# On the number of modes of Gaussian kernel density estimators

arXiv: https://arxiv.org/abs/2412.09080 · HTML: https://mlsanity.com/paper/2412.09080/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 10 of its 34 references that are in the map.
- First posted on arXiv 12 Dec 2024.

## Builds on (most cited first, 10 of 10)

- [A mathematical perspective on Transformers](https://mlsanity.com/paper/2312.10794/) (2023)
- [The emergence of clusters in self-attention dynamics](https://mlsanity.com/paper/2305.05465/) (2023)
- [Sinkformers: Transformers with Doubly Stochastic Attention](https://mlsanity.com/paper/2110.11773/) (2021)
- [Dynamic metastability in the self-attention model](https://mlsanity.com/paper/2410.06833/) (2024)
- [Emergence of meta-stable clustering in mean-field transformer models](https://mlsanity.com/paper/2410.23228/) (2024)
- [Clustering in Causal Attention Masking](https://mlsanity.com/paper/2411.04990/) (2024)
- [Measure-to-measure interpolation using Transformers](https://mlsanity.com/paper/2411.04551/) (2024)
- [A review of mean-shift algorithms for clustering](https://mlsanity.com/paper/1503.00687/) (2015)
- [Synchronization on circles and spheres with nonlinear interactions](https://mlsanity.com/paper/2405.18273/) (2024)
- [Landscape Complexity for the Empirical Risk of Generalized Linear Models](https://mlsanity.com/paper/1912.02143/) (2019)

## Built on by (most cited first, 4 of 4)

- [The Mean-Field Dynamics of Transformers](https://mlsanity.com/paper/2512.01868/) (2025)
- [Dense Associative Memory with Epanechnikov Energy](https://mlsanity.com/paper/2506.10801/) (2025)
- [Perceptrons and localization of attention's mean-field landscape](https://mlsanity.com/paper/2601.21366/) (2026)
- [Random Quadratic Form on a Sphere: Synchronization by Common Noise](https://mlsanity.com/paper/2603.06187/) (2026)

## Abstract (arXiv)

We consider the Gaussian kernel density estimator with bandwidth β-frac12 of n iid Gaussian samples. Using the Kac-Rice formula and an Edgeworth expansion, we prove that the expected number of modes on the real line scales as Theta(√βlogβ) as β,n→∞ provided nclesssim βlesssim n2-c for some constant c>0. An impetus behind this investigation is to determine the number of clusters to which Transformers are drawn in a metastable state.