# Risk Bounds for Mixture Density Estimation on Compact Domains via the h-Lifted Kullback--Leibler Divergence

arXiv: https://arxiv.org/abs/2404.12586 · HTML: https://mlsanity.com/paper/2404.12586/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 5 of its 58 references that are in the map.
- First posted on arXiv 19 Apr 2024.

## Builds on (most cited first, 5 of 5)

- [A non-asymptotic approach for model selection via penalization in high-dimensional mixture of experts models](https://mlsanity.com/paper/2104.02640/) (2021)
- [Generalized Dirichlet-process-means for f-separable distortion measures](https://mlsanity.com/paper/1901.11331/) (2019)
- [Convex optimization on Banach Spaces](https://mlsanity.com/paper/1401.0334/) (2014)
- [Convergence and rate of convergence of some greedy algorithms in convex optimization](https://mlsanity.com/paper/1412.3297/) (2014)
- [Universal Approximation on the Hypersphere](https://mlsanity.com/paper/2004.06328/) (2020)

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

- [Dendrograms of Mixing Measures for Softmax-Gated Gaussian Mixture of Experts: Consistency Without Model Sweeps](https://mlsanity.com/paper/2510.12744/) (2025)
- [Revisiting Incremental Stochastic Majorization-Minimization Algorithms with Applications to Mixture of Experts](https://mlsanity.com/paper/2601.19811/) (2026)
- [Fast Model Selection and Stable Optimization for Softmax-Gated Multinomial-Logistic Mixture of Experts Models](https://mlsanity.com/paper/2602.07997/) (2026)

## Abstract (arXiv)

We consider the problem of estimating probability density functions based on sample data, using a finite mixture of densities from some component class. To this end, we introduce the h-lifted Kullback--Leibler (KL) divergence as a generalization of the standard KL divergence and a criterion for conducting risk minimization. Under a compact support assumption, we prove an O(1/√n) bound on the expected estimation error when using the h-lifted KL divergence, which extends the results of Rakhlin et al. (2005, ESAIM: Probability and Statistics, Vol. 9) and Li and Barron (1999, Advances in Neural Information ProcessingSystems, Vol. 12) to permit the risk bounding of density functions that are not strictly positive. We develop a procedure for the computation of the corresponding maximum h-lifted likelihood estimators (h-MLLEs) using the Majorization-Maximization framework and provide experimental results in support of our theoretical bounds.