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Asymptotic Normality of Spectrum-Aware Debiasing Beyond Right-Rotationally Invariant Designs

Mathematical StatisticsOptimization & Variational MethodsInformation Theory

Sourced from the work of Yufan Li, Pragya Sur

§ Problem Statement

Setup

For each nn, observe (y,X)(y,X) from the high-dimensional linear model

y=Xβ0+ε,X∈Rn×p, y∈Rn, β0∈Rp, ε∈Rn,y=X\beta_0+\varepsilon,\qquad X\in\mathbb R^{n\times p},\ y\in\mathbb R^n,\ \beta_0\in\mathbb R^p,\ \varepsilon\in\mathbb R^n,

with proportional asymptotics p=pnp=p_n, pn/n→γ∈(0,∞)p_n/n\to\gamma\in(0,\infty). Let

Sn:=1nX⊤X=Vn diag(λ1,n,…,λp,n) Vn⊤S_n:=\frac1nX^\top X=V_n\,\mathrm{diag}(\lambda_{1,n},\dots,\lambda_{p,n})\,V_n^\top

be the spectral decomposition of the sample covariance.

Fully explicit definition of the operator Pn\mathcal P_n: for any measurable scalar transfer function ℘n:[0,∞)→R\wp_n:[0,\infty)\to\mathbb R, define the spectral-functional operator

Pn(Sn):=Vn diag ⁣(℘n(λ1,n),…,℘n(λp,n)) Vn⊤∈Rp×p.\mathcal P_n(S_n):=V_n\,\mathrm{diag}\!\big(\wp_n(\lambda_{1,n}),\dots,\wp_n(\lambda_{p,n})\big)\,V_n^\top\in\mathbb R^{p\times p}.

Equivalently, Pn\mathcal P_n applies ℘n\wp_n to each eigenvalue of SnS_n and keeps the same eigenvectors (standard matrix functional calculus).

Given an initial estimator β^init\hat\beta^{\mathrm{init}}, define the one-step spectrum-aware debiased estimator by

β^SAD=β^init+ηn Pn ⁣(Sn)1nX⊤ ⁣(y−Xβ^init).\hat\beta^{\mathrm{SAD}}=\hat\beta^{\mathrm{init}}+\eta_n\,\mathcal P_n\!\left(S_n\right)\frac1nX^\top\!\left(y-X\hat\beta^{\mathrm{init}}\right).

In the baseline right-rotationally invariant SAD formula of Li & Sur (2025), this reduces to a scalar-rescaled gradient correction (the paper's spectrum-aware adjustment), i.e. a special case of the above with scalar preconditioning.

This setup follows Li & Sur (2025).

Detailed known theorem under right-rotationally invariant designs (source Theorem 3.1/Corollary 3.2/Theorem 6.2, notation of arXiv v6): assume

  1. Right-rotationally invariant design X=Q⊤DOX=Q^\top D O with O∼Haar(O(p))O\sim\mathrm{Haar}(\mathcal O(p)), O⊥εO\perp\varepsilon, empirical singular-value distribution converging in W2W_2, and bounded operator norm of X⊤XX^\top X.

  2. Gaussian noise ε∼N(0,σ2In)\varepsilon\sim N(0,\sigma^2 I_n) and signal regularity as in the source (deterministic or random-independent with empirical W2W_2 limit).

  3. Convex penalty regularity and fixed-point existence assumptions of the source.

  4. (If σ2\sigma^2 is unknown) the source's nondegeneracy condition enabling consistent noise-level estimation.

Then the paper proves:

  1. Empirical-distribution CLT:
τ^∗,n−1/2(β^SAD−β0) →W2 N(0,1)\hat\tau_{*,n}^{-1/2}\big(\hat\beta^{\mathrm{SAD}}-\beta_0\big)\ \xrightarrow[W_2]{}\ \mathcal N(0,1)

a.s. as n,p→∞n,p\to\infty.

  1. Finite-dimensional (hence coordinatewise) CLT under the source exchangeability condition on β0\beta_0: for any fixed finite index set I\mathcal I,
β^ISAD−β0,Iτ^∗,n⇒N ⁣(0,I∣I∣),\frac{\hat\beta^{\mathrm{SAD}}_{\mathcal I}-\beta_{0,\mathcal I}}{\sqrt{\hat\tau_{*,n}}} \Rightarrow \mathcal N\!\big(0,I_{|\mathcal I|}\big),

so in particular for fixed coordinate jj,

β^jSAD−β0,jτ^∗,n⇒N(0,1).\frac{\hat\beta^{\mathrm{SAD}}_j-\beta_{0,j}}{\sqrt{\hat\tau_{*,n}}}\Rightarrow\mathcal N(0,1).
  1. Consistent estimation of centering/scale quantities used by the inference procedure (including the asymptotic variance):
γ^n→γ∗,η^n→η∗,τ^∗,n→τ∗,τ^∗∗,n→τ∗∗,\hat\gamma_n\to\gamma_*,\quad \hat\eta_n\to\eta_*,\quad \hat\tau_{*,n}\to\tau_*,\quad \hat\tau_{**,n}\to\tau_{**},

and (when unknown) σ^n2→σ2\hat\sigma_n^2\to\sigma^2, a.s.

