# Reconstruction of Fragmented Trajectories of Collective Motion using Hadamard Deep Autoencoders

arXiv: https://arxiv.org/abs/2110.10428 · HTML: https://mlsanity.com/paper/2110.10428/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 8 of its 41 references that are in the map.
- First posted on arXiv 20 Oct 2021.

## Builds on (most cited first, 8 of 8)

- [TensorFlow: Large-Scale Machine Learning on Heterogeneous Distributed Systems](https://mlsanity.com/paper/1603.04467/) (2016)
- [Multiple Object Tracking: A Literature Review](https://mlsanity.com/paper/1409.7618/) (2014)
- [A Nonlinear Dimensionality Reduction Framework Using Smooth Geodesics](https://mlsanity.com/paper/1707.06757/) (2017)
- [Bounded Manifold Completion](https://mlsanity.com/paper/1912.09026/) (2019)
- [Identifying manifolds underlying group motion in Vicsek agents](https://mlsanity.com/paper/1508.02809/) (2015)
- [Network Topology Mapping from Partial Virtual Coordinates and Graph Geodesics](https://mlsanity.com/paper/1809.03319/) (2018)
- [Detecting phase transitions in collective behavior using manifold's curvature](https://mlsanity.com/paper/1509.07078/) (2015)
- [Blind Image Denoising and Inpainting Using Robust Hadamard Autoencoders](https://mlsanity.com/paper/2101.10876/) (2021)

## Built on by (most cited first, 6 of 6)

- [Recurrent Neural Networks for Dynamical Systems: Applications to Ordinary Differential Equations, Collective Motion, and Hydrological Modeling](https://mlsanity.com/paper/2202.07022/) (2022)
- [Geodesic Gramian Denoising Applied to the Images Contaminated With Noise Sampled From Diverse Probability Distributions](https://mlsanity.com/paper/2203.02600/) (2022)
- [Real-time Forecasting of Time Series in Financial Markets Using Sequentially Trained Many-to-one LSTMs](https://mlsanity.com/paper/2205.04678/) (2022)
- [Fraud Detection Using Optimized Machine Learning Tools Under Imbalance Classes](https://mlsanity.com/paper/2209.01642/) (2022)
- [Efficient Image Denoising by Low-Rank Singular Vector Approximations of Geodesics' Gramian Matrix](https://mlsanity.com/paper/2209.13094/) (2022)
- [Efficient and Robust Remote Sensing Image Denoising Using Randomized Approximation of Geodesics' Gramian on the Manifold Underlying the Patch Space](https://mlsanity.com/paper/2504.10820/) (2025)

## Abstract (arXiv)

Learning dynamics of collectively moving agents such as fish or humans is an active field in research. Due to natural phenomena such as occlusion and change of illumination, the multi-object methods tracking such dynamics might lose track of the agents where that might result fragmentation in the constructed trajectories. Here, we present an extended deep autoencoder (DA) that we train only on fully observed segments of the trajectories by defining its loss function as the Hadamard product of a binary indicator matrix with the absolute difference between the outputs and the labels. The trajectories of the agents practicing collective motion is low-rank due to mutual interactions and dependencies between the agents that we utilize as the underlying pattern that our Hadamard deep autoencoder (HDA) codes during its training. The performance of our HDA is compared with that of a low-rank matrix completion scheme in the context of fragmented trajectory reconstruction.