A fast derivation of Karhunen-Loeve transform kernel for first-order autoregressive discrete process

Abstract

Karhunen-Loève Transform (KLT), also called principal component analysis (PCA) or factor analysis, based signal processing methods have been successfully used in applications spanning from eigenfiltering to recommending systems. KLT is a signal dependent transform and comprised of three major steps where each has its own computational requirement. Namely, statistical measurement of random data is performed to populate its covariance matrix. Then, eigenvectors (eigenmatrix) and eigenvalues are calculated for the given covariance matrix. Last, incoming random data vector is mapped onto the eigenspace (subspace) by using the calculated eigenmatrix. The recently developed method by Torun and Akansu offers an efficient derivation of the explicit eigenmatrix for the covariance matrix of first-order autoregressive, AR(1), discrete stochastic process. It is the second step of the eigenanalysis implementation as summarized in the paper. Its computational complexity is investigated and compared with the currently used techniques. It is shown that the new method significantly outperforms the others, in particular, for very large matrix sizes that are common in big data applications.