Research map / Learning theory and algorithms / Optimal transport
Variational Regularized Unbalanced Optimal Transport: Single Network, Least Action
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Proposes Var-RUOT, which derives first-order optimality conditions of the regularized unbalanced OT (WFR-type) action and learns only a single scalar field lambda with a neural network, trained with reconstruction, HJB and action losses, plus a growth penalty choice tied to biological priors.
Problem
Recovering continuous single-cell dynamics from a few unnormalized snapshots under the regularized unbalanced optimal transport (RUOT) framework, where existing solvers often fail to satisfy least-action optimality and converge unstably.
Approach
Use the Euler-Lagrange optimality conditions of the ITI RUOT action to parameterize velocity and growth directly from one scalar potential lambda (u = grad lambda, alpha psi'(g) = lambda), add an HJB residual loss and a particle-based action loss, and choose a concave-second-derivative growth penalty (g^{2/15}) to match the biological prior that growth decreases along trajectories.
Key results
- On the three-gene simulation, Var-RUOT reports path action 1.1105 vs 1.4058 for DeepRUOT and 1.2442 for TIGON, with best W1 at t=1-4 in most columns.
- Var-RUOT reaches OT-loss threshold 0.30 in 27.60 epochs on the simulation (vs 172-228 for RUOT/TIGON baselines) and 5.20 epochs on EMT.
- On mouse hematopoiesis, Var-RUOT (standard WFR) achieves path action 3.1491 vs 5.5887 for DeepRUOT, and the g^{2/15} modified metric yields decreasing growth along the flow.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Parameterize dynamics through one scalar field and enforce RUOT optimality conditions | RUOT dynamics · Three-gene simulation | Path Action: 1.4058 -> \textbf{1.1105} $\pm$0.0515 | Helped |
| Parameterize dynamics through one scalar field and enforce RUOT optimality conditions | RUOT dynamics · EMT | Path Action: 0.4857 -> \textbf{0.3544} $\pm$0.0019 | Helped |
| Use the modified growth penalty $\psi_2(g)=g^{2/15}$ instead of standard WFR | Var-RUOT · Mouse blood hematopoiesis | W2: 0.1459 $\pm$0.0038 -> 0.3117 $\pm$0.0323 | Hurt |
| Parameterize dynamics through one scalar field and enforce RUOT optimality conditions | RUOT training · Simulation Gene | Epochs to OT loss threshold: 172.00 $\pm$229.11 -> \textbf{27.60} $\pm$5.75 | Helped |
| Parameterize dynamics through one scalar field and enforce RUOT optimality conditions | RUOT training · EMT | Epochs to OT loss threshold: 110.40 $\pm$193.37 -> \textbf{5.20} $\pm$1.26 | Helped |
| Use the modified growth penalty $\psi_2(g)=g^{2/15}$ | Var-RUOT growth dynamics · Mouse blood hematopoiesis | Growth rate direction along velocity | Helped |
Surprising findings
- The modified growth penalty g^{2/15} gives worse distribution fit (W1 ~0.295 vs ~0.120 at t=1) than the standard WFR penalty, which the authors acknowledge as a limitation.
Limitations
- Neural optimization only reaches a local minimum; HJB loss acts as a regularizer rather than a hard constraint.
- Modified growth penalty degrades distribution-matching fit.
- Choice of psi(g) depends on biological priors and is not automated.
Experimental setup
Datasets: three-gene simulation dataset EMT dataset Mouse Blood Hematopoiesis dataset
Benchmarks: three-gene simulation EMT Mouse Blood Hematopoiesis
Metrics: W1 distance W2 distance path action epochs to OT loss threshold wall time
Baselines: SF2M PISDE MIO Flow Action Matching OTCFM UOTCFM WLF TIGON DeepRUOT
Builds on: DeepRUOT Action Matching WFR metric Schrödinger bridge
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