Schrödinger Bridges

Audio-visual deepfake localization demands interval-level outputs that serve as temporal evidence. Despite recent progress, symmetric fusion under single-sided or asynchronous forgeries propagates cross-modal noise, degrading high-precision localization. We present IaMSB, an inconsistency-aware multimodal Schrödinger Bridge (SB) that jointly estimates cross-modal consistency and performs interval-level localization. Unlike diffusion models, SB minimizes path-distribution discrepancy and yields consistency scores without explicit noise injection or denoising. With the Schrödinger Bridge (SB), IaMSB unifies consistency estimation, cross-modal information selection, and bridge-step scheduling in one framework. Specifically, a lightweight coarse bridge first proposes candidate intervals and estimates cross-modal consistency; these statistics select cross-modal witness signals and allocate bridge steps asymmetrically across modalities. A refinement bridge then performs step-tuned fusion and outputs refined, time-aligned intervals. IaMSB anticipates single-sided and asynchronous forgeries and, using bottlenecked cross-modal interaction with step allocation, suppresses noise transfer, avoids unnecessary iterations. Across benchmarks, IaMSB stabilizes strict-IoU boundary precision, raising [email protected] by 3%~10%, and yields improved high-precision localization, particularly for single-sided forgeries.

For a fixed flow-based generative model under a small inference budget, sample quality can depend strongly on where the sampler spends its few function evaluations. Flow matching and Schrödinger bridges define probability paths, yet their inference grids are usually heuristic or inherited from one-endpoint diffusion. We derive a conditional-marginal entropy-rate objective for bridge-aware discretization, separating endpoint-conditioned bridge geometry from marginal flow evolution, and use it to build a training-free entropic inference-time scheduler from first principles. For Gaussian Brownian bridges this rate is closed-form and U-shaped, motivating boundary-heavy nonuniform grids. On trained two-dimensional bridge/flow models, the estimated profile recovers the predicted shape and improves 10-step ODE-Heun MMD over linear by 18.1%, with a paired 22.7% SDE-Heun improvement in the same low-NFE sweep. On EDM/CIFAR-10, the entropic time-discretization gives the best tested five-step FID (186.3 \pm 4.0 versus 200.5 \pm 2.9 for linear and 238.0 \pm 5.3 for cosine). On AlphaFlow protein generation, entropic conditional-marginal (cond-marg) scheduling shows advantage in low-NFE regimes on both CAMEO22 and ATLAS benchmarks. These results support entropy-rate scheduling as a practical low-budget allocation signal for high-dimensional bridge and flow samplers.

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.

Learning generative models in settings where the source and target distributions are only specified through unpaired samples is gaining in importance. Here, one frequently-used model are Schrödinger bridges (SB), which represent the most likely evolution between both endpoint distributions. To accelerate training, simulation-free SBs avoid the path simulation of the original SB models. However, learning simulation-free SBs requires paired data; a coupling of the source and target samples is obtained as the solution of the entropic optimal transport (OT) problem. As obtaining the optimal global coupling is infeasible in many practical cases, the entropic OT problem is iteratively solved on minibatches instead. Still, the repeated cost remains substantial and the locality can distort the global transport geometry. We propose quantized diffusion Schrödinger bridges (QDSB), which compute the endpoint coupling on anchor-quantized endpoint distributions and lift the resulting plan back to original data points through cell-wise sampling. We show that the regularized optimal coupling is stable w.r.t. anchor quantization, with an error controlled by the quality of the anchor approximation. In real-world experiments, QDSB matches the sample quality of existing baselines, requiring substantially less time. Code and data are available at github.com/mathefuchs/qdsb.

We consider a mutual information (MI) regularized optimal density control of a discrete-time linear system. MI optimal control has been utilized for exploration in reinforcement learning and privacy protection in control. MI regularization induces stochasticity in the policy, which poses challenges for applications of MI optimal control in safety-critical scenarios. To remedy this situation, we impose Gaussian density constraints at specified times to directly control state uncertainty. For this MI optimal density control problem, we propose an alternating optimization algorithm and investigate its convergence properties. In addition, we reveal a relationship between the MI optimal density control problem and a so-called generalized Schrödinger bridge problem associated with the discrete-time linear system. Based on the results of MI optimal density control and this relationship, we also investigate alternating optimization for the Schrödinger bridge problem and its convergence properties.

