Kernel Density Estimation

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Kernel density estimation builds a probability distribution from data by placing a kernel around each sample and combining their contributions. A central challenge is choosing the kernel bandwidth: fixed rules and cross-validation are being complemented by adaptive, pretrained, and amortized selectors, as well as criteria tailored to tasks such as calibration. Scale-space analysis, geometry-aware kernels, and density constraints extend the method to settings such as image thresholding, probability simplices, and missing-data imputation. For large or streaming datasets, sketches and random features can reduce computation, while a key tension remains between scalable estimation and reliable accuracy across different sample sizes and distribution shapes.

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Signed random Fourier features for fast density estimation with indefinite kernels
Aug 29, 2026Xie Wang, Nicolas Langrené, Wen ChenKernel Density Estimation Random Feature Methods
Mathematical Institute, University of Oxford, Oxford, OX2 6GG, UK · Guangdong Provincial/Zhuhai Key Laboratory of Interdisciplinary Research and Application for Data Science, Beijing Normal-Hong Kong Baptist University, Zhuhai 519087, China.
Amortized Bandwidth Learning for Kernel Density Estimation under Logarithmic Score
Aug 20, 2026Junyi Liang, Hailiang DuDensity Estimation Kernel Density Estimation
School of Mathematics, East China University of Science and Technology, 200237 Shanghai, China · Department of Mathematical Sciences; Institute of Hazard, Risk and Resilience, Durham University, DH1 3LE Durham, United Kingdom · Data Science Institute; Global School of Sustainability, The London School of Economics and Political Science, WC2A 2AE London, United Kingdom
Distributionally Faithful Imputation via Positive Semi-Definite Kernel Density Estimation
Jul 8, 2026Andrea Basteri, Carlo Ciliberto, Alessandro RudiIncomplete Data Imputation Kernel Density Estimation
Inria - Ecole Normale Supériore, Paris, France · Department of Computer Science - UCL, London, UK · SDA Bocconi School of Management, Milan, Italy
Geometry-Aware Bayesian Quantification via Compositional Data Analysis
Jul 6, 2026Alejandro Moreo, Pablo González, Juan José del CozLabel Shift Bayesian Inference
Istituto di Scienza e Tecnologie dell’Informazione Consiglio Nazionale delle Ricerche Pisa, Italy · Artificial Intelligence Center University of Oviedo Asturias, Spain
Bandwidth Selection in Kernel Density Estimation for Model Calibration
Jun 29, 2026Han Zhou, Teodora Popordanoska, Matthew BlaschkoModel Calibration Kernel Density Estimation
Department ESAT, Center for Processing Speech and Images, KU Leuven, Leuven, Belgium
Cascade-KDE: Robust Time-Series Restoration under Out-of-Distribution Impulse Corruptions
May 22, 2026Yuefeng Liu, Ning Yang, Ziyu YangDistribution Shift Robustness Time Series Imputation
School of Digital and Intelligent Industry (School of Cyber Science and Technology), Inner Mongolia University of Science and Technology, Baotou, China
Adaptive Kernel Density Estimation with Pre-training
May 13, 2026Ruitong Zhang, Ke DengKernel Density Estimation Density Estimation
Department of Statistics and Data Science Tsinghua University Beijing, China
Raising the Ceiling: Better Empirical Fixation Densities for Saliency Benchmarking
May 5, 2026Susmit Agrawal, Jannis Hollman, Matthias KümmererBenchmark Design Eye Tracking
Tübingen AI Center, University of Tübingen, Germany
On Imbalanced Regression with Hoeffding Trees
Feb 25, 2026Pantia-Marina Alchirch, Dimitrios I. DiochnosDecision Tree Learning Kernel Density Estimation
University of Oklahoma
Estimation of Confidence Bounds in Binary Classification using Wilson Score Kernel Density Estimation
Feb 24, 2026Thorbjørn Mosekjær Iversen, Zebin Duan, Frederik HagelskjærBinary Classification Confidence Region Estimation
SDU Robotics, The Maersk Mc-Kinney Moller Institute, University of Southern
Attention in Geometry: Scalable Spatial Modeling via Adaptive Density Fields and FAISS-Accelerated Kernels
Jan 5, 2026Zhaowen Fan, Yunxiang HanKernel Density Estimation Spatial Attention
School of Computer, SICHUAN UNIVERSITY, CHENGDU, CHINA, 610065
Sublinear Sketches for Approximate Nearest Neighbor and Kernel Density Estimation
Oct 27, 2025Ved Danait, Srijan Das, Sujoy BhoreRandomized Sketching Kernel Density Estimation
Department of Computer Science & Engineering, Indian Institute of Technology Bombay, Mumbai, India. · Department of Computer Science & Engineering and Centre for Machine Intelligence & Data Science (CMINDS), Indian Institute of Technology Bombay, Mumbai, India.
Multiclass histogram-based thresholding using kernel density estimation and scale-space representations
Feb 10, 2022S. Korneev, J. Gilles, I. BattiatoUnsupervised Image Segmentation Image Segmentation

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Signed random Fourier features for fast density estimation with indefinite kernels

Xie Wang, Nicolas Langrené, Wen Chen

Mathematical Institute, University of Oxford, Oxford, OX2 6GG, UK · Guangdong Provincial/Zhuhai Key Laboratory of Interdisciplinary Research and Application for Data Science, Beijing Normal-Hong Kong Baptist University, Zhuhai 519087, China.

Kernel Density Estimation Random Feature Methods Kernel Methods

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Kernel density estimation (KDE) is one of the most fundamental statistical estimators of density functions. Its direct implementation on a dataset of

N

points incurs an

\mathcal{O}(N^{2})

computational cost, which is prohibitive for large-scale datasets. Kernel approximation techniques can be applied to bring the computational cost down to

\mathcal{O}(N)

. The random Fourier features (RFF) technique, based on sampling from the spectral density of the kernel function, has become popular to speed up kernel estimators for machine learning applications. Unfortunately, it is restricted to positive definite kernels, while the majority of kernel functions popular in KDE, such as the parabolic kernel, do not satisfy this property. To overcome this limitation, this article introduces the signed random Fourier features (SRFF) technique. It is a generalization of RFF compatible with indefinite kernels whose inverse Fourier transform is absolutely integrable. The motivation for introducing this method is to speed up KDE in the case of multivariate compact kernels, which are generally not positive definite. We detail how to implement SRFF for both product kernels and isotropic kernels. For the class of Kuttner-Golubov kernels

K(\boldsymbol{x}_{i},\boldsymbol{x}_{j})=(1-\left\Vert \boldsymbol{x}_{i}-\boldsymbol{x}_{j}\right\Vert ^α)^β\mathbf{1}_{\{\left\Vert \boldsymbol{x}_{i}-\boldsymbol{x}_{j}\right\Vert \leq1\}}

where

\boldsymbol{x}_{i}\in\mathbb{R}^{d}

\boldsymbol{x}_{j}\in\mathbb{R}^{d}

α>0

β>0

, which includes the triangular, parabolic, biweight, triweight, and other kernel functions of interest for KDE as particular examples, we provide an explicit acceptance-rejection algorithm to sample from its signed spectral density. Our numerical tests on a dataset of one million points confirm the computational efficiency and accuracy of SRFF for large-scale KDE.