From this video, it is clear that we are effectively tracking the dynamics of the submanifold, and keeping the representation parsimonious so the number of subsets used by our model is proportional to the curvature of the submanifold.
That attendant paper is: Multiscale Online Tracking of Manifolds by Yao Xie, Jiaji Huang, Rebecca Willett
This paper describes a Multiscale Online Union of SubSpaces Estimation (MOUSSE) algorithm for online tracking ofa time-varying manifold. MOUSSE uses linear subsets of lowdimensional hyperplanes to approximate a manifold embedded ina high-dimensional space. Each subset corresponds to the leafnode in a binary tree which encapsulates the multiresolution analysis underlying the proposed algorithm. The tree structure andparameters of the subsets are estimated and sequentially updatedusing a stream of noisy samples. For each update, MOUSSE requires only simple linear computations. The update of each hyperplane in the estimate is computed via gradient descent on theGrassmannian manifold. Numerical simulations demonstrate thestrong performance of MOUSSE in tracking a time-varying manifold.
and a related one: Changepoint detection for high-dimensional time series with missing data by Yao Xie, Jiaji Huang, Rebecca Willett
This paper describes a novel approach to change-point detection when the observed high-dimensional data may have missing elements. The performance of classical methods for change-point detection typically scales poorly with the dimensionality of the data, so that a large number of observations are collected after the true change-point before it can be reliably detected. Furthermore, missing components in the observed data handicap conventional approaches. The proposed method addresses these challenges by modeling the dynamic distribution underlying the data as lying close to a time-varying low-dimensional submanifold embedded within the ambient observation space. Specifically, streaming data is used to track a submanifold approximation, measure deviations from this approximation, and calculate a series of statistics of the deviations for detecting when the underlying manifold has changed in a sharp or unexpected manner. The approach described in this paper leverages several recent results in the field of high-dimensional data analysis, including subspace tracking with missing data, multiscale analysis techniques for point clouds, online optimization, and change-point detection performance analysis. Simulations and experiments highlight the robustness and efficacy of the proposed approach in detecting an abrupt change in an otherwise slowly varying low-dimensional manifold.
The project page for MOUSSE and attendant implementation can be found here.
Yao Xie made a presentation of it for ITA, here it is:
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