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I am trying to solve the ridge regression problem given by

$D_{s} = min_{D_{s}} \hspace{2mm} || X_{s} - D_{s}Y_{s}||_F^{2} + \lambda \hspace{2mm} ||D_{s}||_{F}^{2} \hspace{10mm} s.t. \hspace{5mm} ||d_{i}||_{2}^{2} \leq 1$

What extra information is this $||d_{i}||_{2}^{2} \leq 1$ giving and how to incorporate this in the objective function.

Actually, I tried to incorporate $||d_{i}||_{2}^{2} \leq 1$ by writing it as $||D_{s}||_{F}^{2} \leq 1$. Am I correct ?

$D_{s}$ is the matrix that need to be learnt.

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    $\begingroup$ Could you clarify your notation and the type of problem you are addressing? It looks like a multivariate regression model where you have more than one dependent variable for each observation. However using the standard notation of $Y$ for the dependent variables and $X$ for the independent variables, then your sum of squared errors would be $\|Y-XD\|$ instead of $\|X-YD\|$. Also could you clarify if $d_i$ is a row or column of $D_s$? Finally, presumably the constraint is supposed to hold for all $i$. $\endgroup$
    – josliber
    Mar 4, 2020 at 14:29

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It looks like you might be confusing the Penalized and Lagrangian forms of ridge regression, and trying to do both at once. See e.g. One-to-one correspondence between penalty parameters of equivalent formulations of penalised regression methods

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