# Fast Inference and Update of Probabilistic Density Estimation on Trajectory Prediction

arXiv: https://arxiv.org/abs/2308.08824 · HTML: https://mlsanity.com/paper/2308.08824/

## Key facts

- Cited by 27 papers in the map (data through 14 Oct 2026).
- Builds on 38 of its 74 references that are in the map.
- First posted on arXiv 17 Aug 2023.

## Builds on (most cited first, 30 of 38)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [Diffusion Models Beat GANs on Image Synthesis](https://mlsanity.com/paper/2105.05233/) (2021)
- [Generative Adversarial Networks](https://mlsanity.com/paper/2203.00667/) (2022)
- [Variational Inference with Normalizing Flows](https://mlsanity.com/paper/1505.05770/) (2015)
- [Density estimation using Real NVP](https://mlsanity.com/paper/1605.08803/) (2016)
- [Glow: Generative Flow with Invertible 1x1 Convolutions](https://mlsanity.com/paper/1807.03039/) (2018)
- [DiffWave: A Versatile Diffusion Model for Audio Synthesis](https://mlsanity.com/paper/2009.09761/) (2020)
- [NICE: Non-linear Independent Components Estimation](https://mlsanity.com/paper/1410.8516/) (2014)
- [Social GAN: Socially Acceptable Trajectories with Generative Adversarial Networks](https://mlsanity.com/paper/1803.10892/) (2018)
- [Masked Autoregressive Flow for Density Estimation](https://mlsanity.com/paper/1705.07057/) (2017)
- [Invertible Residual Networks](https://mlsanity.com/paper/1811.00995/) (2018)
- [AgentFormer: Agent-Aware Transformers for Socio-Temporal Multi-Agent Forecasting](https://mlsanity.com/paper/2103.14023/) (2021)
- [Social-STGCNN: A Social Spatio-Temporal Graph Convolutional Neural Network for Human Trajectory Prediction](https://mlsanity.com/paper/2002.11927/) (2020)
- [MADE: Masked Autoencoder for Distribution Estimation](https://mlsanity.com/paper/1502.03509/) (2015)
- [SoPhie: An Attentive GAN for Predicting Paths Compliant to Social and Physical Constraints](https://mlsanity.com/paper/1806.01482/) (2018)
- [Multimodal Trajectory Predictions for Autonomous Driving using Deep Convolutional Networks](https://mlsanity.com/paper/1809.10732/) (2018)
- [Residual Flows for Invertible Generative Modeling](https://mlsanity.com/paper/1906.02735/) (2019)
- [It Is Not the Journey but the Destination: Endpoint Conditioned Trajectory Prediction](https://mlsanity.com/paper/2004.02025/) (2020)
- [Human Motion Trajectory Prediction: A Survey](https://mlsanity.com/paper/1905.06113/) (2019)
- [Convolutional Social Pooling for Vehicle Trajectory Prediction](https://mlsanity.com/paper/1805.06771/) (2018)
- [Spatio-Temporal Graph Transformer Networks for Pedestrian Trajectory Prediction](https://mlsanity.com/paper/2005.08514/) (2020)
- [Social Attention: Modeling Attention in Human Crowds](https://mlsanity.com/paper/1710.04689/) (2017)
- [Stochastic Trajectory Prediction via Motion Indeterminacy Diffusion](https://mlsanity.com/paper/2203.13777/) (2022)
- [PRECOG: PREdiction Conditioned On Goals in Visual Multi-Agent Settings](https://mlsanity.com/paper/1905.01296/) (2019)
- [From Goals, Waypoints & Paths To Long Term Human Trajectory Forecasting](https://mlsanity.com/paper/2012.01526/) (2020)
- [Stochastic Normalizing Flows](https://mlsanity.com/paper/2002.06707/) (2020)
- [MotionNet: Joint Perception and Motion Prediction for Autonomous Driving Based on Bird's Eye View Maps](https://mlsanity.com/paper/2003.06754/) (2020)
- [GRIP++: Enhanced Graph-based Interaction-aware Trajectory Prediction for Autonomous Driving](https://mlsanity.com/paper/1907.07792/) (2019)
- [SurVAE Flows: Surjections to Bridge the Gap between VAEs and Flows](https://mlsanity.com/paper/2007.02731/) (2020)

