Guaranteed bounds on the Kullback-Leibler divergence of univariate mixtures using piecewise log-sum-exp inequalities (1606.05850v2)

Abstract: Information-theoretic measures such as the entropy, cross-entropy and the Kullback-Leibler divergence between two mixture models is a core primitive in many signal processing tasks. Since the Kullback-Leibler divergence of mixtures provably does not admit a closed-form formula, it is in practice either estimated using costly Monte-Carlo stochastic integration, approximated, or bounded using various techniques. We present a fast and generic method that builds algorithmically closed-form lower and upper bounds on the entropy, the cross-entropy and the Kullback-Leibler divergence of mixtures. We illustrate the versatile method by reporting on our experiments for approximating the Kullback-Leibler divergence between univariate exponential mixtures, Gaussian mixtures, Rayleigh mixtures, and Gamma mixtures.

• The paper introduces a novel method for obtaining guaranteed lower and upper bounds on the Kullback-Leibler divergence between univariate mixture distributions.
• This method utilizes piecewise log-sum-exp inequalities to derive computationally efficient, algorithmically closed-form bounds.
• The approach achieves significant efficiency gains, offering bounds for univariate Gaussian mixtures in O(k log k + k' log k') time, making it practical for various applications.

Guaranteed Bounds on the Kullback-Leibler Divergence of Univariate Mixtures

The paper "Guaranteed bounds on the Kullback-Leibler divergence of univariate mixtures using piecewise log-sum-exp inequalities" by Frank Nielsen and Ke Sun explores the computational challenges and solutions related to estimating the Kullback-Leibler (KL) divergence for mixture models. This divergence is pivotal in many signal processing and machine learning applications, where quantifying the distance between probability distributions is essential.

Key Contributions

The authors present an innovative method for bounding the KL divergence between univariate mixture distributions, which traditionally lacks closed-form solutions. They propose a computationally efficient approach that provides deterministic lower and upper bounds on the KL divergence, entropy, and cross-entropy. This novel technique leverages piecewise log-sum-exp inequalities to create algorithmically closed-form expressions that approximate these information measures.

Technical Insights

1. Mixture Models and KL Divergence: Mixture models are prevalent in various domains due to their ability to capture complex distributions. However, calculating the KL divergence between two mixtures is complex because it doesn't admit a closed-form expression. Traditional methods, such as Monte Carlo simulations, are computationally expensive and provide only estimated solutions.
2. Log-sum-exp Inequalities: The core of the proposed method lies in bounding the logarithmic sum of exponential functions, allowing the derivation of the KL divergence bounds. These inequalities help handle the non-linearity and non-analytic nature of the KL divergence for mixtures.
3. Computational Complexity: For specific cases like univariate Gaussian mixtures, the authors achieve significant efficiency. The proposed method offers bounds within an additive factor of $\log k + \log k'$, where $k$ and $k'$ are the number of components in the mixtures. This is achieved in $O(k \log k + k' \log k')$ time for non-adaptive bounds, and further improved adaptive bounds (data-dependent) demonstrate enhanced tightness.
4. Numerical Examples: The paper provides extensive empirical results validating the method across various types of mixtures—Exponential, Rayleigh, Gaussian, and Gamma mixtures. The results show how the proposed bounds efficiently approximate the true KL divergence.

Implications and Future Directions

The provision of efficient deterministic bounds on the KL divergence significantly enhances the practical utility of mixture models in scenarios where computational resources or time are constrained. This method paves the way for faster algorithms in histogram comparison, image segmentation, and other applications relying on mixture models.

Theoretically, this work opens potential research avenues in extending these techniques to multivariate mixtures, although this presents additional complexity. The ability to derive closed-form or bounded estimates for other statistical divergences, such as the Jensen-Shannon divergence or R\'enyi divergence, also constitutes a promising research direction.

Concluding Remarks

In summary, this paper addresses the long-standing challenge of efficiently estimating the KL divergence between mixture distributions by providing an elegant and computationally feasible bounding approach using log-sum-exp inequalities. The work stands as a testament to the authors' expertise in information geometry and computational methods, offering both theoretical insights and practical tools for advanced applications in signal processing and machine learning.