conditional normalizing flows

Conditional Flow Matching (CFM) suffers from high variance in vector field estimation and limited generation quality when applied to highly correlated data such as time series. To address this, we propose an implicit stochastic path-based conditional probability flow modeling framework grounded in Bayesian decision theory. Our approach is the first to integrate Gaussian processes (GPs) into the CFM paradigm, leveraging the key property that the derivative of a GP remains a GP—enabling analytical, simulation-free sampling. The method supports joint multi-point modeling and principled incorporation of prior knowledge. Empirical evaluation demonstrates substantial reduction in marginal vector field estimation variance. On benchmark datasets including handwritten digit generation, our method achieves significant improvements in FID and LPIPS scores, confirming concurrent enhancement in both sample fidelity and generalization capability.

Existing conditional flow-based generative models employ generic unimodal noise priors, leading to unnecessarily long source-to-target mapping paths and inefficient sampling. To address this, we propose the **Conditional Centralized Prior (CCP)**: a conditional encoder maps textual or other prompts to modality-specific centers in data space, from which class-adaptive Gaussian priors are constructed—marking the first dynamic prior customization within the flow matching framework. CCP significantly accelerates training convergence and reduces sampling steps while achieving superior performance across FID, KID, and CLIP Score versus baselines, balancing generation quality and efficiency. Our core contribution lies in departing from conventional fixed-prior paradigms by integrating prior design into the conditional modeling process, thereby enhancing both the geometric plausibility and computational efficiency of flow-based generative models.

Existing expert prior elicitation methods struggle to model complex dependency structures and flexibly specify joint distributions. Method: We propose the first end-to-end, nonparametric joint prior learning framework based on normalizing flows. It transforms expert heuristic judgments into a differentiable density estimation task, employs deep normalizing flows to capture high-dimensional nonlinear dependencies, and integrates simulation-based inference for likelihood-free prior calibration. Contribution/Results: This work is the first to systematically introduce normalizing flows into expert elicitation, unifying support for both parametric and nonparametric, as well as independent and joint prior modeling; it further introduces a multi-stage diagnostic evaluation pipeline. Four simulation experiments demonstrate substantial improvements in prior density fidelity and expert interpretability, establishing a more powerful and transparent paradigm for Bayesian prior learning.

Normalized flows (NFs) remain underexploited for density estimation and generative modeling due to architectural complexity and limited scalability. This paper proposes TarFlow—a scalable NF architecture built upon a direction-alternating autoregressive Transformer that directly models pixel-level distributions within image patches. To enhance robustness and sample quality, we introduce Gaussian noise injection during training, post-training denoising, and a unified conditional/unconditional guidance mechanism. TarFlow is the first single-flow model to significantly surpass prior state-of-the-art methods on standard image likelihood estimation benchmarks, while simultaneously achieving sample fidelity and diversity on par with diffusion models. The implementation is publicly available.

This work addresses the challenge that conditional flow matching (CFM) struggles to accurately recover the true data distribution dynamics when modeling probability paths. To overcome this limitation, the authors propose a novel approach that introduces a partial differential equation characterizing the discrepancy between learned and ground-truth probability paths. They formulate a joint objective that simultaneously optimizes both the flow field and its divergence, and for the first time establish a theoretical upper bound linking the total variation error of the probability path to the CFM loss and the divergence loss. This method enables concurrent matching of the flow field and its divergence, achieving significant performance improvements over standard CFM on tasks including dynamical systems modeling, DNA sequence generation, and video synthesis, while preserving computational efficiency during generation.

This work addresses the topological mismatch between standard normal latent variables and complex data distributions, which hinders the training efficiency and generative performance of normalizing flows. To mitigate this issue, the paper introduces, for the first time, a mixture of probabilistic principal component analyzers (MPPCA) as a learnable low-rank latent prior within the normalizing flow framework. This formulation effectively alleviates topological obstructions, simplifies the flow transformation architecture, and enables efficient initialization. The model is trained end-to-end by integrating the expectation-maximization (EM) algorithm with KL divergence minimization. Empirical evaluations on both tabular and image datasets demonstrate that the proposed approach significantly outperforms baseline methods, achieving faster convergence and superior sample quality.

Existing flow matching approaches struggle to jointly model forward generation and reverse classification of multivariate data, lacking consistency in conditional inference. This work proposes a Joint Flow Matching (JFM) framework that assigns symmetric roles to variables at temporal endpoints, thereby constructing a shared joint distribution such that forward and backward integrations naturally correspond to conditional forms of the same joint distribution. JFM is the first method to enable consistent bidirectional conditional inference within continuous normalizing flows, inherently supporting confidence calibration without post-processing and providing an interpretable foundation for discriminative–generative tasks. Experiments demonstrate that JFM achieves competitive classification accuracy on conditional datasets, generates samples highly consistent with the classifier, and yields natively calibrated confidence scores.

This study addresses the challenge of preserving the full conditional distribution of predictors given a response variable in dimension reduction. To this end, it proposes a likelihood-based sufficient dimension reduction (SDR) framework that introduces conditional normalizing flows to the SDR literature for the first time. The method jointly learns a linear projection and a flexible conditional density by maximizing the conditional log-likelihood, employing monotonic rational quadratic spline flows to model complex conditional distributions. The approach is grounded in an interpretable mutual information objective and complemented by a neural Gaussian SDR variant as an auxiliary model. Theoretical analysis establishes Fisher consistency, and empirical evaluations across diverse simulation settings and the UTKFace age prediction task demonstrate accurate recovery of the central subspace, significantly outperforming existing SDR methods and neural Gaussian baselines.

This work addresses the limitations of conventional flow matching methods, which rely on Gaussian noise as auxiliary variables and thus lack flexibility in designing probability paths tailored to diverse generative tasks. The authors propose AuxPath-FM, a novel framework that, for the first time, permits auxiliary variables to follow arbitrary distributions—such as uniform, Laplace, or Rademacher—enabling more general probabilistic paths. Built upon conditional flow matching theory, the method integrates the continuity equation with marginal consistency and introduces learnable time-scaling functions \(a(t)\), \(b(t)\), and \(c(t)\) to modulate the generation trajectory. Experiments demonstrate that AuxPath-FM achieves high-quality generation across various priors and effectively supports structured semantic tasks like label-guided synthesis, confirming both its theoretical generality and practical adaptability.