Generative Modeling

We develop a kernel method for generative modeling within the stochastic interpolant framework, replacing neural network training with linear systems. The drift of the generative SDE is $\hat b_t(x) = \nablaφ(x)^\topη_t$, where $η_t \in \mathbb{R}^P$ solves a $P\times P$ system computable from data, with $P$ independent of the data dimension $d$. Since estimates are inexact, the diffusion coefficient $D_t$ affects sample quality; the optimal $D_t^*$ from Girsanov diverges at $t=0$, but this poses no difficulty and we develop an integrator that handles it seamlessly. The framework accommodates diverse feature maps: scattering transforms, pretrained generative models, etc, enabling generation and model combination without neural network training. We demonstrate the approach on financial time series, turbulence, and image generation.

Many high-dimensional datasets concentrate near a low-dimensional structure embedded in the ambient space. Generative models for such data must control off-support mass while remaining computationally practical. Diffusion models use iterative denoising at inference, whereas standard normalizing flows require invertible, dimension-preserving maps. We propose MAGT (Manifold-Aligned Generative Transport), a direct transport from a low-dimensional base distribution to the data space. Its core objective compares the data and generator-induced scores at a selected Gaussian smoothing level. A posterior identity expresses this score through a latent conditional mean, which is approximated by self-normalized importance sampling over a finite anchor set. After training, generation requires one evaluation of the transport, whose image also carries an intrinsic density with respect to manifold volume. We establish a minimax-optimal Wasserstein convergence rate for an explicitly constructed localized spline-RePU coordinate transport estimator, and treat finite-anchor approximation separately. Experiments on synthetic, image, and tabular benchmarks compare fidelity, support alignment, and sampling cost with diffusion, flow-matching, and adversarial baselines.

In this paper, we present Laplacian multiscale flow matching (LapFlow), a novel framework that enhances flow matching by leveraging multi-scale representations for image generative modeling. Our approach decomposes images into Laplacian pyramid residuals and processes different scales in parallel through a mixture-of-transformers (MoT) architecture with causal attention mechanisms. Unlike previous cascaded approaches that require explicit renoising between scales, our model generates multi-scale representations in parallel, eliminating the need for bridging processes. The proposed multi-scale architecture not only improves generation quality but also accelerates the sampling process and promotes scaling flow matching methods. Through extensive experimentation on CelebA-HQ and ImageNet, we demonstrate that our method achieves superior sample quality with fewer GFLOPs and faster inference compared to single-scale and multi-scale flow matching baselines. The proposed model scales effectively to high-resolution generation (up to 1024$\times$1024) while maintaining lower computational overhead.

AI training datasets will inevitably contain AI-generated examples, leading to ``feedback'' in which the output of one model impacts the training of another. It is known that such iterative feedback can lead to model collapse, yet the mechanisms underlying this degeneration remain poorly understood. Here we show that a broad class of feedback processes converges to a low-dimensional invariant structure in latent space, a phenomenon we call neural resonance. By modeling iterative feedback as a Markov Chain, we show that two conditions are needed for this resonance to occur: ergodicity of the feedback process and directional contraction of the latent representation. By studying diffusion models on MNIST and ImageNet, as well as CycleGAN and an audio feedback experiment, we map how local and global manifold geometry evolve, and we introduce an eight-pattern taxonomy of collapse behaviors. Neural resonance provides a unified explanation for long-term degenerate behavior in generative models and provides practical diagnostics for identifying, characterizing, and eventually mitigating collapse.

Large-scale flow matching models have achieved strong performance across generative tasks such as text-to-image, video, 3D, and speech synthesis. However, aligning their outputs with human preferences and task-specific objectives remains challenging. Flow-GRPO extends Group Relative Policy Optimization (GRPO) to generation models, enabling stable reinforcement learning alignment for generative systems. Since its introduction, Flow-GRPO has triggered rapid research growth, spanning methodological refinements and diverse application domains. This survey provides a comprehensive review of Flow-GRPO and its subsequent developments. We organize existing work along two primary dimensions. First, we analyze methodological advances beyond the original framework, including reward signal design, credit assignment, sampling efficiency, diversity preservation, reward hacking mitigation, and reward model construction. Second, we examine extensions of GRPO-based alignment across generative paradigms and modalities, including text-to-image, video generation, image editing, speech and audio, 3D modeling, embodied vision-language-action systems, unified multimodal models, autoregressive and masked diffusion models, and restoration tasks. By synthesizing theoretical insights and practical adaptations, this survey highlights Flow-GRPO as a general alignment framework for modern generative models and outlines key open challenges for scalable and robust reinforcement-based generation.

