Multi-class semi-supervised kernel minimum squared error for face recognition

doi:10.1016/j.ijleo.2015.08.181

Optik

Volume 126, Issue 23, December 2015, Pages 3496-3500

https://doi.org/10.1016/j.ijleo.2015.08.181 Get rights and content

Abstract

Kernel Minimum Squared Error (KMSE) has become a hot topic in machine learning and pattern recognition in the past years. However, KMSE is essentially a binary classifier and one-against-all and one-against-one strategies are usually employed to deal with multi-class problems. In this situation, KMSE needs to resolve multiple equations with the high computation complexity. Meanwhile, labeled examples are usually insufficient and unlabeled ones are abundant in many real-world applications. Therefore in this paper, we introduce a novel multi-class semi-supervised KMSE algorithm, called multi-class Laplacian regularized KMSE (McLapKMSE). Compared to KMSE and semi-supervised KMSE, we need resolve one equation at once and therefore the method has the lower complexity. The experiments on face recognition are conducted to illustrate that our algorithm can achieve the comparable performance and lower complexity in contrast to the other supervised and semi-supervised methods.

Introduction

During the past years, face recognition has become one of the most successful applications of pattern recognition and machine learning [1], [2], [3], [4], [5]. One of the most widely used methods for face recognition is appearance-based methods [6], [7], [8], [9]. In order to cover the appearance variations, multiple images associated to different types, such as poses and illuminations, should be gathered before training a classifier. However, in real-world applications, there are only a small amount of labeled face images per person. In this situation, the face recognition performance may not be robust to face variations since the images do not well cover all the face variations only using a small amount of labeled data. And unlabeled face images are easy to collect and often abundant in real world. For instance, in a surveillance system in a factory or entrance access system used by a company, most of the collected face images in the working stage will be unlabeled data but still belong to one of the known classes. Consequently, semi-supervised learning may be a useful tool which attempts to train a better classifier using both labeled and unlabeled data.

Meanwhile, kernel methods have been receiving more and more attention in nonlinear learning [10]. KMSE, which is one of the kernel methods, has become a hot topic due to its higher computational efficiency in the training phase. Gan el al. [11] proposed a semi-supervised KMSE algorithm, called Laplacian regularized KMSE (LapKMSE), which explicitly exploits the manifold structure of both labeled and unlabeled data. Experimental results on benchmark datasets and face recognition have illustrated the effectiveness of LapKMSE. However, KMSE and LapKMSE are originally designed for binary classification problem. How to effectively deal with multi-class problems is still an open problem [12]. Up to now, a widely used technique is that how to design and combining several binary classifiers. There are usually two ways: one-against-all (OAA) [13] and one-against-one (OAO) [14]. OAA and OAO construct c and c(c − 1)/2 binary classifiers where c is the number of classes, respectively. When the number of classes is large, it will be very time-consuming.

In this paper, we propose a multi-class LapKMSE (McLapKMSE) which is applied to face recognition. We directly consider the training samples in one optimization equation and resolve the equation to obtain the decision function of multi-class classification. Because our algorithm resolves one equation at once not multiple ones, McLapKMSE has lower complexity than LapKMSE. And the experiments conducted on face recognition demonstrate that McLapKMSE can achieve the comparable results but be more efficient.

Section snippets

Naïve KMSE

Let X = {(x₁, y₁), ⋯ , (x_l, y_l)} be a training set of size l, where $x_{i} \in ℝ^{D}$ and $y_{i} \in ℝ$ . For the binary classification problem, y_i = −1 if x_i ∈ ω₁ or y_i = 1 if x_i ∈ ω₂. By using a nonlinear mapping function Φ, a training sample is transformed into a new feature space Φ(x_i) from the original feature space. The task of KMSE is to build a linear model in the new feature. The outputs of the training samples obtained by the linear model are desired to be equal to the labels. $Φ W = Y$ where $Φ = [\begin{matrix} 1 & Φ {(x_{1})}^{T} \\ ⋮ & ⋮ \\ 1 & Φ {(x_{l})}^{T} \end{matrix}], W = [\begin{matrix} α_{0} \\ w \end{matrix}], and Y = [\begin{matrix} y_{1} \\ ⋮ \end{matrix}]$

Method

In this section, we will present our method in details.

Assuming the dataset is given as 2.2 with y_i ∈ {1, 2, ⋯ , c}, we define r_i = [0, 0, ⋯ , 1, 0, ⋯ , 0]_1×c to encode the sample label where r_ij = 1 if y_i = j. K is divided into two subsets as follows: $K = [\begin{matrix} K^{l} \\ K^{u} \end{matrix}]$ where K^l is the first l rows and K^u is the latter u rows of K.

First, we can rewrite Eq.(12) as: $J_{r} (α) = \sum_{i = 1}^{l} {(f (x_{i}) - y_{i})}^{2} + γ_{A} α^{T} α + \frac{γ_{I}}{2} \sum_{i, j = 1}^{n} {(f (x_{i}) - f (x_{j}))}^{2} w_{ij}$

In the above equation, α is a vector and the equation can only construct a binary classifier. When it

Experimental results

In this section, a series of experiments on face recognition are conducted to evaluate our algorithm performance. We compare the performance of our algorithm with KMSE, LapRLS, LapSVM and LapKMSE. For a fair comparison, the four methods employ the OAA strategy for multi-class problems. Two face databases are used in the experiments: Yale face database ¹ and ORL face database ². The

Conclusion

In this paper, we introduce a multi-class semi-supervised learning algorithm. The algorithm extends LapKSME to the multi-class situation. Compared to the traditional used strategy, our algorithm needs only resolve one equation at once while OAO or OAA needs construct multiple binary classifiers and resolve multiple equations. A series of experiments are carried out on the two face databases and the results show the effectiveness and efficiency of our algorithm.

Acknowledgements

We would like to express our sincere thanks to Yale University and AT&T Laboratories Cambridge for offering Yale and ORL face databases, respectively. The work is supported by the National Natural Science Foundation of China under grant No. 61271328.

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View full text

Multi-class semi-supervised kernel minimum squared error for face recognition

Abstract

Introduction

Section snippets

Naïve KMSE

Method

Experimental results

Conclusion

Acknowledgements

Image Vis. Comput.

Optik

Dictionaries for image and video-based face recognition

J. Opt. Soc. Am. A

An adaptive approximation image reconstruction method for single sample problem in face recognition using FLDA

Multimed. Tools Appl.

Decision optimization for face recognition based on an alternate correlation plane quantification metric

Opt. Lett.

Face recognition using laplacian faces

IEEE Trans. Pattern Anal. Mach. Intell.

Discriminative multimanifold analysis for face recognition from a single training sample per person

IEEE Trans. Pattern Anal. Mach. Intell.

Appearance-based face recognition and light-fields

IEEE Trans. Pattern Anal. Mach. Intell.

Visual learning and recognition of 3-d objects from appearance

Int. J. Comput. Vis.

Face recognition by exploring information jointly in space, scale and orientation

IEEE Trans. Image Process.