Stable Spectral Learning Based On Schur Decomposition

Spectral methods are a powerful tool for inferring the parameters of certain classes of probability distributions by means of standard eigenvalue-eigenvector decompositions. Spectral algorithms can be orders of magnitude faster than log-likelihood based and related iterative methods, and, thanks to the uniqueness of the spectral decomposition, they enjoy global optimality guarantees. In practice, however, the applicability of spectral methods is limited due to their sensitivity to model misspecification, which can cause instability issues in the case of non-exact models. We present a new spectral approach that is based on the Schur triangularization of an observable matrix, and we carry out the corresponding theoretical analysis. Our main result is a bound on the estimation error that is shown to depend linearly on the condition number of the ground-truth conditional probability matrix and inversely on the eigengap of an observable matrix. Numerical experiments show that the proposed method is more stable, and performs better in general, than the classical spectral approach using direct matrix diagonalization.