Deep Generative models (DGMs) play two key roles in modern machine learning: (i) producing new information (e.g., image synthesis) and (ii) reducing dimensionality. However, traditional architectures often rely on auxiliary networks such as encoders in Variational Autoencoders (VAEs) or discriminators in Generative Adversarial Networks (GANs), which introduce training instability, computational overhead, and risks like mode collapse. We present NeuroSQL, a new generative paradigm that eliminates the need for auxiliary networks by learning low-dimensional latent representations implicitly. NeuroSQL leverages an asymptotic approximation that expresses the latent variables as the solution to an optimal transportation problem. Specifically, NeuroSQL learns the latent variables by solving a linear assignment problem and then passes the latent information to a standalone generator. We benchmark its performance against GANs, VAEs, and a budget-matched diffusion baseline on four datasets: handwritten digits (MNIST), faces (CelebA), animal faces (AFHQ), and brain images (OASIS). Compared to VAEs, GANs, and diffusion models: (1) in terms of image quality, NeuroSQL achieves overall lower mean pixel distance between synthetic and authentic images and stronger perceptual/structural fidelity; (2) computationally, NeuroSQL requires the least training time; and (3) practically, NeuroSQL provides an effective solution for generating synthetic data with limited training samples. By embracing quantile assignment rather than an encoder, NeuroSQL provides a fast, stable, and robust way to generate synthetic data with minimal information loss.

Equivariance is central to graph generative models, as it ensures the model respects the permutation symmetry of graphs. However, strict equivariance can increase computational cost due to added architectural constraints, and can slow down convergence because the model must be consistent across a large space of possible node permutations. We study this trade-off for graph generative models. Specifically, we start from an equivariant discrete flow-matching model, and relax its equivariance during training via a controllable symmetry modulation scheme based on sinusoidal positional encodings and node permutations. Experiments first show that symmetry-breaking can accelerate early training by providing an easier learning signal, but at the expense of encouraging shortcut solutions that can cause overfitting, where the model repeatedly generates graphs that are duplicates of the training set. On the contrary, properly modulating the symmetry signal can delay overfitting while accelerating convergence, allowing the model to reach stronger performance with $19\%$ of the baseline training epochs.

For multivariate co-generation in scientific applications, we advocate pairwise block rather than joint modeling of all variables. This design mitigates the computational burden and data imbalance. To this end, we propose an Annealed Co-Generation (ACG) framework that replaces high-dimensional diffusion modeling with a low-dimensional diffusion model, which enables multivariate co-generation by composing pairwise variable generations. We first train an unconditional diffusion model over causal variables that are disentangled into pairs. At inference time, we recover the joint distribution by coupling these pairwise models through shared common variables, enabling coherent multivariate generation without any additional training. By employing a three-stage annealing process-Consensus, Heating, and Cooling-our method enforces consistency across shared common variables and progressively constrains each pairwise data distribution to lie on a learnable manifold, while maintaining high likelihood within each pair. We demonstrate the framework's flexibility and efficacy on two distinct scientific tasks: flow-field completion and antibody generation. All datasets and code will be made publicly available upon publication.

Fully unsupervised deep generative modeling (FU-DGM) offers significant potential for compressively sampled magnetic resonance imaging (CS-MRI) reconstruction. Representative FU-DGM formulations, such as deep image prior (DIP) and implicit neural representation (INR), employ architectural bias to induce a low-dimensional manifold in the image space that aligns with the forward observation. However, as the underlying inverse system is highly ill-posed, prolonged iterative fitting in FU-DGM typically leads to poor efficiency and noise amplification. In this paper, guided by the cognitive principle of easy-to-hard learning, we propose CogGen, an FU-DGM framework that reformulates CS-MRI reconstruction as a staged inversion problem. Specifically, CogGen implements an self-paced curriculum learning (SPCL)-driven progressive scheduling strategy through an MRI-aware dual-threshold weighting criterion, which adaptively regulates k-space measurement participation. The data-consistency residual thresholding evaluates the fitting reliability of the current generator, while the k-space radius thresholding controls stage-wise measurement exposure, thereby avoiding uniform fitting throughout optimization. Theoretically, our analysis shows that, when early stages favor easy-to-fit measurements, CogGen yields a reduced local sufficient-iteration bound and a smaller cumulative noise-amplification bound, explaining the improved convergence behavior and reconstruction fidelity of CogGen within a finite iteration budget. Numerical experiments demonstrate that both CogGen instantiations, CogGen-DIP and CogGen-INR, achieve superior performance over prevailing CS-MRI reconstruction techniques, including unsupervised and supervised pipelines.

