0
$\begingroup$

Let $M$ be a $m \times n$ matrix, $x$ a $n$-vector, $y$ a $m$-vector, and $\|\cdot\|_2$ represent the $L_2$ norm (i.e., Euclidean norm). Given $M,y$, the goal is to find $x$ that minimizes the expression

$$\Psi(x) = \|x\|^2 + \|Mx-y\|^2.$$

Is there a closed-form expression for the optimal $x$? Without the $\|x\|^2$ term, this would be an instance of ordinary least squares linear regression and there would be a nice solution, but I'm not sure happens with this additional term.

$\endgroup$
1
$\begingroup$

Your problem is still a linear least squares problem. You can write $\Psi(x)$ as

$\Psi(x)=\| Hx - g \|_{2}^{2}$

where

$H=\left[ \begin{array}{c} I \\ M \end{array} \right] $

and

$g=\left[ \begin{array}{c} 0 \\ y \end{array} \right]$.

Using the normal equations,

$x=(H^{T}H)^{-1}H^{T}g=(M^{T}M+I)^{-1} M^{T}y$.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.