We study algorithms for approximating pairwise similarity matrices that arise in natural language processing. Generally, computing a matrix n data points requires Omega(n^2) computations. This quadratic scaling is significant bottleneck, especially when similarities are computed via expensive functions, e.g., transformer models. Approximation methods reduce this complexity, often by using small...

We obtain necessary and sufficient conditions for the existence of positive solutions for a class of sublinear Dirichlet quasilinear elliptic systems.