Two-dimensional singular value decomposition

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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).

SVD[edit]

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 .

2DSVD[edit]

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.

References[edit]

  • 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.