# Mixture of experts models for multilevel data: modelling framework and approximation theory

arXiv: https://arxiv.org/abs/2209.15207 · HTML: https://mlsanity.com/paper/2209.15207/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Placed in MoE theory and gating functions, one of 12 directions in Mixture-of-experts.
- Builds on 4 of its 49 references that are in the map.
- First posted on arXiv 30 Sep 2022.

## Placed in

- Mixture-of-experts / [MoE theory and gating functions](https://mlsanity.com/moe/moe-theory-and-gating-functions/)

## Builds on (most cited first, 4 of 4)

- [Scaling Vision with Sparse Mixture of Experts](https://mlsanity.com/paper/2106.05974/) (2021)
- [DSelect-k: Differentiable Selection in the Mixture of Experts with Applications to Multi-Task Learning](https://mlsanity.com/paper/2106.03760/) (2021)
- [A Universal Approximation Theorem for Mixture of Experts Models](https://mlsanity.com/paper/1602.03683/) (2016)
- [Robust mixture of experts modeling using the t distribution](https://mlsanity.com/paper/1701.07429/) (2016)

## Built on by (most cited first, 3 of 3)

- [Mixture of Experts Provably Detect and Learn the Latent Cluster Structure in Gradient-Based Learning](https://mlsanity.com/paper/2506.01656/) (2025)
- [Mixture-of-Experts in Remote Sensing: A Survey](https://mlsanity.com/paper/2604.03342/) (2026)
- [Structure Aware Neural Architecture Search for Mixture of Experts](https://mlsanity.com/paper/2608.29817/) (2026)

## Abstract (arXiv)

Multilevel data are prevalent in many real-world applications. However, it remains an open research problem to identify and justify a class of models that flexibly capture a wide range of multilevel data. Motivated by the versatility of the mixture of experts (MoE) models in fitting regression data, in this article we extend upon the MoE and study a class of mixed MoE (MMoE) models for multilevel data. Under some regularity conditions, we prove that the MMoE is dense in the space of any continuous mixed effects models in the sense of weak convergence. As a result, the MMoE has a potential to accurately resemble almost all characteristics inherited in multilevel data, including the marginal distributions, dependence structures, regression links, random intercepts and random slopes. In a particular case where the multilevel data is hierarchical, we further show that a nested version of the MMoE universally approximates a broad range of dependence structures of the random effects among different factor levels.