Tikhonov Regularization for Complex Matrices

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?

• In most cases your matrix B will be "well behaved". What about it requires regularization ? Jul 7, 2021 at 7:21
• Im solving an inverse problem where the condition number of B will be very high and the problem will always be severly ill-conditioned for certain frequencies in application. In my application, this leads to very high amplification of noise and divergence of the iterative procedures used to solve the problem. Consequently, I need to regularize the complex matrix to avoid high amplification of noise and to stabilize the system. Jul 7, 2021 at 7:54

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.

• I liked your answer, is there a systematic way to choose $\lambda$?
– Bob
Aug 1, 2021 at 20:15
• Yes - for example the "L-Curve" Method where you plot the regularization parameter vs the error of the solution. Aug 2, 2021 at 7:00
• I have a follow up question: As far as I understand this regularization changes the phase in the elements of $x$. Is there an easy way to manipulate the matrix in such a way that the Amplitude in $x$ changes but the phase of the elements in $x$ are the same? Sep 14, 2021 at 8:42