Online Approximation Multilinear Principal Component Analysis Of Tensor Objects

Principal component analysis(PCA) is a well-known unsupervised linear technique for dimensionality reduction. The central idea behind PCA is to reduce the dimensionality of a data set consisting of a larger number of interrelated variables, while retaining as much as possible the original data variation. This is achieved by transforming to a new set of variables, the so-called principal components(pcs), which are uncorrelated and ordered.Na?ve application of PCA to tensor objects requires their reshaping into vectors with high dimensionality(vectorization), which obviously results in high processing cost in terms of increased computational and memory demands. Besides, it is well understood that reshaping breaks the natural structure and correlation in the original data, removing redundancies and/or higher order dependencies present in the original data set and losing potentially more compact or useful representations that can be obtained in the original form. Therefore, a dimensionality reduction algorithm operating directly on a tensor object is developed, named multilinear principal component analysis(MPCA).Furthermore, MPCA generally employ off-line traditionally learning to deal with new added samples, which aggravates the computational cost. To address this problem, online approximation learning algorithms of tensor objects are proposed in this paper, so called on-line approximation multilinear principal component analysis(OLAMPCA).This paper shows the relationship between MPCA and OLAMPCA, and proves the graph embedding framework theoretically and derives the on line learning procedure to add single sample and multiple samples in detail.Experiments are conducted on four benchmarking databases including twelve tensor datasets to compare the performance of MPCA and OLAMPCA. We evaluate the dimensionality reduced, dimensionality reduction scatter and dimensionality reduction time of MPCA and OLAMPCA, and the performances of the data after dimensionality reduction combined with the support higher-order tensor machine(SHTM).The experiment results show that OLAMPCA algorithm is superior to MPCA algorithm on dimensionality reduction for tensor objects; Meanwhile, both of them have the similar test accuracy for support higher order tensor machine. In conclusion, OLAMPCA algorithm is available.