Based on literature review about existing diffusion models and flow matching with a neural network to predict a predefined target from noisy data, a unified representation is first proposed for these models using two simple linear equations in this paper. Theoretical analysis of the proposed model is then presented. Our theoretical analysis shows that the correlation between the noisy data and the predicted target is sometimes weak in the existing diffusion models and flow matching. This might affect the prediction (or learning) process which plays a crucial role in all models.

Training large-scale generative models is resource-intensive and relies heavily on heuristic dataset weighting. We address two fundamental questions: Can we train Large Language Models (LLMs) modularly, combining small, domain-specific experts to match monolithic performance, and can we do so robustly for any data mixture, eliminating heuristic tuning? We present a theoretical framework for modular generative modeling where a set of pre-trained experts are combined via a gating mechanism. We define the space of normalized gating functions $\mathcal{G}_{1}$ and formulate the problem as a minimax game to find a single robust gate that minimizes divergence to the worst-case data mixture. We prove the existence of such a robust gate using Kakutani's fixed-point theorem and show that modularity acts as a strong regularizer, with generalization bounds scaling with the lightweight gate's complexity. Furthermore, we prove that this modular approach can theoretically outperform models retrained on aggregate data, with the gap characterized by the Jensen-Shannon Divergence. Finally, we introduce a scalable Stochastic Primal-Dual algorithm and a Structural Distillation method for efficient inference. Empirical results on synthetic and real-world datasets confirm that our modular architecture effectively mitigates gradient conflict and can robustly outperform monolithic baselines.

Controlling the longitudinal laser pulse shape in photoinjectors of Free-Electron Lasers is a powerful lever for optimizing electron beam quality, but systematic exploration of the vast design space is limited by the cost of brute-force pulse propagation simulations. We present a generative modeling framework based on Wasserstein Autoencoders to learn a differentiable latent interface between pulse shaping and downstream beam dynamics. Our empirical findings show that the learned latent space is continuous and interpretable while maintaining high-fidelity reconstructions. Pulse families such as higher-order Gaussians trace coherent trajectories, while standardizing the temporal pulse lengths shows a latent organization correlated with pulse energy. Analysis via principal components and Gaussian Mixture Models reveals a well behaved latent geometry, enabling smooth transitions between distinct pulse types via linear interpolation. The model generalizes from simulated data to real experimental pulse measurements, accurately reconstructing pulses and embedding them consistently into the learned manifold. Overall, the approach reduces reliance on expensive pulse-propagation simulations and facilitates downstream beam dynamics simulation and analysis.

Generating samples from limited information is a fundamental problem across scientific domains. Classical maximum entropy methods provide principled uncertainty quantification from moment constraints but require sampling via MCMC or Langevin dynamics, which typically exhibit exponential slowdown in high dimensions. In contrast, generative models based on diffusion and flow matching efficiently transport noise to data but offer limited theoretical guarantees and can overfit when data is scarce. We introduce Moment Guided Diffusion (MGD), which combines elements of both approaches. Building on the stochastic interpolant framework, MGD samples maximum entropy distributions by solving a stochastic differential equation that guides moments toward prescribed values in finite time, thereby avoiding slow mixing in equilibrium-based methods. We formally obtain, in the large-volatility limit, convergence of MGD to the maximum entropy distribution and derive a tractable estimator of the resulting entropy computed directly from the dynamics. Applications to financial time series, turbulent flows, and cosmological fields using wavelet scattering moments yield estimates of negentropy for high-dimensional multiscale processes.

