Two-dimensional singular-value decomposition

In linear algebra, 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

Let matrix $X=[\mathbf {x} _{1},\ldots ,\mathbf {x} _{n}]$ contains the set of 1D vectors which have been centered. In PCA/SVD, we construct covariance matrix $F$ and Gram matrix $G$

$F=XX^{\mathsf {T}}$ , $G=X^{\mathsf {T}}X,$

and compute their eigenvectors $U=[\mathbf {u} _{1},\ldots ,\mathbf {u} _{n}]$ and $V=[\mathbf {v} _{1},\ldots ,\mathbf {v} _{n}]$. Since $VV^{\mathsf {T}}=I$ and $UU^{\mathsf {T}}=I$ we have

$X=UU^{\mathsf {T}}XVV^{\mathsf {T}}=U\left(U^{\mathsf {T}}XV\right)V^{\mathsf {T}}=U\Sigma V^{\mathsf {T}}.$

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

2DSVD

Here we deal with a set of 2D matrices $(X_{1},\ldots ,X_{n})$. Suppose they are centered ${\textstyle \sum _{i}X_{i}=0}$. We construct row–row and column–column covariance matrices

$F=\sum _{i}X_{i}X_{i}^{\mathsf {T}}$ and $G=\sum _{i}X_{i}^{\mathsf {T}}X_{i}$

in exactly the same manner as in SVD, and compute their eigenvectors $U$ and $V$. We approximate $X_{i}$ as

$X_{i}=UU^{\mathsf {T}}X_{i}VV^{\mathsf {T}}=U\left(U^{\mathsf {T}}X_{i}V\right)V^{\mathsf {T}}=UM_{i}V^{\mathsf {T}}$

in identical fashion as in SVD. This gives a near optimal low-rank approximation of $(X_{1},\ldots ,X_{n})$ with the objective function

$J=\sum _{i=1}^{n}\left|X_{i}-LM_{i}R^{\mathsf {T}}\right|^{2}$

Error bounds similar to Eckard–Young theorem also exist.

2DSVD is mostly used in image compression and representation.

References

• 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.