A Tensor Approximation Approach to Dimensionality Reduction

Abstract

Dimensionality reduction has recently been extensively studied for computer vision applications. We present a novel multilinear algebra based approach to reduced dimensionality representation of multidimensional data, such as image ensembles, video sequences and volume data. Before reducing the dimensionality we do not convert it into a vector as is done by traditional dimensionality reduction techniques like PCA. Our approach works directly on the multidimensional form of the data (matrix in 2D and tensor in higher dimensions) to yield what we call a Datum-as-Is representation. This helps exploit spatio-temporal redundancies with less information loss than image-as-vector methods. An efficient rank-R tensor approximation algorithm is presented to approximate higher-order tensors. We show that rank-R tensor approximation using Datum-as-Is representation generalizes many existing approaches that use image-as-matrix representation, such as generalized low rank approximation of matrices (GLRAM) (Ye, Y. in Mach. Learn. 61:167–191, 2005), rank-one decomposition of matrices (RODM) (Shashua, A., Levin, A. in CVPR’01: Proceedings of the 2001 IEEE computer society conference on computer vision and pattern recognition, p. 42, 2001) and rank-one decomposition of tensors (RODT) (Wang, H., Ahuja, N. in ICPR ’04: ICPR ’04: Proceedings of the 17th international conference on pattern recognition (ICPR’04), vol. 1, pp. 44–47, 2004). Our approach yields the most compact data representation among all known image-as-matrix methods. In addition, we propose another rank-R tensor approximation algorithm based on slice projection of third-order tensors, which needs fewer iterations for convergence for the important special case of 2D image ensembles, e.g., video. We evaluated the performance of our approach vs. other approaches on a number of datasets with the following two main results. First, for a fixed compression ratio, the proposed algorithm yields the best representation of image ensembles visually as well as in the least squares sense. Second, proposed representation gives the best performance for object classification.