Polynomial based approaches, such as the Mat-Dot and entangled polynomial codes (EPC) have been used extensively within coded matrix computations to obtain schemes with good recovery 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 EPC scheme by combining the idea of gradient coding along with the basic EPC encoding. Our technique 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：基于多项式的方法（例如Mat-Dot和纠缠多项式编码）已被广泛应用于编码矩阵计算中，以获得具有良好恢复阈值的方案。然而，这些方案公认存在解码数值稳定性差的问题。此外，这些方案的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。对于实际相关的稀疏输入矩阵情况，这可能导致工作节点计算时间显著增加的不良后果。本文通过将梯度编码思想与基本EPC编码相结合，提出了一种EPC方案的泛化方法。我们的技术能够降低编码权重，并得到数值稳定性显著改善的方案；这是以牺牲更差的阈值为代价实现的。通过适当设置方案中的参数，我们恢复了文献中若干经典方案。仿真结果表明，与EPC和Mat-Dot码相比，我们的方案在稀疏输入矩阵下展现出优异的数值稳定性和更快的计算速度。