Generative AI presents an unprecedented challenge to our understanding of knowledge and its production. Unlike previous technological transformations, where engineering understanding preceded or accompanied deployment, generative AI operates through mechanisms whose epistemic character remains obscure, and without such understanding, its responsible integration into science, education, and institutional life cannot proceed on a principled basis. This paper argues that the missing account must begin with a paradigmatic break that has not yet received adequate philosophical attention. In the Turing-Shannon-von Neumann tradition, information enters the machine as encoded binary vectors, and semantics remains external to the process. Neural network architectures rupture this regime: symbolic input is instantly projected into a high-dimensional space where coordinates correspond to semantic parameters, transforming binary code into a position in a geometric space of meanings. It is this space that constitutes the active epistemic condition shaping generative production. Drawing on four structural properties of high-dimensional geometry concentration of measure, near-orthogonality, exponential directional capacity, and manifold regularity the paper develops an Indexical Epistemology of High-Dimensional Spaces. Building on Peirce semiotics and Papert constructionism, it reconceptualizes generative models as navigators of learned manifolds and proposes navigational knowledge as a third mode of knowledge production, distinct from both symbolic reasoning and statistical recombination.

While deep generative models excel at capturing graph data distributions, they struggle to satisfy complex, hard constraints. In unconstrained settings, these models typically produce valid topologies; yet imposing strict compositional rules, like those in drug discovery, creates an out-of-distribution (OOD) setting where purely neural methods frequently fail. Because these neural approaches rely on soft conditioning and post-hoc filtering on such tasks, they cannot provide the formal guarantees needed for high-stakes domains. To address this, we introduce Neuro-Symbolic Graph Generative Modeling (NSGGM), a framework built on the principle of Neural Proposals, Symbolic Guarantees. NSGGM decouples generation: an autoregressive model proposes structural scaffolds, and a Satisfiability Modulo Theories (SMT) solver handles the final discrete assembly of the proposed substructures. Empirically, NSGGM is competitive with state-of-the-art methods on unconstrained tasks. To evaluate logical-constraint satisfaction inspired by real drug discovery workflows, we introduce MolSAT, a benchmark for hard compositional rules. On MolSAT, purely neural baselines completely fail OOD (0% satisfaction with zero training support), while NSGGM achieves >95% satisfaction in-distribution and 64-86% with zero training support.

As artificial intelligence (AI)-generated content proliferates, models are increasingly trained on their own outputs, risking progressive degradation or collapse. In this article, we provide the first positive, rigorous theoretical results, to the best of our knowledge, showing that under model-agnostic mild conditions, the model converges to the true data-generating distribution. The convergence rate is the minimum of the model's intrinsic rate and the fraction of real data at each training iteration, revealing a phase transition between data-limited and model-limited regimes. We further show that, for biased real data, correcting the bias prevents the persistence and amplification of early bias over training iteration. Extensive experiments across simulations, real images and texts validate our theoretical framework, establishing quantitative conditions for long-term AI stability in contaminated environments.

Many generative tasks in chemistry and science involve distributions invariant to group symmetries (e.g., permutation and rotation). A common strategy enforces invariance and equivariance through architectural constraints such as equivariant denoisers and invariant priors. In this paper, we challenge this tradition through the alternative canonicalization perspective: first map each sample to an orbit representative with a canonical pose or order, train an unconstrained (non-equivariant) diffusion or flow model on the canonical slice, and finally recover the invariant distribution by sampling a random symmetry transform at generation time. Building on a formal quotient-space perspective, our work provides a comprehensive theory of canonical diffusion by proving: (i) the correctness, universality and superior expressivity of canonical generative models over invariant targets; (ii) canonicalization accelerates training by removing diffusion score complexity induced by group mixtures and reducing conditional variance in flow matching. We then show that aligned priors and optimal transport act complementarily with canonicalization and further improves training efficiency. We instantiate the framework for molecular graph generation under $S_n \times SE(3)$ symmetries. By leveraging geometric spectra-based canonicalization and mild positional encodings, canonical diffusion significantly outperforms equivariant baselines in 3D molecule generation tasks, with similar or even less computation. Moreover, with a novel architecture Canon, CanonFlow achieves state-of-the-art performance on the challenging GEOM-DRUG dataset, and the advantage remains large in few-step generation.

