Sparse PCA is the optimization problem obtained from PCA by adding a sparsity constraint on the principal components. Sparse PCA is NP-hard and hard to approximate even in the single-component case. In this paper we settle the computational complexity of sparse PCA with respect to the rank of the covariance matrix. We show that, if the rank of the covariance matrix is a fixed value, then there is an algorithm that solves sparse PCA to global optimality, whose running time is polynomial in the number of features. We also prove a similar result for the version of sparse PCA which requires the principal components to have disjoint supports.
翻译:通过增加主要成分的宽度限制,从五氯苯甲醚获得的优化问题为松散的五氯苯甲醚; 松散的五氯苯甲醚是硬的,即使在单一成分的情况下也很难估计。 在本文件中,我们解决了稀散的五氯苯甲醚相对于共变矩阵等级的计算复杂性。 我们表明,如果共变矩阵的等级是一个固定值,那么有一种算法可以解决稀散的五氯苯甲醚与全球最佳性的关系,其运行时间在特性数量上是多等的。 对于稀少的五氯苯甲醚的版本,我们也证明一个类似的结果,它需要主要成分的断开支持。