Cross Subspace Alignment Codes for Coded Distributed Batch Computation

Coded distributed batch computation distributes 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 approaches are based on either exclusively matrix-partitioning (Entangled Polynomial (EP) Codes for matrix multiplication), or exclusively batch processing (Lagrange Coded Computing (LCC)). We present three related classes of codes, based on the idea of Cross-Subspace Alignment (CSA) which was introduced originally in the context of 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 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 over strictly download-limited settings. Next, we introduce Generalized CSA (GCSA) codes for matrix multiplication that bridge the extremes of matrix-partitioning and batch processing approaches. 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 achieving significantly lower downloads than LCC codes due to cross-subspace alignment. Generalizations of $N$-CSA codes to include $X$-secure data and $B$-byzantine servers are also obtained.