Cross Subspace Alignment Codes for Coded Distributed Batch Computation

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.