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A Biconvex Formulation for Stable Transport of Mixture Models with a Unique Solution
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Proposes Optimal Mixture Transport (OMT), an entropic OT solver between mixtures of exponential-family components that is strictly biconvex with a unique single-iteration global optimum and carries stability guarantees.
Problem
Scaling optimal transport to large datasets while keeping transport plans interpretable and stable under perturbations of the marginals; existing mini-batch, low-rank, and neural OT methods lack uniqueness or stability guarantees.
Approach
Transport between mixture models rather than individual samples: the coupling is a mixture of component-wise entropic OT plans, with mixing-matrix and component-plan entropies making the objective strictly biconvex, so a GOP-style alternating scheme reaches a unique global optimum in one step. Exponential-family components (p,q in {1,2}) yield closed-form component maps, so cost scales with the nu
Key results
- OMT shows stronger stability than amortized W2-OT, ExNOT, ENOT, P-OT and GMM-OT under additive Gaussian and dropout noise on the W2-Benchmark across d in {2,16,64,128,256}.
- On large MERFISH slices (85,958 and 84,172 cells), OMT outperforms low-rank OT and GMM-OT on cosine similarity and transport cost across 5 marker genes.
- OMT consistently outperforms non-neural baselines across sci-Plex treatment conditions (reported in tables).
Surprising findings
- Biconvex objective with entropic regularization gives a unique global minimizer reached in a single GOP iteration, contrary to the general non-uniqueness of biconvex problems.
Limitations
- Less suited for high-resolution image generation, per the authors.
- Limited to balanced OT; unbalanced extension left for future work.
- Static transport plans only; dynamic transport maps not addressed.
Experimental setup
Datasets: W2-Benchmark sci-Plex mouse brain scRNA-seq development (E11.5-P28) mouse brain aging scRNA-seq Vizgen MERFISH Mouse Brain Receptor Map ImageNet MNIST CIFAR-10
Benchmarks: W2-Benchmark sci-Plex Vizgen MERFISH Mouse Brain Receptor Map ImageNet alignment MNIST CIFAR-10
Metrics: relative MSE degradation under noise cosine similarity of imputed gene expression transport cost
Baselines: amortized W2-OT ExNOT ENOT P-OT GMM-OT HiRef low-rank OT (LOT)
Builds on: entropic optimal transport Sinkhorn algorithm Global Optimization Algorithm (GOP) GMM-OT / aggregated Wasserstein distance Progressive Optimal Transport (P-OT)
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