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K$^2$IE: Kernel Method-based Kernel Intensity Estimators for Inhomogeneous Poisson Processes

ICML 2025 · Probabilistic and scientific modeling › Uncertainty and Gaussian processes · approach: Sparse variational Gaussian processes · 0 citations (OpenAlex)

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Proposes K²IE, a penalized least-squares RKHS intensity estimator for inhomogeneous Poisson processes whose representer theorem gives unit dual coefficients, so it coincides with a classical kernel intensity estimator with equivalent RKHS smoothing kernels and avoids dual optimization.

Problem

Estimating intensity functions of inhomogeneous Poisson processes: classical kernel intensity estimators (KIEs) are cheap but lack the RKHS framework, while kernel method-based estimators (FIE, Flaxman's model) need costly dual coefficient optimization.

Approach

Penalized least-squares loss over an RKHS intensity yields a representer theorem with dual coefficient equal to one, so the estimator is a sum of equivalent RKHS kernels (a KIE-style smoother); this allows closed-form edge correction via random Fourier features and closed-form integrated squared intensity for cross-validation.

Key results

• On 2D synthetic data lambda^1.0, K2IE achieved L2 of 53.0 vs 56.47 for FIE and 63.3 for KIE, with CPU time 0.16 s vs 1.54 s for FIE.
• On 1D data, K2IE matched FIE's predictive performance while CPU time stayed near 0.01 s regardless of data size, versus FIE's 1.77 s on 10x lambda^1.
• On bei real-world data, K2IE achieved the best test least squares loss (-6.16) but KIE had the best test count log-likelihood (267 vs 279 for K2IE).

Measured effects reported in the paper

Surprising findings

• K2IE can produce negative intensity values; on 2D data about 5.9% of grid values were negative on average across trials.
• KIE outperformed K2IE in predictive performance on 1D datasets lambda^1 and lambda^3 in L2 and |L|.

Limitations

• Does not guarantee non-negativity of the estimated intensity; about 5.9% of grid values negative on 2D data.
• Evaluated only on synthetic data and one small real-world dataset.
• Non-negativity constraint extension to nonlinear link functions is analytically hard.

Experimental setup

Datasets: 1D synthetic intensity functions (lambda^1, lambda^2, lambda^3 and 10x scaled) 2D synthetic sigmoidal Gaussian Cox process data bei (R package spatsta) tropical rain forest tree locations

Benchmarks: 1D synthetic data 2D synthetic data bei real-world dataset

Metrics: integrated squared error L2 integrated absolute error |L| rho (fraction better than KIE) CPU time test least squares loss test negative log-likelihood of counts

Baselines: KIE FIE variational Bayesian Gaussian Cox process with quadratic link

Builds on: Flaxman's kernel method-based intensity estimator kernel intensity estimator with edge correction Nystrom approximation random Fourier features

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