High-dimensional linear regression inference via $\ell^2$ weak convergence

Koji Tsukuda; Kou Fujimori

arxiv: 2602.07480 · v4 · submitted 2026-02-07 · 🧮 math.ST · stat.TH

High-dimensional linear regression inference via ell² weak convergence

Kou Fujimori , Koji Tsukuda This is my paper

Pith reviewed 2026-05-16 06:21 UTC · model grok-4.3

classification 🧮 math.ST stat.TH

keywords high-dimensional regressionweak convergenceHilbert spaceasymptotic normalitylinear hypothesessimultaneous inferenceconfidence bandssparsity

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The pith

High-dimensional regression coefficient estimators converge weakly in a separable Hilbert space even as sparsity diverges.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proves that suitably centered and scaled estimators of the coefficient vector in high-dimensional linear regression converge weakly to a Gaussian random element inside the space of square-summable sequences. This convergence holds when only finitely many coefficients are of moderate size while the number of tiny nonzero coefficients is permitted to grow without bound. Because the limit lives in a separable Hilbert space, the continuous mapping theorem applies directly to a wide class of functionals, producing limiting distributions for tests and estimators. The resulting tools cover both fixed finite sets of linear hypotheses and simultaneous inference over infinitely many hypotheses, including global tests and confidence bands for the regression function itself. All limiting laws are weighted sums of independent chi-squared random variables whose critical values can be approximated by plug-in estimators.

Core claim

We prove weak convergence in a separable Hilbert space for estimators of high-dimensional regression coefficients, which yields asymptotic normality and enables direct use of standard asymptotic tools such as the continuous mapping theorem. The approach permits diverging sparsity with many small nonzero coefficients, while requiring that only finitely many have moderate magnitude. As applications, we develop a test for finitely many linear hypotheses and, via a Scheffé-type approach, simultaneous inference for infinitely many linear hypotheses, yielding both a global test and simultaneous confidence bands for the regression function. The limiting distributions are given by weighted sums of独立

What carries the argument

Weak convergence of the scaled regression-coefficient estimator to a Gaussian element in the separable Hilbert space ℓ²

If this is right

Any fixed finite collection of linear hypotheses on the coefficients admits an asymptotically correct test.
A Scheffé-type procedure produces asymptotically valid simultaneous tests and confidence intervals for all linear functionals of the coefficient vector.
Simultaneous confidence bands for the entire regression function can be constructed with correct asymptotic coverage.
Critical values obtained by plugging consistent estimators into the weighted-chi-squared limits achieve the nominal asymptotic size.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same Hilbert-space argument may extend to other high-dimensional M-estimators once the corresponding weak-convergence result is verified.
The finite number of moderate coefficients effectively determines the dimension of the limiting problem, suggesting that inference remains feasible even when the nominal dimension is infinite.
Bootstrap or resampling methods could be justified by showing they mimic the same ℓ² weak limit under the stated sparsity condition.

Load-bearing premise

Only finitely many regression coefficients have moderate nonzero magnitude while the rest may be zero or arbitrarily small.

What would settle it

A sequence of regression problems in which the number of moderate-sized coefficients grows slowly with sample size and the proposed test statistics fail to converge in distribution to the claimed weighted sums of chi-squared variables would falsify the central claim.

read the original abstract

We prove weak convergence in a separable Hilbert space for estimators of high-dimensional regression coefficients, which yields asymptotic normality and enables direct use of standard asymptotic tools such as the continuous mapping theorem. The approach permits diverging sparsity with many small nonzero coefficients, while requiring that only finitely many have moderate magnitude. As applications, we develop a test for finitely many linear hypotheses and, via a Scheff\'{e}-type approach, simultaneous inference for infinitely many linear hypotheses, yielding both a global test and simultaneous confidence bands for the regression function. The limiting distributions are given by weighted sums of independent chi-squared variables, and plug-in critical values achieve asymptotically correct size.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper proves weak convergence in ℓ² for high-dimensional regression estimators under a mixed sparsity regime with finitely many moderate coefficients and diverging small ones, but the tightness of the small-coefficient tail is not clearly secured in the abstract.

read the letter

The main point is that they establish weak convergence of the scaled coefficient estimator in the separable Hilbert space ℓ². This works when only finitely many coefficients have moderate size while the rest can be small but nonzero with diverging sparsity. From there they get asymptotic normality and apply the continuous mapping theorem to build tests for linear hypotheses and simultaneous confidence bands via a Scheffé-type argument. The limiting objects are weighted sums of chi-squares with plug-in critical values that achieve correct asymptotic size. That framing is new relative to the usual fixed-sparsity or lasso-focused results, and the applications to global testing plus bands for the regression function are concrete and usable. The approach handles a practically common case where most signals are tiny rather than exactly zero. The soft spot is the one the stress-test flags. Weak convergence in ℓ² requires tightness, which in turn needs the sum of variances over the small-coefficient tail to stay bounded. The abstract states the finite-moderate assumption but gives no explicit rate or summability condition on how small or how numerous the tail terms can be. If the small coefficients are order n^{-1/4} and their count grows faster than n^{1/2}, the variance sum can diverge and tightness fails. Without the full proof it is impossible to see whether they added a hidden bound or simply overlooked the issue. The citation pattern looks standard and the math is formally stated, but the gap on tail control is load-bearing. This is for readers working on asymptotic inference in high-dimensional linear models who want tools beyond fixed-sparsity assumptions. A serious referee should see it; the central claim is interesting enough to check the details, especially the tightness argument, even if revisions are needed.

