Learning Interaction Kernels for Agent Systems on Riemannian Manifolds

2021-01-30 · Mauro Maggioni, Jason Miller, Hongda Qiu, Ming Zhong

Interacting agent and particle systems are extensively used to model complex phenomena in science and engineering. We consider the problem of learning interaction kernels in these dynamical systems constrained to evolve on Riemannian manifolds from given trajectory data. The models we consider are based on interaction kernels depending on pairwise Riemannian distances between agents, with agents interacting locally along the direction of the shortest geodesic connecting them. We show that our estimators converge at a rate that is independent of the dimension of the state space, and derive bounds on the trajectory estimation error, on the manifold, between the observed and estimated dynamics. We demonstrate the performance of our estimator on two classical first order interacting systems: Opinion Dynamics and a Predator-Swarm system, with each system constrained on two prototypical manifolds, the $2$-dimensional sphere and the Poincar\'e disk model of hyperbolic space.

📄 PDF Abstract BibTeX arXiv:2102.00327

Code (0)

등록된 구현이 없습니다.

Methods 이 논문이 사용한 방법론

Dense Connections Dense Connections, or Fully Connected Connections, are a type of layer in a deep neural network that use a linear operation where every input is connected to every output…

Feedforward Network A Feedforward Network, or a Multilayer Perceptron (MLP), is a neural network with solely densely connected layers. This is the classic neural network architecture of the…

PNAS Progressive Neural Architecture Search, or PNAS, is a method for learning the structure of convolutional neural networks (CNNs). It uses a sequential model-based…

Vector-valued Gaussian Processes on Riemannian Manifolds via Gauge Independent Projected Kernels

2021-10-27 · NeurIPS 2021 12 · Michael Hutchinson, Alexander Terenin, Viacheslav Borovitskiy, So Takao 외

Gaussian processes are machine learning models capable of learning unknown functions in a way that represents uncertainty, thereby facilitating construction of optimal decision-making systems. Motivated by a desire to de…

BIG-bench Machine Learning Decision Making Gaussian Processes Variational Inference

Kernel Methods on Riemannian Manifolds with Gaussian RBF Kernels

2014-11-30 · Sadeep Jayasumana, Richard Hartley, Mathieu Salzmann, Hongdong Li 외

In this paper, we develop an approach to exploiting kernel methods with manifold-valued data. In many computer vision problems, the data can be naturally represented as points on a Riemannian manifold. Due to the non-Euc…

Riemannian Coding and Dictionary Learning: Kernels to the Rescue

2015-06-01 · CVPR 2015 6 · Mehrtash Harandi, Mathieu Salzmann

While sparse coding on non-flat Riemannian manifolds has recently become increasingly popular, existing solutions either are dedicated to specific manifolds, or rely on optimization problems that are difficult to solve, …

Dictionary Learning

Optimizing Over Radial Kernels on Compact Manifolds

2014-12-13 · CVPR 2014 6 · Sadeep Jayasumana, Richard Hartley, Mathieu Salzmann, Hongdong Li 외

We tackle the problem of optimizing over all possible positive definite radial kernels on Riemannian manifolds for classification. Kernel methods on Riemannian manifolds have recently become increasingly popular in compu…

General Classification

Strong Uniform Consistency with Rates for Kernel Density Estimators with General Kernels on Manifolds

2020-07-13 · Hau-Tieng Wu, Nan Wu

When analyzing modern machine learning algorithms, we may need to handle kernel density estimation (KDE) with intricate kernels that are not designed by the user and might even be irregular and asymmetric. To handle this…

Density Estimation