We study nonparametric estimation of Schrödinger bridge (SB) drifts from i.i.d.\ data observed on a single time interval. Starting from the conditional-ratio form of the Schrödinger bridge time-series (SBTS) drift formula, we analyze a direct Nadaraya--Watson plug-in estimator built from kernelized numerator and denominator terms. Unlike recent SB analyses based on entropic-OT potentials, Sinkhorn iterations, or iterative bridge solvers, our approach works directly at the drift level and isolates statistical error from optimization, approximation, and discretization error. Under Hölder regularity, a marginal-density floor, and bounded support, we prove a uniform non-asymptotic bound for admissible bandwidth pairs, a pointwise CLT under genuine undersmoothing, and an adaptive bandwidth selector satisfying an oracle inequality. We also prove a pivot-local minimax lower bound which, through an explicit uniform pivot, yields a global minimax lower bound under transparent compatibility conditions; hence the adaptive selector is minimax-rate optimal up to logarithmic factors. Synthetic experiments provide theorem-targeted diagnostics for finite-sample scaling, Gaussian approximation, and adaptive behavior.

Most modern bridge-diffusion methods achieve finite-time transport by specifying an interpolation, Schrödinger-bridge, or stochastic-control objective and then learning the associated score or drift field with a neural network. In contrast, we identify a restricted but sufficiently broad and analytically solvable class in which the score, intermediate marginals, and protocol gradients are available in closed form without inner stochastic simulation loops and without neural networks in the optimization loop. We recast the classical linear--quadratic--Gaussian (LQG) stochastic-control structure as a transport problem of the Path Integral Diffusion (PID) type. In classical LQG control, linear dynamics, Gaussian noise, and quadratic costs lead to Riccati equations and closed-form optimal feedback. In LQ-GM-PID, we retain the linear--quadratic stochastic-control backbone, but replace terminal state regulation by a prescribed terminal probability density and allow both the initial and terminal laws to be Gaussian Mixtures (GM). Moreover, LQ-GM-PID turns bridge diffusion from a tool for terminal target matching alone into a tool for path shaping. We demonstrate this on a 2D corridor task, a 2D multi-entrance transport task, and a high-dimensional scaling study with $d=32$ and $M=16$ Gaussian-mixture terminal modes, all with sub-50,ms analytic precompute on a laptop. We position LQ-GM-PID as an analytically solvable reference model for the state-of-the-art neural bridge-diffusion and generative-transport methods: a controlled setting in which neural approximations, score estimates, path-shaping objectives, and protocol-learning procedures can be tested against exact quantities.

Inferring cellular trajectories from destructive snapshots is complicated by the challenges of stochasticity and non-conservative mass dynamics such as cell proliferation and apoptosis. Existing unbalanced Optimal Transport (OT) methods treat mass as a continuous fluid, performing inference at the population level. However, this macroscopic view often fails to capture the discrete, jump-like nature of birth-death events at single-cell resolution, which is essential for understanding lineage branching and fate decisions. We present Unbalanced Schrödinger Bridge (USB), a simulation-free framework for learning underlying dynamics that effectively integrates both stochastic and unbalanced effects which also models the discrete, jump-like birth-death dynamics at single-cell resolution. Theoretically, USB provides a tractable solution to the Branching Schrödinger Bridge (BSB) problem, offering a rigorous microscopic interpretation where individual cells undergo both Brownian motion and discrete birth-death jumps. Technically, the method implements an efficient solver by introducing a simulation-free training objective that effectively scales to high-dimensional omics data. Empirically, we demonstrate on both simulated and real-world datasets that USB not only achieves trajectory reconstruction performance better than or comparable to deterministic baselines but also uniquely enables realistic discrete simulation of birth-death dynamics at single-cell resolution.

Solutions to the Schrödinger bridge problem and its generalizations yield feedback control policies for optimal density steering over a controlled diffusion. To numerically compute the same, the dynamic Sinkhorn recursion has become a standard approach. The mathematical engine behind this approach is the Hopf-Cole transform that recasts the conditions for optimality into a system of boundary-coupled linear PDEs. Recent works pointed out that for the control-affine Schrödinger bridge problem, this exact linearity via Hopf-Cole transform, and thus the standard Sinkhorn recursion, apply only if the control and noise channels are proportional. When the channels do not match, the Hopf-Cole-transformed PDEs remain nonlinear, and no algorithm is available to solve the same. We advance the state-of-the-art by designing a Sinkhorn recursion with memory that leverages the structure of these nonlinear PDEs, and demonstrate how it solves the control-affine Schrödinger bridge problem with input and noise channel mismatch. We prove the local stability of the proposed algorithm.

