歧管测绘机：一种非线性分类新框架

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"歧管测绘机，也称为Manifold Mapping Machine (M3)，是一种用于非线性分类的机器学习框架，由Ping Hea、Xiaohua Xu和Ling Chen等人提出。该研究论文主要探讨了如何将线性分类器转化为非线性分类器的方法，以解决传统线性模型在处理非线性数据时的局限性。" 在过去的几十年中，非线性分类一直是机器学习领域的一项重要任务。由于很多实际问题的数据分布往往具有复杂的非线性结构，传统的基于内积的线性分类器在处理这类问题时往往表现不佳。近年来，核方法的出现为这个问题提供了一种解决方案，它能够通过将数据映射到高维或无限维特征空间来实现非线性分类。然而，这种隐式的空间变换和不可观察的潜在特征空间使得人们对核机器的工作机制理解困难。论文中提出的歧管测绘机（M3）框架，旨在提供一个直观且可理解的非线性分类器设计方法。M3的核心思想是将任何线性分类器扩展到非线性环境，这样既保留了线性模型的简单性和可解释性，又能处理非线性问题。M3通过映射数据到流形（manifold）上，利用流形的特性来揭示数据内在的非线性结构，从而实现有效的非线性分类。具体来说，M3利用谱分析技术在数据的流形空间中进行操作。谱空间分析能够揭示数据点之间的拓扑关系，帮助识别出隐藏在复杂数据集中的低维流形结构。这种方法允许M3在保持对分类器工作原理理解的同时，适应非线性数据的复杂性。文章详细讨论了M3的设计原理、算法实现以及可能的应用场景。通过实例和实验结果，作者证明了M3在非线性分类任务上的有效性，并与其他流行的非线性分类方法进行了比较。此外，M3的灵活性使得它可以与其他机器学习算法结合，为解决更广泛的机器学习问题提供了新的思路。 "歧管测绘机"是一项重要的研究进展，它为理解和设计非线性分类器提供了一个直观的途径，有助于推动机器学习领域的理论发展和实践应用。

new manifold. For example, a classiﬁer with rectangle decision

boundaries is not suitable if the new manifold is a hypersphere.

Second, the kernel classiﬁers are deprecated, because there is no

reason to use a more complex classiﬁer than necessary, besides

it makes no sense to increase the data dimension after just

reducing it.

2.3. Out-of-sample extension

To extend the built classiﬁer to the out-of-sample examples, M

ﬁrst maps the test data X

onto the same manifold as where Z

locates, and then classiﬁes the transformed test data Z

with the

classiﬁer built in the classiﬁer construction stage.

Fig. 1 illustrates the diagram of the manifold mapping of out-of-

sample examples. Different from the training data mapped from

the original manifold M to the target manifold H via the supervised

manifold mapping function f, the test data fails to be mapped in the

same way, because it does not have the supervised information that

f requires. To extend the supervised manifold mapping of f to the

out-of-sample test data, we construct an intermediate manifold N

between M and H as their bridge to establish an unsupervised

connection between them. The prerequisite is that there exist a

manifold mapping

from M to N , a manifold mapping

from H to

N and its inverse mapping

1

. By forcing the image of M on N to

be consistent with the image of H on N , the unsupervised

relationship between M and H can be formulated with the

mapping of

and

1

The following Consistent Mapping Principle provides the deter-

mination of the best map point of arbitrary test data from manifold

M to manifold H based on the idea described above.

Deﬁnition 2 (Consistent Mapping Principle). Suppose xA X

is an

arbitrary test data point on manifold M, the best map point of x on

manifold H, denoted as z



, should be



¼ arg min

ðxÞ

ðzÞJ

ð5Þ

where

ðxÞ :

ð6Þ

is the relationship between X

and x A M, and

ðzÞ :

ð7Þ

is the relationship between Z

and an arbitrary data point zA H.

By deﬁning the intermediate manifold N as the local coordinate

space of the test data, Consistent Mapping Principle minimizes the

difference between local coordinate of the test data on H and its

local coordinate on M. Therefore, the best map point of xA X

onto

H is the one that can best preserve the relationship of x with the

training data on M.

Although in theory the functions

and

should reﬂect the

manifold structure of M and H respectively, the fact is in most

cases little is known about the structure of M, while relatively

more information can be obtained about the structure of H from the

supervised manifold mapping function f. As a result,

can choose

some metric which is universally applicable to different data sets,

while

must be selected under the restriction of the known

structure of H.

3. Spectral method

Spectral method is proposed on the foundation of spectral graph

theory [27]. Its basic idea is to transform the graph segmentation

problem to the eigenvalue problem of Laplacian matrix. The

advantage of spectral method lies in its outstanding graph partition

ability without estimating any explicit model of the data

distribution.

Consider an undirected weighted graph G ¼ðV, E, WÞ with V the

vertex set, E the edge set, and W the similarity set. The problem of

graph segmentation refers to dividing G into p unconnected

subgraphs fG

, ..., G

g, so that their inner-cluster similarity can

be maximized while their outer-cluster similarity is minimized.

Four well-known graph partition criteria have been proposed,

including Min Cut [28], Ratio Cut [29] , Normalized Cut [30] and

Min-Max Cut [31]. Their objectives and solutions are summarized

in Table 1, in which V

represents the vertex set of G

cutðV

, V

Þ¼

i A V

, j A V

ð8Þ

is the similarity between V

and V

¼ V=V

, jV

j is the number of the

vertices in V

Deﬁnition 3 ( Laplacian matrix). The Laplacian matrix of graph G is

L ¼

def

DW ð9Þ

where W ¼ðw

nn

D ¼ diagðW1

ð10Þ

with

j ¼ 1

ð11Þ

Deﬁnition 4 (Normalized Laplacian matrix). The normalized

Laplacian matrix of graph G is

L ¼

def

1=2

¼ ID

1=2

ð12Þ

Property 5. The Laplacian matrix L satisﬁes that 8 f A

Lf ¼

i, j ¼ 1

ðf

f

ð13Þ

Property 6. The normalized Laplacian matrix L satisﬁes that 8 f A

Lf ¼

i, j ¼ 1

ﬃﬃﬃﬃ



ﬃﬃﬃﬃ

ð14Þ

The eigenvector solutions of the multiway-cut criteria are

usually collected in an indicating matrix F for post-processing,

such as k-means clustering.

F ¼½v

 v

ð15Þ

By far, there are three different graph construction methods that

are most regularly used in the spectral method.

Table 1

Objectives and solutions of four multiway-cut graph partition criteria.

Criteria Minimization objective Relaxed solution

Min Cut

i ¼ 1

cutðV

, V

min

,..., v

Lv ¼

Ratio Cut

i ¼ 1

cutðV

, V

min

,..., v

Lv ¼

Normalized Cut

i ¼ 1

cutðV

, V

cutðV

, VÞ

min

,..., v

Lv ¼

Min–Max Cut

i ¼ 1

cutðV

, V

cutðV

, V

min

,..., v

Lv ¼

P. He et al. / Neurocomputing 74 (2011) 1450–1466 1453

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歧管测绘机：一种非线性分类新框架

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