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<p>Widely used divergences like Kullback–Leibler (KL) divergence, Bhattacharyya divergence and Cauchy–Schwarz divergence have no closed form expression in the case of Gaussian mixture models (GMMs). This led to computationally expensive methods like numerical approximations, componentwise methods which fails to capture the mixture structure. In this paper, we develop a divergence approximation through the embedding of GMMs into the manifold of symmetric positive definite (SPD) matrices. The main result is that for regular divergences on compact set of non-degenerate GMM parameters, uniform equivalence is obtained for the divergence between two GMMs and computationally tractable corresponding divergence between their centered multivariate normal distribution representations in the SPD manifold. This is an extension of the work presented in GSI’25 conference. We further prove a stability theorem showing that the uniform equivalence degrades only linearly under spectral perturbations of the Hessian, ensuring robustness in numerical implementation. Experiments on the UIUC and KTH-TIPS texture recognition benchmarks evaluate three divergence measures computed via the SPD embedding: symmetric KL divergence, Bhattacharyya divergence and Cauchy–Schwarz divergence.</p>

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A general framework for divergence approximation on Gaussian mixture models

  • 期刊论文

  • 发表日期：

    2026年5月7日

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  • Amit Vishwakarma,
  • K. S. Subrahamanian Moosath

摘要

Widely used divergences like Kullback–Leibler (KL) divergence, Bhattacharyya divergence and Cauchy–Schwarz divergence have no closed form expression in the case of Gaussian mixture models (GMMs). This led to computationally expensive methods like numerical approximations, componentwise methods which fails to capture the mixture structure. In this paper, we develop a divergence approximation through the embedding of GMMs into the manifold of symmetric positive definite (SPD) matrices. The main result is that for regular divergences on compact set of non-degenerate GMM parameters, uniform equivalence is obtained for the divergence between two GMMs and computationally tractable corresponding divergence between their centered multivariate normal distribution representations in the SPD manifold. This is an extension of the work presented in GSI’25 conference. We further prove a stability theorem showing that the uniform equivalence degrades only linearly under spectral perturbations of the Hessian, ensuring robustness in numerical implementation. Experiments on the UIUC and KTH-TIPS texture recognition benchmarks evaluate three divergence measures computed via the SPD embedding: symmetric KL divergence, Bhattacharyya divergence and Cauchy–Schwarz divergence.