Polynomial based approaches, such as the Mat-Dot and entangled polynomial (EP) codes have been used extensively within coded matrix computations to obtain schemes with good thresholds. However, these schemes are well-recognized to suffer from poor numerical stability in decoding. Moreover, the encoding process in these schemes involves linearly combining a large number of input submatrices, i.e., the encoding weight is high. For the practically relevant case of sparse input matrices, this can have the undesirable effect of significantly increasing the worker node computation time. In this work, we propose a generalization of the EP scheme by combining the idea of gradient coding along with the basic EP encoding. Our scheme allows us to reduce the weight of the encoding and arrive at schemes that exhibit much better numerical stability; this is achieved at the expense of a worse threshold. By appropriately setting parameters in our scheme, we recover several well-known schemes in the literature. Simulation results show that our scheme provides excellent numerical stability and fast computation speed (for sparse input matrices) as compared to EPC and Mat-Dot codes.

翻译：多项式方法（如Mat-Dot和纠缠多项式（EP）码）已被广泛应用于编码矩阵计算中，以获取具有良好阈值的方案。然而，这些方案在解码时众所周知的数值稳定性差，且编码过程涉及大量输入子矩阵的线性组合（即编码权重较高）。在实际相关的稀疏输入矩阵场景中，这会显著增加工作节点计算时间，产生不良影响。本文通过结合梯度编码思想与基本EP编码，提出了一种EP方案的泛化方法。我们的方案能够降低编码权重，并得到数值稳定性显著更好的方案，但这是以牺牲阈值为代价的。通过适当设置方案中的参数，我们恢复了文献中若干已知方案。仿真结果表明，与EPC和Mat-Dot码相比，本方案（针对稀疏输入矩阵）具有优异的数值稳定性和快速的计算速度。