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) ϵ, reduced computing cost (γ; the fraction of subfunctions each server must compute), and reduced communication cost (δ; 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 ∈ R K×L — 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 overall δ-sparse and γ-sparse matrices D ∈ R K×NT and E ∈ R NT ×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 computationvs-communication corner points (γ, δ) 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 ϵ 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.