Tikhonov Regularization for Complex Matrices

Question

Tikhonov regularization is used to regularize ill-posed inverse problems if the matrix $A \in \mathbb{R}^{n,m}$ to be inversed has a high condition number. For example

$$ A=\begin{bmatrix}1&1\\ 1&1 \end{bmatrix} $$

As far as I understand it, one simply adds a multiple of the identity matrix

$$ \lambda I=\lambda\begin{bmatrix}1&0\\ 0&1 \end{bmatrix} $$

to the matrix A to be inverted in order to decrease its condition number. With $\lambda \in \mathbb{R}$. Now I want to apply regularization to a complex matrix $B\in\mathbb{C}^{n,m}$

$$ B=\begin{bmatrix}b_{1,1} e^{i\phi_{1,1}} & b_{1,2}e^{i\phi_{1,2}}\\ b_{2,1}e^{i\phi_{2,1}}&b_{2,2}e^{i\phi_{2,2}} \end{bmatrix} $$

with $i$ being the imaginary number $\phi_{n,m}$ being some angle and $b_{n,m} \in \mathbb{R}$. To apply Regularization to $B$ - can I increase the amplitudes $b_{1,1}$ and $b_{2,2}$ on the diagonal by multiplying them with some factor $1<c<\infty$?

Answer 1

Usually Tikhonov Regularization is applied in the following form:

$$ \arg \min_{\boldsymbol{x}} \frac{1}{2} {\left\| A \boldsymbol{x} - \boldsymbol{y} \right\|}_{2}^{2} + \frac{\lambda}{2} {\left\| \boldsymbol{x} \right\|}_{2}^{2} $$

This formulation can be seen as:

1. MAP Estimator with the prior of $ \boldsymbol{x} \sim N \left( 0, {\sigma}_{x}^{2} \right) $.
2. Limiting the ability of $ \boldsymbol{x} $ having high values (Sensitive -> Regularization).

This formulation works for $ A $ in Real or Complex domain as:

$$ \boldsymbol{x} = {\left( {A}^{H} A + \lambda I \right)}^{-1} {A}^{H} \boldsymbol{y} $$

Pay attention that the identity matrix over the complex domain is still the same matrix as over the real domain. Hence the solution above is valid for your case as well.

Pay attention that it is not done by multiplication of the matrix but adding a diagonal matrix.