Dimension-free Concentration Bounds on Hankel Matrices for Spectral Learning

Abstract: Learning probabilistic models over strings is an important issue for many applications. Spectral methods propose elegant solutions to the problem of inferring weighted automata from finite samples of variable-length strings drawn from an unknown target distribution. These methods rely on a singular value decomposition of a matrix $H_S$, called the Hankel matrix, that records the frequencies of (some of) the observed strings. The accuracy of the learned distribution depends both on the quantity of information embedded in $H_S$ and on the distance between $H_S$ and its mean $H_r$. Existing concentration bounds seem to indicate that the concentration over $H_r$ gets looser with the size of $H_r$, suggesting to make a trade-off between the quantity of used information and the size of $H_r$. We propose new dimension-free concentration bounds for several variants of Hankel matrices. Experiments demonstrate that these bounds are tight and that they significantly improve existing bounds. These results suggest that the concentration rate of the Hankel matrix around its mean does not constitute an argument for limiting its size.