Large language models often improve reasoning by generating explicit chain-of-thought (CoT), demonstrating the importance of intermediate computation. However, textual CoT forces this computation through a discrete, serial, and communication-oriented token stream: each reasoning step must be verbalized before the model can proceed, even when the underlying update is semantic, uncertain, or only partially formed. Latent reasoning offers a higher-bandwidth alternative by performing intermediate computation in compact continuous states before committing to text. Yet existing latent-reasoning methods often sacrifice key advantages that make CoT effective in autoregressive language models, including native left-to-right generation, probabilistic sampling, compatibility with KV-cache decoding, and tractable likelihood estimation. We propose NF-CoT, a latent reasoning framework that preserves these advantages by modeling continuous thoughts with normalizing flows. NF-CoT instantiates a TARFlow-style normalizing flow inside the LLM backbone, defining a tractable probability model over compact continuous thoughts distilled from explicit CoT. Continuous-thought positions are generated by an NF head, while text positions are generated by the standard LM head within the same causal stream. This design provides exact likelihoods for latent thoughts, enables probabilistic left-to-right decoding with the original KV cache, and supports direct policy-gradient optimization in the latent reasoning space. On code-generation benchmarks, NF-CoT improves pass rates over explicit-CoT and prior latent-reasoning baselines while substantially reducing intermediate-reasoning cost.
Normalizing flows are a powerful class of generative models for continuous random variables, showing both strong model flexibility and the potential for non-autoregressive generation. These benefits are also desired when modeling discrete random variables such as text, but directly applying normalizing flows to discrete sequences poses significant additional challenges. We propose a VAE-based generative model which jointly learns a normalizing flow-based distribution in the latent space and a stochastic mapping to an observed discrete space. In this setting, we find that it is crucial for the …
Department of Electrical and Computer Engineering（电气与计算机工程系）；Duke University（杜克大学）
We introduce Projected Latent Markov Chain Monte Carlo (PL-MCMC), a technique for sampling from the high-dimensional conditional distributions learned by a normalizing flow. We prove that a Metropolis-Hastings implementation of PL-MCMC asymptotically samples from the exact conditional distributions associated with a normalizing flow. As a conditional sampling method, PL-MCMC enables Monte Carlo Expectation Maximization (MC-EM) training of normalizing flows from incomplete data. Through experimental tests applying normalizing flows to missing data tasks for a variety of data sets, we demonstrat…
University of California, Santa Barbara（加州大学圣巴巴拉分校）
Recurrent Variational Autoencoder has been widely used for language modeling and text generation tasks. These models often face a difficult optimization problem, also known as the Kullback-Leibler (KL) term vanishing issue, where the posterior easily collapses to the prior, and the model will ignore latent codes in generative tasks. To address this problem, we introduce an improved Wasserstein Variational Autoencoder (WAE) with Riemannian Normalizing Flow (RNF) for text modeling. The RNF transforms a latent variable into a space that respects the geometric characteristics of input space, which…
We study a normalizing flow in the latent space of a top-down generator model, in which the normalizing flow model plays the role of the informative prior model of the generator. We propose to jointly learn the latent space normalizing flow prior model and the top-down generator model by a Markov chain Monte Carlo (MCMC)-based maximum likelihood algorithm, where a short-run Langevin sampling from the intractable posterior distribution is performed to infer the latent variables for each observed example, so that the parameters of the normalizing flow prior and the generator can be updated with …
Normalizing flow (NF) has gained popularity over traditional maximum likelihood based methods due to its strong capability to model complex data distributions. However, the standard approach, which maps the observed data to a normal distribution, has difficulty in handling data distributions with multiple relatively isolated modes. To overcome this issue, we propose a new framework based on variational latent representation to improve the practical performance of NF. The idea is to replace the standard normal latent variable with a more general latent representation, jointly learned via Variat…
McGill University（麦吉尔大学）；Mila（米拉研究所）；University of Toronto（多伦多大学）；Vector Institute（向量研究所）
The choice of approximate posterior distributions plays a central role in stochastic variational inference (SVI). One effective solution is the use of normalizing flows \cut{defined on Euclidean spaces} to construct flexible posterior distributions. However, one key limitation of existing normalizing flows is that they are restricted to the Euclidean space and are ill-equipped to model data with an underlying hierarchical structure. To address this fundamental limitation, we present the first extension of normalizing flows to hyperbolic spaces. We first elevate normalizing flows to hyperbolic …
In this work, we propose a new family of generative flows on an augmented data space, with an aim to improve expressivity without drastically increasing the computational cost of sampling and evaluation of a lower bound on the likelihood. Theoretically, we prove the proposed flow can approximate a Hamiltonian ODE as a universal transport map. Empirically, we demonstrate state-of-the-art performance on standard benchmarks of flow-based generative modeling.
