# Robust Kernel Density Estimation by Scaling and Projection in Hilbert Space

arXiv: https://arxiv.org/abs/1411.4378 · HTML: https://mlsanity.com/paper/1411.4378/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 17 Nov 2014.

## Built on by (most cited first, 2 of 2)

- [Designing Robust Transformers using Robust Kernel Density Estimation](https://mlsanity.com/paper/2210.05794/) (2022)
- [Beyond Smoothness: Incorporating Low-Rank Analysis into Nonparametric Density Estimation](https://mlsanity.com/paper/2204.00930/) (2022)

## Abstract (arXiv)

While robust parameter estimation has been well studied in parametric density estimation, there has been little investigation into robust density estimation in the nonparametric setting. We present a robust version of the popular kernel density estimator (KDE). As with other estimators, a robust version of the KDE is useful since sample contamination is a common issue with datasets. What "robustness" means for a nonparametric density estimate is not straightforward and is a topic we explore in this paper. To construct a robust KDE we scale the traditional KDE and project it to its nearest weighted KDE in the L2 norm. This yields a scaled and projected KDE (SPKDE). Because the squared L2 norm penalizes point-wise errors superlinearly this causes the weighted KDE to allocate more weight to high density regions. We demonstrate the robustness of the SPKDE with numerical experiments and a consistency result which shows that asymptotically the SPKDE recovers the uncontaminated density under sufficient conditions on the contamination.