Reconstruction of Agents’ Corrupted Trajectories of Collective Motion Using Low-rank Matrix Completion

Abstract

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.

Recurrent Neural Networks for Dynamical Systems: Applications to Ordinary Differential Equations, Collective Motion, and Hydrological Modeling

“…We generate 30 agents using Eqns. ( 10), ( 11), (12), and (15) with 201 for the time-steps (T ), a rectangular domain with periodic boundary conditions. We set two for the radius of interaction (r d ), 0.05 for the speed of the particles (v (t) i for all t and i), 0.05 for the noise on the orientation ( (t) i for all t and i), and one for the time-step size (δ), see the first row of Fig.…”

“…Our focus is either reconstruct or forecast short-term or long-term trajectories of a dynamical system of interest with given initial conditions. We use three exemplary datasets that are generated from three dynamical systems, namely, the Lorenz attractor 11 , a generalized Vicsek model 12 , and a streamflow model, where those represent three diverse fields, ordinary differential equations, collective motion, and hydrological modeling, respectively. For given orbits that are generated from a Lorenz system with a formulation error, we train an RNN to eliminate the formulation error and reconstruct the correct system's responses.…”

“…For given orbits that are generated from a Lorenz system with a formulation error, we train an RNN to eliminate the formulation error and reconstruct the correct system's responses. Then, we generate a collective motion dataset using the generalized version in 12 , of the classic Vicsek model in 13 , and create two copies of that, one by imposing some additive Gaussian noise and the other by deleting some segments of the trajectories. Here, our goals in these experiments are to eliminate the additive noise from the noisy trajectories.…”