Abstract
This paper considers the problem of factorizing a matrix with missing components into a product of two smaller matrices, also known as principal component analysis with missing data (PCAMD). The Wiberg algorithm is a numerical algorithm developed for the problem in the community of applied mathematics. We argue that the algorithm has not been correctly understood in the computer vision community. Although there are many studies in our community, almost every one of which refers to the Wiberg study, as far as we know, there is no literature in which the performance of the Wiberg algorithm is investigated or the detail of the algorithm is presented. In this paper, we present derivation of the algorithm along with a problem in its implementation that needs to be carefully considered, and then examine its performance. The experimental results demonstrate that the Wiberg algorithm shows a considerably good performance, which should contradict the conventional view in our community, namely that minimization-based algorithms tend to fail to converge to a global minimum relatively frequently. The performance of the Wiberg algorithm is such that even starting with random initial values, it converges in most cases to a correct solution, even when the matrix has many missing components and the data are contaminated with very strong noise. Our conclusion is that the Wiberg algorithm can also be used as a standard algorithm for the problems of computer vision.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Belhumeur, P.N. and Kriegman, D.J. 1998. What is the set of images of an object under all possible illumination conditions? International Journal of Computer Vision, 28(3):245–260.
Buchanan, A.M. and Fitzgibbon, A.W. 2005. Damped newton algorithms for matrix factorization with missing data. In Proceedings of IEEE Computer Vision and Pattern Recognition.
Buchanan, A.M. 2004. Investigation into matrix factorization when elements are unknown. Technical report, University of Oxford, http://www.robots.ox.ac.uk/~amb.
Chen, P. and Suter, D. 2004. Recovering the missing components in a large noisy low-rank matrix: Application to sfm. IEEE Transaction on Pattern Analysis and Machine Intelligence, 26(8):1051–1063.
Epstein, R. Yuille, A.L., and Belhumeur, P.N. 1996. Learning object representations from lighting variations. In Object Representation in Computer Vision II, J. Ponce, A. Zisserman, and M. Herbert (Eds.) Springer Lecture Notes in Computer Science, Vol. 1144, pp. 179–199.
Hayakawa, H. 1994. Photometric stereo under a light source with arbitrary motion. Journal of the Optical Society of America A, 11(11):3079–3089.
Jacobs, D.W. 2001. Linear fitting with missing data for structure-from-motion. Computer Vision and Image Understanding, 82(1):57–81.
Ruhe, A. and Wedin, P. ÅA. 1980. Algorithms for separable nonlinear least squares problems. SIAM Review, 22(3):318–337.
Shum, H. Ikeuchi, K. and Reddy, R. 1995. Principal component analysis with missing data and its application to polyhedral object modeling. IEEE Transaction on Pattern Analysis and Machine Intelligence, 17(9):855–867.
Tomasi, C. and Kanade, T. 1992. Shape and motion from image streams under orthography: A factorization method. International Journal of Computer Vision, 9(2):137–154.
Wiberg, T. 1976. Computation of principal components when data are missing. In Proceedings of the Second Symposium of Computational Statistics. Berlin, pp. 229–326.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Okatani, T., Deguchi, K. On the Wiberg Algorithm for Matrix Factorization in the Presence of Missing Components. Int J Comput Vision 72, 329–337 (2007). https://doi.org/10.1007/s11263-006-9785-5
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11263-006-9785-5