Block matrix pseudoinverse

Consider a column-wise partitioned matrix:

${\displaystyle {\begin{bmatrix}\mathbf {A} &\mathbf {B} \end{bmatrix}},\quad \mathbf {A} \in \mathbb {R} ^{m\times n},\quad \mathbf {B} \in \mathbb {R} ^{m\times p},\quad m\geq n+p.}$

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

${\displaystyle {\begin{aligned}{\begin{bmatrix}\mathbf {A} &\mathbf {B} \end{bmatrix}}^{+}&=\left({\begin{bmatrix}\mathbf {A} &\mathbf {B} \end{bmatrix}}^{\textsf {T}}{\begin{bmatrix}\mathbf {A} &\mathbf {B} \end{bmatrix}}\right)^{-1}{\begin{bmatrix}\mathbf {A} &\mathbf {B} \end{bmatrix}}^{\textsf {T}},\\{\begin{bmatrix}\mathbf {A} ^{\textsf {T}}\\\mathbf {B} ^{\textsf {T}}\end{bmatrix}}^{+}&={\begin{bmatrix}\mathbf {A} &\mathbf {B} \end{bmatrix}}\left({\begin{bmatrix}\mathbf {A} &\mathbf {B} \end{bmatrix}}^{\textsf {T}}{\begin{bmatrix}\mathbf {A} &\mathbf {B} \end{bmatrix}}\right)^{-1}.\end{aligned}}}$

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]

${\displaystyle {\begin{aligned}{\begin{bmatrix}\mathbf {A} &\mathbf {B} \end{bmatrix}}^{+}&={\begin{bmatrix}\mathbf {P} _{B}^{\perp }\mathbf {A} \left(\mathbf {A} ^{\textsf {T}}\mathbf {P} _{B}^{\perp }\mathbf {A} \right)^{-1}\\\mathbf {P} _{A}^{\perp }\mathbf {B} \left(\mathbf {B} ^{\textsf {T}}\mathbf {P} _{A}^{\perp }\mathbf {B} \right)^{-1}\end{bmatrix}}={\begin{bmatrix}\left(\mathbf {P} _{B}^{\perp }\mathbf {A} \right)^{+}\\\left(\mathbf {P} _{A}^{\perp }\mathbf {B} \right)^{+}\end{bmatrix}},\\{\begin{bmatrix}\mathbf {A} ^{\textsf {T}}\\\mathbf {B} ^{\textsf {T}}\end{bmatrix}}^{+}&={\begin{bmatrix}\mathbf {P} _{B}^{\perp }\mathbf {A} \left(\mathbf {A} ^{\textsf {T}}\mathbf {P} _{B}^{\perp }\mathbf {A} \right)^{-1},\quad \mathbf {P} _{A}^{\perp }\mathbf {B} \left(\mathbf {B} ^{\textsf {T}}\mathbf {P} _{A}^{\perp }\mathbf {B} \right)^{-1}\end{bmatrix}}={\begin{bmatrix}\left(\mathbf {A} ^{\textsf {T}}\mathbf {P} _{B}^{\perp }\right)^{+}&\left(\mathbf {B} ^{\textsf {T}}\mathbf {P} _{A}^{\perp }\right)^{+}\end{bmatrix}},\end{aligned}}}$

where orthogonal projection matrices are defined by

${\displaystyle {\begin{aligned}\mathbf {P} _{A}^{\perp }&=\mathbf {I} -\mathbf {A} \left(\mathbf {A} ^{\textsf {T}}\mathbf {A} \right)^{-1}\mathbf {A} ^{\textsf {T}},\\\mathbf {P} _{B}^{\perp }&=\mathbf {I} -\mathbf {B} \left(\mathbf {B} ^{\textsf {T}}\mathbf {B} \right)^{-1}\mathbf {B} ^{\textsf {T}}.\end{aligned}}}$

