# Model-free Estimation of Latent Structure via Multiscale Nonparametric Maximum Likelihood

arXiv: https://arxiv.org/abs/2410.22248 · HTML: https://mlsanity.com/paper/2410.22248/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 13 of its 59 references that are in the map.
- First posted on arXiv 29 Oct 2024.

## Builds on (most cited first, 13 of 13)

- [Structure Learning in Graphical Modeling](https://mlsanity.com/paper/1606.02359/) (2016)
- [Learning Gaussian Mixtures Using the Wasserstein-Fisher-Rao Gradient Flow](https://mlsanity.com/paper/2301.01766/) (2023)
- [DBSCAN++: Towards fast and scalable density clustering](https://mlsanity.com/paper/1810.13105/) (2018)
- [On the nonparametric maximum likelihood estimator for Gaussian location mixture densities with application to Gaussian denoising](https://mlsanity.com/paper/1712.02009/) (2017)
- [Self-regularizing Property of Nonparametric Maximum Likelihood Estimator in Mixture Models](https://mlsanity.com/paper/2008.08244/) (2020)
- [Identifiability of Nonparametric Mixture Models and Bayes Optimal Clustering](https://mlsanity.com/paper/1802.04397/) (2018)
- [A framework for benchmarking clustering algorithms](https://mlsanity.com/paper/2209.09493/) (2022)
- [Consistent procedures for cluster tree estimation and pruning](https://mlsanity.com/paper/1406.1546/) (2014)
- [Uniform Consistency in Nonparametric Mixture Models](https://mlsanity.com/paper/2108.14003/) (2021)
- [On Efficient and Scalable Computation of the Nonparametric Maximum Likelihood Estimator in Mixture Models](https://mlsanity.com/paper/2208.07514/) (2022)
- [Fully adaptive density-based clustering](https://mlsanity.com/paper/1409.8437/) (2014)
- [Bayesian Clustering via Fusing of Localized Densities](https://mlsanity.com/paper/2304.00074/) (2023)
- [Tight Bounds on the Hardness of Learning Simple Nonparametric Mixtures](https://mlsanity.com/paper/2203.15150/) (2022)

## Abstract (arXiv)

Multivariate distributions often carry latent structures that are difficult to identify and estimate, and which better reflect the data generating mechanism than extrinsic structures exhibited simply by the raw data. In this paper, we propose a model-free approach for estimating such latent structures whenever they are present, without assuming they exist a priori. Given an arbitrary density p0, we construct a multiscale representation of the density and propose data-driven methods for selecting representative models that capture meaningful discrete structure. Our approach uses a nonparametric maximum likelihood estimator to estimate the latent structure at different scales and we further characterize their asymptotic limits. By carrying out such a multiscale analysis, we obtain coarseto-fine structures inherent in the original distribution, which are integrated via a model selection procedure to yield an interpretable discrete representation of it. As an application, we design a clustering algorithm based on the proposed procedure and demonstrate its effectiveness in capturing a wide range of latent structures.