Asymptotic errors for convex penalized linear regression beyond Gaussian matrices
This work addresses the theoretical understanding of regression errors for a broad class of random matrices, which is incremental but extends previous results beyond i.i.d. matrices.
The paper tackles the problem of learning a coefficient vector from noisy linear observations in high-dimensional settings, providing a rigorous derivation of an explicit formula for the asymptotic mean squared error of convex penalized regression estimators like LASSO, with predictions that match numerical results well even for moderate sizes.
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 dimensional limit $M,N$ to infinity with $α=M/N$ fixed. We provide a rigorous derivation of an explicit formula -- first conjectured using heuristic methods from statistical physics -- for the asymptotic mean squared error obtained by penalized convex regression estimators such as the LASSO or the elastic net, for a class of very generic random matrices corresponding to rotationally invariant data matrices with arbitrary spectrum. The proof is based on a convergence analysis of an oracle version of vector approximate message-passing (oracle-VAMP) and on the properties of its state evolution equations. Our method leverages on and highlights the link between vector approximate message-passing, Douglas-Rachford splitting and proximal descent algorithms, extending previous results obtained with i.i.d. matrices for a large class of problems. We illustrate our results on some concrete examples and show that even though they are asymptotic, our predictions agree remarkably well with numerics even for very moderate sizes.