A related situation arises when an auxiliary process is introduced to aid training but modelling its dynamics at generation time is unnecessary or difficult, as in Billera et al. [2025b] and Kim et al. [2025]. In each of these works, the projection result and its associated loss are derived on a case-by-case basis, and all theorems are restricted to marginalization over a discrete component of the extended state space. We introduce a general framework that removes these restrictions: given a time-inhomogeneous Feller process (Yt)0 t 1 on an arbitrary state space Y and a map Φ: Y X, one may learn a linear parametrisation of the generator of a Feller process on X whose one-time marginals coincide with those of (Φ(Yt))0 t 1. For Y = X Z and Φthe projection onto the first coordinate, this subsumes these prior works as special cases, allowing for a general class of latent processes (Zt)0 t 1 in a nearly arbitrary state space Z, using the formalism of generator matching to allow for continuous, discrete, or manifold-valued processes. In particular, the learnt process at t = 1 samples from the distribution of Φ(Y1), which is the desired data distribution. We give sufficient conditions for a loss function to be valid in this general setting, recovering the results of the works cited above as corollaries. This result has broad applicability, enabling the construction of a wide array of new flow matching schemes by allowing for a more general class of latent spaces. As a concrete new application, we outline a non-projection Φ: Y X with manifold-valued latents for protein structure generation that separates chain-level rigid-body motion from internal flexibility ( 4), where the particular chain-level versus residue-level or internal state is latent, and the model only sees the world state, which we plan to implement in future work. 2 EARLIERWORK Several recent generative models train with the aid of a latent stochastic process that is marginalised out at generation time.

Toevaluate theseGANs,weusedagroupofrealdatasets to set-up a benchmarking system and implemented three of the most recent techniques. For comparison purposes, we created two baseline methodsusingBayesiannetworks.

In this paper, we propose a general framework for testing the conditional distribution equality in a two-sample problem, which is most relevant to covariate shift and causal discovery. Our framework is built on neural network-based generative methods and sample splitting techniques by transforming the conditional testing problem into an unconditional one. We introduce the generative classification accuracy-based conditional distribution equality test (GCA-CDET) to illustrate the proposed framework. We establish the convergence rate for the learned generator by deriving new results related to the recently-developed offset Rademacher complexity and prove the testing consistency of GCA-CDET under mild conditions.Empirically, we conduct numerical studies including synthetic datasets and two real-world datasets, demonstrating the effectiveness of our approach. Additional discussions on the optimality of the proposed framework are provided in the online supplementary material.

Mental health disorders, particularly cognitive disorders defined by deficits in cognitive abilities, are described in detail in the DSM-5, which includes definitions and examples of signs and symptoms. A simplified, machine-actionable representation was developed to assess the similarity and separability of these disorders, but it is not suited for the most complex cases. Generating or applying a full binary matrix for similarity calculations is infeasible due to the vast number of symptom combinations. This research develops an alternative representation that links the narrative form of the DSM-5 with the binary matrix representation and enables automated generation of valid symptom combinations. Using a strict pre-defined format of lists, sets, and numbers with slight variations, complex diagnostic pathways involving numerous symptom combinations can be represented. This format, called the symptom profile generator (or simply generator), provides a readable, adaptable, and comprehensive alternative to a binary matrix while enabling easy generation of symptom combinations (profiles). Cognitive disorders, which typically involve multiple diagnostic criteria with several symptoms, can thus be expressed as lists of generators. Representing several psychotic disorders in generator form and generating all symptom combinations showed that matrix representations of complex disorders become too large to manage. The MPCS (maximum pairwise cosine similarity) algorithm cannot handle matrices of this size, prompting the development of a profile reduction method using targeted generator manipulation to find specific MPCS values between disorders. The generators allow easier creation of binary representations for large matrices and make it possible to calculate specific MPCS cases between complex disorders through conditional generators.

This brief note clarifies that, in Generator Matching (which subsumes a large family of flow matching and diffusion models over continuous, manifold, and discrete spaces), both the Bregman divergence loss and the linear parameterization of the generator can depend on both the current state $X_t$ and the time $t$, and we show that the expectation over time in the loss can be taken with respect to a broad class of time distributions. We also show this for Edit Flows, which falls outside of Generator Matching. That the loss can depend on $t$ clarifies that time-dependent loss weighting schemes, often used in practice to stabilize training, are theoretically justified when the specific flow or diffusion scheme is a special case of Generator Matching (or Edit Flows). It also often simplifies the construction of $X_1$-predictor schemes, which are sometimes preferred for model-related reasons. We show examples that rely upon the dependence of linear parameterizations, and of the Bregman divergence loss, on $t$ and $X_t$.

Diffusion and flow matching approaches to generative modeling have shown promise in domains where the state space is continuous, such as image generation or protein folding & design, and discrete, exemplified by diffusion large language models. They offer a natural fit when the number of elements in a state is fixed in advance (e.g. images), but require ad hoc solutions when, for example, the length of a response from a large language model, or the number of amino acids in a protein chain is not known a priori. Here we propose Branching Flows, a generative modeling framework that, like diffusion and flow matching approaches, transports a simple distribution to the data distribution. But in Branching Flows, the elements in the state evolve over a forest of binary trees, branching and dying stochastically with rates that are learned by the model. This allows the model to control, during generation, the number of elements in the sequence. We also show that Branching Flows can compose with any flow matching base process on discrete sets, continuous Euclidean spaces, smooth manifolds, and `multimodal' product spaces that mix these components. We demonstrate this in three domains: small molecule generation (multimodal), antibody sequence generation (discrete), and protein backbone generation (multimodal), and show that Branching Flows is a capable distribution learner with a stable learning objective, and that it enables new capabilities.

Generating high-quality synthetic text under differential privacy (DP) is critical for training and evaluating language models without compromising user privacy. Prior work on synthesizing DP datasets often fail to preserve key statistical attributes, suffer utility loss from the noise required by DP, and lack fine-grained control over generation. To address these challenges, we make two contributions. First, we introduce a hierarchical framework that decomposes DP synthetic text generation into two subtasks: feature learning and conditional text generation. This design explicitly incorporates learned features into the generation process and simplifies the end-to-end synthesis task. Through systematic ablations, we identify the most effective configuration: a rich tabular schema as feature, a DP tabular synthesizer, and a DP fine-tuned conditional generator, which we term ACTG (Attribute-Conditioned Text Generation). Second, we propose Anchored RL (ARL), a post-training method that improves the instruction-following ability of ACTG for conditional generation. ARL combines RL to boost control with an SFT anchor on best-of-$N$ data to prevent reward hacking. Together, these components form our end-to-end algorithm ACTG-ARL, which advances both the quality of DP synthetic text (+20% MAUVE over prior work) and the control of the conditional generator under strong privacy guarantees.

Recent developments in deep generative models have led to a wealth of possibilities.