Deep generative models have achieved great success in producing high-quality samples, making them a central tool across machine learning applications. Beyond sample quality, an important yet less systematically studied question is whether trained generative models faithfully capture the diversity of the underlying data distribution. In this work, we address this question by directly comparing the diversity of samples generated by state-of-the-art models with that of test samples drawn from the target data distribution, using recently proposed reference-free entropy-based diversity scores, Vendi and RKE. Across multiple benchmark datasets, we find that test data consistently attains substantially higher Vendi and RKE diversity scores than the generated samples, suggesting a systematic downward diversity bias in modern generative models. To understand the origin of this bias, we analyze the finite-sample behavior of entropy-based diversity scores and show that their expected values increase with sample size, implying that diversity estimated from finite training sets could inherently underestimate the diversity of the true distribution. As a result, optimizing the generators to minimize divergence to empirical data distributions would induce a loss of diversity. Finally, we discuss potential diversity-aware regularization and guidance strategies based on Vendi and RKE as principled directions for mitigating this bias, and provide empirical evidence suggesting their potential to improve the results.

Leveraging representation encoders for generative modeling offers a path for efficient, high-fidelity synthesis. However, standard diffusion transformers fail to converge on these representations directly. While recent work attributes this to a capacity bottleneck proposing computationally expensive width scaling of diffusion transformers we demonstrate that the failure is fundamentally geometric. We identify Geometric Interference as the root cause: standard Euclidean flow matching forces probability paths through the low-density interior of the hyperspherical feature space of representation encoders, rather than following the manifold surface. To resolve this, we propose Riemannian Flow Matching with Jacobi Regularization (RJF). By constraining the generative process to the manifold geodesics and correcting for curvature-induced error propagation, RJF enables standard Diffusion Transformer architectures to converge without width scaling. Our method RJF enables the standard DiT-B architecture (131M parameters) to converge effectively, achieving an FID of 3.37 where prior methods fail to converge. Code: https://github.com/amandpkr/RJF

We study generation in separable metric instance spaces. We extend the language generation framework from Kleinberg and Mullainathan [2024] beyond countable domains by defining novelty through metric separation and allowing asymmetric novelty parameters for the adversary and the generator. We introduce the $(\varepsilon,\varepsilon')$-closure dimension, a scale-sensitive analogue of closure dimension, which yields characterizations of uniform and non-uniform generatability and a sufficient condition for generation in the limit. Along the way, we identify a sharp geometric contrast. Namely, in doubling spaces, including all finite-dimensional normed spaces, generatability is stable across novelty scales and invariant under equivalent metrics. In general metric spaces, however, generatability can be highly scale-sensitive and metric-dependent; even in the natural infinite-dimensional Hilbert space $\ell^2$, all notions of generation may fail abruptly as the novelty parameters vary.

Sampling from unnormalized probability densities is a central challenge in computational science. Boltzmann generators are generative models that enable independent sampling from the Boltzmann distribution of physical systems at a given temperature. However, their practical success depends on data-efficient training, as both simulation data and target energy evaluations are costly. To this end, we propose off-policy log-dispersion regularization (LDR), a novel regularization framework that builds on a generalization of the log-variance objective. We apply LDR in the off-policy setting in combination with standard data-based training objectives, without requiring additional on-policy samples. LDR acts as a shape regularizer of the energy landscape by leveraging additional information in the form of target energy labels. The proposed regularization framework is broadly applicable, supporting unbiased or biased simulation datasets as well as purely variational training without access to target samples. Across all benchmarks, LDR improves both final performance and data efficiency, with sample efficiency gains of up to one order of magnitude.

Modern generative models based on diffusion or flow matching transport noise toward the data manifold, without representing the data manifold explicitly. We propose Distance Marching that explores an alternative design by explicitly modeling a distance field of the data manifold. A distance field offers a geometric view of generation as a descent process toward the data manifold using only the current position. The negative gradient of the distance field provides the direction, and the distance value can optionally determine the step size. However, sparse training data in high-dimensional space make it difficult to identify the closest points on the data manifold needed to train the distance field. To address this challenge, we design new loss functions that emphasize denoising targets from closer data points. Our analysis shows that, without time input, emphasizing closer targets reduces denoising ambiguity and the bias toward the data mean. This distance-field perspective naturally motivates two principled deterministic samplers, showing fast convergence in inference. As a result, our model is the only one without time input that remains competitive with time-conditioned models across datasets and architectures. Moreover, our distance prediction is also helpful for early stopping during sampling and for out-of-distribution detection. We hope distance-field modeling can open a promising direction for exploring generative modeling from a geometric perspective.

