# Can ADMM be applied to "latently coupled" variables?

I've been studying a paper where the authors employ the ADMM in a way that has left me somewhat perplexed. The paper focuses on addressing a robust principal component analysis (RPCA) problem, enhanced with an additional prior. This problem is formulated as: $$\underset{L,E}{\min} f_1(L) + f_2(L) + g(E) \quad \text{s.t.} \quad X = L + E$$

To maintain focus on the central query, I've simplified the problem to avoid any unnecessary details. In this equation, $$X$$ represents a constant matrix. All functions are convex.

To separate the variables in the objective function, the authors introduced auxiliary variables $$Z$$ and $$Y$$, reformulating the problem as: $$\underset{Z,E,Y}{\min} f_1(Z) + f_2(Y) + g(E) \quad \text{s.t.} \quad X = L + E, Z = L, Y = L$$

Subsequently, they presented the augmented Lagrange function as follows: \begin{aligned} L(Z,E,Y,\Lambda_1,\Lambda_2, \Lambda_3) &= f_1(Z) + f_2(Y) + g(E) \\ &+ \langle L+E-X, \Lambda_1 \rangle + \lambda_1 \| L+E-X \| \\ &+ \langle Y - L, \Lambda_2 \rangle + \lambda_2 \| Y - L \| \\ &+ \langle Z - L, \Lambda_3 \rangle + \lambda_3 \| Z - L \|. \end{aligned} The paper details an iterative approach to solve the subproblems for $$Z$$, $$Y$$, $$L$$, $$E$$, followed by updating the dual variables.

However, a point of confusion arises here. While $$Z$$ and $$Y$$ appear to be decoupled, $$Z$$-$$L$$ and $$Y$$-$$L$$ remain coupled. In other words, the variables are "latently coupled". The authors assert that since $$Z$$ and $$Y$$ are decoupled, it's feasible to iteratively solve for $$Z$$, $$Y$$, and then address the $$L$$-subproblem (which is quite easy as what remains about $$L$$ are norms and inner products). My problems are:

1. Is their solution reasonable, or does it lean more towards a heuristic approach?
2. Could there be a more efficient method to tackle this problem?