The latent space of normalizing flows must be of the same dimensionality as their output space. This constraint presents a problem if we want to learn low-dimensional, semantically meaningful representations. Recent work has provided compact representations by fitting flows constrained to manifolds, but hasn't defined a density off that manifold. In this work we consider flows with full support in data space, but with ordered latent variables. Like in PCA, the leading latent dimensions define a sequence of manifolds that lie close to the data. We note a trade-off between the flow likelihood an…
Fractal Flow: Hierarchical and Interpretable Normalizing Flow via Topic Modeling and Recursive Strategy
School of Mathematics Harbin Institute of Technology（哈尔滨工业大学数学学院）；School of Mathematics and Institute for Artificial Intelligence Harbin Instiute of Technology（哈尔滨工业大学数学学院和人工智能研究院）；School of Earth and Space Sciences and Institute for Artificial Intelligence Peking University（北京大学地球和空间科学学院和人工智能研究院）
Anomaly detection in time-series via inductive biases in the latent space of conditional normalizing flows
Department of Computer Science Norwegian University of Science and Technology（挪威科学技术大学计算机科学系）；BCAM – Basque Center for Applied Mathematics University of Coimbra, CISUC/LASI, DEI（巴斯克应用数学中心，科英布拉大学，CISUC/LASI，DEI）；IKERBASQUE — Basque Foundation for Science（巴斯克科学基金会）
Normalizing flows transform a latent distribution through an invertible neural network for a flexible and pleasingly simple approach to generative modelling, while preserving an exact likelihood. We propose FlowGMM, an end-to-end approach to generative semi supervised learning with normalizing flows, using a latent Gaussian mixture model. FlowGMM is distinct in its simplicity, unified treatment of labelled and unlabelled data with an exact likelihood, interpretability, and broad applicability beyond image data. We show promising results on a wide range of applications, including AG-News and Ya…
We propose a hierarchical normalizing flow model for generating molecular graphs. The model produces new molecular structures from a single-node graph by recursively splitting every node into two. All operations are invertible and can be used as plug-and-play modules. The hierarchical nature of the latent codes allows for precise changes in the resulting graph: perturbations in the top layer cause global structural changes, while perturbations in the consequent layers change the resulting molecule marginally. The proposed model outperforms existing generative graph models on the distribution l…
Data augmentation is a widely adopted technique for avoiding overfitting when training deep neural networks. However, this approach requires domain-specific knowledge and is often limited to a fixed set of hard-coded transformations. Recently, several works proposed to use generative models for generating semantically meaningful perturbations to train a classifier. However, because accurate encoding and decoding are critical, these methods, which use architectures that approximate the latent-variable inference, remained limited to pilot studies on small datasets. Exploiting the exactly reversi…
Université Paris Cité（巴黎西岱大学）；INRIA（法国国家信息与自动化研究所）；Inserm（法国国家健康与医学研究院）；Centre de Recherche des Cordeliers（科德利耶研究中心）
This paper introduces a new latent variable generative model able to handle high dimensional longitudinal data and relying on variational inference. The time dependency between the observations of an input sequence is modelled using normalizing flows over the associated latent variables. The proposed method can be used to generate either fully synthetic longitudinal sequences or trajectories that are conditioned on several data in a sequence and demonstrates good robustness properties to missing data. We test the model on 6 datasets of different complexity and show that it can achieve better l…
Entropy-Informed Weighting Channel Normalizing Flow for Deep Generative Models
School of Mathematics, South China University of Technology, Guangzhou, China（华南理工大学数学学院）；Shenzhen International Graduate School, Tsinghua University, Shenzhen, China（清华大学深圳国际研究生院）；School of Electronic and Information Engineering, South China University of Technology, Guangzhou, China（华南理工大学电子信息学院）；Department of Electrical Engineering, Columbia University, New York, NY USA（哥伦比亚大学电气工程系）
Hierarchical Gaussian Mixture Normalizing Flow Modeling for Unified Anomaly Detection
School of Electronic Information and Electrical Engineering, Shanghai Jiao Tong University（上海交通大学电子信息与电气工程学院）；School of Intelligent Manufacturing, Wuxi Vocational College of Science and Technology（无锡科技职业学院智能制造学院）；MoE Key Lab of Artificial Intelligence, AI Institute, Shanghai Jiao Tong University（上海交通大学人工智能研究院、教育部人工智能重点实验室）
University of California at Berkeley（加州大学伯克利分校）；International Computer Science Institute（国际计算机科学研究所）；University of Queensland（昆士兰大学）
We introduce stochastic normalizing flows, an extension of continuous normalizing flows for maximum likelihood estimation and variational inference (VI) using stochastic differential equations (SDEs). Using the theory of rough paths, the underlying Brownian motion is treated as a latent variable and approximated, enabling efficient training of neural SDEs as random neural ordinary differential equations. These SDEs can be used for constructing efficient Markov chains to sample from the underlying distribution of a given dataset. Furthermore, by considering families of targeted SDEs with prescr…
Yandex（扬德克斯）；National Research University Higher School of Economics（国立研究型大学高等经济学院）
