  We study the problem of dictionary learning for signals that can be
represented as polynomials or polynomial matrices, such as convolutive signals
with time delays or acoustic impulse responses. Recently, we developed a method
for polynomial dictionary learning based on the fact that a polynomial matrix
can be expressed as a polynomial with matrix coefficients, where the
coefficient of the polynomial at each time lag is a scalar matrix. However, a
polynomial matrix can be also equally represented as a matrix with polynomial
elements. In this paper, we develop an alternative method for learning a
polynomial dictionary and a sparse representation method for polynomial signal
reconstruction based on this model. The proposed methods can be used directly
to operate on the polynomial matrix without having to access its coefficients
matrices. We demonstrate the performance of the proposed method for acoustic
impulse response modeling.
