Roughning matrix format for the first-order Tikhonov Regularization (inverse problem)

Question

I have been trying to solve the regularized least square problem of

min||Gm-d||^2 + a ||Lm||^2

using first order Tikhonov regularization method.

the general form of L for calculating the first derivative of m is

L1={{-1,1,0,0,...,0},{0,-1,1,0,0,...,0}, ...,{0,0,...,-1,1}

Given that the matrix G is a 50x50 lower triangular matrix, what would be the components of L1? I mean obviously it cannot be a 50x50 matrix, as the last row will be {0,0,...,-1}.

Answer 1

The answer is L should be 49x50 matrix

Answer 2

(sorry to revive an old topic, don't mind to open old wounds but I stumbled on it and think I can help)

Your $L$ matrix has to be $50\times 50$.

First order Tikhonov regularization as you wrote it (watch out as $L$ is commonly used to denote a specific kind of norm, this is a bit misleading), $|| L \mathbf{m}||^2$ will give the $\ell_2$-norm squared of the first order (discrete) derivative of $\mathbf{m}$. That's why it is $\{1, -1, 0, 0, ..\}$, for a constant signal it will give zero.

Now you have a problem at the boundaries, and there you have to make a choice on what boundary conditions you'd use. The two typical choices are:

1. symmetric condition, you still have to decide where exactly you put the limit
2. periodic, you assume that your signal repeats itself. In this case, the last row of your matrix $L$ should be $\{1, 0, 0, ..., 0, -1\}$

Hope it helps.