# Optimal Transport

On 5,357 structurally resolved small molecules from DrugBank 5.1.10, Schrödinger Bridge Onsager Flow drives the transport from a uniform chemical library toward a drug-likeness distribution to a total variation distance of 0.0141, against 0.1753 for a Sinkhorn baseline at its best setting across a ten-value regularization sweep. The method keeps one scalar potential per molecule updated by mirror descent instead of an all-pairs coupling matrix; its neural amortization, SBOFNet, returns a total variation distance of 0.0380 in a single forward pass with 1.6M parameters, at lower accuracy than the analytic solver.

On 5,357 structurally resolved small molecules from DrugBank 5.1.10, Schrödinger Bridge Onsager Flow drives the transport from a uniform chemical library toward a drug-likeness distribution to a total variation distance of 0.0141, against 0.1753 for a Sinkhorn baseline at its best setting across a ten-value regularization sweep. The method keeps one scalar potential per molecule updated by mirror descent instead of an all-pairs coupling matrix; its neural amortization, SBOFNet, returns a total variation distance of 0.0380 in a single forward pass with 1.6M parameters, at lower accuracy than the analytic solver.

Spatial transcriptomics sequencing is destructive, so each time point yields unaligned cell snapshots and the intermediate trajectories and cross-scale intercellular interactions stay unobserved. stMIND decomposes each cell's expression drift into a cell-intrinsic program, a local graph-difference operator over relative expression differences and spatial geometry within a radius, and a mass-weighted nonlocal mean-field attention operator outside it, with neighborhoods co-evolving as cells migrate. On axolotl telencephalon regeneration and zebrafish embryogenesis benchmarks it reaches the highest gene-expression accuracy, anatomical consistency and spatial coherence; ablations show both operators are needed, and the learned fields locate a regenerative transition at 10–15 days post-injury.

Spatial transcriptomics sequencing is destructive, so each time point yields unaligned cell snapshots and the intermediate trajectories and cross-scale intercellular interactions stay unobserved. stMIND decomposes each cell's expression drift into a cell-intrinsic program, a local graph-difference operator over relative expression differences and spatial geometry within a radius, and a mass-weighted nonlocal mean-field attention operator outside it, with neighborhoods co-evolving as cells migrate. On axolotl telencephalon regeneration and zebrafish embryogenesis benchmarks it reaches the highest gene-expression accuracy, anatomical consistency and spatial coherence; ablations show both operators are needed, and the learned fields locate a regenerative transition at 10–15 days post-injury.

Similarity between multimodal large language model representations and human visual cortex shows that the two are alike but not which semantics they share. SAIL recasts voxel interpretation as a correspondence between each voxel and multiple sparse semantic units: it trains a unified cross-layer sparse autoencoder on several model layers, matches each voxel to several units, and compares the semantic similarity of their preferred image sets with optimal transport. Across eight subjects in the Natural Scenes Dataset, joint cross-layer training reduced unit response redundancy and raised semantic similarity over layer-wise training, in both whole cortex and selective regions of interest.

Similarity between multimodal large language model representations and human visual cortex shows that the two are alike but not which semantics they share. SAIL recasts voxel interpretation as a correspondence between each voxel and multiple sparse semantic units: it trains a unified cross-layer sparse autoencoder on several model layers, matches each voxel to several units, and compares the semantic similarity of their preferred image sets with optimal transport. Across eight subjects in the Natural Scenes Dataset, joint cross-layer training reduced unit response redundancy and raised semantic similarity over layer-wise training, in both whole cortex and selective regions of interest.

Single-cell perturbation experiments are costly and low-throughput, yet computational prediction has been hampered by inconsistent evaluation and by models that do not reliably beat simple baselines. DCAF combines data pairing, an evaluation criterion and a prediction model in one framework: entropy-regularized optimal transport recovers a pre-perturbation state for each perturbed cell, log fold changes of non-silenced genes drive scoring, and a JEPA structure encodes control and perturbed cells jointly. Across cross-gene, cross-cell-type and cross-region settings, DCAF-model beats GEARS, scGPT, linear and mean baselines on R² and MSE of log fold change for non-silenced genes.

Single-cell perturbation experiments are costly and low-throughput, yet computational prediction has been hampered by inconsistent evaluation and by models that do not reliably beat simple baselines. DCAF combines data pairing, an evaluation criterion and a prediction model in one framework: entropy-regularized optimal transport recovers a pre-perturbation state for each perturbed cell, log fold changes of non-silenced genes drive scoring, and a JEPA structure encodes control and perturbed cells jointly. Across cross-gene, cross-cell-type and cross-region settings, DCAF-model beats GEARS, scGPT, linear and mean baselines on R² and MSE of log fold change for non-silenced genes.

BLOOM frames transcriptomic virtual-embryo modeling as conditional generation: given an observed window of cell populations, a requested developmental stage and cell-state information, it generates the expression distribution at that stage. Pseudo-trajectories link independently sampled populations, while developmental age, cell-identity hierarchy, anatomical context, gene programmes and transcription-factor–target networks condition both a Gaussian base and a flow-matching generator. Across prenatal, visual-cortex and multi-organ endothelial mouse single-cell atlases, BLOOM ranks first for stage-averaged gene and cell correlations against 22 time-series forecasting baselines.

BLOOM frames transcriptomic virtual-embryo modeling as conditional generation: given an observed window of cell populations, a requested developmental stage and cell-state information, it generates the expression distribution at that stage. Pseudo-trajectories link independently sampled populations, while developmental age, cell-identity hierarchy, anatomical context, gene programmes and transcription-factor–target networks condition both a Gaussian base and a flow-matching generator. Across prenatal, visual-cortex and multi-organ endothelial mouse single-cell atlases, BLOOM ranks first for stage-averaged gene and cell correlations against 22 time-series forecasting baselines.

