# Koopman Methods for Estimation of Animal Motions over Unknown Submanifolds

arXiv: https://arxiv.org/abs/2203.05646 · HTML: https://mlsanity.com/paper/2203.05646/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 9 of its 52 references that are in the map.
- First posted on arXiv 10 Mar 2022.

## Builds on (most cited first, 9 of 9)

- [Modern Koopman Theory for Dynamical Systems](https://mlsanity.com/paper/2102.12086/) (2021)
- [Structural-RNN: Deep Learning on Spatio-Temporal Graphs](https://mlsanity.com/paper/1511.05298/) (2015)
- [Deep representation learning for human motion prediction and classification](https://mlsanity.com/paper/1702.07486/) (2017)
- [Data-driven approximation of the Koopman generator: Model reduction, system identification, and control](https://mlsanity.com/paper/1909.10638/) (2019)
- [On Convergence of Extended Dynamic Mode Decomposition to the Koopman Operator](https://mlsanity.com/paper/1703.04680/) (2017)
- [Eigendecompositions of Transfer Operators in Reproducing Kernel Hilbert Spaces](https://mlsanity.com/paper/1712.01572/) (2017)
- [A review on locomotion robophysics: the study of movement at the intersection of robotics, soft matter and dynamical systems](https://mlsanity.com/paper/1602.04712/) (2016)
- [Learning Theory for Estimation of Animal Motion Submanifolds](https://mlsanity.com/paper/2003.13811/) (2020)
- [Intrinsic and Extrinsic Approximation of Koopman Operators over Manifolds](https://mlsanity.com/paper/2004.05327/) (2020)

## Abstract (arXiv)

This paper introduces a data-dependent approximation of the forward kinematics map for certain types of animal motion models. It is assumed that motions are supported on a low-dimensional, unknown configuration manifold Q that is regularly embedded in high dimensional Euclidean space X:=ℝd. This paper introduces a method to estimate forward kinematics from the unknown configuration submanifold Q to an n-dimensional Euclidean space Y:=ℝn of observations. A known reproducing kernel Hilbert space (RKHS) is defined over the ambient space X in terms of a known kernel function, and computations are performed using the known kernel defined on the ambient space X. Estimates are constructed using a certain data-dependent approximation of the Koopman operator defined in terms of the known kernel on X. However, the rate of convergence of approximations is studied in the space of restrictions to the unknown manifold Q. Strong rates of convergence are derived in terms of the fill distance of samples in the unknown configuration manifold, provided that a novel regularity result holds for the Koopman operator. Additionally, we show that the derived rates of convergence can be applied in some cases to estimates generated by the extended dynamic mode decomposition (EDMD) method. We illustrate characteristics of the estimates for simulated data as well as samples collected during motion capture experiments.