Generative models have enjoyed widespread success in a variety of applications. However, they encounter inherent mathematical limitations in modeling distributions where samples are constrained by equalities, as is frequently the setting in scientific domains. In this work, we develop a computationally cheap, mathematically justified, and highly flexible distributional modification for combating known pitfalls in equality-constrained generative models. We propose perturbing the data distribution in a constraint-aware way such that the new distribution has support matching the ambient space dimension while still implicitly incorporating underlying manifold geometry. Through theoretical analyses and empirical evidence on several representative tasks, we illustrate that our approach consistently enables data distribution recovery and stable sampling with both diffusion models and normalizing flows.

Diffusion models achieve remarkable generation quality, yet face a fundamental challenge known as memorization, where generated samples can replicate training samples exactly. We develop a theoretical framework to explain this phenomenon by showing that the empirical score function (the score function corresponding to the empirical distribution) is a weighted sum of the score functions of Gaussian distributions, in which the weights are sharp softmax functions. This structure causes individual training samples to dominate the score function, resulting in sampling collapse. In practice, approximating the empirical score function with a neural network can partially alleviate this issue and improve generalization. Our theoretical framework explains why: In training, the neural network learns a smoother approximation of the weighted sum, allowing the sampling process to be influenced by local manifolds rather than single points. Leveraging this insight, we propose two novel methods to further enhance generalization: (1) Noise Unconditioning enables each training sample to adaptively determine its score function weight to increase the effect of more training samples, thereby preventing single-point dominance and mitigating collapse. (2) Temperature Smoothing introduces an explicit parameter to control the smoothness. By increasing the temperature in the softmax weights, we naturally reduce the dominance of any single training sample and mitigate memorization. Experiments across multiple datasets validate our theoretical analysis and demonstrate the effectiveness of the proposed methods in improving generalization while maintaining high generation quality.

Tactile memory, the ability to store and retrieve touch-based experience, is critical for contact-rich tasks such as key insertion under uncertainty. To replicate this capability, we introduce Tactile Memory with Soft Robot (TaMeSo-bot), a system that integrates a soft wrist with tactile retrieval-based control to enable safe and robust manipulation. The soft wrist allows safe contact exploration during data collection, while tactile memory reuses past demonstrations via retrieval for flexible adaptation to unseen scenarios. The core of this system is the Masked Tactile Trajectory Transformer (MAT$^\text{3}$), which jointly models spatiotemporal interactions between robot actions, distributed tactile cues, force-torque measurements, and proprioceptive signals. Through masked token prediction, MAT$^\text{3}$ learns rich spatiotemporal representations by inferring missing sensory information from context, autonomously extracting task-relevant features without explicit subtask segmentation. We validate our approach on peg-in-hole tasks with diverse pegs and conditions in real-robot experiments. Our extensive evaluation demonstrates that MAT$^\text{3}$ achieves higher success rates than the baselines over all conditions and shows remarkable capability to adapt to unseen pegs and conditions.

Accurate and efficient detector simulation is essential for modern collider experiments. To reduce the high computational cost, various fast machine learning surrogate models have been proposed. Traditional surrogate models for calorimeter shower modeling train separate networks for each particle species, limiting scalability and reuse. We introduce AllShowers, a unified generative model that simulates calorimeter showers across multiple particle types using a single generative model. AllShowers is a continuous normalizing flow model with a Transformer architecture, enabling it to generate complex spatial and energy correlations in variable-length point cloud representations of showers. Trained on a diverse dataset of simulated showers in the highly granular ILD detector, the model demonstrates the ability to generate realistic showers for electrons, photons, and charged and neutral hadrons across a wide range of incident energies and angles without retraining. In addition to unifying shower generation for multiple particle types, AllShowers surpasses the fidelity of previous single-particle-type models for hadronic showers. Key innovations include the use of a layer embedding, allowing the model to learn all relevant calorimeter layer properties; a custom attention masking scheme to reduce computational demands and introduce a helpful inductive bias; and a shower- and layer-wise optimal transport mapping to improve training convergence and sample quality. AllShowers marks a significant step towards a universal model for calorimeter shower simulations in collider experiments.

