Random-Bridges as Stochastic Transports for Generative Models

At the core of modern generative modeling frameworks, including diffusion models, score-based models, and flow matching, is the task of transforming a simple prior distribution into a complex target distribution through stochastic paths in probability space. Schrödinger bridges provide a unifying principle underlying these approaches, framing the problem as determining an optimal stochastic bridge between marginal distribution constraints with minimal-entropy deviations from a pre-defined reference process. This guide develops the mathematical foundations of the Schrödinger bridge problem, drawing on optimal transport, stochastic control, and path-space optimization, and focuses on its dynamic formulation with direct connections to modern generative modeling. We build a comprehensive toolkit for constructing Schrödinger bridges from first principles, and show how these constructions give rise to generalized and task-specific computational methods.

Modern generative models can be understood as probability transport from a simple base distribution to a target data distribution. Deterministic transport models offer tractable velocity-field parameterizations, whereas stochastic generative models capture richer density evolution through drift and diffusion. Yet when stochastic dynamics are described through deterministic velocity fields, the effects of drift and diffusion are often compressed into a single effective field, obscuring the distinct roles of deterministic evolution and stochastic fluctuation. In this work, we show that the deterministic field $b_t$ of a stochastic generative process admits a natural transport--osmotic decomposition that separates deterministic transport from stochastic, diffusion-induced effects: $b_t = u_t + d_t$, where $u_t$ governs marginal probability transport and $d_t$ captures an osmotic effect induced by diffusion and determined by the marginal score. Based on this decomposition, we propose Bridge Matching, a flow-based framework for learning decomposed generative dynamics through both marginal and conditional formulations. In generative modeling experiments, we recombine the learned components as $b_t = u_t + λ_d d_t$, showing that the proposed decomposition enables interpretable and controllable sampling by adjusting the osmotic contribution in probability transport.

We introduce Action-Inspired Generative Models (AGMs), a dual-network generative framework motivated by the observation that existing bridge-matching methods assign uniform regression weight to every stochastic transition in the transport landscape, regardless of whether a given bridge sample lies along a structurally coherent trajectory or a degenerate one. We address this by introducing a lightweight learned scalar potential $V_φ$ that scores bridge samples online and modulates the drift objective via importance weights derived through a stop-gradient barrier -- preventing adversarial feedback between the two networks whilst preserving $V_φ$'s guiding signal. Crucially, $V_φ$ comprises only $\sim$1.4% of the primary drift network's parameter count, adds no overhead to the inference graph, and requires no iterative half-bridge fitting or auxiliary stochastic differential equation (SDE) solvers: it is a plug-and-play enhancement to any bridge-matching training loop. At inference, $V_φ$ is discarded entirely, leaving standard Euler-Maruyama integration of the exponential moving average (EMA) drift. We demonstrate that selectively penalising uninformative transport paths through the learned potential yields consistent improvements in generation quality across fidelity and coverage metrics.