Multi-view reconstructive preserving embedding for dimension reduction

Abstract

With the development of feature extraction technique, one sample always can be represented by multiple features which are located in different high-dimensional spaces. Because multiple features can reflect one same sample from various perspectives, there must be compatible and complementary information among the multiple views. Therefore, it’s natural to learn information from multiple views to obtain better performance. However, most multi-view dimension reduction methods cannot handle multiple features from nonlinear space with high dimensions. To address this problem, we propose a novel multi-view dimension reduction method named multi-view reconstructive preserving embedding (MRPE) in this paper. MRPE reconstructs each sample by utilizing its k nearest neighbors. The similarities between each sample and its neighbors are mapped into lower-dimensional space in order to preserve the underlying neighborhood structure of the original manifold. MRPE fully exploits correlations between each sample and its neighbors from multiple views by linear reconstruction. Furthermore, MRPE constructs an optimization problem and derives an iterative procedure to obtain the low-dimensional embedding. Various evaluations based on the applications of document classification, face recognition and image retrieval demonstrate the effectiveness of our proposed approach on multi-view dimension reduction.

Appendix

This appendix shows how to obtain Eqs. 5 from 4 by matrix operations.

$$\begin{aligned} \begin{aligned}&\sum _{v=1}^m \sum _{i=1}^n \alpha ^{(v)} ||y_i - \sum _{j=1}^k w_{ij}^{(v)} y_{ij}||^2 \\&\quad = \sum _{v=1}^m \sum _{i=1}^n \alpha ^{(v)} ||Y I_i - Y W_i^{(v)}||^2 \\ \end{aligned} \end{aligned}$$

(10)

where $Y = [y_1, \dots , y_i, \dots , y_n] \in {\mathbb {R}}^{d \times n}$ is the final low-dimensional representations, $I_i = [0, \dots , 1, \dots , 0] \in {\mathbb {R}}^{n \times 1}$ is a vector with value 1 of the ith position, and $W_i^{(v)} \in {\mathbb {R}}^{n \times 1}$ is extended by filling number 0 of the vector $\sum _j^k w_{ij}^{(v)} \in {\mathbb {R}}^{k \times 1}$.

$$\begin{aligned}&\sum _{v=1}^m \sum _{i=1}^n \alpha ^{(v)} ||y_i - \sum _{j=1}^k w_{ij}^{(v)} y_{ij}||^2 \nonumber \\&\quad = \sum _{v=1}^m \sum _{i=1}^n \alpha ^{(v)} ||Y I_i - Y W_i^{(v)}||^2 \nonumber \\&\quad = \sum _{v=1}^m \sum _{i=1}^n \alpha ^{(v)} (Y I_i - Y W_i^{(v)})(Y I_i - Y W_i^{(v)})^T \nonumber \\&\quad = \sum _{v=1}^m \alpha ^{(v)} tr((Y I - YW^{(v)})(Y I - YW^{(v)})^T) \nonumber \\&\quad = \sum _{v=1}^m \alpha ^{(v)} tr(Y(I-W^{(v)})(I-W^{(v)})^T Y^T) \nonumber \\&\quad = \sum _{v=1}^m \alpha ^{(v)} tr(YM^{(v)}Y^T) \end{aligned}$$

(11)

where $M^{(v)} = (I-W^{(v)})(I-W^{(v)})^T \in {\mathbb {R}}^{n \times n}$ and $W^{(v)} = [W_1^{(v)}, \dots , W_n^{(v)}] \in {\mathbb {R}}^{n \times n}$.