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We consider the problem of learning a coefficient vector $x_{0}$ in $R^{N}$from noisy linear observations $y=Fx_{0}+w$ in $R^{M}$ in the high dimensionallimit $M,N$ to infinity with $\alpha=M/N$ fixed. We provide a rigorousderivation of an explicit formula -- first conjectured using heuristic methodsfrom statistical physics -- for the asymptotic mean squared error obtained bypenalized convex regression estimators such as the LASSO or the elastic net,for a class of very generic random matrices corresponding to rotationallyinvariant data matrices with arbitrary spectrum. The proof is based on aconvergence analysis of an oracle version of vector approximate message-passing(oracle-VAMP) and on the properties of its state evolution equations. Ourmethod leverages on and highlights the link between vector approximatemessage-passing, Douglas-Rachford splitting and proximal descent algorithms,extending previous results obtained with i.i.d. matrices for a large class ofproblems. We illustrate our results on some concrete examples and show thateven though they are asymptotic, our predictions agree remarkably well withnumerics even for very moderate sizes.

Asymptotic errors for convex penalized linear regression beyond Gaussian matrices