基于分而治之的多维标度算法

摘要

作为一种典型的多元统计分析方法，多维标度法( MDS)广泛应用于降维和可视化研究中。 MDS从n个样本间的距离距阵出发，求取它们在低维欧氏空间的坐标。经典MDS算法( CMDS)的时间复杂度为Θ( n3)，影响MDS的速度。文中基于分而治之的思想提出一种新的MDS算法。首先将距离矩阵沿对角线分成若干子矩阵，然后对每个子矩阵求解，最后通过正交变换和平移变换整合各子矩阵的解，从而得到原距离矩阵的全局解。该算法的结果与CMDS完全一致。当样本维数远小于样本个数时，其时间复杂度仅为Θ( nlgn)。与CMDS 算法相比，该算法的速度大大提高，从而使MDS可应用于更大规模数据集。%As a typical method for multivariate statistical analysis, multidimensional scaling(MDS) has been widely used in dimensionality reduction and visualization. Based on a distance matrix of n samples, the coordinates of MDS in a low-dimensional Euclidean space are obtained. The time complexity for classical MDS algorithm( CMDS) is Θ( n3 ) , which affects the speed of MDS. A new MDS algorithm based on the idea of divide-and-conquer is proposed. The distance matrix is divided into several submatrices along its diagonal, then the submatrices are solved respectively. With orthogonal transformation and translation transformation, all of these subproblem solutions are aligned to get the global solution to the original distance matrix. The result of the proposed algorithm is completely the same as that of CMDS. When the dimensionality of sample space is far less than the number of samples, the time complexity of the proposed algorithm is only Θ( nlgn) . Thus, compared with CMDS, the speed of the proposed algorithm is greatly improved, which makes MDS applicable to larger datasets.