基于 Riemann 流形几何的降维学习框架

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“黎曼流形学习” 黎曼流形学习（Riemannian Manifold Learning）是机器学习和数据分析领域中的一种重要技术。它的主要思想是将高维数据降维到低维空间中，从而保留原数据的几何结构。这种方法广泛应用于模式识别、数据分析和机器学习等领域。在黎曼流形学习中，数据点被看作是一个黎曼流形的采样点。黎曼流形是一种数学对象，它可以描述复杂的几何结构。为了学习黎曼流形，我们需要构建一个坐标图（coordinate chart），以便将高维数据映射到低维空间中。在这篇论文中，作者提出了一种新的黎曼流形学习框架，称为黎曼流形学习（RML）。该框架的主要思想是将降维问题转化为黎曼几何学中的经典问题，即如何构建一个黎曼流形的坐标图。作者实现了黎曼_normal 坐标图（Riemannian normal coordinate chart），这是黎曼几何学中最广泛使用的坐标图。在该框架中，作者首先使用高效的 simplicial 重构算法来估计输入参数，即邻域大小 k 和 intrinsic 维数 d。然后，作者使用 normal 坐标来将高维数据映射到低维空间中。实验结果表明，该算法可以学习到数据的 intrinsic 几何结构。黎曼流形学习有很多实际应用。例如，在计算机视觉领域中，可以使用黎曼流形学习来降维图像数据，从而进行图像分类和图像检索。同时，在数据分析领域中，也可以使用黎曼流形学习来降维高维数据，从而进行数据可视化和数据挖掘。黎曼流形学习是一种强大的降维技术，它可以学习到数据的 intrinsic 几何结构，并将其应用于各种实际领域中。

This manifold can be conceived as the earth running on a

circular orbit in the four-dimensional space-time.

For face images of multiple persons, a vector bundle is

suitable to model the relation between interperson and

intraperson variation (Fig. 1b). Each fiber (shown in Fig. 1b

as a curve for visualization) represents the face data manifold

of one person, and the base manifold M (shown in Fig. 1b as a

surface) collects the face images of all persons in a standard

setting (such as the frontal view, frontal illumination, and

natural facial expression). In face recognition, a new test

sample can be projected from the total space E (formed by

gluing together all the face image manifolds of each person)

onto the base M in order to remove intraperson variation,

provided that the projection map  : E ! M has been

learned. It is important to point out that both the total

space E and the base M are manifolds. Therefore, manifold

learning algorithms can be applied to a set of face image data

of multiple persons in face recognition.

In general, the manifoldassumption may not hold forsome

complex data sets acquired from the real world. Nevertheless,

this assumption has been widely exploited in practice. First, it

is still possible to approximate the true data model as a

manifold by considering only a small set of significant factors.

Second, the popular manifold assumption can greatly

simplify the problem by exploiting the classical frameworks

of differentiable manifolds and Riemannian geometry.

In the remainder of this paper, we assume that the input

data lie on a d-dimensional Riemannian manifold M

embedded in a euclidean space R

. The Riemannian metric

on M is induced from the standard euclidean inner product

in R

. Mathematically, Nash proved that every Riemannian

manifold M can be isometrically embedded into some

euclidean space R

[35]. This theorem reduces the study of

Riemannian manifolds to the study of submanifolds of

euclidean spaces.

1.3 The Proposed Framework

In this paper, we propose a general framework called

Riemannian manifold learning

(RML). The dimensionality

reduction problem is formulated as constructing coordinate

charts for a Riemannian manifold. One of the most widely

used is the (Riemannian) normal coordinate chart, which was

introduced by Riemann in his famous 1854 lecture [28].

Riemannian normal coordinates are a vital tool for calcula-

tions in Riemannian geometry. One advantage is that normal

coordinates can preserve geodesic metrics to a certain extent

if the complete preservation of the metrics is not possible. The

metric preserving property is a cornerstone in geometry and

also a desirable feature for dimensionality reduction. In

supervised or unsupervised classification, better preserving

the metrics often leads to better results in the reduced low-

dimensional space. Particularly, in normal coordinates,

geodesic distances on each radial geodesic curve emanating

from the coordinate origin can be faithfully preserved. On the

other hand, distances between two radial geodesic curves can

freely stretch or shrink to fit this mapping. Fig. 2a provides an

example of radial geodesic curves, and Fig. 2b presents the

basic idea of normal coordinates. These radial geodesic

curves play a key role in unfolding a curved manifold onto a

flat space. This is very similar to the role of an umbrella

skeleton when we open an umbrella. Because of this set of

radial geodesic curves, which is a “skeleton” of the manifold,

normal coordinates can preserve the intrinsic geometric

structures with minimal distortions.

This paper presents a novel algorithm for manifold

learning based on the calculations of Riemannian normal

coordinates. This algorithm has a number of desirable

properties and overcomes several problems in previous work

reported in the literature:

1. The metric preserving problem. ISOMAP attempts to

preserve geodesic distances of all pairs of points on

the manifold. For intrinsically flat manifolds of zero

Gaussian curvature, such as the well-known “Swiss

roll” data (Fig. 3a), ISOMAP yield s the i deal

embedding by isometrically unfolding the curved

data onto a planar region (Fig. 3b). However, for

general manifolds with a nonzero Gaussian curva-

ture (for example, a sphere), ISOMAP fails to

perform the isometric mapping onto a flat euclidean

space, as Gaussian curvature is isometry invariant.

On the contrary, global metric information is totally

lost in those spectral embedding methods under the

unit variance constraint. For the example of Swiss

roll data, spectral methods tend to generate an

embedding into a square region (Fig. 3c) or even

sometimes yield unpredictable irregular results

(Fig. 3d). For pattern recognition applications, large

differences in global shape and aspect ratios pose

serious threats to any classifier (for example, (KNN)),

because n eighborhood relationship is changed

greatly. Mathematically, the metric preserving pro-

blem is of paramount importance in Gaussian

intrinsic geometry and Riemannian geometry. Com-

pared with ISOMAP and spectral methods, the

proposed RML can preserve radial geodesic dis-

tances, give a more faithful representation of the

data’s glo bal structure, a nd achieve a t rade-o ff

between the rigid constraint of isometry and the

deficiency of global metrics.

2. The cost averaging problem. Previous methods based

on global cost optimization attempt to average the cost

among all data points, thus preventing the correct

unfolding in large deformation areas. For instance, the

boundary areas of a punctured sphere (Fig. 4a) often

shrink into a circular curve by using Hessian eigen-

maps or LTSA (Fig. 4b). This shrinkage significantly

decreases the global cost but yields incorrect over-

lapping. Our algorithm computes the embedding by

798 IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, VOL. 30, NO. 5, MAY 2008

2. Although several existing methods mentioned Riemannian manifolds

in their papers, their proposed algorithms essentially had little connection

with Riemannian geometry. This work, together with [32], points out that

the manifold learning problem can be translated into the mathematical

problem of constructing coordinate charts for the manifold, thus unifying

various ideas within a common mathematical framework. In this paper,

only the Riemannian normal coordinate chart is implemented, but other

coordinate charts can also be employed.

Fig. 2. (a) Radial geodesic curves on a surface. (b) Charting a sphere

with Riemannian normal coordinates on the tangent plane.

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