2. Local tangent coordinates system

A manifold is a topological space that locally resembles Euclidean space near every point. For example, around each point, there is a neighborhood that is topologically the same as the open unit ball in ℝ<sup>D</sup>. The simplest manifold is a linear manifold, usually called a hyperplane. There exists a tangent space at each point of a nonlinear manifold. The tangent space is a linear manifold which locally approximates the manifold. Suppose there are N points {x1;…;xN} in ℝ<sup>D</sup> residing on a smooth manifold M⊂ℝ<sup>D</sup>, which is the image of a coordinate space Y⊂ℝ<sup>d</sup> under a smooth mapping <sup>ψ</sup> : <sup>Y</sup> ! <sup>ℝ</sup><sup>D</sup>, where <sup>d</sup>≪D. The mapping <sup>ψ</sup> is assumed as a locally isometric embedding. The aim of a NLDR algorithm is to acquire the corresponding lowdimensional representation yi ∈Y of each xi∈M and preserve certain intrinsic structures of data at the same time. Suppose M is smooth such that the tangent space TxðMÞ is well defined at every point x∈M. We can regard the local tangent space as a d-dimensional affine subspace of ℝ<sup>D</sup> which is tangent to M at x. Thus, the tangent space has the natural inner product induced by the embedding M⊂ℝ<sup>D</sup>. Within some neighborhood of x, each point x∈M has a sole closest point in TxðMÞ, and therefore, an orthonormal coordinate system from the corresponding local coordinates on M can be associated with the tangent space.

A manifold can be represented by its coordinates. While the current research of differential geometry focuses on the characterization of the global properties of manifolds, NLDR algorithms, which try to find the coordinate representations of data, only need the local properties of manifolds. In this chapter, we use local coordinates associated with the tangent space to estimate the local objects over the manifold. To acquire the local tangent coordinates, we first perform Principal Component Analysis (PCA) [12] on the points in N ðxiÞ ¼ {xi; xi<sup>1</sup> ;…; xik } that is the local patch built by the point xi and its k nearest neighborhoods, and get d leading PCA eigenvectors <sup>V</sup><sup>i</sup> <sup>¼</sup> {v<sup>i</sup> 1;vi 2;…;vi <sup>d</sup>} which correspond to an orthogonal basis of Txi ðMÞ (the orthogonal basis can be seen as a d-dimensional affine subspace of ℝ<sup>D</sup> which is tangent to M at xi). For high-dimensional data, we employ the trick presented by Turk and Pentland for EigenFaces [13]. Then, we obtain the local tangent coordinates <sup>U</sup><sup>i</sup> <sup>¼</sup> {0;ui 1;…;ui <sup>k</sup>} of the neighborhood N ðxiÞ by projecting the local neighborhoods to this tangent subspace:

$$\mu\_{\dot{j}}^i = \left(V^i\right)^T \left(\mathbf{x}\_{\dot{i}\_{\dot{j}}} - \mathbf{x}\_i\right) \tag{1}$$

An illustration of the local tangent space at xi and the corresponding tangent coordinates system (i.e., the point xij 's local tangent coordinate is u<sup>i</sup> j ) is shown in Figure 1.

Figure 1. Local tangent space and tangent coordinates system.

[2], and local coordinates alignment with global preservation [3]. Local methods mainly include Laplacian eigenmaps (LEM) [4], locally linear embedding (LLE) [5], Hessian eigenmaps (HLLE) [6], local tangent space alignment (LTSA) [7], local linear transformation

Different local approaches try to learn different geometric information of the underlying manifold, since they are developed based on the knowledge and experience of experts for their own purposes [11]. Therefore, only partial information from the true underlying manifold is learned by each existing local manifold learning method. Thus, to better discover the underlying manifold structure, it is more informative and essential to provide a common framework for synthesizing the geometric information extracted from different local methods. In this chapter, we propose an interesting method to unify the local manifold learning algorithms (e. g., LEM, LLE, HLLE, and LTSA). Inspired by HLLE which employs local tangent coordinates to compute the local Hessian, we propose to utilize local tangent coordinates to estimate the local objects defined in different local methods. Then, we employ the truncation function from differential manifold to connect the local objects with a global functional. Finally, we develop an alternating optimization-based algorithm to discover the global coordinate system of lower

A manifold is a topological space that locally resembles Euclidean space near every point. For example, around each point, there is a neighborhood that is topologically the same as the open unit ball in ℝ<sup>D</sup>. The simplest manifold is a linear manifold, usually called a hyperplane. There exists a tangent space at each point of a nonlinear manifold. The tangent space is a linear manifold which locally approximates the manifold. Suppose there are N points {x1;…;xN} in ℝ<sup>D</sup> residing on a smooth manifold M⊂ℝ<sup>D</sup>, which is the image of a coordinate space Y⊂ℝ<sup>d</sup> under a smooth mapping <sup>ψ</sup> : <sup>Y</sup> ! <sup>ℝ</sup><sup>D</sup>, where <sup>d</sup>≪D. The mapping <sup>ψ</sup> is assumed as a locally isometric embedding. The aim of a NLDR algorithm is to acquire the corresponding low-

at the same time. Suppose M is smooth such that the tangent space TxðMÞ is well defined at every point x∈M. We can regard the local tangent space as a d-dimensional affine subspace of ℝ<sup>D</sup> which is tangent to M at x. Thus, the tangent space has the natural inner product induced by the embedding M⊂ℝ<sup>D</sup>. Within some neighborhood of x, each point x∈M has a sole closest point in TxðMÞ, and therefore, an orthonormal coordinate system from the corresponding

A manifold can be represented by its coordinates. While the current research of differential geometry focuses on the characterization of the global properties of manifolds, NLDR algorithms, which try to find the coordinate representations of data, only need the local properties of manifolds. In this chapter, we use local coordinates associated with the tangent space to estimate the local objects over the manifold. To acquire the local tangent coordinates, we first perform Principal Component Analysis (PCA) [12] on the points in N ðxiÞ ¼ {xi; xi<sup>1</sup> ;…; xik } that

local coordinates on M can be associated with the tangent space.

∈Y of each xi∈M and preserve certain intrinsic structures of data

embedding [8], stable local approaches [9], and maximal linear embedding [10].

dimensionality.

134 Manifolds - Current Research Areas

2. Local tangent coordinates system

dimensional representation yi