Controlling the evolution of a many-body stochastic system from a disordered reference state to a structured target ensemble, characterized empirically through samples, arises naturally in non-equilibrium statistical mechanics and stochastic control. The natural relaxation of such a system - driven by diffusion - runs from the structured target toward the disordered reference. The natural question is then: what is the minimum-work stochastic process that reverses this relaxation, given a pathwise cost functional combining spatial penalties and control effort? Computing this optimal process requires knowledge of trajectories that already sample the target ensemble - precisely the object one is trying to construct. We resolve this by establishing a time-reversal duality: the value function governing the hard backward dynamics satisfies an equivalent forward-in-time HJB equation, whose solution can be read off directly from the tractable forward relaxation trajectories. Via the Cole-Hopf transformation and its associated Feynman-Kac representation, this forward potential is computed as a path-space free energy averaged over these forward trajectories - the same relaxation paths that are easy to simulate - without any backward simulation or knowledge of the target beyond samples. The resulting framework provides a physically interpretable description of stochastic transport in terms of path-space free energy, risk-sensitive control, and spatial cost geometry. We illustrate the theory with numerical examples that visualize the learned value function and the induced controlled diffusions, demonstrating how spatial cost fields shape transport geometry analogously to Fermat's Principle in inhomogeneous media. Our results establish a unifying connection between stochastic optimal control, Schrödinger bridge theory, and non-equilibrium statistical mechanics.

We study the problem of generating synthetic time series that reproduce both marginal distributions and temporal dynamics, a central challenge in financial machine learning. Existing approaches typically fail to jointly model drift and stochastic volatility, as diffusion-based methods fix the volatility while martingale transport models ignore drift. We introduce the Schrödinger-Bass Bridge for Time Series (SBBTS), a unified framework that extends the Schrödinger-Bass formulation to multi-step time series. The method constructs a diffusion process that jointly calibrates drift and volatility and admits a tractable decomposition into conditional transport problems, enabling efficient learning. Numerical experiments on the Heston model demonstrate that SBBTS accurately recovers stochastic volatility and correlation parameters that prior SchrödingerBridge methods fail to capture. Applied to S&P 500 data, SBBTS-generated synthetic time series consistently improve downstream forecasting performance when used for data augmentation, yielding higher classification accuracy and Sharpe ratio compared to real-data-only training. These results show that SBBTS provides a practical and effective framework for realistic time series generation and data augmentation in financial applications.

The mean-field Schrödinger bridge (MFSB) problem concerns designing a minimum-effort controller that guides a diffusion process with nonlocal interaction to reach a given distribution from another by a fixed deadline. Unlike the standard Schrödinger bridge, the dynamical constraint for MFSB is the mean-field limit of a population of interacting agents with controls. It serves as a natural model for large-scale multi-agent systems. The MFSB is computationally challenging because the nonlocal interaction makes the problem nonconvex. We propose a generalization of the Hopf-Cole transform for MFSB and, building on it, design a Sinkhorn-type recursive algorithm to solve the associated system of integro-PDEs. Under mild assumptions on the interaction potential, we discuss convergence guarantees for the proposed algorithm. We present numerical examples with repulsive and attractive interactions to illustrate the theoretical contributions.

Visual navigation is a core challenge in Embodied AI, requiring autonomous agents to translate high-dimensional sensory observations into continuous, long-horizon action trajectories. While generative policies based on diffusion models and Schrödinger Bridges (SB) effectively capture multimodal action distributions, they require dozens of integration steps due to high-variance stochastic transport, posing a critical barrier for real-time robotic control. We propose Rectified Schrödinger Bridge Matching (RSBM), a framework that exploits a shared velocity-field structure between standard Schrödinger Bridges ($\varepsilon=1$, maximum-entropy transport) and deterministic Optimal Transport ($\varepsilon\to 0$, as in Conditional Flow Matching), controlled by a single entropic regularization parameter $\varepsilon$. We prove two key results: (1) the conditional velocity field's functional form is invariant across the entire $\varepsilon$-spectrum (Velocity Structure Invariance), enabling a single network to serve all regularization strengths; and (2) reducing $\varepsilon$ linearly decreases the conditional velocity variance, enabling more stable coarse-step ODE integration. Anchored to a learned conditional prior that shortens transport distance, RSBM operates at an intermediate $\varepsilon$ that balances multimodal coverage and path straightness. Empirically, while standard bridges require $\geq 10$ steps to converge, RSBM achieves over 94% cosine similarity and 92% success rate in merely 3 integration steps -- without distillation or multi-stage training -- substantially narrowing the gap between high-fidelity generative policies and the low-latency demands of Embodied AI.

