4.1 algorithm, as shown in document "Randomized LU Decomposition" in:
First, the input matrix A projected on a random matrix G, to obtain a matrix Y, reflects a large part of this matrix A range of the input matrix;
and , decomposition of the matrix Y RRLU application, the substrate to give L K and the U- K , and again enter the projection matrix a, matrix B obtained;
then, again the LU column matrix B is selected from principal component decomposition to obtain a new substrate L B and the U- B ;
and finally, to obtain L = L K L B , the U-= the U- B .
In the process of implementing the algorithm needs to be considered in step 3 RRLU decomposition is how to achieve, so we find the following articles based on citations of this article, and a study.
RRLU decomposition: full name of the rank-revealing LU factorizations, to realize the contents of the algorithm used are:
- Compute the matrix volume (Volume)
figure 3.1 shows the matrix defines the volume of the definition is the product of all the singular values.
Then, the paper gives the definition of a local maximum and the volume of the matrix: row select k (or k-th line) from the original sub-matrix constituting the matrix A in B, and B is such that the maximum volume of the matrix.
Specifically original definition, as shown below:
In practice, in order to increase the operability, to avoid falling into an infinite loop, the FIG inequality (3.2) is often replaced by the inequality (3.3)
noted when using inequality (3.3), corresponding to the matrix B the volume is no longer solely dependent on the local, but also on the coefficients selected mju therefore where B is the matrix volume (Vol (B)) is also known as partial mju maximum volume. FIG follows:
The idea of this document is to find RRLU decomposition of the matrix A, it is to find two permutation matrices of the matrix may take a B (i.e. the volume of the matrix is a partial matrix mju maximum), then on the basis of partial mju The maximum elect k row. To achieve the above-mentioned idea of taking a subset of the matrix, the authors used the Gaussian elimination.
RRLU algorithm:
