TY - JOUR
T1 - A tensor approximation approach to dimensionality reduction
AU - Wang, Hongcheng
AU - Ahuja, Narendra
N1 - Funding Information:
Acknowledgements The support of the Office of Naval Research under grant N00014-03-1-0107 is gratefully acknowledged. We would like to thank the anonymous reviewers for their constructive comments. Thanks go as well to Qing Qu and Yizhou Yu for the helpful discussion that strengthened this paper.
PY - 2008/3
Y1 - 2008/3
N2 - 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.
AB - 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.
KW - Dimensionality reduction
KW - Multilinear analysis
KW - Object recognition
KW - Rank-R tensor approximation
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U2 - 10.1007/s11263-007-0053-0
DO - 10.1007/s11263-007-0053-0
M3 - Article
AN - SCOPUS:38349091931
SN - 0920-5691
VL - 76
SP - 217
EP - 229
JO - International Journal of Computer Vision
JF - International Journal of Computer Vision
IS - 3
ER -