Electroencephalography (EEG) provides a non-invasive insight into the brain's cognitive and emotional dynamics. However, modeling how these states evolve in real time and quantifying the energy required for such transitions remains a major challenge. The Schrödinger Bridge Problem (SBP) offers a principled probabilistic framework to model the most efficient evolution between the brain states, interpreted as a measure of cognitive energy cost. While generative models such as GANs have been widely used to augment EEG data, it remains unclear whether synthetic EEG preserves the underlying dynamical structure required for transition-based analysis. In this work, we address this gap by using SBP-derived transport cost as a metric to evaluate whether GAN-generated EEG retains the distributional geometry necessary for energy-based modeling of cognitive state transitions. We compare transition energies derived from real and synthetic EEG collected during Stroop tasks and demonstrate strong agreement across group and participant-level analyses. These results indicate that synthetic EEG preserves the transition structure required for SBP-based modeling, enabling its use in data-efficient neuroadaptive systems. We further present a framework in which SBP-derived cognitive energy serves as a control signal for adaptive human-machine systems, supporting real-time adjustment of system behavior in response to user cognitive and affective state.

We study the problem of identifying an optimal coupling between input-output distributional data generated by a causal dynamical system. The coupling is required to satisfy prescribed marginal distributions and a causality constraint reflecting the temporal structure of the system. We formulate this problem as a Schr"odinger Bridge, which seeks the coupling closest - in Kullback-Leibler divergence - to a given prior while enforcing both marginal and causality constraints. For the case of Gaussian marginals and general time-dependent quadratic cost functions, we derive a fully tractable characterization of the Sinkhorn iterations that converges to the optimal solution. Beyond its theoretical contribution, the proposed framework provides a principled foundation for applying causal optimal transport methods to system identification from distributional data.

Classical on-policy algorithms such as PPO and mirror descent policy optimization provide stable proximal policy updates through tractable action likelihoods, but are typically instantiated with simple Gaussian policies whose expressiveness can be limited in complex continuous-control tasks. Generative policies based on diffusion and flow models provide more expressive action distributions, but they naturally define distributions over multi-step denoising paths whose terminal action density is often intractable, creating a mismatch with likelihood-based on-policy proximal updates. To address this mismatch, we introduce GSB-MDPO (Generalized Schrödinger Bridge Mirror Descent Policy Optimization), which formulates on-policy generative policy optimization as a Generalized Schrödinger Bridge problem over state-conditioned generation paths and instantiates the resulting path-measure update through mirror descent policy optimization. The key insight is that the GSB path-space KL plays the role of the proximal term in MDPO while upper-bounding the terminal action KL, enabling direct control of the executed action distribution without explicit terminal action likelihood evaluation. Experiments on 14 continuous-control tasks across Playground and Gym-MuJoCo demonstrate the empirical effectiveness of GSB-MDPO and support path-space regularization as a principled proximal update for multi-step generative policies.

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.

This work studies the Schrödinger bridge problem for the kinematic equation on a compact connected Lie group. The objective is to steer a controlled diffusion between given initial and terminal densities supported over the Lie group while minimizing the control effort. We develop a coordinate-free formulation of this stochastic optimal control problem that respects the underlying geometric structure of the Lie group, thereby avoiding limitations associated with local parameterizations or embeddings in Euclidean spaces. We establish the existence and uniqueness of solution to the corresponding Schrödinger system. Our results are constructive in that they derive a geometric controller that optimally interpolates probability densities supported over the Lie group. To illustrate the results, we provide numerical examples on $\mathsf{SO}(2)$ and $\mathsf{SO}(3)$. The codes and animations are publicly available at https://github.com/gradslab/SbpLieGroups.git .

