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Is Score Matching Suitable for Estimating Point Processes?
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Shows existing score matching estimators for point processes (Poisson and Hawkes) are incomplete because integration-by-parts boundary terms do not vanish, and proposes (autoregressive) weighted score matching with weight functions vanishing at boundaries, with consistency and convergence-rate proofs.
Problem
Existing implicit score matching objectives for point processes (Poisson, Hawkes) omit boundary terms from integration by parts, so they fail to consistently estimate parameters for general point processes, while MLE needs costly intensity-integral computation.
Approach
Introduce weighted score matching (WSM) and autoregressive WSM (AWSM) for point processes, using a weight function that vanishes at the boundaries of the integration domain so boundary terms vanish; the near-optimal weight h^0 is the distance to the support boundary. Also proves consistency and a non-asymptotic error bound.
Key results
- On synthetic parametric models, (A)WSM achieves MAE comparable to MLE (e.g., Poisson phi 0.07 vs MLE -0.02) while implicit (A)SM has large errors (Poisson phi 1.56).
- On deep Hawkes models SAHP/THP, AWSM TLL is close to MLE on StackOverflow, Taobao and Retweets, and AWSM beats MLE on Half-Sin (SAHP 1.703 vs 1.542).
- ASM completely fails on deep point process models and its results are not reported.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Replace implicit (A)SM with (A)WSM | Poisson and Hawkes parameter estimation · Synthetic Poisson, exponential Hawkes, and Gaussian Hawkes | MAE | Helped |
| Train SAHP with AWSM instead of MLE | SAHP · Half-Sin | TLL: 1.542 -> 1.703 | Helped |
| Train THP with AWSM instead of MLE | THP · StackOverflow | TLL: -2.368 -> -2.508 | Hurt |
| Train SAHP with AWSM instead of MLE | SAHP · Taobao | ACC: 0.572 -> 0.455 | Hurt |
| Train SAHP with AWSM instead of MLE | SAHP · Retweet | ACC: 0.454 -> 0.411 | Hurt |
| Use near-optimal weight h⁰ instead of natural weight h¹ or square-root weight h² (ablation) | AWSM · Exponential Hawkes synthetic data | MAE | Helped |
| Use natural weight h¹ instead of near-optimal weight h⁰ (ablation) | AWSM · Exponential Hawkes synthetic data | MAE | Hurt |
| Use square-root weight h² instead of near-optimal weight h⁰ (ablation) | AWSM · Exponential Hawkes synthetic data | MAE | Hurt |
Surprising findings
- Implicit ASM objectives used in prior work fail even on simple Hawkes processes with exponential kernels because the boundary terms depend on theta.
- DSM with noise variance sigma>0 gives inconsistent estimates and performs much worse than MLE and AWSM on deep models.
Limitations
- Requires knowledge of the observation endpoint T for the weight function; real datasets with unknown T need approximations or truncation.
- Weight function choice is only near-optimal; the optimality of h^0 is not established for the error bound ratio.
- Experiments are limited to small synthetic models and a few real datasets; deep-model comparisons use a single 3-seed setup.
Experimental setup
Datasets: Poisson synthetic Exp-Hawkes synthetic Gaussian-Hawkes synthetic Half-Sin Hawkes synthetic StackOverflow Retweet Taobao
Benchmarks: StackOverflow Retweet Taobao Half-Sin Hawkes
Metrics: MAE test log-likelihood event type accuracy
Baselines: MLE (A)SM DSM
Builds on: score matching autoregressive score matching weighted score matching denoising score matching Hawkes process
Closest papers by meaning
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- Fast Bayesian Estimation of Point Process Intensity as Function of Covariates (NeurIPS 2022 · similarity 0.82)
- Beyond Point Prediction: Score Matching-based Pseudolikelihood Estimation of Neural Marked Spatio-Temporal Point Process (ICML 2024 · similarity 0.81)
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