Fix λ>0 and consider a sequence of high-dimensional linear models
y=Xθ0+w,
where X∈Rn×d, n/d→δ∈(0,∞), the coordinates of w are independent N(0,σ(n)2) random variables with σ(n)/n→σ0, and the empirical distribution d−1∑j=1dδθ0,j converges in Wasserstein-k distance to the law of a random variable Θ. Define the Lasso estimator
θ(λ)=argminθ∈Rd{2n1∥y−Xθ∥22+λ∥θ∥1},
the soft-thresholding function
η(u;a)=argminv∈R{21(u−v)2+a∣v∣}=sign(u)(∣u∣−a)+,
and, for a>0, the debiased estimator
θd(a,λ)=θ(λ)+nλaXT(y−Xθ(λ)).
For a design matrix with independent standard Gaussian entries, let (α∗,τ∗)∈(0,∞)2 be the unique solution of
λ=α∗[1−δ1P(∣Θ+τ∗Z∣≥α∗)],τ∗2=σ02+δ1E[(η(Θ+τ∗Z;α∗)−Θ)2],
where Z∼N(0,1) is independent of Θ. In this Gaussian case, for every pseudo-Lipschitz function ψ:R2→R of order k,
d1j=1∑dψ(θ0,j,θjd(α∗,λ))a.s.E[ψ(Θ,Θ+τ∗Z)].
Characterize the random-matrix ensembles with non-independent or non-identically distributed entries for which this same asymptotic characterization remains valid. For ensembles for which the Gaussian prediction is not unchanged, determine the corresponding generalized asymptotic law of the Lasso and its debiased estimator.