Normalizing flows are a powerful class of generative models demonstrating strong performance in several speech and vision problems. In contrast to other generative models, normalizing flows are latent variable models with tractable likelihoods and allow for stable training. However, they have to be carefully designed to represent invertible functions with efficient Jacobian determinant calculation. In practice, these requirements lead to overparameterized and sophisticated architectures that are inferior to alternative feed-forward models in terms of inference time and memory consumption. In t…
CONTRA: Conformal Prediction Region via Normalizing Flow Transformation
Department of Statistics and Actuarial Science（统计与精算科学系）；The University of Iowa（爱荷华大学）；Department of Applied Mathematics（应用数学系）；The Hong Kong Polytechnic University（香港理工大学）
Normalizing flows map an independent set of latent variables to their samples using a bijective transformation. Despite the exact correspondence between samples and latent variables, their high level relationship is not well understood. In this paper we characterize the geometric structure of flows using principal manifolds and understand the relationship between latent variables and samples using contours. We introduce a novel class of normalizing flows, called principal manifold flows (PF), whose contours are its principal manifolds, and a variant for injective flows (iPF) that is more effic…
How to generate conditional synthetic data for a domain without utilizing information about its labels/attributes? Our work presents a solution to the above question. We propose a transfer learning-based framework utilizing normalizing flows, coupled with both maximum-likelihood and adversarial training. We model a source domain (labels available) and a target domain (labels unavailable) with individual normalizing flows, and perform domain alignment to a common latent space using adversarial discriminators. Due to the invertible property of flow models, the mapping has exact cycle consistency…
A large part of the expressive speech synthesis literature focuses on learning prosodic representations of the speech signal which are then modeled by a prior distribution during inference. In this paper, we compare different prior architectures at the task of predicting phoneme level prosodic representations extracted with an unsupervised FVAE model. We use both subjective and objective metrics to show that normalizing flow based prior networks can result in more expressive speech at the cost of a slight drop in quality. Furthermore, we show that the synthesized speech has higher variability,…
Real-world data with underlying structure, such as pictures of faces, are hypothesized to lie on a low-dimensional manifold. This manifold hypothesis has motivated state-of-the-art generative algorithms that learn low-dimensional data representations. Unfortunately, a popular generative model, normalizing flows, cannot take advantage of this. Normalizing flows are based on successive variable transformations that are, by design, incapable of learning lower-dimensional representations. In this paper we introduce noisy injective flows (NIF), a generalization of normalizing flows that can go acro…
We introduce CAFLOW, a new diverse image-to-image translation model that simultaneously leverages the power of auto-regressive modeling and the modeling efficiency of conditional normalizing flows. We transform the conditioning image into a sequence of latent encodings using a multi-scale normalizing flow and repeat the process for the conditioned image. We model the conditional distribution of the latent encodings by modeling the auto-regressive distributions with an efficient multi-scale normalizing flow, where each conditioning factor affects image synthesis at its respective resolution sca…
Normalizing flows are prominent deep generative models that provide tractable probability distributions and efficient density estimation. However, they are well known to fail while detecting Out-of-Distribution (OOD) inputs as they directly encode the local features of the input representations in their latent space. In this paper, we solve this overconfidence issue of normalizing flows by demonstrating that flows, if extended by an attention mechanism, can reliably detect outliers including adversarial attacks. Our approach does not require outlier data for training and we showcase the effici…
Rice University（莱斯大学）；NVIDIA（英伟达）；University of Toronto（多伦多大学）；California Institute of Technology（加州理工学院）
Continuous Normalizing Flows (CNFs) have emerged as promising deep generative models for a wide range of tasks thanks to their invertibility and exact likelihood estimation. However, conditioning CNFs on signals of interest for conditional image generation and downstream predictive tasks is inefficient due to the high-dimensional latent code generated by the model, which needs to be of the same size as the input data. In this paper, we propose InfoCNF, an efficient conditional CNF that partitions the latent space into a class-specific supervised code and an unsupervised code that shared among …
IRCAM（声学与音乐研究与协作所）；CNRS（法国国家科学研究中心）；Sorbonne Université（索邦大学）；Laboratorio d’Informatica Musicale (LIM)（音乐信息学实验室）；UNIMI（米兰大学）
