Approximation by finite mixtures of continuous density functions that vanish at infinity

Abstract

Given sufficiently many components, it is often cited that finite mixture models can approximate any other probability density function (pdf) to an arbitrary degree of accuracy. Unfortunately, the nature of this approximation result is often left unclear. We prove that finite mixture models constructed from pdfs in ${\mathcal{C}}_0$ can be used to conduct approximation of various classes of approximands in a number of different modes. That is, we prove approximands in ${\mathcal{C}}_0$ can be uniformly approximated, approximands in ${\mathcal{C}}_b$ can be uniformly approximated on compact sets, and approximands in ${\mathcal{L}}_p$ can be approximated with respect to the ${\mathcal{L}}_p$, for $p\in \left[1,\infty \right)$. Furthermore, we also prove that measurable functions can be approximated, almost everywhere.

References in corpus (2)

• Model-based Clustering
• Mixtures of Experts Models

Cited by in corpus (15)

• An Asymptotically MSE-Optimal Estimator based on Gaussian Mixture Models
• Evidence for the evolution of black hole mass function with redshift
• GAS: A Gaussian Mixture Distribution-Based Adaptive Sampling Method for PINNs
• Channel Estimation based on Gaussian Mixture Models with Structured Covariances
• A Versatile Low-Complexity Feedback Scheme for FDD Systems via Generative Modeling
• Approximations of conditional probability density functions in Lebesgue spaces via mixture of experts models
• Selecting the number of clusters, clustering models, and algorithms. A unifying approach based on the quadratic discriminant score
• Gaussian Mixture Reduction with Composite Transportation Divergence
• Predicting exoplanet mass from radius and incident flux: A Bayesian mixture model
• Regression-based heterogeneity analysis to identify overlapping subgroup structure in high-dimensional data
• Not Eliminate but Aggregate: Post-Hoc Control over Mixture-of-Experts to Address Shortcut Shifts in Natural Language Understanding
• A non-asymptotic approach for model selection via penalization in high-dimensional mixture of experts models
• Subtractive Mixture Models via Squaring: Representation and Learning
• Non-parametric identification of single-lined binary candidates in young clusters using single-epoch spectroscopy
• Dealing with overdispersion in multivariate count data