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 ∈ RK × 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\|²F overall δ-sparse and γ-sparse matrices D∈ RK × NT and E ∈ RNT × 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.