  Spectral methods are popular in detecting global structures in the given data
that can be represented as a matrix. However when the data matrix is sparse or
noisy, classic spectral methods usually fail to work, due to localization of
eigenvectors (or singular vectors) induced by the sparsity or noise. In this
work, we propose a general method to solve the localization problem by learning
a regularization matrix from the localized eigenvectors. Using matrix
perturbation analysis, we demonstrate that the learned regularizations suppress
down the eigenvalues associated with localized eigenvectors and enable us to
recover the informative eigenvectors representing the global structure. We show
applications of our method in several inference problems: community detection
in networks, clustering from pairwise similarities, rank estimation and matrix
completion problems. Using extensive experiments, we illustrate that our method
solves the localization problem and works down to the theoretical detectability
limits in different kinds of synthetic data. This is in contrast with existing
spectral algorithms based on data matrix, non-backtracking matrix, Laplacians
and those with rank-one regularizations, which perform poorly in the sparse
case with noise.
