Smoothing a topological manifold

Local Smoothing for Manifold Learning

We propose methods for outlier handling and noise reduction using weighted local linear smoothing for a set of noisy points sampled from a nonlinear manifold. The methods can be used by manifold learning methods such as Isomap, LLE and LTSA as a preprocessing step to obtain a more accurate reconstruction of the underlying nonlinear manifolds. Weighted PCA is used as a building block for our met...

Bernhard Brehm Topological Stable Manifold

Local Linear Smoothing for Nonlinear Manifold Learning

In this paper, we develop methods for outlier removal and noise reduction based on weighted local linear smoothing for a set of noisy points sampled from a nonlinear manifold. The methods can be used by manifold learning methods such as Isomap, LLE and LTSA as a preprocessing procedure so as to obtain a more accurate reconstruction of the underlying nonlinear manifolds. Weighted principal compo...

Accurate Integration of Aerosol Predictions by Smoothing on a Manifold

Accurately measuring the aerosol optical depth (AOD) is essential for our understanding of the climate. Currently, AOD can be measured by (i) satellite instruments, which operate on a global scale but have limited accuracies; and (ii) ground-based instruments, which are more accurate but not widely available. Recent approaches focus on integrating measurements from these two sources to compleme...

Consistent Manifold Representation for Topological Data Analysis

For data sampled from an arbitrary density on a manifold embedded in Euclidean space, we introduce the Continuous k-Nearest Neighbors (CkNN) graph construction. We prove that CkNN is the unique unweighted construction that is consistent with the underlying manifold topology in the limit of large data, for compact Riemannian manifolds and a large class of non-compact manifolds. In contrast to pe...