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When I see their formulation, it is the same. In SALSA, the formulation is:

\begin{equation*} \min_{x} \phi(x) \text{ subject to } \frac{1}{2} \|Ax-y\|_F \leq \epsilon \end{equation*} This constrained problem is then being transformed into an unconstrained by adding the indicator function of feasible set: $x: \|Bx-y\| \leq \epsilon$ to the objective function. The resulting unconstrained problem is then transformed into a different constrained problem, by the application of a variable splitting operation. The final obtained constrained problem is then attacked by augmented Lagrangian scheme. $\phi(x)$ in my case is $\|x\|_1 \leq k$

The formulation for LASSO is similar:

\begin{equation*} \min_{x} \frac{1}{N} \Sigma_{i=1}^{N}\|y_i - Ax_i\|_F^2 \text{ subject to } \|x_i\|_1 \leq k \end{equation*}

N here is the number of cases. You can ignore it and set it equal to 1 in my case. I'm working on a cube data, but I want to simplify the answer to 2 dimensions before applying it to 3D.

nguyen545
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  • By "SALSA," do you mean the technique described at https://arxiv.org/abs/1602.00287? – whuber Jun 12 '18 at 17:56
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    Hi, I mean SALSA in this paper: https://arxiv.org/abs/0910.4887 But I'm sure they are the same. Also, if anyone downvote, please tell me why so I'll learn not to do it next time. This is a genuine question that I couldn't find or understand. – nguyen545 Jun 12 '18 at 18:02
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    I suspect the downvote might have been a protest against the lack of information in your question. Could you perhaps quote the two formulations of these procedures that you are referring to? – whuber Jun 12 '18 at 18:04
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    Please add some context for this question. I'm not sure it's answerable at present. – gung - Reinstate Monica Jun 12 '18 at 18:05
  • Hi all, I have adjusted the question. Tell me if you need any thing else. I'm not sure what is the different between the 2 approaches in solving this problem. – nguyen545 Jun 12 '18 at 18:38
  • Yes, it is $x_i$ but you can ignore it. I just want it to work in 2D first so setting $N = 1$ – nguyen545 Jun 12 '18 at 18:50

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