The ubiquity of sound synthesizers has reshaped music production and even entirely defined new music genres. However, the increasing complexity and number of parameters in modern synthesizers make them harder to master. Hence, the development of methods allowing to easily create and explore with synthesizers is a crucial need. Here, we introduce a novel formulation of audio synthesizer control. We formalize it as finding an organized latent audio space that represents the capabilities of a synthesizer, while constructing an invertible mapping to the space of its parameters. By using this formu…
Despite their popularity, to date, the application of normalizing flows on categorical data stays limited. The current practice of using dequantization to map discrete data to a continuous space is inapplicable as categorical data has no intrinsic order. Instead, categorical data have complex and latent relations that must be inferred, like the synonymy between words. In this paper, we investigate \emph{Categorical Normalizing Flows}, that is normalizing flows for categorical data. By casting the encoding of categorical data in continuous space as a variational inference problem, we jointly op…
Normalizing flows transform a simple base distribution into a complex target distribution and have proved to be powerful models for data generation and density estimation. In this work, we propose a novel type of normalizing flow driven by a differential deformation of the Wiener process. As a result, we obtain a rich time series model whose observable process inherits many of the appealing properties of its base process, such as efficient computation of likelihoods and marginals. Furthermore, our continuous treatment provides a natural framework for irregular time series with an independent a…
Duke University（杜克大学）；Carnegie Mellon University（卡内基梅隆大学）
Normalizing flows are a widely used class of latent-variable generative models with a tractable likelihood. Affine-coupling (Dinh et al, 2014-16) models are a particularly common type of normalizing flows, for which the Jacobian of the latent-to-observable-variable transformation is triangular, allowing the likelihood to be computed in linear time. Despite the widespread usage of affine couplings, the special structure of the architecture makes understanding their representational power challenging. The question of universal approximation was only recently resolved by three parallel papers (Hu…
Fair representation learning is an attractive approach that promises fairness of downstream predictors by encoding sensitive data. Unfortunately, recent work has shown that strong adversarial predictors can still exhibit unfairness by recovering sensitive attributes from these representations. In this work, we present Fair Normalizing Flows (FNF), a new approach offering more rigorous fairness guarantees for learned representations. Specifically, we consider a practical setting where we can estimate the probability density for sensitive groups. The key idea is to model the encoder as a normali…
Variational Autoencoder with Normalizing flow for X-ray spectral fitting
Department of Physics & Astronomy University of Southampton（物理与天文学系 英国南安普顿大学）；Laboratoire d’astrophysique EPFL（天体物理学实验室 EPFL）；Harvard-Smithsonian Center for Astrophysics（哈佛-史密松天体物理中心）
Helsinki Institute for Information Technology HIIT（赫尔辛基信息技术研究所（HIIT））；University of Helsinki（赫尔辛基大学）
Variational approximations are increasingly based on gradient-based optimization of expectations estimated by sampling. Handling discrete latent variables is then challenging because the sampling process is not differentiable. Continuous relaxations, such as the Gumbel-Softmax for categorical distribution, enable gradient-based optimization, but do not define a valid probability mass for discrete observations. In practice, selecting the amount of relaxation is difficult and one needs to optimize an objective that does not align with the desired one, causing problems especially with models havi…
Normalizing flows are bijective mappings between inputs and latent representations with a fully factorized distribution. They are very attractive due to exact likelihood valuation and efficient sampling. However, their effective capacity is often insufficient since the bijectivity constraint limits the model width. We address this issue by incrementally padding intermediate representations with noise. We precondition the noise in accordance with previous invertible units, which we describe as cross-unit coupling. Our invertible glow-like modules increase the model expressivity by fusing a dens…
Simplifying Flow Matching Transformations with Low-Rank Mixture Models
Stanford Intelligent Systems Laboratory in the Department of Aeronautics and Astronautics at Stanford University（斯坦福大学航空航天系智能系统实验室）；Industrial and Systems Engineering Department at King Fahd University of Petroleum and Minerals（科威特石油矿物大学工业与系统工程系）
ReGuLaR: Variational Latent Reasoning Guided by Rendered Chain-of-Thought
Gaoling School of Artificial Intelligence, Renmin University of China（中国人民大学 Gallagher人工智能学院）；Beijing Key Laboratory of Research on Large Models（北京大型模型研究关键实验室）；Engineering Research Center of Next-Generation Intelligent Search（下一代智能搜索工程研究中心）
Generative adversarial networks (GANs) have proven to be surprisingly efficient for image editing by inverting and manipulating the latent code corresponding to an input real image. This editing property emerges from the disentangled nature of the latent space. In this paper, we identify that the facial attribute disentanglement is not optimal, thus facial editing relying on linear attribute separation is flawed. We thus propose to improve semantic disentanglement with supervision. Our method consists in learning a proxy latent representation using normalizing flows, and we show that this lead…
