Cost Sensitive Semi-Supervised Canonical Correlation Analysis for Multi-view Dimensionality Reduction

Abstract

To deal with the cost sensitive and semi-supervised learning problems in Multi-view Dimensionality Reduction (MDR), we propose a Cost Sensitive Semi-Supervised Canonical Correlation Analysis $(\hbox {CS}^{3}\hbox {CCA}). \hbox {CS}^{3}\hbox {CCA}$ first uses the $L_2$ norm approach to obtain the soft label for each unlabeled data, and then embed the misclassification cost into the framework of Canonical Correlation Analysis (CCA). Compared with existing CCA based methods, $\hbox {CS}^{3}\hbox {CCA}$ has the following advantages: (1) It uses the $L_2$ norm approach to infer the soft label for unlabeled data, which is computationally efficient and effective, especially for cost sensitive face recognition. (2) The objective function of $\hbox {CS}^{3}\hbox {CCA}$ not only maximizes the soft cost sensitive within-class correlations and minimizes the soft cost sensitive between-class correlations in the inter-view, but also considers the class imbalance problem simultaneously. With the discriminant projections learned by $\hbox {CS}^{3}\hbox {CCA}$, we employ it for cost sensitive face recognition. The experimental results on four well-known face data sets, including AR, Extended Yale B, PIE and ORL, demonstrate the effectiveness of $\hbox {CS}^{3}\hbox {CCA}$.

Appendix 1

In this section, we introduce to use SVD to solve the optimal problems of CCA, LPbSCCA and our $\hbox {CS}^{3}\hbox {CCA}$, which are defined by Eqs. (1), (4) and (11), respectively. Note that, they can also be solved by computing a generalized eigenvalue decomposition problem, i.e., Eq. (2). As the form of their objective functions are similar, the only difference between them is the definitions of $\tilde{M}_{xy}, \tilde{M}_{xx}$ and $\tilde{M}_{yy}$, they can be unified written as follows:

$$\begin{aligned} \begin{aligned} \max ~&W_x^T\tilde{M}_{xy} W_y\\ s.t. ~&W_x^T\tilde{M}_{xx}W_x=1,~~W_y^T\tilde{M}_{yy}W_y=1. \end{aligned} \end{aligned}$$

(12)

Firstly, we define the Lagrange function of Eq. (12) as follows:

$$\begin{aligned} L(\lambda _1,\lambda _2,W_x,W_y)=W_x^T\tilde{M}_{xy} W_y-\frac{\lambda _1}{2}(W_x^T\tilde{M}_{xx} W_x-1)-\frac{\lambda _2}{2}(W_y^T\tilde{M}_{yy} W_y-1), \end{aligned}$$

(13)

and get its partial derivatives:

$$\begin{aligned} \left\{ \begin{aligned}&\partial L/\partial W_x=\tilde{M}_{xy}W_y-\lambda _1\tilde{M}_{xx}W_x=0,\\&\partial L/\partial W_y=\tilde{M}_{yx}W_x-\lambda _2\tilde{M}_{yy}W_y=0. \end{aligned}\right. \end{aligned}$$

(14)

By simple derivation, we can prove $\lambda _1=\lambda _2$ and get:

$$\begin{aligned} \left\{ \begin{aligned}&\tilde{M}_{xy}\tilde{M}_{yy}^{-1}\tilde{M}_{yx}W_x=\lambda ^2\tilde{M}_{xx}W_x,\\&\tilde{M}_{yx}\tilde{M}_{xx}^{-1}\tilde{M}_{xy}W_y=\lambda ^2\tilde{M}_{yy}W_y. \end{aligned}\right. \end{aligned}$$

(15)

In the following, we use the SVD to find the solution. Let $H=\tilde{M}_{xx}^{-1/2}\tilde{M}_{xy}\tilde{M}_{yy}^{-1/2}, U=\tilde{M}_{xx}^{1/2}W_x, V=\tilde{M}_{yy}^{1/2}W_y$, Eq. (15) becomes:

$$\begin{aligned} \left\{ \begin{aligned}&HH^TU=\lambda ^2U,\\&H^THV=\lambda ^2V. \end{aligned}\right. \end{aligned}$$

(16)

Observing Eq. (16), we discover that it only needs to do SVD of the matrix $H=UDV^T=\sum _{i=1}^d {\lambda _i u_iv_i^T}, U=[u_1,\ldots ,u_d], V=[v_1,\ldots ,v_d]$, and then the projection directions of $W_x$ and $W_y$ can be got respectively.

$$\begin{aligned} \left\{ \begin{aligned}&W_x=\tilde{M}_{xx}^{-1/2}U,\\&W_y=\tilde{M}_{yy}^{-1/2}V. \end{aligned}\right. \end{aligned}$$

(17)