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Robust Gaussian Mixture Modeling: A <inline-formula><tex-math notation="LaTeX">$\mathcal{K}$</tex-math></inline-formula>-Divergence Based Approach

Authors :
Kenig, Ori
Todros, Koby
Adali, Tulay
Source :
IEEE Transactions on Signal Processing; 2024, Vol. 72 Issue: 1 p3578-3594, 17p
Publication Year :
2024

Abstract

This paper addresses the problem of robust Gaussian mixture modeling in the presence of outliers. We commence by introducing a general expectation-maximization (EM)-like scheme, called &lt;inline-formula&gt;&lt;tex-math notation=&quot;LaTeX&quot;&gt;$\mathcal{K}$&lt;/tex-math&gt;&lt;/inline-formula&gt;-BM, for iterative numerical computation of the minimum &lt;inline-formula&gt;&lt;tex-math notation=&quot;LaTeX&quot;&gt;$\mathcal{K}$&lt;/tex-math&gt;&lt;/inline-formula&gt;-divergence estimator (M&lt;inline-formula&gt;&lt;tex-math notation=&quot;LaTeX&quot;&gt;$\mathcal{K}$&lt;/tex-math&gt;&lt;/inline-formula&gt;DE). This estimator leverages Parzen&#39;s non-parametric &lt;inline-formula&gt;&lt;tex-math notation=&quot;LaTeX&quot;&gt;$\mathcal{K}$&lt;/tex-math&gt;&lt;/inline-formula&gt;ernel density estimate to down-weight low density regions associated with outlying measurements. Akin to the conventional EM, the &lt;inline-formula&gt;&lt;tex-math notation=&quot;LaTeX&quot;&gt;$\mathcal{K}$&lt;/tex-math&gt;&lt;/inline-formula&gt;-BM involves successive &lt;underline&gt;M&lt;/underline&gt;aximizations of lower &lt;underline&gt;B&lt;/underline&gt;ounds on the objective function of the M&lt;inline-formula&gt;&lt;tex-math notation=&quot;LaTeX&quot;&gt;$\mathcal{K}$&lt;/tex-math&gt;&lt;/inline-formula&gt;DE. However, differently from EM, these bounds are not exclusively reliant on conditional expectations. The &lt;inline-formula&gt;&lt;tex-math notation=&quot;LaTeX&quot;&gt;$\mathcal{K}$&lt;/tex-math&gt;&lt;/inline-formula&gt;-BM algorithm is applied to robust parameter estimation of a finite-order multivariate Gaussian mixture model (GMM). We proceed by introducing a new robust variant of the Bayesian information criterion (BIC) that penalizes the M&lt;inline-formula&gt;&lt;tex-math notation=&quot;LaTeX&quot;&gt;$\mathcal{K}$&lt;/tex-math&gt;&lt;/inline-formula&gt;DE&#39;s objective function. The proposed criterion, called &lt;inline-formula&gt;&lt;tex-math notation=&quot;LaTeX&quot;&gt;$\mathcal{K}$&lt;/tex-math&gt;&lt;/inline-formula&gt;-BIC, is conveniently applied for robust GMM order selection. In the paper, we also establish a data-driven procedure for selection of the kernel&#39;s bandwidth parameter. This procedure operates by minimizing an empirical asymptotic approximation of the mean-integrated-squared-error (MISE) between the underlying density and the estimated GMM density. Lastly, the &lt;inline-formula&gt;&lt;tex-math notation=&quot;LaTeX&quot;&gt;$\mathcal{K}$&lt;/tex-math&gt;&lt;/inline-formula&gt;-BM, the &lt;inline-formula&gt;&lt;tex-math notation=&quot;LaTeX&quot;&gt;$\mathcal{K}$&lt;/tex-math&gt;&lt;/inline-formula&gt;-BIC, and the MISE based selection of the kernel&#39;s bandwidth are combined into a unified framework for joint order selection and parameter estimation of a GMM. The advantages of the &lt;inline-formula&gt;&lt;tex-math notation=&quot;LaTeX&quot;&gt;$\mathcal{K}$&lt;/tex-math&gt;&lt;/inline-formula&gt;-divergence based framework over other robust approaches are illustrated in simulation studies involving synthetic and real data.

Details

Language :
English
ISSN :
1053587X
Volume :
72
Issue :
1
Database :
Supplemental Index
Journal :
IEEE Transactions on Signal Processing
Publication Type :
Periodical
Accession number :
ejs67220019
Full Text :
https://doi.org/10.1109/TSP.2024.3426965