Abstract
In many data analysis tasks, one is often confronted with very high dimensional data. The manifold assumption, which states that the data is sampled from a submanifold embedded in much higher dimensional Euclidean space, has been widely adopted by many researchers. In the last 15 years, a large number of manifold learning algorithms have been proposed. Many of them rely on the evaluation of the geometrical and topological of the data manifold. In this paper, we present a review of these methods on a novel geometric perspective. We categorize these methods by three main groups: Laplacian-based, Hessian-based, and parallel field-based methods. We show the connection and difference between these three groups on their continuous and discrete counterparts. The discussion is focused on the problem of dimensionality reduction and semi-supervised learning.
Highlights
In many data analysis tasks, one is often confronted with very high dimensional data
A linear map X : C∞(M) → R is called a derivation at p if it satisfies: X(fg) = f (p)Xg + g(p)Xf for all smooth functions f, g ∈ C∞(M)
Previous studies focus on using differential operators on the manifold to construct a regularization term on the unlabeled data. These methods can be roughly classified into three categories: Laplacian regularization, Hessian regularization, and parallel field regularization
Summary
In many data analysis tasks, one is often confronted with very high dimensional data. Estimating and extracting the low-dimensional manifold structure, or the intrinsic topological and geometrical properties of the data manifold, become a crucial problem These problems are often referred to as manifold learning (Belkin and Niyogi 2007). PCA finds the directions along which the data has maximum variance These linear methods may fail to recover the intrinsic manifold structure when the data manifold is not a low-dimensional subspace or an affine manifold. Vector diffusion maps (VDM; Singer and Wu 2012) and parallel field embedding (PFE; Lin et al 2013) are much recent works which employ the vector fields to study the metric of the manifold Among many of these methods, the core ideas of learning the manifold structure are based on differential operators. We try to give a rigorous derivation of these methods and provide some new insights for future work
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