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  X=(x_1,...,x_n) contains the set of 1D vectors which have been centered. In PCA/SVD, we construct covariance matrix  F and Gram matrix  G

 F=X X^T ,  G=X^T X ,

and compute their eigenvectors  U = (u_1, ..., u_n) and  V=(v_1,..., v_n) . Since  VV^T=I, UU^T=I , we have

 X = UU^T X VV^T = U (U^T XV) V^T = U \Sigma V^T

If we retain only  K principal eigenvectors in  U , V, this gives low-rank approximation of  X .

2DSVD[edit]

Here we deal with a set of 2D matrices  (X_1,...,X_n) . Suppose they are centered  \sum_i X_i =0 . We construct row–row and column–column covariance matrices

 F=\sum_i X_i X_i^T ,  G=\sum_i X_i^T X_i

in exactly the same manner as in SVD, and compute their eigenvectors  U and  V. We approximate  X_i as

 X_i = U U^T X_i  V V^T = U (U^T X_i V) V^T = U M_i V^T

in identical fashion as in SVD. This gives a near optimal low-rank approximation of  (X_1,...,X_n) with the objective function

 J=  \sum_i | X_i - L M_i R^T| ^2

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.