The goal of coded distributed computation is to efficiently distribute a computation task, such as matrix multiplication, $N$ -linear computation, or multivariate polynomial evaluation, across $S$ servers through a coding scheme, such that the response from any $R$ servers ( $R$ is called the recovery threshold) is sufficient for the user to recover the desired computed value. Current state-of-art approaches are based on either exclusively matrix-partitioning (Entangled Polynomial (EP) Codes for matrix multiplication), or exclusively batch processing (Lagrange Coded Computing (LCC) for $N$ -linear computations or multivariate polynomial evaluations). We present three related classes of codes, based on the idea of Cross-Subspace Alignment (CSA) which was introduced originally in the context of secure and private information retrieval. CSA codes are characterized by a Cauchy-Vandermonde matrix structure that facilitates interference alignment along Vandermonde terms, while the desired computations remain resolvable along the Cauchy terms. These codes are shown to unify, generalize and improve upon the state-of-art codes for distributed computing. First we introduce CSA codes for matrix multiplication, which yield LCC codes as a special case, and are shown to outperform LCC codes in general in download-limited settings. While matrix-partitioning approaches (EP codes) for distributed matrix multiplication have the advantage of flexible server computation latency, batch processing approaches (CSA, LCC) have significant advantages in communication costs as well as encoding and decoding complexity per matrix multiplication. In order to combine the benefits of these approaches, we introduce Generalized CSA (GCSA) codes for matrix multiplication that bridge the extremes of matrix-partitioning and batch processing approaches and demonstrate synergistic gains due to cross subspace alignment. Finally, we introduce $N$ -CSA codes for $N$ -linear distributed batch computations and multivariate batch polynomial evaluations. $N$ -CSA codes include LCC codes as a special case, and are in general capable of outperforming LCC codes in download-constrained settings by upto a factor of $N$ . Generalizations of $N$ -CSA codes to include $X$ -secure data and $B$ -byzantine servers are also provided.

中文翻译：

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跨子空间对齐代码，用于编码的分布式批处理计算

编码分布式计算的目标是有效地分配计算任务，例如矩阵乘法， $ N $ 线性计算或多元多项式求值 $ S $ 服务器通过一种编码方案，使得来自任何 $ R $ 服务器（ $ R $ 称为恢复阈值）足以使用户恢复所需的计算值。当前最先进的方法基于专门用于矩阵分割（用于矩阵乘法的纠缠多项式（EP）代码）或专门用于批处理（用于Lagrange编码计算（LCC））的方法。 $ N $ -线性计算或多元多项式求值）。基于跨子空间对齐（CSA）的概念，我们介绍了三类相关的代码，该概念最初是在安全和私有信息检索的背景下引入的。CSA代码的特征在于Cauchy-Vandermonde矩阵结构，该结构有助于沿Vandermonde项进行干扰对齐，而所需的计算沿Cauchy项仍可解析。这些代码显示为统一，概括和改进了分布式计算的最新代码。首先，我们介绍用于矩阵乘法的CSA码，这在特殊情况下会产生LCC码，并且在下载受限的设置中通常表现出优于LCC码。虽然用于分布式矩阵乘法的矩阵划分方法（EP代码）具有灵活的服务器计算延迟优势，但批处理方法（CSA，LCC）在通信成本以及每个矩阵乘法的编码和解码复杂性方面均具有显着优势。为了结合这些方法的优点，我们引入了用于矩阵乘法的通用CSA（GCSA）代码，该代码桥接了矩阵划分和批处理方法的极限，并展示了由于交叉子空间对齐而带来的协同收益。最后，我们介绍 我们介绍了用于矩阵乘法的通用CSA（GCSA）代码，该代码桥接了矩阵划分和批处理方法的极限，并演示了由于交叉子空间对齐而产生的协同收益。最后，我们介绍 我们介绍了用于矩阵乘法的通用CSA（GCSA）代码，该代码桥接了矩阵划分和批处理方法的极限，并演示了由于交叉子空间对齐而产生的协同收益。最后，我们介绍 $ N $ -CSA代码 $ N $ -线性分布式批处理计算和多元批多项式求值。 $ N $ -CSA代码在特殊情况下包括LCC代码，并且通常在下载受限的情况下，其性能优于LCC代码，最高可达 $ N $ 。的概括 $ N $ -要包含的CSA代码 $ X $ -安全数据和 $ B $ 还提供了-byzantine服务器。