Referee Report

1 major / 2 minor

Summary. The paper proves weak convergence in the separable Hilbert space ℓ² for suitably scaled estimators of high-dimensional linear regression coefficients, under an assumption allowing diverging sparsity provided only finitely many coefficients have moderate magnitude. This yields asymptotic normality for linear functionals of the coefficients and permits direct application of the continuous mapping theorem. Applications include a test for finitely many linear hypotheses and a Scheffé-type procedure for simultaneous inference over infinitely many hypotheses, producing both a global test and simultaneous confidence bands for the regression function. Limiting distributions are weighted sums of independent chi-squared random variables, with plug-in critical values shown to achieve asymptotically correct size.

Significance. If the weak convergence holds, the result supplies a flexible framework for high-dimensional inference that accommodates many small nonzero coefficients while still delivering standard asymptotic tools in infinite dimensions. This is a strength relative to stricter exact-sparsity assumptions common in the literature. The direct use of the continuous mapping theorem and the practical plug-in critical values for the weighted-chi-squared limits enhance applicability to both finite and infinite collections of linear hypotheses.

major comments (1)

[Proof of the main weak-convergence theorem] The central claim of weak convergence in ℓ² (stated in the abstract and proved in the main theorem) rests on the finite-moderate-magnitude assumption. Tightness in ℓ² additionally requires that the sum of variances contributed by the diverging small nonzero coefficients remains bounded. The manuscript does not appear to impose or verify an explicit rate condition preventing, for example, a number of n^{-1/4}-sized coefficients large enough to make the tail variance diverge. Please add the precise summability condition used in the tightness argument and confirm it is implied by the stated assumption alone.

minor comments (2)

[Abstract] The abstract refers to 'weighted sums of independent chi-squared variables' without specifying how the weights are determined from the design matrix or the estimator; an explicit formula or reference to the relevant equation would improve clarity.
[Section on plug-in critical values] Notation for the plug-in estimators of the weights in the limiting distribution should be introduced consistently when the critical-value procedure is described.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and constructive feedback on our manuscript. The single major comment concerns the tightness argument underlying the main weak-convergence result. We address it directly below and will incorporate the necessary clarification in the revised version.

read point-by-point responses

Referee: [Proof of the main weak-convergence theorem] The central claim of weak convergence in ℓ² (stated in the abstract and proved in the main theorem) rests on the finite-moderate-magnitude assumption. Tightness in ℓ² additionally requires that the sum of variances contributed by the diverging small nonzero coefficients remains bounded. The manuscript does not appear to impose or verify an explicit rate condition preventing, for example, a number of n^{-1/4}-sized coefficients large enough to make the tail variance diverge. Please add the precise summability condition used in the tightness argument and confirm it is implied by the stated assumption alone.

Authors: We agree with the referee that the finite-moderate-magnitude assumption (only finitely many coefficients bounded away from zero) controls the bias terms but does not by itself guarantee that the sum of the coordinate-wise asymptotic variances remains finite. In the tightness proof we implicitly rely on the additional requirement that ∑_j v_j < ∞, where v_j is the asymptotic variance of the j-th coordinate of the scaled estimator. This summability condition is not implied by the finite-moderate-magnitude assumption alone, as the referee’s counter-example with sufficiently many n^{-1/4}-sized coefficients demonstrates. We will revise the manuscript to state the condition explicitly (as part of the assumption set preceding Theorem 1), verify its use in the tightness argument, and add a brief remark noting that it is satisfied under standard high-dimensional regimes where the effective number of coordinates with non-negligible variance does not grow faster than order n. revision: yes

Circularity Check

0 steps flagged

No circularity: direct proof of ℓ² weak convergence under stated assumptions

full rationale

The paper states a direct proof of weak convergence in the separable Hilbert space ℓ² for the scaled high-dimensional regression estimator. The central result is established from the model assumptions (finite number of moderate-magnitude coefficients, diverging but small nonzero tail) and standard tightness arguments in Hilbert space, without reducing the target weak-convergence statement to a fitted parameter, a self-citation chain, or an ansatz smuggled from prior work. The subsequent applications (tests and simultaneous bands via continuous mapping) follow from the proved convergence and do not feed back into the convergence claim itself. No load-bearing equation equates the result to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies on standard mathematical properties of separable Hilbert spaces and weak convergence together with domain assumptions on the regression model and design matrix; no free parameters or invented entities are indicated in the abstract.

axioms (2)

standard math Properties of separable Hilbert spaces and weak convergence in metric spaces
Invoked to establish the main convergence result.
domain assumption Regularity conditions on the design matrix, errors, and coefficient vector in high-dimensional linear regression
Required for the estimator to satisfy the stated weak convergence.

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discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

We prove weak convergence in a separable Hilbert space for estimators of high-dimensional regression coefficients... Rn →d R in ℓ²
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

Assumption 4: n s_{0,n} η_n² →0 ... implies (θ₀,n,0,...) ∈ ℓ² uniformly

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