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).
Let matrix contains the set of 1D vectors which have been centered. In PCA/SVD, we construct covariance matrix and Gram matrix
and compute their eigenvectors and . Since and we have
If we retain only principal eigenvectors in , this gives low-rank approximation of .
Here we deal with a set of 2D matrices . Suppose they are centered . We construct row–row and column–column covariance matrices
in exactly the same manner as in SVD, and compute their eigenvectors and . We approximate as
in identical fashion as in SVD. This gives a near optimal low-rank approximation of with the objective function
Error bounds similar to Eckard–Young theorem also exist.
2DSVD is mostly used in image compression and representation.