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Exact, Fast and Expressive Poisson Point Processes via Squared Neural Families
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Proposes squared neural Poisson point processes (SNEPPPs), parameterising the intensity as the squared norm of a two-layer network so the integrated intensity has a closed form, and shows convex MLE/MAP in the final layer under a fixed hidden layer.
Problem
Learning nonnegative intensity functions for Poisson point processes that are expressive, admit tractable (exact) integrated intensity, and allow tractable optimisation; kernel/GP intensity methods scale cubically in the number of events.
Approach
Model the intensity as alpha times the squared norm of a two-layer neural network with learnable hidden layer; the integral reduces to a trace against a neural network kernel (NNK), giving closed-form integrals and, with fixed hidden layer, a strictly convex NLL in M = V^T V solvable by projected gradient descent. Also enumerates new closed-form NNKs (exponential family type, homogeneous activatio
Key results
- On the synthetic R-shaped dataset, SNEPPP achieves the best NLL (MC) of 2.87 ± 0.07 and RMSE of 10.56 ± 0.74, outperforming naive kernel (RMSE 10.79) and RFM methods.
- On bei, SNEPPP reaches exact NLL -0.56 ± 0.01 versus -0.07 for log-linear mixture, and trains in 6.20 s versus 18.48 s for the naive kernel method.
- SNEPPP fits 100 million spatio-temporal wildfire events in about 8 hours of computation, whereas the naive method runs out of memory beyond 5.5% of the data.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use SNEPPP instead of the log-linear mixture | Synthetic PPP · Synthetic PPP | NLL (exact): 2.94 ± 0.06 -> 2.91 ± 0.07 | Helped |
| Use SNEPPP instead of Nyström | Synthetic PPP · Synthetic PPP | NLL (MC): 2.86 ± 0.06 -> 2.87 ± 0.07 | Mixed |
| Use SNEPPP instead of Nyström | Synthetic PPP · Synthetic PPP | RMSE: 10.78 ± 0.45 -> 10.56 ± 0.74 | Helped |
| Use SNEPPP instead of Nyström | Synthetic PPP · Synthetic PPP | Time (seconds): 4.19 ± 0.24 -> 3.09 ± 0.09 | Helped |
| Use SNEPPP instead of the strongest listed baseline | Bei · Bei | NLL (exact): -0.07 ± 0.02 -> -0.56 ± 0.01 | Helped |
| Use SNEPPP instead of Nyström | Bei · Bei | NLL (MC): -0.52 ± 0.02 -> -0.62 ± 0.01 | Helped |
| Use SNEPPP instead of the log-linear mixture | Bei · Bei | Count percent error: 0.02 ± 0.02 -> 0.04 ± 0.03 | Hurt |
| Use SNEPPP instead of Nyström | Bei · Bei | Time (seconds): 4.74 ± 0.33 -> 6.20 ± 0.18 | Hurt |
| Use SNEPPP instead of the log-linear mixture | Clmfires · Clmfires | NLL (exact): -0.08 ± 0.01 -> -0.78 ± 0.03 | Helped |
| Use SNEPPP instead of Nyström | Clmfires · Clmfires | NLL (MC): -0.35 ± 0.03 -> -0.84 ± 0.03 | Helped |
| Use SNEPPP instead of RFM | Clmfires · Clmfires | Time (seconds): 4.17 ± 0.31 -> 7.07 ± 0.17 | Hurt |
| Use SNEPPP instead of the log-linear mixture | Copper · Copper | NLL (exact): 0.19 ± 0.02 -> 0.85 ± 0.90 | Hurt |
Surprising findings
- On the small copper dataset (N=67), SNEPPP performs worse than simple log-linear models with NLL 0.85 ± 0.90 versus 0.21 for log-linear, which the authors attribute to dataset size.
- Exact and Monte Carlo NLL estimates strongly disagree for the log-linear models on copper.
Limitations
- Theory of generalisation error for PPPs is not provided; the authors leave it for future work.
- Convexity holds only when the hidden layer is fixed; learning the hidden layer is non-convex.
- Results on the small copper dataset show SNEPPP underperforms simple baselines.
Experimental setup
Datasets: bei clmfires copper synthetic R-shaped intensity NASA wildfire data
Benchmarks: bei clmfires copper synthetic R-shaped intensity
Metrics: test NLL (exact) test NLL (Monte Carlo) RMSE count percent error training time (seconds)
Baselines: Log-linear Log-linear mixture RFM Nystrom Naive kernel method
Builds on: squared neural families neural network kernels augmented permanental point processes random Fourier features
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