Simultaneously Learning Neighborship and Projection Matrix for Supervised Dimensionality Reduction
This addresses a specific bottleneck in dimensionality reduction for classification tasks, offering an incremental improvement over existing methods.
The paper tackles the problem of suboptimal neighbor selection and similarity measurement in supervised dimensionality reduction due to the curse of dimensionality, proposing a method that simultaneously learns similarity and projection matrix in low-dimensional space, with experimental validation on YALE B, COIL-100, and MNIST datasets showing effectiveness.
Explicitly or implicitly, most of dimensionality reduction methods need to determine which samples are neighbors and the similarity between the neighbors in the original highdimensional space. The projection matrix is then learned on the assumption that the neighborhood information (e.g., the similarity) is known and fixed prior to learning. However, it is difficult to precisely measure the intrinsic similarity of samples in high-dimensional space because of the curse of dimensionality. Consequently, the neighbors selected according to such similarity might and the projection matrix obtained according to such similarity and neighbors are not optimal in the sense of classification and generalization. To overcome the drawbacks, in this paper we propose to let the similarity and neighbors be variables and model them in low-dimensional space. Both the optimal similarity and projection matrix are obtained by minimizing a unified objective function. Nonnegative and sum-to-one constraints on the similarity are adopted. Instead of empirically setting the regularization parameter, we treat it as a variable to be optimized. It is interesting that the optimal regularization parameter is adaptive to the neighbors in low-dimensional space and has intuitive meaning. Experimental results on the YALE B, COIL-100, and MNIST datasets demonstrate the effectiveness of the proposed method.