Blind Separation of Sparse Sources Using Jeffrey's Inverse Prior and the EM Algorithm

In this paper we study the properties of the Jeffrey’s inverse prior for blind separation of sparse sources. This very sparse prior was previously used for Wavelet-based image denoising. In this paper we consider separation of 3×3 and 2×3 noisy mixtures of audio signals, decomposed on a MDCT basis. The hierarchical formulation of the inverse prior allows for EM-based computation of MAP estimates. This procedure happens to be fast when compared to a standard more complex Markov chain Monte Carlo method using the flexible Student t prior, with competitive results obtained. Blind Source Separation (BSS) consists of estimating n signals (the sources) from the sole observation of m mixtures of them (the observations). If many efficient approaches exist for (over)determined (m ≥ n) non-noisy linear instantaneous, in particular within the field of Independent Component Analysis, the general linear instantaneous case, with mixtures possibly noisy and/or underdetermined (m < n) is still a very challenging problem. A now common approach for BSS, in particular for underdetermined mixtures, consists of exploiting source sparsity assumptions. Sparsity means that only “few” expansion coefficients of the sources on a given basis are significantly different from zero and its use to handle source separation problem (possibly underdetermined) was introduced in the seminal papers [1,2]. In [3,4] we modeled the expansion coefficients of the sources by identically and independently distributed (i.i.d) Student t processes and a Gibbs sampler (a standard MCMC simulation method) was proposed to sample from the posterior distribution of the mixing matrix, the input noise variance, the source coefficients and hyperparameters of the Student t distributions. The method was successfully applied to determined and underdetermined noisy audio mixtures, decomposed on a MDCT basis (a local cosine basis). In this paper, we give the source coefficients the Jeffrey’s inverse prior p(x) ∝ 1/|x|. This prior was used for image denoising and sparse regression in [5,6]. It provides very sparse signal estimates and, as shown in [5] in the context of denoising, is good compromise between soft and hard thresholding. Though Jeffrey’s prior corresponds to an improper probability density function, it admits a hierarchical formulation which leads to proper posterior densities, and allows for efficient EM-based computaJ. Rosca et al. (Eds.): ICA 2006, LNCS 3889, pp. 593–600, 2006. c © Springer-Verlag Berlin Heidelberg 2006 594 C. Févotte and S.J. Godsill tion of Maximum A Posteriori (MAP) estimates of the source coefficients, the mixing matrix and the noise variance. The paper is organized as follows: Section 1 introduces notations and assumptions. Section 2 presents the different priors used for the source coefficients, the mixing matrix, and the input noise variance. Section 3 gives the EM updates of each of the latter parameters. Section 4 provides separation results for determined and underdetermined mixtures of audio sources. The Jeffrey’s prior is shown to have good denoising properties, and the proposed method happens to be fast when compared to the more complex MCMC approach using the flexible Student t prior proposed in [4], with competitive results obtained. Section 5 draws some conclusions.