Tessellated Distributed Computing

The work considers the N-server distributed computing scenario with K users requesting functions that are linearly-decomposable over an arbitrary basis of L real (potentially non-linear) subfunctions. In our problem, the aim is for each user to receive their function outputs, allowing for reduced reconstruction error (distortion) epsilon , reduced computing cost ( gamma ; the fraction of subfunctions each server must compute), and reduced communication cost ( delta ; the fraction of users each server must connect to). For any given set of K requested functions - which is here represented by a coefficient matrix F is an element of R-KxL - our problem is made equivalent to the open problem of sparse matrix factorization that seeks - for a given parameter T, representing the number of shots for each server - to minimize the reconstruction distortion 1/KL||F -DE||(2)(F) over all $\delta $ -sparse and delta -sparse matrices D is an element of R-Kx NT and E is an element of R-NT x L . With these matrices respectively defining which servers compute each subfunction, and which users connect to each server, we here design our D,E by designing tessellated-based and SVD-based fixed support matrix factorization methods that first split F into properly sized and carefully positioned submatrices, which we then approximate and then decompose into properly designed submatrices of D and E. For the zero-error case and under basic dimensionality assumptions, the work reveals achievable computation-vs-communication corner points (gamma ,delta) which, for various cases, are proven optimal over a large class of D,E by means of a novel tessellations-based converse. Subsequently, for large N, and under basic statistical assumptions on F, the average achievable error epsilon is concisely expressed using the incomplete first moment of the standard Marchenko-Pastur distribution, where this performance is shown to be optimal over a large class of D and E. In the end, the work also reveals that the overall achieved gains over baseline methods are unbounded.