Unsolved Problem

The documented open extension (Appendix D, Conjecture D.1) asks whether an analogous coordinatewise CLT and consistent variance estimation continue to hold for the paper's ellipsoidal design model class.

Precisely, for the ellipsoidal models in Conjecture D.1, define the design as

X=Qn⊤DnOnΣn1/2,X=Q_n^\top D_n O_n\Sigma_n^{1/2},

where Σn∈Rp×p\Sigma_n\in\mathbb R^{p\times p} is observed and nonsingular, Qn∈Rn×nQ_n\in\mathbb R^{n\times n} and On∈Rp×pO_n\in\mathbb R^{p\times p} are orthogonal, Dn∈Rn×pD_n\in\mathbb R^{n\times p} is diagonal (rectangular), and On∼Haar(O(p))O_n\sim\mathrm{Haar}(\mathcal O(p)) is independent of (ε,Qn,Dn)(\varepsilon,Q_n,D_n).

Allow a (possibly non-separable) proper closed convex penalty Hn:Rp→R∪{+∞}\mathcal H_n:\mathbb R^p\to\mathbb R\cup\{+\infty\} and initial estimator

β^init∈arg⁡min⁡b∈Rp 12∥y−Xb∥22+Hn(b).\hat\beta^{\mathrm{init}}\in\arg\min_{b\in\mathbb R^p}\ \frac12\|y-Xb\|_2^2+\mathcal H_n(b).

The ellipsoidal SAD correction considered in Appendix D is

β^ell=β^init+1αn Σn−1X⊤ ⁣(y−Xβ^init),\hat\beta^{\mathrm{ell}}=\hat\beta^{\mathrm{init}}+\frac1{\alpha_n}\,\Sigma_n^{-1}X^\top\!\left(y-X\hat\beta^{\mathrm{init}}\right),

where αn\alpha_n solves the fixed-point equation

1p∑i=1p1di,n2−αnp Tr ⁣(αnIp+Σn−1∇2Hn(β^init))−1+1=1,\frac1p\sum_{i=1}^p\frac{1}{\frac{d_{i,n}^2-\alpha_n}{p}\,\mathrm{Tr}\!\left(\alpha_n I_p+\Sigma_n^{-1}\nabla^2\mathcal H_n(\hat\beta^{\mathrm{init}})\right)^{-1}+1}=1,

with di,n2d_{i,n}^2 the eigenvalues of X⊤XX^\top X (and with the same extension convention for non-smooth penalties used in the source).

Question: prove or disprove that for each fixed coordinate jj there exist a centering term bj,nb_{j,n} and a consistent variance estimator τ^j,n2\hat\tau_{j,n}^2 such that

β^jell−β0,j−bj,nτ^j,n ⇒n,p→∞ N(0,1),τ^j,n2/τj,n2→p1,\frac{\hat\beta^{\mathrm{ell}}_j-\beta_{0,j}-b_{j,n}}{\hat\tau_{j,n}}\ \xRightarrow[n,p\to\infty]{}\ \mathcal N(0,1), \qquad \hat\tau_{j,n}^2/\tau_{j,n}^2\overset{p}\to 1,

with nondegenerate limit scale τj,n2\tau_{j,n}^2, without right-rotational invariance.

§ Discussion

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§ Significance & Implications

Appendix D records this as an explicit open extension (ellipsoidal models). Resolving it would determine whether the proven right-rotationally-invariant CLT extends at least to that specific non-RRI class.

§ Known Partial Results

  • Li et al. (2025): The paper proves asymptotic normality (with proper centering/scaling) and consistent variance estimation for right-rotationally invariant designs; Appendix D states Conjecture D.1 for ellipsoidal models as open. No definitive published proof or counterexample resolving Conjecture D.1 is identified in the cited sources.

§ References

[1]
[2]

Spectrum-Aware Debiasing: A Modern Inference Framework with Applications to Principal Components Regression

Yufan Li, Pragya Sur (2023)

📍 Appendix D (Conjectures for Ellipsoidal Models), Conjecture D.1, with the immediately following sentence: "We leave the proof of Conjecture D.1 as an open problem"; pp. 56-57 (arXiv v6).

Preprint version used for exact appendix citation.

Link ↗arXiv ↗

§ Tags

high-dimensional-inferencedebiasingrotational-invarianceellipsoidal-modelscentral-limit-theorem
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