Optimal Transport

Unlike the original Monge formulation requiring a deterministic one-to-one mapping, the Kantorovich relaxation allows probabilistic couplings where mass from a single source cell can be split across multiple target cells. This is critical for batch correction because cell populations often have different sizes across batches. The solution is a transport plan—a joint probability matrix—that specifies how much mass flows between each pair of cells, minimizing the total cost while respecting marginal constraints.

The Earth Mover's Distance (EMD), also known as the Wasserstein distance, quantifies the minimal cost to transform one distribution into another. In batch correction:

• Cost matrix: Pairwise distances between cells in gene expression space
• Interpretation: EMD captures both the magnitude and geometry of distributional shift
• Advantage over KL divergence: EMD respects the underlying metric space and remains well-defined even when distributions have non-overlapping supports

Exact optimal transport computation scales as O(n³ log n), making it prohibitive for large single-cell datasets. Entropic regularization, introduced by Cuturi (2013), adds an entropy penalty to the objective:

• Smooths the transport plan, making the problem strictly convex
• Enables the Sinkhorn algorithm, reducing complexity to O(n²)
• The regularization parameter λ controls the trade-off between precision and speed
• Small λ yields sparse, precise couplings; large λ produces diffuse, faster solutions

When batches are measured in different feature spaces—such as integrating scRNA-seq with scATAC-seq—standard optimal transport fails because cells don't share a common coordinate system. Gromov-Wasserstein (GW) optimal transport compares distributions using intra-domain distances:

• Aligns cells by preserving pairwise similarity structures within each modality
• Matches cells that have analogous relational profiles to their respective neighbors
• Enables translation between fundamentally different molecular measurement technologies

Standard optimal transport assumes exact mass preservation—every cell must be matched. Unbalanced optimal transport relaxes this constraint using divergence penalties on marginal violations:

• Accounts for cell types present in one batch but absent in another
• Uses Kullback-Leibler divergence or total variation penalties to allow mass creation/destruction
• Critical for realistic scenarios where biological conditions genuinely alter cell-type proportions, not just their distributions

Fused Gromov-Wasserstein (FGW) combines the strengths of both classical and GW optimal transport by optimizing a weighted sum of:

• Feature-level cost: Direct comparison of shared features (like Wasserstein)
• Structure-level cost: Comparison of intra-domain geometries (like GW)
• The trade-off parameter α balances feature fidelity against topological preservation
• Particularly effective when batches share some features but have distinct relational structures, such as integrating spatial transcriptomics with dissociated scRNA-seq