Variational optimality of generalized Tweedie via Schrödinger bridge control

Connection

The stochastic control formulation of the Schrödinger bridge problem (Chen, Georgiou & Pavon 2014) proves that the optimal drift perturbation to a prior diffusion is $u^{\ast }(x,t)=\nabla \ln \phi (x,t)$, where $\phi$ is a positive space-time harmonic function solving ${\partial }_t\phi +b_{+}^P\cdot \nabla \phi +\frac{1}{2}\Delta \phi =0$. This correction is optimal in the sense of minimizing the KL divergence $H(Q,P)$ on path space, which Girsanov’s theorem converts to the expected integrated kinetic energy $E_Q[{\int }_0^1\frac{1}{2}\Vert {\beta }_{+}^Q-{\beta }_{+}^P{\Vert }^{2}dt]$.

The generalized Tweedie’s formula derived in Appendix C of Shelley & Mengütürk (2025) computes the same score correction algebraically. Proposition C.1 gives $E[\varphi (\eta )|Z=z]=({\nabla }_z\log f(z)-E[{\nabla }_z\log h(z,\eta )|Z=z])c^{-1}$ for the generalised exponential family, which in the Gaussian case (Remark C.3) reduces to the classical Tweedie formula $E[\mu |Z=z]=z+{\sigma }^2{\nabla }_z\log f(z)$. The term ${\sigma }^2\nabla \log f(z)$ is exactly the score correction $\nabla \ln \phi$ that Chen et al. derive variationally.

The deep insight is that these are two views of the same object: Chen et al. provide the variational justification (this score correction is the unique minimizer of KL divergence on path space), while the generalised Tweedie formula provides the algebraic computation (it gives a closed-form expression for the score correction for any generalised exponential family conditional). The Tweedie formula thus generalizes the stochastic control result beyond Brownian diffusions to the full generalised exponential family, including Poisson random bridges, where $h(z,\eta )$ is no longer independent of $\eta$ and the correction acquires an additional posterior-dependent term $E[{\nabla }_z\log h(z,\eta )|Z=z]$.

Bridged Concepts

From Chen et al. (2014)

• Schrödinger bridge: Optimal drift = prior drift + $\nabla \ln \phi$ (eq 29), derived by minimizing $H(Q,P)$
• Fisher information: The osmotic kinetic energy $\frac{1}{8}\Vert \nabla \ln \rho {\Vert }^2$ separates SB from OT
• optimal transport with prior: The zero-noise limit recovers OT with prior velocity, where optimal correction is $\nabla \psi$ (eq 64)
• Nelson’s osmotic velocity: Time-symmetric formulation decomposes KL into current + osmotic kinetic energies

From Shelley & Mengütürk (2025)

• generalized Tweedie’s formula: Proposition C.1 computes posterior natural parameter for generalised exponential family
• Brownian random bridge: Score correction ${\sigma }^2\nabla \log p_t$ is the Gaussian Tweedie correction
• Poisson random bridge: Intensity ${\lambda }_{tT}=(E[Y|{\xi }_t]-{\xi }_t)/(T-t)$ is the Poisson analogue of the score correction, requiring the generalised (not classical) Tweedie formula

Why It Matters

This connection establishes that the bridge drift corrections used throughout the generative modelling literature are not just heuristically motivated but are provably optimal perturbations in the KL sense. For Brownian bridges, this was implicit in the classical theory; but for Poisson bridges and other non-Gaussian drivers, the generalised Tweedie formula in Appendix C is the only available tool for computing these corrections, and Chen et al.’s framework confirms that they inherit the same variational optimality.

This also suggests a path toward a stochastic control formulation for Lévy bridges: extending Chen et al.’s Girsanov-based argument to jump-diffusion reference measures would provide the variational justification for the generalised Tweedie corrections in the Poisson case, and potentially yield new bridge constructions for other Lévy drivers.

Potential Directions

1. Stochastic control for jump-diffusion bridges: Extend Chen et al.’s KL minimization to reference processes with jumps (compound Poisson, stable processes). The relative entropy decomposition via Girsanov for jump processes should yield an analogue of $u^{\ast }=\nabla \ln \phi$, with $\phi$ solving an integro-differential (not just PDE) harmonic equation.
2. Optimal transport with Lévy prior: Formulate the zero-noise limit of Lévy-driven Schrödinger bridges as a generalised OT problem with prior. The cost function $c(x,y)=\inf {\int }_0^{1}L(t,x,\dot{x})dt$ would involve the Lévy-Khintchine Lagrangian rather than the quadratic one.
3. Finite-sample optimality: Chen et al.’s result is population-level. Connect to Efron’s empirical Bayes perspective (Tweedie’s Formula and Selection Bias) to obtain finite-sample guarantees: the learned score network approximates $\nabla \ln \phi$, and empirical Bayes information provides the rate of convergence.

Evidence

• Chen et al. (2014), eq 29: optimal SB drift is $b_{+}^{Q\ast }=b_{+}^P+\nabla \ln \phi$ where $\phi$ is space-time harmonic (On the Relation Between Optimal Transport and Schrödinger Bridges)
• Chen et al. (2014), eq 37: the KL functional $I(Q)=E_Q[{\int }_0^1\frac{1}{2}\Vert {\beta }_{+}^Q-b_{+}^P-\nabla \ln \phi {\Vert }^{2}dt]$, showing $\nabla \ln \phi$ is the unique minimizer (On the Relation Between Optimal Transport and Schrödinger Bridges)
• Shelley & Mengütürk (2025), Proposition C.1: $E[\varphi (\eta )|Z=z]=({\nabla }_z\log f(z)-E[{\nabla }_z\log h(z,\eta )|Z=z])c^{-1}$ (Connecting Brownian and Poisson Random Bridges with Rectified Flows)
• Shelley & Mengütürk (2025), Remark C.3: Gaussian recovery gives $E[\mu |Z=z]=z+{\sigma }^2{\nabla }_z\log f(z)$, matching $\nabla \ln \phi$ (Connecting Brownian and Poisson Random Bridges with Rectified Flows)
• Nelson’s duality (eq 28 in Chen et al.): $b_{-}^P=b_{+}^P-\nabla \ln \rho$ confirms osmotic velocity = score (Nelson’s osmotic velocity)

Suggested Papers

• Stochastic Control and Schrödinger Bridges on Riemannian Manifolds — extension to curved spaces, relevant if bridges on non-Euclidean domains
• Relative Entropy Minimization over Path Space for Jump Processes — Girsanov theory for jump processes, needed for direction 1

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