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Theoretical Error Analysis of Entropy Approximation for Gaussian Mixtures

Takashi Furuya, Hiroyuki Kusumoto, Koichi Taniguchi, Naoya Kanno, Kazuma Suetake

TL;DR

The paper addresses the challenge of computing the entropy of Gaussian mixtures by analyzing the analytic approximation $\widetilde{H}[q]$ formed from the entropies of individual components. It derives rigorous upper and lower bounds on the entropy error $|H[q]-\widetilde{H}[q]|$ that depend on component separation measures $\alpha_{\{k,k'\}}$ and $\alpha_{k,k'}$, and shows the error decays exponentially as these separations grow or when mixture weights concentrate; it also provides probabilistic bounds for high-dimensional regimes and derivative error bounds with respect to model parameters. In the special case of coincident covariances, the authors obtain explicit forms and sharper bounds, including a necessary/sufficient condition for convergence, and confirm through experiments that the approximation becomes highly accurate in high dimensions. These results offer theoretical guarantees that the simple, dimension-agnostic entropy approximation is reliable for high-dimensional problems, such as neural networks with many parameters, and inform choices in variational inference and uncertainty estimation.

Abstract

Gaussian mixture distributions are commonly employed to represent general probability distributions. Despite the importance of using Gaussian mixtures for uncertainty estimation, the entropy of a Gaussian mixture cannot be calculated analytically. In this paper, we study the approximate entropy represented as the sum of the entropies of unimodal Gaussian distributions with mixing coefficients. This approximation is easy to calculate analytically regardless of dimension, but there is a lack of theoretical guarantees. We theoretically analyze the approximation error between the true and the approximate entropy to reveal when this approximation works effectively. This error is essentially controlled by how far apart each Gaussian component of the Gaussian mixture is. To measure such separation, we introduce the ratios of the distances between the means to the sum of the variances of each Gaussian component of the Gaussian mixture, and we reveal that the error converges to zero as the ratios tend to infinity. In addition, the probabilistic estimate indicates that this convergence situation is more likely to occur in higher-dimensional spaces. Therefore, our results provide a guarantee that this approximation works well for high-dimensional problems, such as neural networks that involve a large number of parameters.

Theoretical Error Analysis of Entropy Approximation for Gaussian Mixtures

TL;DR

The paper addresses the challenge of computing the entropy of Gaussian mixtures by analyzing the analytic approximation formed from the entropies of individual components. It derives rigorous upper and lower bounds on the entropy error that depend on component separation measures and , and shows the error decays exponentially as these separations grow or when mixture weights concentrate; it also provides probabilistic bounds for high-dimensional regimes and derivative error bounds with respect to model parameters. In the special case of coincident covariances, the authors obtain explicit forms and sharper bounds, including a necessary/sufficient condition for convergence, and confirm through experiments that the approximation becomes highly accurate in high dimensions. These results offer theoretical guarantees that the simple, dimension-agnostic entropy approximation is reliable for high-dimensional problems, such as neural networks with many parameters, and inform choices in variational inference and uncertainty estimation.

Abstract

Gaussian mixture distributions are commonly employed to represent general probability distributions. Despite the importance of using Gaussian mixtures for uncertainty estimation, the entropy of a Gaussian mixture cannot be calculated analytically. In this paper, we study the approximate entropy represented as the sum of the entropies of unimodal Gaussian distributions with mixing coefficients. This approximation is easy to calculate analytically regardless of dimension, but there is a lack of theoretical guarantees. We theoretically analyze the approximation error between the true and the approximate entropy to reveal when this approximation works effectively. This error is essentially controlled by how far apart each Gaussian component of the Gaussian mixture is. To measure such separation, we introduce the ratios of the distances between the means to the sum of the variances of each Gaussian component of the Gaussian mixture, and we reveal that the error converges to zero as the ratios tend to infinity. In addition, the probabilistic estimate indicates that this convergence situation is more likely to occur in higher-dimensional spaces. Therefore, our results provide a guarantee that this approximation works well for high-dimensional problems, such as neural networks that involve a large number of parameters.
Paper Structure (18 sections, 14 theorems, 109 equations, 4 figures)

This paper contains 18 sections, 14 theorems, 109 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Related work
  3. Entropy approximation for Gaussian mixtures
  4. Theoretical error analysis of the entropy approximation
  5. General covariance case
  6. Coincident covariance case
  7. Experiment
  8. Limitations and future work
  9. Proofs in Section \ref{['entropy approximation']}
  10. Proof of Theorem \ref{['main_result_1']}
  11. Proof of Corollary \ref{['prob_ineq_cor']}
  12. Proof of Theorem \ref{['derivative-entropy']}
  13. Proof of Proposition \ref{['explicit form 1']}
  14. Details of Section \ref{['sec:experiment-error']}
  15. Application example: Variational inference to BNNs with Gaussian mixtures
  16. ...and 3 more sections

Key Result

Theorem 4.2

Let $s \in [0,1)$. Then where the coefficient $c_{k,k^{\prime}}$ is defined by and the set $\mathbb{R}^{m}_{k,k^{\prime}}$ is defined by Moreover, the same upper bound holds for $\alpha_{\{k,k'\}}$ instead of $\alpha_{k,k'}$:

Figures (4)

  • Figure 1: Illustration of $\alpha=\alpha_{\{k,k'\}}$ ($m=2$, isotropic)
  • Figure 2: Illustration of $\alpha=\alpha_{k,k'}$ ($m=2$, anisotropic, $\sigma=\|\Sigma_k^{-\frac{1}{2}}\Sigma_{k'}^{\frac{1}{2}}\|_{\rm op})$
  • Figure 3: Relative error $|H[q] - \widetilde{H}_*[q]|/|H[q]|$ for the true entropy $H[q]$ and the approximation ones $\widetilde{H}_*[q]$. Each line indicates the mean value for 500 samples and the filled region indicates the min--max interval. $c$ denotes the same symbol in Corollary \ref{['prob_ineq_common']}. Methods of Ours, Taylor ($2$nd), Taylor ($0$th), and Lower bound denote $\widetilde{H}_{{\rm ours}}[q]$, $\widetilde{H}_{{\rm Huber}(2)}[q]$, $\widetilde{H}_{{\rm Huber}(0)}[q]$, and $\widetilde{H}_{{\rm Bonilla}}[q]$ in Appendix \ref{['app:experiment-error']}, respectively. A method of MC denotes the Monte Carlo integration with $1000$ sampling points.
  • Figure 4: Uncertainty estimation on the toy $1$D regression task. Each line indicates one component of the ensemble or mixture, and the filled region indicates the mean value of the prediction with the standard deviation$\times(\pm 2)$. As for BNN-GMs, the stronger the color intensity of the line, the larger the mixture coefficients of the model.

Theorems & Definitions (29)

  • Definition 4.1
  • Theorem 4.2
  • Remark 4.3
  • Remark 4.4
  • Remark 4.5
  • Remark 4.6
  • Corollary 4.7
  • Theorem 4.8
  • Proposition 4.9
  • Theorem 4.10
  • ...and 19 more