The above formulas are not necessarily valid if ${\displaystyle {\begin{bmatrix}\mathbf {A} &\mathbf {B} \end{bmatrix}}}$ does not have full rank – for example, if ${\displaystyle \mathbf {A} \neq 0}$, then

${\displaystyle {\begin{bmatrix}\mathbf {A} &\mathbf {A} \end{bmatrix}}^{+}={\frac {1}{2}}{\begin{bmatrix}\mathbf {A} ^{+}\\\mathbf {A} ^{+}\end{bmatrix}}\neq {\begin{bmatrix}\left(\mathbf {P} _{A}^{\perp }\mathbf {A} \right)^{+}\\\left(\mathbf {P} _{A}^{\perp }\mathbf {A} \right)^{+}\end{bmatrix}}=0}$

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

Suppose a solution ${\displaystyle \mathbf {x} ={\begin{bmatrix}\mathbf {x} _{1}\\\mathbf {x} _{2}\\\end{bmatrix}}}$ solves an over-determined system:

${\displaystyle {\begin{bmatrix}\mathbf {A} ,&\mathbf {B} \end{bmatrix}}{\begin{bmatrix}\mathbf {x} _{1}\\\mathbf {x} _{2}\\\end{bmatrix}}=\mathbf {d} ,\quad \mathbf {d} \in \mathbb {R} ^{m\times 1}.}$

Using the block matrix pseudoinverse, we have

${\displaystyle \mathbf {x} ={\begin{bmatrix}\mathbf {A} ,&\mathbf {B} \end{bmatrix}}^{+}\,\mathbf {d} ={\begin{bmatrix}\left(\mathbf {P} _{B}^{\perp }\mathbf {A} \right)^{+}\\\left(\mathbf {P} _{A}^{\perp }\mathbf {B} \right)^{+}\end{bmatrix}}\mathbf {d} .}$

Therefore, we have a decomposed solution:

${\displaystyle \mathbf {x} _{1}=\left(\mathbf {P} _{B}^{\perp }\mathbf {A} \right)^{+}\,\mathbf {d} ,\quad \mathbf {x} _{2}=\left(\mathbf {P} _{A}^{\perp }\mathbf {B} \right)^{+}\,\mathbf {d} .}$

Row-wise partitioning in under-determined least squares

Suppose a solution ${\displaystyle \mathbf {x} }$ solves an under-determined system:

${\displaystyle {\begin{bmatrix}\mathbf {A} ^{\textsf {T}}\\\mathbf {B} ^{\textsf {T}}\end{bmatrix}}\mathbf {x} ={\begin{bmatrix}\mathbf {e} \\\mathbf {f} \end{bmatrix}},\quad \mathbf {e} \in \mathbb {R} ^{n\times 1},\quad \mathbf {f} \in \mathbb {R} ^{p\times 1}.}$

The minimum-norm solution is given by

${\displaystyle \mathbf {x} ={\begin{bmatrix}\mathbf {A} ^{\textsf {T}}\\\mathbf {B} ^{\textsf {T}}\end{bmatrix}}^{+}\,{\begin{bmatrix}\mathbf {e} \\\mathbf {f} \end{bmatrix}}.}$

Using the block matrix pseudoinverse, we have

${\displaystyle \mathbf {x} ={\begin{bmatrix}\left(\mathbf {A} ^{\textsf {T}}\mathbf {P} _{B}^{\perp }\right)^{+}&\left(\mathbf {B} ^{\textsf {T}}\mathbf {P} _{A}^{\perp }\right)^{+}\end{bmatrix}}{\begin{bmatrix}\mathbf {e} \\\mathbf {f} \end{bmatrix}}=\left(\mathbf {A} ^{\textsf {T}}\mathbf {P} _{B}^{\perp }\right)^{+}\,\mathbf {e} +\left(\mathbf {B} ^{\textsf {T}}\mathbf {P} _{A}^{\perp }\right)^{+}\,\mathbf {f} .}$

• Invertible matrix § Blockwise inversion