Block matrix pseudoinverse

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In mathematics, a block matrix pseudoinverse is a formula for the pseudoinverse of a partitioned matrix. This is useful for decomposing or approximating many algorithms updating parameters in signal processing, which are based on the least squares method.

Derivation[edit]

Consider a column-wise partitioned matrix:

If the above matrix is full rank, the Moore–Penrose inverse matrices of it and its transpose are

This computation of the pseudoinverse requires (n + p)-square matrix inversion and does not take advantage of the block form.

To reduce computational costs to n- and p-square matrix inversions and to introduce parallelism, treating the blocks separately, one derives [1]

where orthogonal projection matrices are defined by

The above formulas are not necessarily valid if does not have full rank – for example, if , then

Application to least squares problems[edit]

Given the same matrices as above, we consider the following least squares problems, which appear as multiple objective optimizations or constrained problems in signal processing. Eventually, we can implement a parallel algorithm for least squares based on the following results.

Column-wise partitioning in over-determined least squares[edit]

Suppose a solution solves an over-determined system:

Using the block matrix pseudoinverse, we have

Therefore, we have a decomposed solution:

Row-wise partitioning in under-determined least squares[edit]

Suppose a solution solves an under-determined system:

The minimum-norm solution is given by

Using the block matrix pseudoinverse, we have

Comments on matrix inversion[edit]

Instead of , we need to calculate directly or indirectly[citation needed][original research?]

In a dense and small system, we can use singular value decomposition, QR decomposition, or Cholesky decomposition to replace the matrix inversions with numerical routines. In a large system, we may employ iterative methods such as Krylov subspace methods.

Considering parallel algorithms, we can compute and in parallel. Then, we finish to compute and also in parallel.

See also[edit]

References[edit]

  1. ^ J.K. Baksalary and O.M. Baksalary (2007). "Particular formulae for the Moore–Penrose inverse of a columnwise partitioned matrix". Linear Algebra Appl. 421: 16–23. doi:10.1016/j.laa.2006.03.031. 

External links[edit]