Two-dimensional singular value decomposition
Two-dimensional singular value decomposition (2DSVD) computes the low-rank approximation of a set of matrices such as 2D images or weather maps in a manner almost identical to SVD (singular value decomposition) which computes the low-rank approximation of a single matrix (or a set of 1D vectors).
Let matrix contains the set of 1D vectors which have been centered. In PCA/SVD, we construct covariance matrix and Gram matrix
- , ,
and compute their eigenvectors and . Since , we have
If we retain only principal eigenvectors in , this gives low-rank approximation of .
Here we deal with a set of 2D matrices . Suppose they are centered . We construct row–row and column–column covariance matrices
in exactly the same manner as in SVD, and compute their eigenvectors and . We approximate as
in identical fashion as in SVD. This gives a near optimal low-rank approximation of with the objective function
Error bounds similar to Eckard-Young Theorem also exist.
2DSVD is mostly used in image compression and representation.
- Chris Ding and Jieping Ye. "Two-dimensional Singular Value Decomposition (2DSVD) for 2D Maps and Images". Proc. SIAM Int'l Conf. Data Mining (SDM'05), pp. 32–43, April 2005. http://ranger.uta.edu/~chqding/papers/2dsvdSDM05.pdf
- Jieping Ye. "Generalized Low Rank Approximations of Matrices". Machine Learning Journal. Vol. 61, pp. 167—191, 2005.