Optimal Approximation of Gaussian Mixtures by Finite Gaussian Mixtures

Core Concepts

The minimum order of finite Gaussian mixtures required to approximate a general Gaussian location mixture within a prescribed accuracy, measured by various f-divergences, is determined up to constant factors for distribution families with compact support or appropriate tail conditions.

Abstract

The paper studies the problem of approximating a general Gaussian location mixture by finite Gaussian mixtures. The key results are as follows:

For compactly supported mixing distributions, the minimum number of components m required to achieve an approximation error ε, measured by various f-divergences (TV, Hellinger, KL, χ2), is shown to be:

If M ≲ (log 1/ε)^(1/2 - δ) for some δ > 0, then m ≍ log(1/ε) / log log(1/ε).
If (log 1/ε)^(1/2) ≲ M ≲ ε^(-c1) for some 0 < c1 < 1, then m ≍ (log 1/ε) / log(1 + 1/√(log 1/ε)).

For distribution families with exponential tail decay (e.g., sub-Gaussian and sub-exponential), the minimum number of components m required to achieve an approximation error ε satisfies:

β^((2+α)/(2α)) ≲ m ≲ (log 1/ε) / log(1 + 1/β log(1/ε))^((α-2)/(2α)), where β is the scale parameter and α characterizes the tail decay.
In particular, for the sub-Gaussian family with c0 ≤ σ ≤ ε^(-c1), m ≍ log(1/ε).

The upper bounds are achieved using local moment matching, while the lower bounds are established by relating the approximation error to the low-rank approximation of certain trigonometric moment matrices, followed by a refined spectral analysis.

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Key Insights Distilled From

On the best approximation by finite Gaussian mixtures

by Yun Ma,Yihon... at arxiv.org 04-16-2024

https://arxiv.org/pdf/2404.08913.pdf

Deeper Inquiries

How do the approximation rates change if the mixing distributions are allowed to have unbounded support but satisfy other moment constraints, such as bounded variance or higher-order moments

When the mixing distributions are allowed to have unbounded support but satisfy other moment constraints, such as bounded variance or higher-order moments, the approximation rates may change. In the context provided, the approximation rates for the family of compactly supported distributions and distribution families with tail conditions were determined within constant factors. For distributions with unbounded support but bounded variance, the approximation rates may depend on the relationship between the variance and the order of the finite mixtures. If the variance is small relative to the order of the mixtures, the complexity level may grow slower with respect to the prescribed accuracy. On the other hand, if the variance is large relative to the order of the mixtures, the complexity level may grow faster as the variance increases. The specific rates would need to be derived based on the specific moment constraints and tail conditions of the mixing distributions.

Can the lower bound techniques be extended to other mixture models beyond Gaussian mixtures, such as exponential mixtures or Poisson mixtures

The lower bound techniques used in the context of Gaussian mixtures can potentially be extended to other mixture models beyond Gaussian mixtures, such as exponential mixtures or Poisson mixtures. The key idea is to relate the best approximation error to the low-rank approximation of certain moment matrices, followed by a refined spectral analysis of their minimum eigenvalue. By adapting this approach to the moment constraints and tail probabilities specific to exponential or Poisson mixtures, it may be possible to establish lower bounds on the approximation error for these models as well. The spectral analysis of moment matrices and the use of orthogonal polynomials can be applied to these different mixture models to determine the complexity of approximating them by finite mixtures.

What are the implications of these approximation results on the statistical complexity of nonparametric density estimation and other related problems involving Gaussian mixtures

The approximation results discussed in the context have implications on the statistical complexity of nonparametric density estimation and related problems involving Gaussian mixtures. These results provide insights into the minimum order of finite mixtures required to achieve a prescribed accuracy when approximating Gaussian location mixtures. By determining the best approximation error and the complexity level of finite mixtures for different mixing distributions, these results contribute to understanding the trade-offs between accuracy and complexity in statistical modeling of heterogeneous populations. The techniques used in deriving these results can be applied to other mixture models, enhancing the understanding of approximation by finite mixtures in various statistical applications.