Diffusion-based models decompose sampling into many small Gaussian denoising steps -- an assumption that breaks down when generation is compressed to a few coarse transitions. Existing few-step methods address this through distillation, consistency training, or adversarial objectives, but sacrifice the likelihood framework in the process. We introduce Normalizing Trajectory Models (NTM), which models each reverse step as an expressive conditional normalizing flow with exact likelihood training. Architecturally, NTM combines shallow invertible blocks within each step with a deep parallel predictor across the trajectory, forming an end-to-end network trainable from scratch or initializable from pretrained flow-matching models. Its exact trajectory likelihood further enables self-distillation: a lightweight denoiser trained on the model's own score produces high-quality samples in four steps. On text-to-image benchmarks, NTM matches or outperforms strong image generation baselines in just four sampling steps while uniquely retaining exact likelihood over the generative trajectory.
Federal University of Ceará（塞阿拉联邦大学）；National Department of Public Security（国家公共安全部）
The task of detecting anomalous data patterns is as important in practical applications as challenging. In the context of spatial data, recognition of unexpected trajectories brings additional difficulties, such as high dimensionality and varying pattern lengths. We aim to tackle such a problem from a probability density estimation point of view, since it provides an unsupervised procedure to identify out of distribution samples. More specifically, we pursue an approach based on normalizing flows, a recent framework that enables complex density estimation from data with neural networks. Our pr…
The University of Texas at Austin（德克萨斯大学奥斯汀分校）；Aurora Innovation（奥罗拉创新公司）
A popular way to plan trajectories in dynamic urban scenarios for Autonomous Vehicles is to rely on explicitly specified and hand crafted cost functions, coupled with random sampling in the trajectory space to find the minimum cost trajectory. Such methods require a high number of samples to find a low-cost trajectory and might end up with a highly suboptimal trajectory given the planning time budget. We explore the application of normalizing flows for improving the performance of trajectory planning for autonomous vehicles (AVs). Our key insight is to learn a sampling policy in a low-dimensio…
fortiss GmbH, Research Institute of the Free State of Bavaria（巴伐利亚自由州研究机构fortiss有限公司）；Technical University of Munich（慕尼黑工业大学）
The future motion of traffic participants is inherently uncertain. To plan safely, therefore, an autonomous agent must take into account multiple possible trajectory outcomes and prioritize them. Recently, this problem has been addressed with generative neural networks. However, most generative models either do not learn the true underlying trajectory distribution reliably, or do not allow predictions to be associated with likelihoods. In our work, we model motion prediction directly as a density estimation problem with a normalizing flow between a noise distribution and the future motion dist…
Institute of Information Science, Academia Sinica（中央研究院资讯科学研究所）；Department of Computer Science and Engineering, Texas A&M University（德州农工大学计算机科学与工程系）
In this paper, we propose an approach to effectively accelerating the computation of continuous normalizing flow (CNF), which has been proven to be a powerful tool for the tasks such as variational inference and density estimation. The training time cost of CNF can be extremely high because the required number of function evaluations (NFE) for solving corresponding ordinary differential equations (ODE) is very large. We think that the high NFE results from large truncation errors of solving ODEs. To address the problem, we propose to add a regularization. The regularization penalizes the diffe…
TrajFlow: A Generative Framework for Occupancy Density Estimation Using Normalizing Flows
Department of Computer Science University of Minnesota（计算机科学系明尼苏达大学）；Department of Civil Environmental and Geo- Engineering University of Minnesota（土木环境与地球工程系明尼苏达大学）
University of Pennsylvania（宾夕法尼亚大学）；MIT CSAIL（麻省理工学院计算机科学与人工智能实验室）
Forecasting complex vehicle and pedestrian multi-modal distributions requires powerful probabilistic approaches. Normalizing flows (NF) have recently emerged as an attractive tool to model such distributions. However, a key drawback is that independent samples drawn from a flow model often do not adequately capture all the modes in the underlying distribution. We propose Likelihood-Based Diverse Sampling (LDS), a method for improving the quality and the diversity of trajectory samples from a pre-trained flow model. Rather than producing individual samples, LDS produces a set of trajectories in…
We present an algorithm for Inverse Reinforcement Learning (IRL) from expert state observations only. Our approach decouples reward modelling from policy learning, unlike state-of-the-art adversarial methods which require updating the reward model during policy search and are known to be unstable and difficult to optimize. Our method, IL-flOw, recovers the expert policy by modelling state-state transitions, by generating rewards using deep density estimators trained on the demonstration trajectories, avoiding the instability issues of adversarial methods. We demonstrate that using the state tr…
MGF: Mixed Gaussian Flow for Diverse Trajectory Prediction
Zhejiang University（浙江大学）；Shanghai AI Laboratory（上海人工智能实验室）；Carnegie Mellon University（卡内基梅隆大学）；The Chinese University of Hong Kong（香港中文大学）；CPII under InnoHK（InnoHK下属CPII）
University of California, Los Angeles（加利福尼亚大学洛杉矶分校）；Baidu Research（百度研究院）；iSee Inc.（iSee公司）
The problem of continuous inverse optimal control (over finite time horizon) is to learn the unknown cost function over the sequence of continuous control variables from expert demonstrations. In this article, we study this fundamental problem in the framework of energy-based model, where the observed expert trajectories are assumed to be random samples from a probability density function defined as the exponential of the negative cost function up to a normalizing constant. The parameters of the cost function are learned by maximum likelihood via an "analysis by synthesis" scheme, which iterat…
