mlsanityData to 7 Oct 2026

On the number of modes of Gaussian kernel density estimators

arXiv 2412.09080 ↗First posted 12 Dec 2024math.ST stat.ML stat.THcited by 4

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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.

Key facts. Cited by 4 papers in the map (data through 7 Oct 2026). Builds on 10 of its 34 references that are in the map. First posted on arXiv 12 Dec 2024.

Builds on

10 of 34 references.

References · 10

  1. 2023A mathematical perspective on Transformerscited by 160
  2. 2021Sinkformers: Transformers with Doubly Stochastic Attentioncited by 136
  3. 2023The emergence of clusters in self-attention dynamicscited by 133
  4. 2024Dynamic metastability in the self-attention modelcited by 42
  5. 2024Emergence of meta-stable clustering in mean-field transformer modelscited by 35
  6. 2024Clustering in Causal Attention Maskingcited by 31
  7. 2024Measure-to-measure interpolation using Transformerscited by 28
  8. 2015A review of mean-shift algorithms for clusteringcited by 18
  9. 2024Synchronization on circles and spheres with nonlinear interactionscited by 17
  10. 2019Landscape Complexity for the Empirical Risk of Generalized Linear Modelscited by 15

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Cited by year (2026 to 7 Oct)

Citing papers per year: 2025: 2; 2026: 2.

4 papers.

  1. 2025The Mean-Field Dynamics of Transformerscited by 24
  2. 2025Dense Associative Memory with Epanechnikov Energycited by 10
  3. 2026Perceptrons and localization of attention's mean-field landscapecited by 7
  4. 2026Random Quadratic Form on a Sphere: Synchronization by Common Noisecited by 2