# Analyzing the Variety Loss in the Context of Probabilistic Trajectory Prediction

arXiv: https://arxiv.org/abs/1907.10178 · HTML: https://mlsanity.com/paper/1907.10178/

## Key facts

- Cited by 31 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 23 Jul 2019.

## Built on by (most cited first, 30 of 31)

- [Large Scale Interactive Motion Forecasting for Autonomous Driving : The Waymo Open Motion Dataset](https://mlsanity.com/paper/2104.10133/) (2021)
- [The Waymo Open Sim Agents Challenge](https://mlsanity.com/paper/2305.12032/) (2023)
- [Trajectory Prediction with Latent Belief Energy-Based Model](https://mlsanity.com/paper/2104.03086/) (2021)
- [Uncertainty-aware State Space Transformer for Egocentric 3D Hand Trajectory Forecasting](https://mlsanity.com/paper/2307.08243/) (2023)
- [End-to-End Trajectory Distribution Prediction Based on Occupancy Grid Maps](https://mlsanity.com/paper/2203.16910/) (2022)
- [Unlimited Neighborhood Interaction for Heterogeneous Trajectory Prediction](https://mlsanity.com/paper/2108.00238/) (2021)
- [Euro-PVI: Pedestrian Vehicle Interactions in Dense Urban Centers](https://mlsanity.com/paper/2106.12442/) (2021)
- [UniFolding: Towards Sample-efficient, Scalable, and Generalizable Robotic Garment Folding](https://mlsanity.com/paper/2311.01267/) (2023)
- [HYPER: Learned Hybrid Trajectory Prediction via Factored Inference and Adaptive Sampling](https://mlsanity.com/paper/2110.02344/) (2021)
- [Diverse Multiple Trajectory Prediction Using a Two-stage Prediction Network Trained with Lane Loss](https://mlsanity.com/paper/2206.08641/) (2022)
- [Intention-aware Denoising Diffusion Model for Trajectory Prediction](https://mlsanity.com/paper/2403.09190/) (2024)
- [SICNav-Diffusion: Safe and Interactive Crowd Navigation with Diffusion Trajectory Predictions](https://mlsanity.com/paper/2503.08858/) (2025)
- [Deep Time Series Forecasting with Shape and Temporal Criteria](https://mlsanity.com/paper/2104.04610/) (2021)
- [Vision-based Multi-future Trajectory Prediction: A Survey](https://mlsanity.com/paper/2302.10463/) (2023)
- [SUPR-GAN: SUrgical PRediction GAN for Event Anticipation in Laparoscopic and Robotic Surgery](https://mlsanity.com/paper/2105.04642/) (2021)
- [ModeSeq: Taming Sparse Multimodal Motion Prediction with Sequential Mode Modeling](https://mlsanity.com/paper/2411.11911/) (2024)
- [Computational Teaching for Driving via Multi-Task Imitation Learning](https://mlsanity.com/paper/2410.01608/) (2024)
- [A Hierarchical Hybrid Learning Framework for Multi-agent Trajectory Prediction](https://mlsanity.com/paper/2303.12274/) (2023)
- [TrajPRed: Trajectory Prediction with Region-based Relation Learning](https://mlsanity.com/paper/2404.06971/) (2024)
- [Conformal Trajectory Prediction with Multi-View Data Integration in Cooperative Driving](https://mlsanity.com/paper/2408.00374/) (2024)
- [Trajectory Forecasting through Low-Rank Adaptation of Discrete Latent Codes](https://mlsanity.com/paper/2405.20743/) (2024)
- [STEP: Structured Training and Evaluation Platform for benchmarking trajectory prediction models](https://mlsanity.com/paper/2509.14801/) (2025)
- [Single Shot Multitask Pedestrian Detection and Behavior Prediction](https://mlsanity.com/paper/2101.02232/) (2021)
- [Reliable Probabilistic Human Trajectory Prediction for Autonomous Applications](https://mlsanity.com/paper/2410.06905/) (2024)
- [Pedestrian motion prediction evaluation for urban autonomous driving](https://mlsanity.com/paper/2410.16864/) (2024)
- [Socially-Informed Reconstruction for Pedestrian Trajectory Forecasting](https://mlsanity.com/paper/2412.04673/) (2024)
- [Probability Trajectory: One New Movement Description for Trajectory Prediction](https://mlsanity.com/paper/2101.10595/) (2021)
- [JACoP: Joint Alignment for Compliant Multi-Agent Prediction](https://mlsanity.com/paper/2605.11385/) (2026)
- [Diverse Yet Consistent: Context-Guided Diffusion with Energy-Based Joint Refinement for Multi-Agent Motion Prediction](https://mlsanity.com/paper/2605.22017/) (2026)
- [Protocol before progress: leakage-aware evaluation of AIS trajectory prediction](https://mlsanity.com/paper/2609.25827/) (2026)

## Abstract (arXiv)

Trajectory or behavior prediction of traffic agents is an important component of autonomous driving and robot planning in general. It can be framed as a probabilistic future sequence generation problem and recent literature has studied the applicability of generative models in this context. The variety or Minimum over N (MoN) loss, which tries to minimize the error between the ground truth and the closest of N output predictions, has been used in these recent learning models to improve the diversity of predictions. In this work, we present a proof to show that the MoN loss does not lead to the ground truth probability density function, but approximately to its square root instead. We validate this finding with extensive experiments on both simulated toy as well as real world datasets. We also propose multiple solutions to compensate for the dilation to show improvement of log likelihood of the ground truth samples in the corrected probability density function.