On prediction with the LASSO when the design is not incoherent

The LASSO estimator is an $\ell_1$-norm penalized least-squares estimator, which was introduced for variable selection in the linear model. When the design matrix satisfies, e.g. the Restricted Isometry Property, or has a small coherence index, the LASSO estimator has been proved to recover, with high probability, the support and sign pattern of sufficiently sparse regression vectors. Under similar assumptions, the LASSO satisfies adaptive prediction bounds in various norms. The present note provides an adaptive prediction bound based on a new index for measuring how favorable is a design matrix for the LASSO estimator. One original feature of our contribution is that the results hold for design matrices with highly correlated columns.