ${\bf A}$ and the second to ${\bf B}$, we establish bounds on the optimal sum rate that allows a receiver to compute an important class of non-linear functions, and in particular bilinear functions, including dot products $\langle {\bf A},{\bf B}\rangle$, and general matrix products ${\bf A}^{\intercal}{\bf B}$ over finite fields. The bounds are tight for large field sizes, for which case we can derive the exact fundamental performance limits for all problem dimensions and a large class of sources. Our achievability scheme involves the design of non-linear transformations of ${\bf A}$ and ${\bf B}$, carefully calibrated to work synergistically with the structured linear encoding scheme by Körner and Marton. The subsequent converses derived here, calibrate the Han-Kobayashi approach and the strong converse of Ahlswede-Gács-Körner to yield relatively tight converses on the sum rate. We exhibit unbounded compression gains over Slepian-Wolf coding, depending on the source correlations. In the end, this work characterizes the fundamental limits of distributed computing for a crucial class of functions, while succinctly capturing the inherent computation structures and source correlations. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2501_00371 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Structured Codes for Distributed Matrix Multiplication Malak, Derya Information Theory Our work addresses the well-known open problem of distributed computing of bilinear functions of two correlated sources ${\bf A}$ and ${\bf B}$. In a setting with two nodes, with the first node having access to ${\bf A}$ and the second to ${\bf B}$, we establish bounds on the optimal sum rate that allows a receiver to compute an important class of non-linear functions, and in particular bilinear functions, including dot products $\langle {\bf A},{\bf B}\rangle$, and general matrix products ${\bf A}^{\intercal}{\bf B}$ over finite fields. The bounds are tight for large field sizes, for which case we can derive the exact fundamental performance limits for all problem dimensions and a large class of sources. Our achievability scheme involves the design of non-linear transformations of ${\bf A}$ and ${\bf B}$, carefully calibrated to work synergistically with the structured linear encoding scheme by Körner and Marton. The subsequent converses derived here, calibrate the Han-Kobayashi approach and the strong converse of Ahlswede-Gács-Körner to yield relatively tight converses on the sum rate. We exhibit unbounded compression gains over Slepian-Wolf coding, depending on the source correlations. In the end, this work characterizes the fundamental limits of distributed computing for a crucial class of functions, while succinctly capturing the inherent computation structures and source correlations. |
| title | Structured Codes for Distributed Matrix Multiplication |
| topic | Information Theory |
| url | https://arxiv.org/abs/2501.00371 |