Abstract
This paper concerns with the problem of approximating a target matrix with a matrix of lower rank with respect to a weighted norm. Weighted norms can arise in several situations: when some of the entries of the matrix are not observed or need not to be treated equally. A gradient flow approach for solving weighted low rank approximation problems is provided. This approach allows the treatment of both real and complex matrices and exploits some important features of the approximation matrix that optimization techniques do not use. Finally, some numerical examples are provided.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Chu, M.T., Del Buono, N., Lopez, L., Politi, T.: On the Low Rank Approximation of Data on the Unit Sphere. Submitted for publication
Del Buono, N., Lopez, L.: Runge-Kutta Type Methods Based on Geodesics for Systems of ODEs on the Stiefel Manifold. BIT 41(5), 912–923 (2001)
Edelman, A., Arias, T.A., Smith, S.T.: The Geometry of Algorithms with Orthogonality Constraints. SIAM J. Matr. Anal. Appl. 20, 303–353 (1998)
Klema, V.C., Laub, A.J.: The singular value decomposition: its computation and some applications. IEEE Trans. Automat. Contr. 25, 164–176 (1980)
Lu, W.S.: Is the SVD-Based Low-Rank Approximation Optimal? In: Proc. 38th Midwest Symp. CAS, Rio, Brazil, pp. 470–473 (1995)
Lu, W.S., Pei, S.C., Wang, P.H.: Weighted Low-Rank Approximation of General Complex Matrices and its Application in the Design of 2-D Digital Filters. IEEE Trans. Circuits Syst. I 44(7), 650–655 (1997)
Lu, W.S., Pei, S.C.: On Optimal Low-Rank Approximation of Multidimensional Discrete Signals. IEEE Trans. Circuits Syst. II 45(3), 417–422 (1998)
Lu, W.S., Antoniou, A.: New Method for Weighted Low-Rank Approximation of Complex-Valued Matrices and its Application for the Design of 2-D Digital Filters. In: ISCAS, vol. III, pp. 694–697 (2003)
Simonsson, L., Eldén, L.: Computing a Partial SVD of a Matrix with Missing Data. In: Numerical Linear Algebra and its Applications: XXI International School and Workshop, Porto Giardino, Monopoli, Italy (2003)
Shpak, D.: A weighted-least-squares matrix decomposition method with applications to the design of two-dimensional digital filters. In: IEEE Thirty Third Midwest Symposium on Circuits and Systems (1990)
Srebro, N., Jaakkola, T.: Generalized Low-Rank Approximations. AI Memo 2003-001, MIT Artificial Intelligence Laboratory (2003)
Srebro, N., Jaakkola, T.: Weighted Low-Rank Approximations. In: Proceedings of Twentieth International Conference on Machine Learning (ICML 2003), Washington DC (2003)
Tenenbaum, J.B., Freeman, W.T.: Separating style and content with bilinear models. Neural Computation 12, 1247–1283 (2000)
Young, G.: Maximum likelihood estimation and factor anlysis. Psychometrika 6, 49–53 (1940)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2004 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Del Buono, N., Politi, T. (2004). A Continuous Technique for the Weighted Low-Rank Approximation Problem. In: Laganá, A., Gavrilova, M.L., Kumar, V., Mun, Y., Tan, C.J.K., Gervasi, O. (eds) Computational Science and Its Applications – ICCSA 2004. ICCSA 2004. Lecture Notes in Computer Science, vol 3044. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-24709-8_104
Download citation
DOI: https://doi.org/10.1007/978-3-540-24709-8_104
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-22056-5
Online ISBN: 978-3-540-24709-8
eBook Packages: Springer Book Archive