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Unlocking Point Processes through Point Set Diffusion
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Introduces Point Set Diffusion, a diffusion-based latent variable model that generates arbitrary point processes on metric spaces by stochastically interpolating data and noise point sets via thinning and superposition, without an intensity function.
Problem
Modeling and sampling general point processes on metric spaces is hard because existing models rely on (conditional) intensity functions, which trade off efficiency against flexibility and are intractable to sample from for complex interactions.
Approach
Define a forward Markov chain that superposes thinned data points with thinned noise points (independent thinning and superposition properties), learn the reverse posterior by predicting retained points with a binary classifier and thinned points with a Gaussian mixture, enabling parallel point-set sampling and conditional generation via binary masks.
Key results
- On STPP density estimation, Point Set Diffusion attains MMD 0.023 on Earthquakes versus 0.064 for the next best DiffSTPP, and best results on all four datasets.
- On SPP conditional imputation, Point Set Diffusion reduces MAE on Earthquakes from 30.419 (Regularized Method) to 4.651.
- Sampling runtime is near constant in the number of points since all points are generated in parallel, while autoregressive baselines scale at least linearly, giving up to orders-of-magnitude speedups.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| use Point Set Diffusion instead of the strongest reported baseline | SPP density estimation · Earthquakes | MMD: 0.214 -> 0.173 | Helped |
| use Point Set Diffusion instead of the strongest reported baseline | SPP density estimation · Covid NJ | MMD: 0.340 -> 0.268 | Helped |
| use Point Set Diffusion instead of the strongest reported baseline | SPP density estimation · Citybike | MMD: 0.336 -> 0.092 | Helped |
| use Point Set Diffusion instead of the strongest reported baseline | SPP density estimation · Pinwheel | MMD: 0.285 -> 0.099 | Helped |
| use Point Set Diffusion instead of the Regularized Method | SPP conditional generation · Earthquakes | WD: 0.162 -> 0.106 | Helped |
| use Point Set Diffusion instead of the Regularized Method | SPP conditional generation · Covid NJ | WD: 0.148 -> 0.119 | Helped |
| use Point Set Diffusion instead of the Regularized Method | SPP conditional generation · Citybike | WD: 0.115 -> 0.085 | Helped |
| use Point Set Diffusion instead of the Regularized Method | SPP conditional generation · Pinwheel | WD: 0.150 -> 0.122 | Helped |
| use Point Set Diffusion instead of the strongest reported baseline | STPP density estimation · Earthquakes | MMD: 0.064 -> 0.023 | Helped |
| use Point Set Diffusion instead of the strongest reported baseline | STPP density estimation · Covid NJ | MMD: 0.146 -> 0.043 | Helped |
| use Point Set Diffusion instead of the strongest reported baseline | STPP density estimation · Citybike | MMD: 0.327 -> 0.020 | Helped |
| use Point Set Diffusion instead of the strongest reported baseline | STPP density estimation · Pinwheel | MMD: 0.055 -> 0.020 | Helped |
Surprising findings
- The autoregressive STPP baseline DeepSTPP outperforms Point Set Diffusion on Covid NJ forecasting in CD/MAE terms.
- Standard NLL evaluation is argued to be insufficient for judging the generative capacity of point process models, since autoregressive baselines struggle to sample realistic unconditional point sets.
Limitations
- Evaluation limited to Euclidean SPPs and STPPs with bounded domains; generality to high-dimensional or non-Euclidean spaces untested.
- Standard NLL-based comparison is replaced by generative metrics, and AutoSTPP was excluded from forecasting due to slow sampling.
Experimental setup
Datasets: Japan Earthquakes New Jersey COVID-19 cases Citibike pickups Pinwheel (multivariate Hawkes synthetic)
Benchmarks: Japan Earthquakes Covid NJ Citybike Pinwheel
Metrics: Sequence Length Wasserstein distance MMD Wasserstein distance Counting Distance MAE
Baselines: Log-Gaussian Cox Process Regularized Method DeepSTPP DiffSTPP AutoSTPP
Builds on: Add-Thin denoising diffusion probabilistic models Log-Gaussian Cox process
Closest papers by meaning
- Learning Registered Point Processes from Idiosyncratic Observations (ICML 2018 · similarity 0.88)
- Neural Jump-Diffusion Temporal Point Processes (ICML 2024 · similarity 0.87)
- Interacting Diffusion Processes for Event Sequence Forecasting (ICML 2024 · similarity 0.86)
- PointInfinity: Resolution-Invariant Point Diffusion Models (CVPR 2024 · similarity 0.85)
- Learning Spatiotemporal Dynamical Systems from Point Process Observations (ICLR 2025 · similarity 0.85)
- Latent variable model for high-dimensional point process with structured missingness (ICML 2024 · similarity 0.85)
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