求解大规模谱聚类的近似加权核k-means算法

Approximate Weighted Kernel k-means for Large-Scale Spectral Clustering

谱聚类将聚类问题转化成图划分问题,是一种基于代数图论的聚类方法.在求解图划分目标函数时,一般利用Rayleigh熵的性质,通过计算Laplacian矩阵的特征向量将原始数据点映射到一个低维的特征空间中,再进行聚类.然而在谱聚类过程中,存储相似矩阵的空间复杂度是O(n2),对Laplacian矩阵特征分解的时间复杂度一般为O(n3),这样的复杂度在处理大规模数据时是无法接受的.理论证明,Normalized Cut图聚类与加权核k-means都等价于矩阵迹的最大化问题.因此,可以用加权核k-means算法来优化Normalized Cut的目标函数,这就避免了对Laplacian矩阵特征分解.不过,加权核k-means算法需要计算核矩阵,其空间复杂度依然是O(n2).为了应对这一挑战,提出近似加权核k-means算法,仅使用核矩阵的一部分来求解大数据的谱聚类问题.理论分析和实验对比表明,近似加权核k-means的聚类表现与加权核k-means算法是相似的,但是极大地减小了时间和空间复杂性.

Spectral clustering is based on algebraic graph theory. It turns the clustering problem into the graph partitioning problem. To solve the graph cut objective function, the properties of the Rayleigh quotient are usually utilized to map the original data points into a lower dimensional eigen-space by calculating the eigenvectors of Laplacian matrix and then conducting the clustering in the new space. However, during the process of spectral clustering, the space complexity of storing similarity matrix is O（n^2）, and the time complexity of the eigen-decomposition of Laplacian matrix is usually O（n^3）. Such complexity is unacceptable when dealing with large-scale data sets. It can be proved that both normalized cut graph clustering and weighted kernel k-means are equivalent to the matrix trace maximization problem, which suggests that weighted kernel k-means algorithm can be used to optimize the objective function of normalized cut without the eigen-decomposition of Laplacian matrix. Nonetheless, weighted kernel k-means algorithm needs to calculate the kernel matrix, and its space complexity is still O（n^2）. To address this challenge, this study proposes an approximate weighted kernel k-means algorithm in which only part of the kernel matrix is used to solve big data spectral clustering problem. Theoretical analysis and experimental comparison show that approximate weighted kernel k-means has similar clustering performance with weighted kernel k-means algorithm, but its time and space complexity is greatly reduced.