# Matrix of Polynomials Model based Polynomial Dictionary Learning Method for Acoustic Impulse Response Modeling

## TL;DR

This paper introduces a novel polynomial dictionary learning method for acoustic impulse response modeling, enabling direct operation on polynomial matrices without decomposing into coefficient matrices, improving polynomial signal reconstruction.

## Contribution

The paper proposes an alternative polynomial dictionary learning approach that operates directly on polynomial matrices, enhancing efficiency in acoustic impulse response modeling.

## Key findings

- Effective polynomial dictionary learning demonstrated on acoustic signals
- Improved polynomial signal reconstruction accuracy
- Method operates directly on polynomial matrices

## Abstract

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.