The multi-user linearly-separable distributed computing problem is considered here, in which $N$ servers help to compute the real-valued functions requested by $K$ users, where each function can be written as a linear combination of up to $L$ (generally non-linear) subfunctions. Each server computes a fraction $γ$ of the subfunctions, then communicates a function of its computed outputs to some of the users, and then each user collects its received data to recover its desired function. Our goal is to bound the ratio between the computation workload done by all servers over the number of datasets. To this end, we here reformulate the real-valued distributed computing problem into a matrix factorization problem and then into a basic sparse recovery problem, where sparsity implies computational savings. Building on this, we first give a simple probabilistic scheme for subfunction assignment, which allows us to upper bound the optimal normalized computation cost as $γ\leq \frac{K}{N}$ that a generally intractable $\ell_0$-minimization would give. To bypass the intractability of such optimal scheme, we show that if these optimal schemes enjoy $γ\leq - r\frac{K}{N}W^{-1}_{-1}(- \frac{2K}{e N r} )$ (where $W_{-1}(\cdot)$ is the Lambert function and $r$ calibrates the communication between servers and users), then they can actually be derived using a tractable Basis Pursuit $\ell_1$-minimization. This newly-revealed connection between distributed computation and compressed sensing opens up the possibility of designing practical distributed computing algorithms by employing tools and methods from compressed sensing.

Submitted to ITW2023. arXiv admin note: text overlap with arXiv:2206.11119