These notes recapitulate the high level mathematical principles behind different techniques for generative modeling. I show the connections between optimal transport and standard techniques such as Schr{ö}dinger bridge and flow matching.

Many modern generative modeling methods, including diffusion models, normalizing flows, and flow matching, estimate transport maps or plans between distributions without explicitly targeting an optimal transport (OT) map. In applications like generative modeling, the transport cost itself is irrelevant, and this makes it natural to target maps which are more tractable from either a statistical or computational standpoint. In this short note, we formalize the task of estimating any valid transport map in a rigorous minimax framework. One consequence of this framing is that it yields sample complexity lower bounds for any method whose learned object is evaluated as a transport map or plan, including flow matching and diffusion-based generative models, in settings where direct analysis would be challenging due to the analytic complexity of the methods and their target maps. We observe that, under standard, though strong, stability assumptions from the OT literature, estimating any valid transport map is statistically as hard as estimating the OT map. We complement these results with some examples showing that when these stability assumptions fail, alternative transport maps can be learned substantially more accurately than the OT map. Our minimax framing provides a rigorous foundation for understanding the statistical limits of modern transport-based generative methods and clarifies when targeting sub-optimal maps can provide real statistical advantages.

Discrete Flow-based Models (DFMs) are powerful generative models for highquality discrete data but typically suffer from slow sampling speeds due to their reliance on iterative decoding processes. This reliance on a multi-step process originates from the factorization approximation of DFMs, which is necessary for handling high-dimensional data. In this paper, we analyze the factorization approximation error using Conditional Total Correlation (TC), and reveal its dependence on the coupling. To address the challenge of efficient few-step generation, we propose Rectified Discrete Flow (ReDi), a novel iterative method that reduces the underlying factorization error (measured as Conditional TC) by rectifying the coupling between source and target distributions. We theoretically prove that each ReDi step guarantees a monotonic decreasing Conditional TC, ensuring its convergence. Empirically, ReDi significantly reduces Conditional TC and enables few-step generation. Moreover, we demonstrate that the rectified couplings are well-suited for training efficient one-step models on image generation. ReDi offers a simple and theoretically grounded approach for tackling the few-step challenge, providing a new perspective on efficient discrete data synthesis.

This paper investigates the connections between rectified flows, flow matching, and optimal transport. Flow matching is a recent approach to learning generative models by estimating velocity fields that guide transformations from a source to a target distribution. Rectified flow matching aims to straighten the learned transport paths, yielding more direct flows between distributions. Our first contribution is a set of invariance properties of rectified flows and explicit velocity fields. In addition, we also provide explicit constructions and analysis in the Gaussian (not necessarily independent) and Gaussian mixture settings and study the relation to optimal transport. Our second contribution addresses recent claims suggesting that rectified flows, when constrained such that the learned velocity field is a gradient, can yield (asymptotically) solutions to optimal transport problems. We study the existence of solutions for this problem and demonstrate that they only relate to optimal transport under assumptions that are significantly stronger than those previously acknowledged. In particular, we present several counterexamples that invalidate earlier equivalence results in the literature, and we argue that enforcing a gradient constraint on rectified flows is, in general, not a reliable method for computing optimal transport maps.

Recent advances in flow-based generative modelling have provided scalable methods for computing the Schr odinger Bridge (SB) between distributions, a dynamic form of entropy-regularised Optimal Transport (OT) for the quadratic cost. The successful Iterative Markovian Fitting (IMF) procedure solves the SB problem via sequential bridge-matching steps, presenting an elegant and practical approach with many favourable properties over the more traditional Iterative Proportional Fitting (IPF) procedure. Beyond the standard setting, optimal transport can be generalised to the multi-marginal case in which the objective is to minimise a cost defined over several marginal distributions. Of particular importance are costs defined over a tree structure, from which Wasserstein barycentres can be recovered as a special case. In this work, we extend the IMF procedure to solve for the tree-structured SB problem. Our resulting algorithm inherits the many advantages of IMF over IPF approaches in the tree-based setting. In the case of Wasserstein barycentres, our approach can be viewed as extending the widely used fixed-point approach to use flow-based entropic OT solvers, while requiring only simple bridge-matching steps at each iteration.

