Dimension-free Concentration Bounds on Hankel Matrices for Spectral Learning

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 HS , 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 HS and on the distance betweenHS and its mean Hr. Existing concentration bounds seem to indicate that the concentration over Hr gets looser with the size of Hr, suggesting to make a tradeoff between the quantity of used information and the size of Hr. We propose new dimensionfree 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.