Ultra Low Field (64 mT) brain MRI improves accessibility but suffers from reduced image quality compared to 3 T. As paired 64 mT - 3 T scans are scarce, we propose an unpaired 64 mT $\rightarrow$ 3 T translation framework that enhances realism while preserving anatomy. Our method builds upon the Unpaired Neural Schrödinge Bridge (UNSB) with multi-step refinement. To strengthen target distribution alignment, we augment the adversarial objective with DMD2-style diffusion-guided distribution matching using a frozen 3T diffusion teacher. To explicitly constrain global structure beyond patch-level correspondence, we combine PatchNCE with an Anatomical Structure Preservation (ASP) regularizer that enforces soft foreground background consistency and boundary aware constraints. Evaluated on two disjoint cohorts, the proposed framework achieves an improved realism structure trade-off, enhancing distribution level realism on unpaired benchmarks while increasing structural fidelity on the paired cohort compared to unpaired baselines.

Diffusion models often yield highly curved trajectories and noisy score targets due to an uninformative, memoryless forward process that induces independent data-noise coupling. We propose Adjoint Schrödinger Bridge Matching (ASBM), a generative modeling framework that recovers optimal trajectories in high dimensions via two stages. First, we view the Schrödinger Bridge (SB) forward dynamic as a coupling construction problem and learn it through a data-to-energy sampling perspective that transports data to an energy-defined prior. Then, we learn the backward generative dynamic with a simple matching loss supervised by the induced optimal coupling. By operating in a non-memoryless regime, ASBM produces significantly straighter and more efficient sampling paths. Compared to prior works, ASBM scales to high-dimensional data with notably improved stability and efficiency. Extensive experiments on image generation show that ASBM improves fidelity with fewer sampling steps. We further showcase the effectiveness of our optimal trajectory via distillation to a one-step generator.

Understanding the continuous evolution of populations from discrete temporal snapshots is a critical research challenge, particularly in fields like developmental biology and systems medicine where longitudinal tracking of individual entities is often impossible. Such trajectory inference is vital for unraveling the mechanisms of dynamic processes. While Schrödinger Bridge (SB) offer a potent framework, their traditional application to pairwise time points can be insufficient for systems defined by multiple intermediate snapshots. This paper introduces Multi-Marginal Schrödinger Bridge Matching (MSBM), a novel algorithm specifically designed for the multi-marginal SB problem. MSBM extends iterative Markovian fitting (IMF) to effectively handle multiple marginal constraints. This technique ensures robust enforcement of all intermediate marginals while preserving the continuity of the learned global dynamics across the entire trajectory. Empirical validations on synthetic data and real-world single-cell RNA sequencing datasets demonstrate the competitive or superior performance of MSBM in capturing complex trajectories and respecting intermediate distributions, all with notable computational efficiency.

The Schrödinger bridge problem is concerned with finding a stochastic dynamical system bridging two marginal distributions that minimises a certain transportation cost. This problem, which represents a generalisation of optimal transport to the stochastic case, has received attention due to its connections to diffusion models and flow matching, as well as its applications in the natural sciences. However, all existing algorithms allow to infer such dynamics only for cases where samples from both distributions are available. In this paper, we propose the first general method for modelling Schrödinger bridges when one (or both) distributions are given by their unnormalised densities, with no access to data samples. Our algorithm relies on a generalisation of the iterative proportional fitting (IPF) procedure to the data-free case, inspired by recent developments in off-policy reinforcement learning for training of diffusion samplers. We demonstrate the efficacy of the proposed data-to-energy IPF on synthetic problems, finding that it can successfully learn transports between multimodal distributions. As a secondary consequence of our reinforcement learning formulation, which assumes a fixed time discretisation scheme for the dynamics, we find that existing data-to-data Schrödinger bridge algorithms can be substantially improved by learning the diffusion coefficient of the dynamics. Finally, we apply the newly developed algorithm to the problem of sampling posterior distributions in latent spaces of generative models, thus creating a data-free image-to-image translation method. Code: https://github.com/mmacosha/d2e-stochastic-dynamics

Unpaired image-to-image translation has seen significant progress since the introduction of CycleGAN. However, methods based on diffusion models or Schrödinger bridges have yet to be widely adopted in real-world applications due to their iterative sampling nature. To address this challenge, we propose Implicit Bridge Consistency Distillation (IBCD), which enables single-step bidirectional unpaired translation without using adversarial loss. IBCD reformulates consistency distillation by using a diffusion implicit bridge model that connects PF-ODE trajectories between distributions, parametrized on a signed time axis whose sign selects the target domain, so that one network translates in both directions in a single step. Additionally, we introduce two key improvements: 1) distribution matching for consistency distillation and 2) adaptive weighting method based on distillation difficulty. Experimental results demonstrate that IBCD achieves state-of-the-art performance on benchmark datasets in a single generation step, and that it extends to brain MRI contrast translation.