This paper motivates the use of random-bridges -- stochastic processes conditioned to take target distributions at fixed timepoints -- in the realm of generative modelling. Herein, random-bridges act as stochastic transports between two probability distributions when appropriately initialized, displaying either Markovian or non-Markovian, and continuous, discontinuous, or hybrid patterns depending on the choice of the driving process. We show how one can start from general probabilistic statements and branch out into specific representations for learning and simulation algorithms. Essentially, our framework formulates stochastic transports as noisy information processes governing the dynamic translation and propagation of information between arbitrary probability measures. Our theoretical and algorithmic presentation details training and simulation frameworks for both Gaussian and Lévy (specifically, Gamma, Poisson and Stable-$1/2$) random-bridge transports. The empirical validation focuses on the Gaussian case for demonstration and comparison purposes, where our construct delivers high-quality samples, achieving competitive Fréchet Inception Distance scores compared to widely used denoising diffusions and flow matching frameworks.

We show that nonequilibrium dynamics can play a constructive role in unsupervised machine learning by inducing the spontaneous emergence of latent-state cycles. We introduce a model in which visible and hidden variables interact through two independently parametrized transition matrices, defining a Markov chain whose steady state is intrinsically out of equilibrium. Likelihood maximization drives this system toward nonequilibrium steady states with finite entropy production, reduced self-transition probabilities, and persistent probability currents in the latent space. These cycles are not imposed by the architecture but arise from training, and models that develop them reproduce the empirical distribution of data classes more faithfully, with a clear correlation between agreement with the data and entropy production. Compared with equilibrium approaches such as restricted Boltzmann machines, our model breaks the detailed balance between the forward and backward conditional transitions and relies on a log-likelihood gradient that depends explicitly on the last two steps of the Markov chain. Hence, this exploration of the interface between nonequilibrium statistical physics and modern machine learning suggests that introducing irreversibility into latent-variable models can improve the fidelity of the generated data distribution.

It has been hypothesized that achieving the data efficiency of human visual perception requires a generative approach in which internal representations result from inverting a decoder. Yet today's most successful vision models are non-generative, relying on an encoder that maps images to representations without decoder inversion. This raises the question of whether generation is necessary for data-efficient machine perception. To address this, we study to what extent generative and non-generative methods can achieve compositional generalization, a hallmark of human data efficiency. Under a compositional generative process, we formally characterize the inductive biases required for compositional generalization in decoder-based (generative) and encoder-based (non-generative) methods. We show theoretically that the inductive biases required for an encoder are substantially more complex and generally infeasible to impose explicitly through architectural constraints or regularization. By contrast, the decoder biases take a simple form that can be enforced directly. These results suggest that compositional generalization may be substantially easier to achieve through a generative paradigm of learning and inverting a decoder rather than learning an encoder directly. We examine our theoretical findings empirically by training a range of generative and non-generative methods on synthetic image data. We find that non-generative methods often fail to generalize compositionally and require large-scale pretraining to improve generalization. By comparison, generative methods yield gains in generalization without requiring additional data.

Most visual generative models compress images into a latent space before applying diffusion or autoregressive modelling. Yet, existing approaches such as VAEs and foundation model aligned encoders implicitly constrain the latent space without explicitly shaping its distribution, making it unclear which types of distributions are optimal for modeling. We introduce Distribution-Matching VAE (DMVAE), which explicitly aligns the encoder's latent distribution with an arbitrary reference distribution via a distribution matching constraint. This generalizes beyond the Gaussian prior of conventional VAEs, enabling alignment with distributions derived from self-supervised features, diffusion noise, or other prior distributions. With DMVAE, we can systematically investigate which latent distributions are more conducive to modeling, and we find that SSL-derived distributions provide an excellent balance between reconstruction fidelity and modeling efficiency, reaching gFID equals 3.2 on ImageNet with only 64 training epochs. Our results suggest that choosing a suitable latent distribution structure (achieved via distribution-level alignment), rather than relying on fixed priors, is key to bridging the gap between easy-to-model latents and high-fidelity image synthesis. Code is avaliable at https://github.com/sen-ye/dmvae.