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Unlocking Point Processes through Point Set Diffusion

David Lüdke, Enric Rabasseda Raventós, Marcel Kollovieh, Stephan Günnemann

TL;DR

Experiments demonstrate that Point Set Diffusion achieves state-of-the-art performance in unconditional and conditional generation of spatial and spatiotemporal point processes while providing up to orders of magnitude faster sampling than autoregressive baselines.

Abstract

Point processes model the distribution of random point sets in mathematical spaces, such as spatial and temporal domains, with applications in fields like seismology, neuroscience, and economics. Existing statistical and machine learning models for point processes are predominantly constrained by their reliance on the characteristic intensity function, introducing an inherent trade-off between efficiency and flexibility. In this paper, we introduce Point Set Diffusion, a diffusion-based latent variable model that can represent arbitrary point processes on general metric spaces without relying on the intensity function. By directly learning to stochastically interpolate between noise and data point sets, our approach enables efficient, parallel sampling and flexible generation for complex conditional tasks defined on the metric space. Experiments on synthetic and real-world datasets demonstrate that Point Set Diffusion achieves state-of-the-art performance in unconditional and conditional generation of spatial and spatiotemporal point processes while providing up to orders of magnitude faster sampling than autoregressive baselines.

Unlocking Point Processes through Point Set Diffusion

TL;DR

Experiments demonstrate that Point Set Diffusion achieves state-of-the-art performance in unconditional and conditional generation of spatial and spatiotemporal point processes while providing up to orders of magnitude faster sampling than autoregressive baselines.

Abstract

Point processes model the distribution of random point sets in mathematical spaces, such as spatial and temporal domains, with applications in fields like seismology, neuroscience, and economics. Existing statistical and machine learning models for point processes are predominantly constrained by their reliance on the characteristic intensity function, introducing an inherent trade-off between efficiency and flexibility. In this paper, we introduce Point Set Diffusion, a diffusion-based latent variable model that can represent arbitrary point processes on general metric spaces without relying on the intensity function. By directly learning to stochastically interpolate between noise and data point sets, our approach enables efficient, parallel sampling and flexible generation for complex conditional tasks defined on the metric space. Experiments on synthetic and real-world datasets demonstrate that Point Set Diffusion achieves state-of-the-art performance in unconditional and conditional generation of spatial and spatiotemporal point processes while providing up to orders of magnitude faster sampling than autoregressive baselines.

Paper Structure

This paper contains 32 sections, 3 theorems, 34 equations, 7 figures, 4 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Background
  3. Point Processes
  4. Diffusion Models
  5. Point Set Diffusion
  6. Forward Process
  7. Reverse Process
  8. Parametrization:
  9. Sampling Procedure
  10. Experiments
  11. Data
  12. Metrics
  13. Spatial Point Processes
  14. Spatio-temporal Point Processes
  15. Other conditioning task
  16. ...and 17 more sections

Key Result

Theorem 1

Let $f_M(\bm{y})$ be a finite mixture of Dirac deltas: where $\bm{x}_1, \dots, \bm{x}_n \in D$ are points in the metric space, and $w_i \in \mathbb{R}$ are weights associated with each Dirac delta function. Then, this finite mixture of Dirac deltas $f_M$ can be approximated by $L^{2}$ functions in $L^2(D, \mu)$.

Figures (7)

  • Figure 1: Illustration of Point Set Diffusion for earthquakes in Japan. The forward process stochastically interpolates between the original data point set $X_0$ and a noise point set $X_T$, progressively removing data points and adding noise points. To generate new samples from the data distribution, we approximate the reverse posterior $q(X_t|X_0, X_{t+1})$ and add approximate data points and remove noise points.
  • Figure 2: The forward process is a Markov Chain $q(X_{t+1} | X_t)$, that stochastically interpolates a data sample $X_0$ with a noise point set $X_{T}$ over $T$ steps by applying a thinning and a noise process.
  • Figure 3: The posterior reverses the stochastic interpolation of $X_0 \to X_T$ of the forward process by adding back thinned points from the thinning process and thinning point added in the noise process.
  • Figure 4: Examples of conditioning masks for $\mathbb{R}_{\geq 0}$ and $\mathbb{R}^2$.
  • Figure 5: SPP conditioning task: top ground truth, middle Regularized Method and bottom Point Set Diffusion.
  • ...and 2 more figures

Theorems & Definitions (4)

  • Definition 1: Dirac delta function
  • Theorem 1
  • Lemma 1
  • Corollary 1