Two-dimensional singular-value decomposition

2DSVD (two-dimensional singular value decomposition) 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) computes the low-rank approximation of a single matrix (or a set of 1D vectors).

Let matrix $X=(x_{1},...,x_{n})$ contains the set of 1D vectors. In SVD, we construct covariance matrix $F=XX^{T}$ and Gram matrix $F=X^{T}X$, and compute their eigenvectors U and V. We approximate X as

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

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

In 2DSVD, we deal with a set of 2D matrices $(X_{1},...,X_{n})$ . We construct $F=\sum _{i}X_{i}^{T}X_{i}$ and $G=\sum _{i}X_{i}^{T}X_{i}$ in exact same manner as in SVD, and compute their eigenvectors U and V. We approximate $X_{i}$ as|

$X_{i}=UU^{T}X_{i}VV^{T}=U(U^{T}X_{i}V)V^{T}=UM_{i}V^{T}$

in almost identical fasion as in SVD. This gives a good low-rank approximation of $(X_{1},...,X_{n})$ with the objective function

$J=\sum _{i}|X_{i}-LM_{i}R^{T}|^{2}$