Plucker matrix

30 year-old Entertainer or Range Artist Wesley from Drumheller, really loves vehicle, property developers properties for sale in singapore singapore and horse racing. Finds inspiration by traveling to Works of Antoni Gaudí. 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 $X=(x_{1},...,x_{n})$ contains the set of 1D vectors which have been centered. In PCA/SVD, we construct covariance matrix $F$ and Gram matrix $G$

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

and compute their eigenvectors $U=(u_{1},...,u_{n})$ and $V=(v_{1},...,v_{n})$. Since $VV^{T}=I,UU^{T}=I$, we have

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

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

2DSVD

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

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

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

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

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.