Polynomial dictionary learning algorithms in sparse representations

Dictionary learning has been extensively studied in sparse representations. However, existing dictionary learning algorithms are developed mainly for standard matrices (i.e. matrices with scalar elements), and little attention has been paid to polynomial matrices, despite their wide use for describing convolutive signals or for modeling acoustic channels in room and underwater acoustics. In this paper, we propose a polynomial dictionary learning technique to deal with signals with time delays. We present two types of polynomial dictionary learning methods based on the fact that a polynomial matrix can be represented either as a polynomial of matrices (i.e. the coefficient in the polynomial corresponding to each time lag is a scalar matrix) or equally as a matrix of polynomial elements (i.e. each element of the matrix is a polynomial). The first method allows one to extend any state-of-the-art dictionary learning method to the polynomial case; and the second method allows one to directly process the polynomial matrix without having to access its coefficient matrices. A sparse coding method is also presented for reconstructing convolutive signals based on a polynomial dictionary. Simulations are provided to demonstrate the performance of the proposed algorithms, e.g. for polynomial signal reconstruction from noisy measurements.