## Built on by (most cited first, 27 of 27)

- [SingularTrajectory: Universal Trajectory Predictor Using Diffusion Model](https://mlsanity.com/paper/2403.18452/) (2024)
- [Can Language Beat Numerical Regression? Language-Based Multimodal Trajectory Prediction](https://mlsanity.com/paper/2403.18447/) (2024)
- [SocialCircle: Learning the Angle-based Social Interaction Representation for Pedestrian Trajectory Prediction](https://mlsanity.com/paper/2310.05370/) (2023)
- [LG-Traj: LLM Guided Pedestrian Trajectory Prediction](https://mlsanity.com/paper/2403.08032/) (2024)
- [Sports-Traj: A Unified Trajectory Generation Model for Multi-Agent Movement in Sports](https://mlsanity.com/paper/2405.17680/) (2024)
- [Physical Plausibility-aware Trajectory Prediction via Locomotion Embodiment](https://mlsanity.com/paper/2503.17267/) (2025)
- [SocialCircle+: Learning the Angle-based Conditioned Interaction Representation for Pedestrian Trajectory Prediction](https://mlsanity.com/paper/2409.14984/) (2024)
- [Continuous Locomotive Crowd Behavior Generation](https://mlsanity.com/paper/2504.04756/) (2025)
- [MGF: Mixed Gaussian Flow for Diverse Trajectory Prediction](https://mlsanity.com/paper/2402.12238/) (2024)
- [Resonance: Learning to Predict Social-Aware Pedestrian Trajectories as Co-Vibrations](https://mlsanity.com/paper/2412.02447/) (2024)
- [CacheFlow: Fast Human Motion Prediction by Cached Normalizing Flow](https://mlsanity.com/paper/2505.13140/) (2025)
- [RealTraj: Towards Real-World Pedestrian Trajectory Forecasting](https://mlsanity.com/paper/2411.17376/) (2024)
- [Who Walks With You Matters: Perceiving Social Interactions with Groups for Pedestrian Trajectory Prediction](https://mlsanity.com/paper/2412.02395/) (2024)
- [EgoCogNav: Cognition-aware Human Egocentric Navigation](https://mlsanity.com/paper/2511.17581/) (2025)
- [Reliable Probabilistic Human Trajectory Prediction for Autonomous Applications](https://mlsanity.com/paper/2410.06905/) (2024)
- [Reverberation: Learning the Latencies Before Forecasting Trajectories](https://mlsanity.com/paper/2511.11164/) (2025)
- [MMCM: Multimodality-aware Metric using Clustering-based Modes for Probabilistic Human Motion Prediction](https://mlsanity.com/paper/2511.15179/) (2025)
- [AGMA: Adaptive Gaussian Mixture Anchors for Prior-Guided Multimodal Human Trajectory Forecasting](https://mlsanity.com/paper/2602.04204/) (2026)
- [DD-MDN: Human Trajectory Forecasting with Diffusion-Based Dual Mixture Density Networks and Uncertainty Self-Calibration](https://mlsanity.com/paper/2602.11214/) (2026)
- [Human Motion Prediction via Test-domain-aware Adaptation with Easily-available Human Motions Estimated from Videos](https://mlsanity.com/paper/2505.07301/) (2025)
- [Selective Social-Interaction via Individual Importance for Fast Human Trajectory Prediction](https://mlsanity.com/paper/2506.18291/) (2025)
- [Sparse Compositional Flow Matching by geometric assembly from motion primitives](https://mlsanity.com/paper/2605.23341/) (2026)
- [PRISA: Proactive Infrastructure LiDAR Framework for Intersection Safety Assessment](https://mlsanity.com/paper/2607.16156/) (2026)
- [A Unified Framework for Trajectory Prediction with Explicit Planning and Reaction Decomposition](https://mlsanity.com/paper/2608.05673/) (2026)
- [Gaussian-Mixture Latent Flow for Stochastic 3D Human Motion Prediction](https://mlsanity.com/paper/2608.21093/) (2026)
- [GEAR: From Dynamic Encoding to Dynamic Activation in Social Trajectory Prediction](https://mlsanity.com/paper/2609.13778/) (2026)
- [PRISM: Predictive Representation of Interaction Style and Motion for Social Robot Navigation](https://mlsanity.com/paper/2609.18125/) (2026)

## Abstract (arXiv)

Safety-critical applications such as autonomous vehicles and social robots require fast computation and accurate probability density estimation on trajectory prediction. To address both requirements, this paper presents a new normalizing flow-based trajectory prediction model named FlowChain. FlowChain is a stack of conditional continuously-indexed flows (CIFs) that are expressive and allow analytical probability density computation. This analytical computation is faster than the generative models that need additional approximations such as kernel density estimation. Moreover, FlowChain is more accurate than the Gaussian mixture-based models due to fewer assumptions on the estimated density. FlowChain also allows a rapid update of estimated probability densities. This update is achieved by adopting the textitnewest observed position and reusing the flow transformations and its log-det-jacobians that represent the textitmotion trend. This update is completed in less than one millisecond because this reuse greatly omits the computational cost. Experimental results showed our FlowChain achieved state-of-the-art trajectory prediction accuracy compared to previous methods. Furthermore, our FlowChain demonstrated superiority in the accuracy and speed of density estimation. Our code is available at urlhttps://github.com/meaten/FlowChain-ICCV2023