# Is there any point putting two simlar regularization terms in the objective function

I am trying to implement a dictionary learning based objective function. I have created two models as follows:

1) $$||X_{s} - Y_{s}D_{s}||_{F}^{2} + \lambda_{1}(||D_{s}||_{F}^{2} - 1)$$

2) $$||X_{s} - Y_{s}D_{s}||_{F}^{2} + \lambda_{1}(||D_{s}||_{F}^{2} - 1) + \lambda_{2}||D_{s}||_{F}^{2}$$

where the first term is the reconstruction loss and the regularization term puts a constraint on values that $$D_{s}$$ may assume.

In my opinion, the last term i.e. $$\lambda_{2}||D_{s}||_{F}^{2}$$ in the second model is extraneous because we have already ensured a similar constraint in the first model using $$\lambda_{1}(||D_{s}||_{F}^{2} - 1)$$.

$$\arg \min ( ||X_s−Y_sD_s||^2_F+λ_1(||D_s||^2_F−1)+λ_2||D_s||^2_F )$$ = $$\arg \min ( ||X_s−Y_sD_s||^2_F+(λ_1+λ_2)||D_s||^2_F-λ_1 )$$ = $$\arg \min ( ||X_s−Y_sD_s||^2_F+(λ_1+λ_2)||D_s||^2_F)$$ = $$\arg \min ( ||X_s−Y_sD_s||^2_F+\tilde{λ}_1||D_s||^2_F )$$
• (1) Why are you maximizing the objective?? (2) What happened to the $-\lambda_1$ term at the end?
• (1) sorry, meant arg min (2) $\lambda_1$ is a constant, it doesn't affect the result of the arg min May 19, 2020 at 12:54
• Thanks for the clarification of (2): I wasn't sure whether you were considering either of the $\lambda_i$ to be variables.