Soft-LOST: EM on a Mixture of Oriented Lines

Abstract

Abstract. Robust clustering of data into overlapping linear subspaces is a common problem. Here we consider one-dimensional subspaces that cross the origin. This problem arises in blind source separation, where the subspaces correspond directly to columns of a mixing matrix. We present an algorithm that identifies these subspaces using an EM procedure, where the E-step calculates posterior probabilities assigning data points to lines and M-step repositions the lines to match the points assigned to them. This method, combined with a transformation into a sparse domain and an L1-norm optimisation, constitutes a blind source separation algorithm for the under-determined case.

Median-based clustering for underdetermined blind signal processing

“…In Table I, the mean of the performance index depending on various parameters is presented. Noting that the mean error when using random (2 3)-matrices with coefficients uniformly taken from [ 1,1] is , we observe good performance, especially for a larger kernel radius and higher approximation parameter , also compared with Soft-LOST's [6]. As an example, in higher mixture dimension, three speech signals are mixed by a column-normalized (3 3)-mixing matrix .…”

“…However, how to do this algorithmically is far from obvious, and although quite a few algorithms have been proposed recently [4]-[6], performance is yet limited. The most commonly used overcomplete algorithms rely on sparse sources (after possible sparsification by preprocessing), which can be identified by clustering, usually by -means or some extension [5], [6]. However, apart from the fact that theoretical justifications have not been found, mean-based clustering only identifies the correct if the data density approaches a delta distribution.…”

“…Assuming statistically independent sources with existing variance and at most one Gaussian component, it is well known that is determined uniquely by [3]. However, how to do this algorithmically is far from obvious, and although quite a few algorithms have been proposed recently [4]- [6], performance is yet limited. The most commonly used overcomplete algorithms rely on sparse sources (after possible sparsification by preprocessing), which can be identified by clustering, usually by -means or some extension [5], [6].…”

Stereo Source Separation and Source Counting with MAP Estimation with Dirichlet Prior Considering Spatial Aliasing Problem

“…Each source is modeled by (7). We also assume that the PDOA for an observed mixture follows a Gaussian mixture model (GMM):…”

“…In order to avoid one cluster being modeled by two or more Gaussians, thus making it possible to estimate the number of sources correctly, we propose utilizing a sparse distribution modeled by the Dirichlet distribution as the prior of the GMM mixture weight. The authors of [6,7] also derived the EM algorithm, however, they still needed to know the number of sources N s in advance. On the other hand, our proposed algorithm does not require information on the source number, thanks to the weight prior.…”