In this paper, we propose a flow-based method for learning all-to-all transfer maps among conditional distributions that approximates pairwise optimal transport. The proposed method addresses the challenge of handling the case of continuous conditions, which often involve a large set of conditions with sparse empirical observations per condition. We introduce a novel cost function that enables simultaneous learning of optimal transports for all pairs of conditional distributions. Our method is supported by a theoretical guarantee that, in the limit, it converges to the pairwise optimal transports among infinite pairs of conditional distributions. The learned transport maps are subsequently used to couple data points in conditional flow matching. We demonstrate the effectiveness of this method on synthetic and benchmark datasets, as well as on chemical datasets in which continuous physical properties are defined as conditions.

Large language model (LLM) watermarks enable authentication of text provenance, curb misuse of machine-generated text, and promote trust in AI systems. Current watermarks operate by changing the next-token predictions output by an LLM. The updated (i.e., watermarked) predictions depend on random side information produced, for example, by hashing previously generated tokens. LLM watermarking is particularly challenging when next-token predictions are near-deterministic. In fact, over 90% of next-token distributions are low-entropy, with more than half of the probability mass on a single token.

We study the sample complexity of Bayesian recovery for solving inverse problems with general prior, forward operator and noise distributions. We consider posterior sampling according to an approximate prior P, and establish sufficient conditions for stable and accurate recovery with high probability. Our main result is a non-asymptotic bound that shows that the sample complexity depends on (i) the intrinsic complexity of P, quantified by its approximate covering number, and (ii) concentration bounds for the forward operator and noise distributions. As a key application, we specialize to generative priors, where P is the pushforward of a latent distribution via a Deep Neural Network (DNN). We show that the sample complexity scales log-linearly with the latent dimension k, thus establishing the efficacy of DNN-based priors. Generalizing existing results on deterministic (i.e., non-Bayesian) recovery for the important problem of random sampling with an orthogonal matrix U, we show how the sample complexity is determined by the coherence of U with respect to the support of P. Hence, we establish that coherence plays a fundamental role in Bayesian recovery as well. Overall, our framework unifies and extends prior work, providing rigorous guarantees for the sample complexity of solving Bayesian inverse problems with arbitrary distributions.

Distributionally Robust Optimization (DRO) offers a promising framework by optimizing worst-case performance over a set of candidate distributions, referred to as the uncertainty set. However, the efficacy of DRO heavily depends on the design of the uncertainty set, and existing methods often perform suboptimally due to an inappropriate or inflexible uncertainty set. In this work, we first propose a novel perspective that casts entropy-regularized Wasserstein DRO as a dynamic process of distributional exploration and semantic alignment, both driven by optimal transport (OT). This unified viewpoint yields two key new techniques: semantic calibration, which bootstraps semantically meaningful transport costs via inverse OT, and adaptive refinement, which adjusts uncertainty set using OT-driven feedback. Together, these components form an exploration-and-feedback system, where the transport costs and uncertainty set evolve jointly during training, enabling the model to better adapt to potential distribution shifts. Moreover, we provide an in-depth analysis of this adaptive process and prove theoretical guarantees of convergence. Finally, we present our experimental results across diverse distribution shift scenarios, which demonstrate that our approach significantly outperforms existing methods, achieving state-ofthe-art robustness.

Understanding complex systems by inferring trajectories from sparse sample snapshots is a fundamental challenge in a wide range of domains, e.g., single-cell biology, meteorology, and economics. Despite advancements in Bridge and Flow matching frameworks, current methodologies rely on pairwise interpolation between adjacent snapshots. This hinders their ability to capture long-range temporal dependencies and potentially affects the coherence of the inferred trajectories. To address these issues, we introduce Momentum Multi-Marginal Schrödinger Bridge Matching (3MSBM), a novel matching framework that learns smooth measure-valued splines for stochastic systems that satisfy multiple positional constraints. This is achieved by lifting the dynamics to phase space and generalizing stochastic bridges to be conditioned on several points, forming a multi-marginal conditional stochastic optimal control problem. The underlying dynamics are then learned by minimizing a variational objective, having fixed the path induced by the multi-marginal conditional bridge. As a matching approach, 3MSBM learns transport maps that preserve intermediate marginals throughout training, significantly improving convergence and scalability. Extensive experimentation in a series of real-world applications validates the superior performance of 3MSBM compared to existing methods in capturing complex dynamics with temporal dependencies, opening new avenues for training matching frameworks in multi-marginal settings.