The Fisher information functional measures the “roughness” of a probability density ρ through the integrated squared score:
It quantifies how much information a sample carries about its location parameter and plays a central role connecting the Schrödinger bridge problem to optimal transport.
In Chen, Georgiou & Pavon (2014), the Fisher information provides the exact quantity separating the fluid dynamic formulations of the two problems. The Benamou-Brenier formula for OT minimizes ∫∫½||v||²ρ dtdx, while the SB fluid dynamic formulation minimizes ∫∫[½||v||² + ⅛||∇ln ρ||²]ρ dtdx — the additional ⅛I(ρ_t) term is the time-integrated Fisher information. This reveals that the Schrödinger bridge penalizes density distributions that are “rough” (have high spatial gradients), preferring smoother transport paths than classical OT.
The Fisher information also connects to Nelson’s osmotic velocity: the osmotic drift u = ½∇ln ρ has kinetic energy ½||u||² = ⅛||∇ln ρ||², which is exactly the Fisher information integrand. The SB problem thus minimizes the total energy of both current (transport) and osmotic (diffusion) velocities.
Key Details
- I(ρ) = ∫||∇ln ρ||²ρ dx = ∫||∇ρ||²/ρ dx
- Separates SB from OT: SB functional = OT functional + ⅛∫I(ρ_t)dt
- Related to osmotic velocity: ½||u||² = ⅛||∇ln ρ||²
- Cramér-Rao bound: Var(T) ≥ 1/I(θ)
- de Bruijn’s identity: ∂H(X_t)/∂t = -½I(X_t) for Brownian diffusion
- Fisher information → 0 corresponds to smooth densities (OT limit)