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.


1 Answer 1


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

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


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


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

Using the normal equations,

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


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