# A Linear Algebraic Approach to Model Parallelism in Deep Learning

arXiv: https://arxiv.org/abs/2006.03108 · HTML: https://mlsanity.com/paper/2006.03108/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 4 Jun 2020.

## Built on by (most cited first, 4 of 4)

- [Model-Parallel Fourier Neural Operators as Learned Surrogates for Large-Scale Parametric PDEs](https://mlsanity.com/paper/2204.01205/) (2022)
- [Learned multiphysics inversion with differentiable programming and machine learning](https://mlsanity.com/paper/2304.05592/) (2023)
- [SciAI4Industry -- Solving PDEs for industry-scale problems with deep learning](https://mlsanity.com/paper/2211.12709/) (2022)
- [Toward efficient resource utilization at edge nodes in federated learning](https://mlsanity.com/paper/2309.10367/) (2023)

## Abstract (arXiv)

Training deep neural networks (DNNs) in large-cluster computing environments is increasingly necessary, as networks grow in size and complexity. Local memory and processing limitations require robust data and model parallelism for crossing compute node boundaries. We propose a linear-algebraic approach to model parallelism in deep learning, which allows parallel distribution of any tensor in the DNN. Rather than rely on automatic differentiation tools, which do not universally support distributed memory parallelism models, we show that parallel data movement operations, e.g., broadcast, sum-reduce, and halo exchange, are linear operators, and by defining the relevant spaces and inner products, we manually develop the adjoint, or backward, operators required for gradient-based training of DNNs. We build distributed DNN layers using these parallel primitives, composed with sequential layer implementations, and demonstrate their application by building and training a distributed DNN using DistDL, a PyTorch and MPI-based distributed deep learning toolkit.