多维尺度分析已经在维度约减和数据挖掘领域得到了广泛应用。MDS的主要缺点是其定义在训练数据上,对于新的测试样本无法直接获得映射结果。另外,MDS基于欧氏距离度量,不适合获取相似数据中的非线性流形结构。将MDS扩展到关联度量空间,称为关联度量多维尺度分析(CMDS)。与传统MDS在训练数据中完成映射,进而缩小空间范围相比,CMDS 能够直接获得测试样本映射结果。此外,CMDS基于关联度量,能够有效学习相似数据中的非线性流形结构。理论分析表明,CMDS可以利用核方法扩展到新特征空间,解决非线性问题。实验结果表明,CMDS及其核形式KG-CMDS性能优于常用传统降维方法。

Multidimensional scaling (MDS) has been applied in many applications such as dimensionality reduction and data mining. However, one of the drawbacks of MDS is that it is only defined on "training" data with no clear extension to out-of-sample points. Furthermore, since MDS is based on Euclidean distance, it is not suitable for detecting the nonlinear manifold structure embedded in the similarities between data points. In this paper, we extend MDS to the correlation measure space (CMDS). In contrast with MDS where the mapping between the input and reduced spaces is implicit, CMDS employs an explicit nonlinear mapping between the both. As a result, CMDS can directly provide predictions for new samples.Correlation is a similarity measure, so the CMDS method can effectively capture the nonlinear manifold structure of data embedded in the similarities between the data points. Theoretical analysis also shows that CMDS has some properties similar to kernel methods and can be extended to the feature space. The effectiveness of our approach is demonstrated by extensive experiments on various data sets, in comparison with existing dimensionality reduction algorithms.