Georgia Institute of Technology College of Sciences

Spectral algorithms, such as principal component analysis and spectral clustering, typically require careful data transformations to be effective: upon observing a matrix A, one may look at the spectrum of ψ(A) for a properly chosen ψ. We propose a simple and generic construction for sparse graphs based on graph powering. It is shown that graph powering regularizes the graph and decontaminates its spectrum in the following sense: (i) If the graph is drawn from the sparse Erd˝os-R´enyi ensemble, which has no spectral gap, it is shown that graph powering produces a “maximal” spectral gap, with the latter justified by establishing an Alon-Boppana result for powered graphs; (ii) If the graph is drawn from the sparse SBM, graph powering is shown to achieve the fundamental limit for weak recovery. (Joint work with E. Abbe, E. Boix, C. Sandon.)