Aligning Manifolds of Double Pendulum Dynamics Under the Influence of Noise

This study presents the results of a series of simulation experiments that evaluate and compare four different manifold alignment methods under the influence of noise. The data was created by simulating the dynamics of two slightly different double pendulums in three-dimensional space. The method of semi-supervised feature-level manifold alignment using global distance resulted in the most convincing visualisations. However, the semi-supervised feature-level local alignment methods resulted in smaller alignment errors. These local alignment methods were also more robust to noise and faster than the other methods.Comment: The final version will appear in ICONIP 2018. A DOI identifier to the final version will be added to the preprint, as soon as it is availabl

Scalable Dense Non-rigid Structure-from-Motion: A Grassmannian Perspective

This paper addresses the task of dense non-rigid structure-from-motion (NRSfM) using multiple images. State-of-the-art methods to this problem are often hurdled by scalability, expensive computations, and noisy measurements. Further, recent methods to NRSfM usually either assume a small number of sparse feature points or ignore local non-linearities of shape deformations, and thus cannot reliably model complex non-rigid deformations. To address these issues, in this paper, we propose a new approach for dense NRSfM by modeling the problem on a Grassmann manifold. Specifically, we assume the complex non-rigid deformations lie on a union of local linear subspaces both spatially and temporally. This naturally allows for a compact representation of the complex non-rigid deformation over frames. We provide experimental results on several synthetic and real benchmark datasets. The procured results clearly demonstrate that our method, apart from being scalable and more accurate than state-of-the-art methods, is also more robust to noise and generalizes to highly non-linear deformations.Comment: 10 pages, 7 figure, 4 tables. Accepted for publication in Conference on Computer Vision and Pattern Recognition (CVPR), 2018, typos fixed and acknowledgement adde

Simultaneous Learning of Nonlinear Manifold and Dynamical Models for High-dimensional Time Series

The goal of this work is to learn a parsimonious and informative representation for high-dimensional time series. Conceptually, this comprises two distinct yet tightly coupled tasks: learning a low-dimensional manifold and modeling the dynamical process. These two tasks have a complementary relationship as the temporal constraints provide valuable neighborhood information for dimensionality reduction and conversely, the low-dimensional space allows dynamics to be learnt efficiently. Solving these two tasks simultaneously allows important information to be exchanged mutually. If nonlinear models are required to capture the rich complexity of time series, then the learning problem becomes harder as the nonlinearities in both tasks are coupled. The proposed solution approximates the nonlinear manifold and dynamics using piecewise linear models. The interactions among the linear models are captured in a graphical model. By exploiting the model structure, efficient inference and learning algorithms are obtained without oversimplifying the model of the underlying dynamical process. Evaluation of the proposed framework with competing approaches is conducted in three sets of experiments: dimensionality reduction and reconstruction using synthetic time series, video synthesis using a dynamic texture database, and human motion synthesis, classification and tracking on a benchmark data set. In all experiments, the proposed approach provides superior performance.National Science Foundation (IIS 0308213, IIS 0329009, CNS 0202067