Suppose I have a system:

$$ Ax = b $$

where $A$ is a $m$ by $n$ matrix which is less than full rank (neither full column nor row rank). In my particular case $m<n$.

I'd like a combination of a minimum norm solution (for elements of $x$ which are not determined and a least squares solution for those that are determined or over determined).

I know I can approximate this with Tikhonov regularization:

$$ \mathop{\text{minimize}}\limits_x \|Ax-b\|^2 + \alpha\|x\|^2. $$

But this is not ideal as the regularizer tugs on all elements of $x$ not just those left undetermined. If I make $\alpha$ too large then my $x$ at determined or over determined elements will not be optimal (w.r.t. $\|Ax-b\|^2$). If $\alpha$ is too small, I imagine I run into conditioning issues.

Is there no way to formulate this correctly using some sort null space/QR decomposition?

Update: See accepted answer, but for completeness here's a full recipe: \begin{align} Ax &= b\\ AE &= QR\\ AE &= Q\left(\begin{array}{c}I\\0\end{array}\right)\left(\begin{array}{cc}R_1&R_2\end{array}\right)\\ \left(\begin{array}{c}R_1^T\\R_2^T\end{array}\right)F &= VT_\text{full}\\ \left(\begin{array}{c}R_1^T\\R_2^T\end{array}\right)F &= V\left(\begin{array}{c}I\\0\end{array}\right)T\\ AE &= Q\left(\begin{array}{c}I\\0\end{array}\right)FT^T\left(\begin{array}{cc}I&0\end{array}\right)V^T \\ x &= A^+b\\ x &= EV\left(\begin{array}{c}I\\0\end{array}\right)T^{-T}F^T\left(\begin{array}{cc}I&0\end{array}\right)Q^Tb \end{align}

Suppose I have a system:

$$ Ax = b $$

where $A$ is a $m$ by $n$ matrix which is less than full rank (neither full column nor row rank). In my particular case $m<n$.

I'd like a combination of a minimum norm solution (for elements of $x$ which are not determined and a least squares solution for those that are determined or over determined).

I know I can approximate this with Tikhonov regularization:

$$ \mathop{\text{minimize}}\limits_x \|Ax-b\|^2 + \alpha\|x\|^2. $$

But this is not ideal as the regularizer tugs on all elements of $x$ not just those left undetermined. If I make $\alpha$ too large then my $x$ at determined or over determined elements will not be optimal (w.r.t. $\|Ax-b\|^2$). If $\alpha$ is too small, I imagine I run into conditioning issues.

Is there no way to formulate this correctly using some sort null space/QR decomposition?

Update: See accepted answer, but for completeness here's a full recipe: \begin{align} Ax &= b\\ AE &= QR &\text{rank-revealing QR decomposition of A}\\ AE &= Q\left(\begin{array}{c}I\\0\end{array}\right)\left(\begin{array}{cc}R_1&R_2\end{array}\right) & \text{only keep $r$ (rank) many rows of $R$}\\ \left(\begin{array}{c}R_1^T\\R_2^T\end{array}\right)F &= VT_\text{full} &\text{rank-revealing QR decomposition of $\left(\begin{array}{c}R_1^T\\R_2^T\end{array}\right)$} \\ \left(\begin{array}{c}R_1^T\\R_2^T\end{array}\right)F &= V\left(\begin{array}{c}I\\0\end{array}\right)T & \text{only keep $r$ many rows of $T$}\\ AE &= Q\left(\begin{array}{c}I\\0\end{array}\right)FT^T\left(\begin{array}{cc}I&0\end{array}\right)V^T \\ x &= A^+b\\ x &= EV\left(\begin{array}{c}I\\0\end{array}\right)T^{-T}F^T\left(\begin{array}{cc}I&0\end{array}\right)Q^Tb \end{align}

Suppose I have a system:

$$ Ax = b $$

where $A$ is a $m$ by $n$ matrix which is less than full rank (neither full column nor row rank). In my particular case $m<n$.

I'd like a combination of a minimum norm solution (for elements of $x$ which are not determined and a least squares solution for those that are determined or over determined).

I know I can approximate this with Tikhonov regularization:

$$ \mathop{\text{minimize}}\limits_x \|Ax-b\|^2 + \alpha\|x\|^2. $$

But this is not ideal as the regularizer tugs on all elements of $x$ not just those left undetermined. If I make $\alpha$ too large then my $x$ at determined or over determined elements will not be optimal (w.r.t. $\|Ax-b\|^2$). If $\alpha$ is too small, I imagine I run into conditioning issues.

Is there no way to formulate this correctly using some sort null space/QR decomposition?

Suppose I have a system:

$$ Ax = b $$

where $A$ is a $m$ by $n$ matrix which is less than full rank (neither full column nor row rank). In my particular case $m<n$.

I'd like a combination of a minimum norm solution (for elements of $x$ which are not determined and a least squares solution for those that are determined or over determined).

I know I can approximate this with Tikhonov regularization:

$$ \mathop{\text{minimize}}\limits_x \|Ax-b\|^2 + \alpha\|x\|^2. $$

But this is not ideal as the regularizer tugs on all elements of $x$ not just those left undetermined. If I make $\alpha$ too large then my $x$ at determined or over determined elements will not be optimal (w.r.t. $\|Ax-b\|^2$). If $\alpha$ is too small, I imagine I run into conditioning issues.

Is there no way to formulate this correctly using some sort null space/QR decomposition?

Update: See accepted answer, but for completeness here's a full recipe: \begin{align} Ax &= b\\ AE &= QR\\ AE &= Q\left(\begin{array}{c}I\\0\end{array}\right)\left(\begin{array}{cc}R_1&R_2\end{array}\right)\\ \left(\begin{array}{c}R_1^T\\R_2^T\end{array}\right)F &= VT_\text{full}\\ \left(\begin{array}{c}R_1^T\\R_2^T\end{array}\right)F &= V\left(\begin{array}{c}I\\0\end{array}\right)T\\ AE &= Q\left(\begin{array}{c}I\\0\end{array}\right)FT^T\left(\begin{array}{cc}I&0\end{array}\right)V^T \\ x &= A^+b\\ x &= EV\left(\begin{array}{c}I\\0\end{array}\right)T^{-T}F^T\left(\begin{array}{cc}I&0\end{array}\right)Q^Tb \end{align}