Comparing two sparse autoencoders (SAEs, dictionaries that decompose model activations into sparse learned components) requires answering three things at once: how much two whole dictionaries differ, which latents account for that difference, and how much is sampling noise from finite documents. One energy-weighted optimal-transport plan connects them: its cost is the dictionary-level discrepancy, the cost decomposes exactly into latent contributions, and transported fractions describe how each latent's activation energy is allocated across counterparts, with joint asymptotic confidence intervals. Across four language models it locates differences between SAEs trained on identical data with different seeds and separates topic-dependent changes into a weight term and a cost-matrix term.

Comparing two sparse autoencoders (SAEs, dictionaries that decompose model activations into sparse learned components) requires answering three things at once: how much two whole dictionaries differ, which latents account for that difference, and how much is sampling noise from finite documents. One energy-weighted optimal-transport plan connects them: its cost is the dictionary-level discrepancy, the cost decomposes exactly into latent contributions, and transported fractions describe how each latent's activation energy is allocated across counterparts, with joint asymptotic confidence intervals. Across four language models it locates differences between SAEs trained on identical data with different seeds and separates topic-dependent changes into a weight term and a cost-matrix term.

The Sinkhorn divergence of entropic optimal transport is fast and statistically well behaved, but it is not a metric and has no Riemannian structure; the earlier tangent space and metric tensor came without covariant derivative, geodesic or parallel transport equations, and inference on them was open. Under finite-smoothness costs the metric tensor is shown Hadamard differentiable along curves of measures, which yields the Levi-Civita covariant derivative and turns geodesics and parallel transport into ODEs with smooth coefficients on a fixed RKHS. Linearizing those solutions with Banach-space ODE theory gives √n Gaussian limits for plug-in estimators of the exponential map, logarithmic map, parallel transport and squared Sinkhorn distance, with a consistent nonparametric bootstrap.

The Sinkhorn divergence of entropic optimal transport is fast and statistically well behaved, but it is not a metric and has no Riemannian structure; the earlier tangent space and metric tensor came without covariant derivative, geodesic or parallel transport equations, and inference on them was open. Under finite-smoothness costs the metric tensor is shown Hadamard differentiable along curves of measures, which yields the Levi-Civita covariant derivative and turns geodesics and parallel transport into ODEs with smooth coefficients on a fixed RKHS. Linearizing those solutions with Banach-space ODE theory gives √n Gaussian limits for plug-in estimators of the exponential map, logarithmic map, parallel transport and squared Sinkhorn distance, with a consistent nonparametric bootstrap.

Metabolomics models trained on one cohort often lose accuracy on another because platform, sample handling and quantification differences shift the data. Using 12,000 plasma NMR samples from the Estonian Biobank to classify heart failure (ICD-10 I50.9), a study evaluated 295 samples re-measured on a different NMR platform and calibrated alignment with up to 108 biospecimens duplicated across both. Harmonization generally beat unaligned data and stabilized as matched pairs grew, yet simple within-domain normalization stayed a strong baseline, reaching balanced accuracy 0.72 and MCC 0.46 with a support vector machine at all 108 pairs.

Metabolomics models trained on one cohort often lose accuracy on another because platform, sample handling and quantification differences shift the data. Using 12,000 plasma NMR samples from the Estonian Biobank to classify heart failure (ICD-10 I50.9), a study evaluated 295 samples re-measured on a different NMR platform and calibrated alignment with up to 108 biospecimens duplicated across both. Harmonization generally beat unaligned data and stabilized as matched pairs grew, yet simple within-domain normalization stayed a strong baseline, reaching balanced accuracy 0.72 and MCC 0.46 with a support vector machine at all 108 pairs.

Embedding probability measures into a Hilbert space before learning a sparse dictionary turns training from repeated Wasserstein distance solves into one autoencoder optimization, cutting reconstruction loss by roughly two to three orders of magnitude against Wasserstein dictionary learning under the same time budget.

Embedding probability measures into a Hilbert space before learning a sparse dictionary turns training from repeated Wasserstein distance solves into one autoencoder optimization, cutting reconstruction loss by roughly two to three orders of magnitude against Wasserstein dictionary learning under the same time budget.

Segmenting cells in 3D fluorescence microscopy has meant choosing between two paths: deep learning is efficient but returns discrete voxel masks, while active surfaces give continuous shape but struggle with multiple objects. Nagini3D joins them, using a 3D U-Net to predict both object locations and spline surface parameters, so each object is one closed smooth parametric surface. On the synthetic CAPS set of non-star-convex objects, the refined Nagini3D+ reaches a mean IoU of 0.859 against StarDist3D's 0.779; on the real BONES set the two are close, but Nagini3D is the fastest tested, at 4.75 seconds on average.

Segmenting cells in 3D fluorescence microscopy has meant choosing between two paths: deep learning is efficient but returns discrete voxel masks, while active surfaces give continuous shape but struggle with multiple objects. Nagini3D joins them, using a 3D U-Net to predict both object locations and spline surface parameters, so each object is one closed smooth parametric surface. On the synthetic CAPS set of non-star-convex objects, the refined Nagini3D+ reaches a mean IoU of 0.859 against StarDist3D's 0.779; on the real BONES set the two are close, but Nagini3D is the fastest tested, at 4.75 seconds on average.