^{1}

^{2}

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Previous works have demonstrated that Laplacian embedding can well preserve the local intrinsic structure. However, it ignores the diversity and may impair the local topology of data. In this paper, we build an objective function to learn the local intrinsic structure that characterizes both the local similarity and diversity of data, and then combine it with global structure to build a scatter difference criterion. Experimental results in face recognition show the effectiveness of our proposed approach.

Local geometric structure has received much attention in dimensionality reduction [

Motivated by LPP and LDA, many local linear discriminant approaches have been developed for image classification, among which the most prevalent ones include MFA (Margin Fisher Analysis) [

In real-world applications, the intrinsic structure of data is often complex, and only local or global structure is not sufficient to represent the underlying intrinsic structures. So, a reasonable approach should be the one that integrates global and local structures into the objective function of dimensionality reduction. Two of the most popular approaches are LapLDA [

In this paper, motivated by [

Given

where

Now considering the problem of mapping the data points into a line, so that the local topology, which characterizes both the similarity and diversity of data, can be well preserved. As it happens, a reasonable criterion for choosing a good map is to optimize the following objective function

where

In order to conveniently analyze the objective function (2), we only consider the nearby data, thus the objective function (2) can be also written as

where

It is easy to see that the first term in (3) is PCA, which preserves the diversity of data, while the second term, which is just LE, preserves the similarity of data. Equation (3), i.e. Equation (2) seeks to find low-dimensional representations of

to see that, our approach well characterizes the local topology preserving at small distance data pairs in circle and thus well preserve the local intrinsic structure, which characterizes both the similarity, diversity of data and improves the stableness of the intra-class representation.

Suppose

where

The aim of LapMMC is to combine the local structure with global structure characterized by LDA. Thus, the objective function of LapMMC can be written as

where:

denote the between-class and within-class scatter matrix, respectively [

The optimal projection vector

Noting that, in many real world applications,

We evaluate the proposed approach LapMMC on image data databases (PIE and COIL20), and compare its performance with classical discriminant approaches including Fisherface [

The CMU PIE database contains 68 subjects with 41368 face images as a whole. The face images were captured by 13 synchronized cameras and 21 flashes, under varying pose, illumination and expression. Each image is manually cropped and resized to

The COIL20 image library [

From

Methods | Fisherface. | LSDA. | MFA. | LapLDA. | SDA. | EFDC. | LapMMC |
---|---|---|---|---|---|---|---|

Recognition Accuracy | 66.90 | 68.69 | 69.17 | 87.78 | 94.72 | 94.72 | 95.97 |

Dimension | 66 | 137 | 125 | 19 | 21 | 21 | 9 |

Methods | Fisherface. | LSDA. | MFA. | LapLDA. | SDA. | EFDC. | LapMMC |
---|---|---|---|---|---|---|---|

Recognition Accuracy | 86.89 | 89.85 | 85.66 | 86.88 | 85.66 | 90.81 | 96.45 |

Dimension | 45 | 64 | 61 | 68 | 64 | 52 | 130 |

geometrical structure characterized by both similarity and variability. Although LapLDA, EFDC and SDA take into consideration both the global and local geometric of data, the local geometric structure preserved by LapLDA and SDA neglect the diversity of data, and EFDC only considers the local diversity of data.

In this paper, we propose a novel linear dimensionality reduction algorithm called LapMMC, which integrates global and local geometrical structures into the objective function. To be specific, we construct an adjacency graph to learn the local intrinsic structure that characterizes both the local similarity and diversity of data, and then combine it with global structure to build a scatter difference criterion for dimensionality reduction. Experimental results on the COIL20 and PIE databases demonstrate the effectiveness of our approach.

Fang Chen,Jing Wang,Quanxue Gao, (2015) Laplacian Maximum Margin Criterion for Image Recognition. Journal of Computer and Communications,03,58-63. doi: 10.4236/jcc.2015.311010