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.

We consider the dictionary learning problem for the analysis model based sparse representation. A novel algorithm is proposed by adapting the synthesis model based simultaneous codeword optimisation (SimCO) algorithm to the analysis model. This algorithm assumes that the analysis dictionary contains unit Ł2-norm atoms and trains the dictionary by the optimisation on manifolds. This framework allows one to update multiple dictionary atoms in each iteration, leading to a computationally efficient optimisation process. We demonstrate the competitive performance of the proposed algorithm using experiments on both synthetic and real data, as compared with three baseline algorithms, Analysis K-SVD, analysis operator learning (AOL) and learning overcomplete sparsifying transforms (LOST), respectively.