Tangent-based manifold approximation with locally linear models

Abstract

a b s t r a c tIn this paper, we consider the problem of manifold approximation with affine subspaces. Our objective is to discover a set of low dimensional affine subspaces that represent manifold data accurately while preserving the manifold's structure. For this purpose, we employ a greedy technique that partitions manifold samples into groups, which are approximated by low dimensional subspaces. We start by considering each manifold sample as a different group and we use the difference of local tangents to determine appropriate group mergings. We repeat this procedure until we reach the desired number of sample groups. The best low dimensional affine subspaces corresponding to the final groups constitute our approximate manifold representation. Our experiments verify the effectiveness of the proposed scheme and show its superior performance compared to state-of-the-art methods for manifold approximation.

A greedy, adaptive approach to learning geometry of nonlinear manifolds

“…In all our experiments, we compare our algorithm with the stateof-the-art algorithm for learning manifold geometry in the ambient space using tangent spaces [11]. Because the method in [11] uses difference of tangents to merge neighboring tangent planes, we dub it as the 'Merging based on Difference of Tangents' (MDOT) method.…”

“…Because the method in [11] uses difference of tangents to merge neighboring tangent planes, we dub it as the 'Merging based on Difference of Tangents' (MDOT) method. To compare the performance of the two algorithms, we sample 1800 data points from different number of half-turns of a swiss roll.…”

“…However, to ensure the fused planes adhere to the local manifold geometry, not every pair of planes in T is eligible for fusion. Our definition of fusibility of tangent planes is inspired by the definition of fusibility of clusters in [11]. Let NK (x) contain the K-nearest neighbors of x ∈ X red in X red with respect to the Euclidean distance metric.…”

Low-Dimensional Data Representation in Data Analysis

“…Note that the defined Data Manifold Reconstruction problem differs from the Manifold approximation problem, in which an unknown manifold embedded in a highdimensional ambient space must be approximated by some geometrical structure with close geometry, without any 'global parameterization' of the structure. For the latter problem, some solutions are known such as approximations by a simplicial complex [21], by finitely many affine subspaces called 'flats' [22], tangential Delaunay complex [23], k-means and k-flats [24], and others. However, the Manifold approximation methods have a common drawback: they do not find a low-dimensional representation (parameterization) of the DM approximation; such parameterization is usually required in Machine Learning tasks with high-dimensional data.…”

Change-Point Detection for High-Dimensional Time Series With Missing Data

“…While manifold learning has received significant attention in the machine learning literature [9]- [16], online learning of a dynamic manifold remains a significant challenge, both algorithmically and statistically. Most existing methods are "batch", in that they are designed to process a collection of independent observations all lying near the same static submanifold, and all data is available for processing simultaneously.…”