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Dimensionality Reduction with Adaptive Graph

Lishan Qiao

Limei Zhang

Songcan Chen

2,*

Department of Mathematics Science, Liaocheng University, 252000, Liaocheng, P.R. China

Department of Computer Science and Engineering, Nanjing University of Aeronautics & Astronautics,

210016, Nanjing, P.R. China

Abstract:

Graph-based dimensionality reduction (DR) methods have

been applied successfully in many practical problems such as face

recognition, where graph plays a crucial role with the aim of modeling the

data distribution or structure. However, the ideal graph is difficult to be

known in practice. Usually, one needs to construct graph empirically

according to various motivations, priors or assumptions, which is

independent of the subsequent DR mapping calculation. In this paper, we

instead attempt to learn a graph closely linking to DR process, and

propose an algorithm called Dimensionality Reduction with Adaptive

Graph (DRAG), whose idea is to, during seeking projection matrix,

simultaneously learn a graph in the neighborhood of a pre-specified one.

Moreover, the pre-specified graph is treated as a noisy observation of the

true one, and the square Frobenius divergence is used to measure their

difference in the objective function. As a result, we achieve an elegant

graph update formula which naturally fuses the original and transformed

data information. In particular, the optimal graph is shown to be a

weighted sum of the pre-defined graph in the original space and a new

graph depending on transformed space. Empirical results on several face

datasets demonstrate the effectiveness of the proposed algorithm.

Keywords:

Dimensionality reduction; Graph construction; Face

recognition

Introduction

High-dimensional data exist pervasively, especially in pattern

recognition, machine learning, and some related fields such as image

Corresponding author: Tel: +86-25-84896481 Ext. 12221; Fax: +86-25-84892400; E-mail: s.chen@nuaa.edu.cn

(S. Chen)

retrieval, text classification, face verification, and so on. High dimensions

not only bring about expensive computation cost, but also result in the

so-called curse of dimensionality [1]. Dimensionality reduction (DR) is a

direct and effective approach for handling these issues by mapping data

into a low-dimensional space. To this day, researchers have developed

abundant DR algorithms [2-7] including linear vs. nonlinear, supervised

vs. unsupervised, global vs. local categories, etc. We recommend the

readers refer to [4, 8-9] for detailed developments of DR methods. In this

paper, we mainly focus on graph-based dimensionality reduction (GDR),

a kind of recently developed DR methods which reduce data dimensions,

while using graph as a tool to preserve some properties of the original

data.

At present, GDR has been becoming increasingly popular due to its

flexibility, effectiveness and tractability. The popular GDR methods

include ISOMAP [10], LLE [11], and Laplacian Eignmap [12], etc.,

which are non-linear and deal with the data lying on or nearby a manifold.

Empirical research [13] has shown that “such nonlinear techniques

perform well on selected artificial tasks, but that this strong performance

does not necessarily extend to real-world tasks”. More recently, there

appear linear versions of GDR methods, such as Locality Preserving

Projections (LPP) [2] and Neighborhood Preserving Embedding (NPE)

[3], which are verified to be more effective than their non-linear

counterparts in many practical applications [14]. In fact, current study [4]

has shown that most DR algorithms even including classical PCA and

LDA, can be unified within a common graph embedding framework

Moreover, different methods only correspond to different intrinsic and

penalty graphs, and thus, a “good” graph is essentially important to their

Of course, there exist many DR methods, such as independent component analysis, self-organizing feature

mapping, principal curve, which currently can not be nicely interpreted under such a framework. However, those

methods go beyond the main focus of this paper.

success.

However, in practice such a graph is generally unknown, and one needs

to specify graph empirically or learn it from data. For example, the

often-used graphs have k nearest neighbor graph and



ball

neighborhood graph, where some parameters, like k and



, are selected

empirically [15] or via neighborhood optimization [16]. More recently,

some researchers addressed graph construction by some novel strategies

including minimal spanning tree [17], b-matching [18], sparse

representation [19-21], and so on, but generally speaking, constructing a

high-quality graph is still an open problem[22].

In addition, a common point in the aforementioned strategies is that the

graph constructing process is independent of the subsequent tasks such as

projection learning for DR here. Instead, a recently-proposed DR

algorithm called Graph-optimized Locality Preserving Projections

(GoLPP) [23] attempts to learn a graph and a corresponding projection

matrix simultaneously in one single objective function, resulting in an

automatically-updated graph. Comparing with classical LPP which is

based on a pre-specified neighborhood graph, GoLPP has been

demonstrated to have superior performance on some public datasets [23].

However, the optimal graph in GoLPP depends heavily on the

transformed data, without making the best use of the original data

information. This easily leads to the instability of performance, especially

when encountering a “bad” projection/transform matrix. Figure 4 in

section 4 gives an intuitive illustration to this point.

To address this problem, in this paper, we propose a novel linear GDR

algorithm called

imensionality

eduction with

daptive

raph

(DRAG). The proposed method shares the graph optimization advantage

of GoLPP, while the original data information is embedded into the graph

optimization process. More concretely, during seeking a projection matrix,

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