An Improved Algorithm for Kernel Principal Component Analysis

Reviewer: Marco Cristani

The kernel principal component analysis (KPCA) is a nonlinear generalization of the principal component analysis (PCA). The drawbacks of the KPCA technique, in its original version, are a high computational effort, which is O(N3), where N is the size of the processed data, and the need for storing the kernel matrix N × N. The authors propose an algebraic trick to reduce the time and space complexity of the original algorithm, based on the block decomposition of the kernel matrix. This leads to an approximate version of the original KPCA technique. The complexity reached is dependent on the ability we have to block-decompose the kernel matrix (let's say M blocks) and on the number of eigenvalues we want to take into account (let's say S). The novel time complexity is max(O(S3),O(N3/M2)), and the space complexity depends on the size of the blocks obtained from the partition of the kernel matrix. The paper is presented with a sufficient experimental validation (even though more comparative qualitative tests using a small dataset with other approximation techniques could have been performed), and the results shown are convincing, in both a quantitative and a qualitative sense. I think that the paper could have been better presented: in the abstract, introduction, and conclusion, no mention of the effective computational gain is given, so the reader cannot immediately appreciate the improvements made with respect to the state of the art. Online Computing Reviews Service