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Unscrambling disease progression at scale: fast inference of event permutations with optimal transport
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Proposes the variational event-based model (vEBM), which infers disease event permutations via Sinkhorn-based optimal transport over the Birkhoff polytope and optimises a variational ELBO, enabling pixel-level disease progression modelling.
Problem
Inferring latent event permutations in discrete disease progression models is combinatorially intractable, limiting models to a few hundred interpretable features.
Approach
Relaxes the latent permutation to a doubly stochastic matrix in the Birkhoff polytope, parameterised with Gumbel-Sinkhorn distributions, so the ELBO is differentiable and optimised with Adam; hard orderings are recovered with the Hungarian algorithm.
Key results
- vEBM is about 1000 times faster than baselines at J=200 features (I=2000).
- vEBM outperforms or matches baselines in Kendall's tau across noise levels and dataset sizes, except I=100, J=1000 at sigma=0.1 (per the text, but baselines are not run there).
- First pixel-level disease event sequences reported for AD (1344 events from TBM maps) and AMD (537 events from OCT).
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use vEBM instead of maximum-likelihood baselines for inference | Synthetic disease progression inference · Synthetic, J=200 | runtime: 1000 times faster | Helped |
| Set τ to 10.0 rather than 0.1 (ablation) | vEBM, τ_prior=1 · Synthetic, I=1000, J=100, σ=0.5 | Kendall's tau: 0.01 -> 0.81 | Helped |
| Set τ to 10.0 rather than 0.1 (ablation) | vEBM, τ_prior=1 · Synthetic, I=2000, J=200, σ=0.5 | Kendall's tau: 0.05 -> 0.54 | Helped |
| Set τ to 0.1 rather than 10.0 (ablation) | vEBM, τ_prior=0.1 · Synthetic, I=2000, J=200, σ=0.5 | Kendall's tau: 0.14 -> 0.5 | Helped |
| Set τ_prior to 10.0 rather than 0.1 (ablation) | vEBM, τ=0.1 · Synthetic, I=1000, J=100, σ=0.5 | Kendall's tau: 0.87 -> 0.1 | Hurt |
| Set τ_prior to 10.0 rather than 0.1 (ablation) | vEBM, τ=0.1 · Synthetic, I=2000, J=200, σ=0.5 | Kendall's tau: 0.5 -> 0.01 | Hurt |
| Set τ to 10.0 rather than 0.1 (ablation) | vEBM, τ_prior=10.0 · Synthetic, I=1000, J=100, σ=0.5 | Kendall's tau: 0.1 -> -0.09 | Hurt |
| Use 100 rather than 1 Sinkhorn-Knopp iteration (ablation) | vEBM, τ=τ_prior=1 · Synthetic, I=100, J=10, σ=0.5 | Kendall's tau: 0.07 -> 1.0 | Helped |
| Use 100 rather than 1 Sinkhorn-Knopp iteration (ablation) | vEBM, τ=τ_prior=1 · Synthetic, I=1000, J=100, σ=0.5 | Kendall's tau: 0.16 -> 0.79 | Helped |
| Use 100 rather than 1 Sinkhorn-Knopp iteration (ablation) | vEBM, τ=τ_prior=1 · Synthetic, I=2000, J=200, σ=0.5 | Kendall's tau: 0.07 -> 0.94 | Helped |
| Use vEBM rather than the baselines under increasing aleatoric noise | Synthetic disease progression inference · Synthetic datasets and noise settings | Kendall's tau | Mixed |
Surprising findings
- The vEBM on synthetic data does not always beat the EBM at low noise with few features, where MCMC can find the global minimum.
Limitations
- Model ordering thousands of events is not fully identifiable with only a few hundred snapshots.
- Memory bottleneck from dense matrix operations; model uncertainty not fully explored.
- Requires image registration preprocessing and does not model feature covariance.
Experimental setup
Datasets: synthetic data ADNI TBM MDT-SC maps ADNI Tadpole cognitive scores Duke University Ophthalmology 2013 OCT
Benchmarks: synthetic data with varying noise and model size
Metrics: Kendall's tau runtime fraction correct
Baselines: event-based model (EBM) ALPACA
Builds on: event-based model Gumbel-Sinkhorn Sinkhorn-Knopp ALPACA
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