# Rate of convergence for geometric inference based on the empirical Christoffel function

arXiv: https://arxiv.org/abs/1910.14458 · HTML: https://mlsanity.com/paper/1910.14458/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 31 Oct 2019.

## Built on by (most cited first, 6 of 6)

- [Leveraging the Christoffel Function for Outlier Detection in Data Streams](https://mlsanity.com/paper/2508.16617/) (2025)
- [Scalable anomaly detection via a univariate Christoffel function](https://mlsanity.com/paper/2606.12483/) (2026)
- [CLOE: Christoffel Loss Autoencoder for Anomaly Detection](https://mlsanity.com/paper/2607.20530/) (2026)
- [Mixtures Closest to a Given Measure: A Semidefinite Programming Approach](https://mlsanity.com/paper/2509.22879/) (2025)
- [Cost Trade-offs in Matrix Inversion Updates for Streaming Outlier Detection](https://mlsanity.com/paper/2603.16697/) (2026)
- [Recovering Lower-Dimensional Semialgebraic Support of a Measure from its Moments](https://mlsanity.com/paper/2609.33420/) (2026)

## Abstract (arXiv)

We consider the problem of estimating the support of a measure from a finite, independent, sample. The estimators which are considered are constructed based on the empirical Christoffel function. Such estimators have been proposed for the problem of set estimation with heuristic justifications. We carry out a detailed finite sample analysis, that allows us to select the threshold and degree parameters as a function of the sample size. We provide a convergence rate analysis of the resulting support estimation procedure. Our analysis establishes that we may obtain finite sample bounds which are comparable to existing rates for different set estimation procedures. Our results rely on concentration inequalities for the empirical Christoffel function and on estimates of the supremum of the Christoffel-Darboux kernel on sets with smooth boundaries, that can be considered of independent interest.