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HollowFlow: Efficient Sample Likelihood Evaluation using Hollow Message Passing
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Introduces HollowFlow, a CNF-based Boltzmann generator using a non-backtracking GNN (Hollow Message Passing) whose Jacobian is block-hollow/block-diagonal, so divergence needs only d backward passes instead of nd.
Problem
Likelihood evaluation in continuous normalizing flows and Boltzmann generators requires O(N) backward passes for the divergence, which scales prohibitively with system size, limiting use on large molecular/particle systems.
Approach
Construct the conditioner as a non-backtracking GNN on the line graph (with pruning of backtracking edges per message passing step) so that each atom's output does not depend on its own coordinates, giving a block-diagonal Jacobian with d-dimensional blocks; any equivariant GNN or attention can be adapted this way.
Key results
- On LJ55 with k=7 the HollowFlow model reaches EffSU_rem of about 93.7 relative to the fully connected baseline (roughly two orders of magnitude).
- On LJ13 the best HollowFlow (k=6) reaches EffSU_rem of about 3.26, with runtime reduced by up to ~15x in the forward/backward breakdown.
- Raw ESS is lower for HollowFlow than for the baseline on both LJ13 and LJ55, reflecting the restricted architecture.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use HollowFlow with k=6 neighbors rather than the fully connected baseline | PaiNN CNF · LJ13 | EffSU_rem: 1 -> 3.260_{3.243}^{3.278} | Helped |
| Use HollowFlow with k=7 neighbors rather than the fully connected baseline | PaiNN CNF · LJ55 | EffSU_rem: 1 -> 93.737_{88.484}^{99.071} | Helped |
| Use HollowFlow rather than the fully connected baseline | PaiNN CNF · LJ13 | ESS_rem (%): 40.73_{40.60}^{40.87} -> 20.20_{20.11}^{20.29} | Hurt |
| Use HollowFlow rather than the fully connected baseline | PaiNN CNF · LJ55 | ESS_rem (%): 2.96_{2.89}^{3.03} -> 0.53_{0.51}^{0.56} | Hurt |
| Change k from 6 to 2 neighbors (ablation) | HollowFlow · LJ13 | EffSU_rem: 3.260_{3.243}^{3.278} -> 1.059_{1.049}^{1.070} | Hurt |
| Change k from 6 to 4 neighbors (ablation) | HollowFlow · LJ13 | EffSU_rem: 3.260_{3.243}^{3.278} -> 2.649_{2.631}^{2.667} | Hurt |
| Change k from 6 to 8 neighbors (ablation) | HollowFlow · LJ13 | EffSU_rem: 3.260_{3.243}^{3.278} -> 2.310_{2.297}^{2.323} | Hurt |
| Change k from 6 to 10 neighbors (ablation) | HollowFlow · LJ13 | EffSU_rem: 3.260_{3.243}^{3.278} -> 1.484_{1.475}^{1.492} | Hurt |
| Change k from 6 to 12 neighbors (ablation) | HollowFlow · LJ13 | EffSU_rem: 3.260_{3.243}^{3.278} -> 1.627_{1.619}^{1.636} | Hurt |
| Change k from 7 to 27 neighbors (ablation) | HollowFlow · LJ55 | EffSU_rem: 93.737_{88.484}^{99.071} -> 9.466_{8.965}^{9.986} | Hurt |
| Change k from 7 to 55 neighbors (ablation) | HollowFlow · LJ55 | EffSU_rem: 93.737_{88.484}^{99.071} -> 4.365_{4.144}^{4.583} | Hurt |
| Use multi-head graph strategy (H=2) instead of single-head kNN (ablation) | HollowFlow · LJ13 | EffSU_rem: 3.260_{3.243}^{3.278} -> 1.613_{1.604}^{1.622} | Hurt |
Surprising findings
- Raw ESS of HollowFlow models is lower than the baseline, but compute-normalized efficiency is far higher.
- On alanine dipeptide, no effective speed-up over the non-hollow baseline was observed despite good ESS_rem for some multi-head variants.
Limitations
- kNN graph imposes a locality assumption that may fail for long-range interactions such as electrostatics.
- Edge removal in the line graph after each message passing step may be problematic at scale.
- Baseline ESS is not state-of-the-art, so absolute performance is limited.
Experimental setup
Datasets: LJ13 Boltzmann samples LJ55 Boltzmann samples Alanine dipeptide (ALA2)
Benchmarks: LJ13 LJ55 ALA2
Metrics: ESS ESS_rem relative ESS effective speed-up (EffSU) runtime
Baselines: fully connected PaiNN CNF baseline
Builds on: HollowNet PaiNN conditional flow matching minibatch optimal transport Boltzmann Generators
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