New recursive preconditioning algorithms solve k-well-conditioned linear systems and regressions in Õ(d² + k^ω) time, matching the conditional lower bound and yielding the first nearly-linear-time nuclear norm approximation.
A Note on Preconditioning by Low-Stretch Spanning Trees
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abstract
Boman and Hendrickson observed that one can solve linear systems in Laplacian matrices in time $\bigO{m^{3/2 + o (1)} \ln (1/\epsilon)}$ by preconditioning with the Laplacian of a low-stretch spanning tree. By examining the distribution of eigenvalues of the preconditioned linear system, we prove that the preconditioned conjugate gradient will actually solve the linear system in time $\softO{m^{4/3} \ln (1/\epsilon)}$.
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Approaching Optimality for Solving Dense Linear Systems with Low-Rank Structure
New recursive preconditioning algorithms solve k-well-conditioned linear systems and regressions in Õ(d² + k^ω) time, matching the conditional lower bound and yielding the first nearly-linear-time nuclear norm approximation.