Module optimal_transport 

Optimal Transport Algorithms

This module provides implementations of optimal transport distances and solvers:

• Sliced Wasserstein Distance: O(n log n) via random 1D projections
• Sinkhorn Algorithm: Log-stabilized entropic regularization
• Gromov-Wasserstein: Cross-space structure comparison

Theory

Optimal transport measures the minimum “cost” to transform one probability distribution into another. The Wasserstein distance (Earth Mover’s Distance) is defined as:

W_p(μ, ν) = (inf_{γ ∈ Π(μ,ν)} ∫∫ c(x,y)^p dγ(x,y))^{1/p}

where Π(μ,ν) is the set of all couplings with marginals μ and ν.

Use Cases in Vector Search

• Cross-lingual document retrieval (comparing embedding distributions)
• Image region matching (comparing feature distributions)
• Time series pattern matching
• Document similarity via word embedding distributions

Structs

GromovWasserstein

Gromov-Wasserstein distance calculator

SinkhornSolver

Log-stabilized Sinkhorn solver for entropic optimal transport

SlicedWasserstein

Sliced Wasserstein distance calculator

TransportPlan

Result of Sinkhorn algorithm

WassersteinConfig

Configuration for Wasserstein distance computation

Traits

OptimalTransport

Trait for optimal transport distance computations