# Regularizing the inverse coefficient matrix in multivariate regression

I'd like to minimize the objective

$$\operatorname{tr}[ (Y-XR^{-1})^T (Y-XR^{-1}) ] + \lambda \sum_{ij} |R_{ij}|$$

wrt to $$R$$ (which is $$P \times P$$ but non-symmetric) where $$Y$$ and $$X$$ are both $$N \times P$$. For my application it would make sense to have the diagonal of $$R$$ constrained to 1 (and possibly that $$\det{(I-R)} \leq 1$$ also). Obviously $$R$$ needs to be invertible.

I'm pretty sure this is non-convex in $$R$$, but can it be converted into a convex problem? e.g. optimize over $$W=R^{-1}$$. (I realize if I put the L1 penalty on $$W$$ rather than $$R$$ I get a lasso-like problem, but I want the sparsity on $$R$$ not $$W$$). I think the issue is $$W=R^{-1}$$ is a non-convex constraint.

Could I use an ADMM approach? e.g. minimize

$$\operatorname{tr}[ (Y-XW)^T (Y-XW) ] + \lambda \sum_{ij} |R_{ij}| + \gamma (\operatorname{tr}(RW) - \log \det R - \log \det W)$$

This is biconvex in $$R$$ and $$W$$, and increasing $$\gamma$$ will force $$W$$ closer to $$R^{-1}$$ (inspired by the graphical lasso).

$$\frac12 \operatorname{tr}[ (Y-XW)^T (Y-XW) ] + \lambda ||R||_1$$
s.t. $$RW = I$$. The augmented Lagrangian is
$$L(R,W,\theta)=\frac12\operatorname{tr}[ (Y-XW)^T (Y-XW) ] + \operatorname{tr}( \theta' (RW-I) ) + \frac12\rho \operatorname{tr}[ (RW-I)^T (RW-I) ] + \lambda ||R||_1$$
where $$\theta$$ is an $$P \times P$$ matrix of Lagrange multipliers. Iterate over optimizing $$W$$ (easy because $$L$$ is quadratic in $$W$$), $$R$$ (can be converted into a standard lasso regression) and updating
$$\theta \leftarrow \theta + \rho(RW-I)$$.