This paper presents an unsupervised learning algorithm for sparse nonnegative matrix factor time⁻frequency deconvolution with optimized fractional β-divergence. The β-divergence is a group of cost functions parametrized by a single parameter β. The Itakura⁻Saito divergence, Kullback⁻Leibler divergence and Least Square distance are special cases that correspond to β=0, 1, 2, respectively. This paper presents a generalized algorithm that uses a flexible range of β that includes fractional values. It describes a maximization⁻minimization (MM) algorithm leading to the development of a fast convergence multiplicative update algorithm with guaranteed convergence. The proposed model operates in the time⁻frequency domain and decomposes an information-bearing matrix into two-dimensional deconvolution of factor matrices that represent the spectral dictionary and temporal codes. The deconvolution process has been optimized to yield sparse temporal codes through maximizing the likelihood of the observations. The paper also presents a method to estimate the fractional β value. The method is demonstrated on separating audio mixtures recorded from a single channel. The paper shows that the extraction of the spectral dictionary and temporal codes is significantly more efficient by using the proposed algorithm and subsequently leads to better source separation performance. Experimental tests and comparisons with other factorization methods have been conducted to verify its efficacy.
Keywords: adaptive signal processing; blind source separation; machine learning, maximization–minimization algorithm, β-divergence, matrix deconvolution; sensors signal processing.