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).

SVD[]

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

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

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

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

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

2DSVD[]

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

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

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

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

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

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

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.