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Enforcing Latent Euclidean Geometry in Single-Cell VAEs for Manifold Interpolation
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Proposes FlatVI, a flattening regulariser on the pullback Fisher-information metric of negative-binomial single-cell VAEs that makes straight latent lines approximate geodesics, improving OT-based trajectory reconstruction and latent interpolation.
Problem
Latent linear interpolations in VAEs trained on single-cell count data (scRNA-seq) do not generally correspond to geodesics on the decoded statistical manifold, which undermines methods that assume Euclidean latent geometry such as OT-based trajectory inference.
Approach
Add a flattening loss that pushes the pullback Fisher-information metric of a negative-binomial decoder toward a scaled identity, so latent straight lines approximate decoded geodesics, while keeping the ELBO for reconstruction.
Key results
- On held-out time points, FlatVI reduces latent 2-Wasserstein on EB from 2.07 (NB-VAE) to 1.54 and on MEF from 2.07 to 1.64.
- On the Pancreas dataset, FlatVI (lambda=1) recovers all six terminal states identified by CellRank, versus four for GAE and five for NB-VAE.
- In simulations, lambda=10 gives the highest 3-NN overlap (0.80) between Euclidean and pullback-geodesic neighbourhoods, versus 0.66 for the unregularised NB-VAE, while mean reconstruction MSE stays roughly unchanged.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| add flattening regularization (λ=1) | NB-VAE with OT-CFM · EB held-out time points | 2-Wasserstein: 2.07 -> 1.54 | Helped |
| add flattening regularization (λ=1) | NB-VAE with OT-CFM · EB held-out time points | MMD: 0.09 -> 0.07 | Helped |
| add flattening regularization (λ=0.1) | NB-VAE with OT-CFM · MEF held-out time points | 2-Wasserstein: 2.07 -> 1.64 | Helped |
| add flattening regularization (λ=0.1) | NB-VAE with OT-CFM · MEF held-out time points | MMD: 0.19 -> 0.16 | Helped |
| set flattening strength to λ=10 instead of λ=0 (ablation) | NB-VAE · Synthetic negative binomial | 3-NN overlap: 0.66 -> 0.80 | Helped |
| set flattening strength to λ=1 instead of λ=0 (ablation) | NB-VAE · Synthetic negative binomial | 3-NN overlap: 0.66 -> 0.63 | Hurt |
| set flattening strength to λ=3 instead of λ=0 (ablation) | NB-VAE · Synthetic negative binomial | 3-NN overlap: 0.66 -> 0.77 | Helped |
| set flattening strength to λ=5 instead of λ=0 (ablation) | NB-VAE · Synthetic negative binomial | 3-NN overlap: 0.66 -> 0.67 | No clear effect |
| set flattening strength to λ=7 instead of λ=0 (ablation) | NB-VAE · Synthetic negative binomial | 3-NN overlap: 0.66 -> 0.72 | Helped |
| increase flattening strength across tested λ values (ablation) | NB-VAE · Synthetic negative binomial | MSE (μ): 15.52; 16.34; 16.35; 14.75; 15.47; 15.41 | Mixed |
| increase flattening strength across tested λ values (ablation) | NB-VAE · Synthetic negative binomial | MSE (θ): 3.10; 5.67; 3.09; 3.20; 3.38; 3.08 | Mixed |
| use FlatVI instead of NB-VAE | OT-CFM representations · Pancreas | identified terminal states | Helped |
Surprising findings
- Regularisation strength has non-monotonic effects; for example lambda=5 gives the lowest mean MSE while lambda=3 yields a higher MSE than lambda=0 in some settings.
Limitations
- Reconstruction likelihood can trade off against flattening on real biological data.
- Locally Euclidean latent geometry is a strong assumption that may fail for cyclic processes such as the cell cycle.
- Assumes geodesic convexity and that local flattening at sampled points yields global geometry.
Experimental setup
Datasets: Embryoid Body (EB) scRNA-seq MEF reprogramming scRNA-seq Pancreatic endocrinogenesis scRNA-seq synthetic negative binomial data
Benchmarks: EB MEF Pancreas
Metrics: MSE 3-NN overlap VoR condition number 2-Wasserstein L2 distance MMD number of terminal macrostates velocity consistency
Baselines: NB-VAE GAE GAGA CellRank
Builds on: VAE OT-CFM Geodesic Autoencoder scVI GAGA
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