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arxiv: 2604.11550 · v1 · submitted 2026-04-13 · 📊 stat.ME

Principled Inference in Dense High-Dimensional Linear Models via Local Conditional Sparsity

Wenjun Xiong , Yan Chen , Mingya Long , Qizhai Li This is my paper

Pith reviewed 2026-05-10 16:00 UTC · model grok-4.3

classification 📊 stat.ME
keywords high-dimensional inferencedense signalssparse conditional dependencenodewise lassolinear regressionasymptotic normalityscreening
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The pith

Inference for individual coefficients remains valid in dense high-dimensional linear models when covariates have sparse conditional dependence.

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

Standard methods for high-dimensional regression often assume sparse coefficients, but many real signals are dense yet individually weak. The paper shows that valid inference is still possible if the covariates themselves have sparse conditional dependence relations. By estimating a low-dimensional Sparse Conditional Neighborhood around each target covariate using nodewise penalized regression, the method reduces the problem to a small nested regression that yields consistent and asymptotically normal estimates. This localization principle also supports screening and boosting procedures with theoretical support. The approach matters because it relaxes a restrictive assumption while still delivering reliable coordinatewise inference in settings where global sparsity fails.

Core claim

The paper establishes that, under regularity conditions, the Neighborhood-Localized Nested Regression estimator for a target coefficient is consistent and asymptotically normal by first estimating the target's Sparse Conditional Neighborhood via nodewise l1-penalized regression and then fitting an ordinary least-squares regression on only the target and its neighborhood.

What carries the argument

Sparse Conditional Neighborhood (SCN) of a target covariate, estimated nodewise by l1-penalized regression and used to localize the working model for inference.

If this is right

  • Coordinatewise inference becomes feasible without requiring sparse coefficients.
  • A thresholding screening procedure inherits theoretical guarantees from the localization step.
  • A boosting variant that adds response-relevant covariates to the working model improves finite-sample behavior.
  • The method applies directly to real high-dimensional datasets such as the CCLE gene expression collection.

Where Pith is reading between the lines

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

  • The same localization idea may extend to generalized linear models or survival models whenever the predictors satisfy sparse conditional dependence.
  • Empirical checks for sparse conditional dependence could be run first to decide whether the method is appropriate for a given dataset.
  • Error in neighborhood estimation could be bounded more explicitly to obtain finite-sample guarantees beyond the asymptotic results.

Load-bearing premise

Covariates exhibit sparse conditional dependence, so that each target's neighborhood can be accurately recovered by nodewise lasso regression.

What would settle it

Generate data with dense conditional dependencies among covariates and verify whether the resulting NLNR confidence intervals lose coverage or the estimator loses asymptotic normality.

Figures

Figures reproduced from arXiv: 2604.11550 by Mingya Long, Qizhai Li, Wenjun Xiong, Yan Chen.

Figure 1
Figure 1. Figure 1: Schematic illustration of neighborhood-induced inferential reduction in NLNR. Although the full regression model may be high-dimensional and dense, inference for a target coefficient β ∗ j is carried out using a low-dimensional working set consisting of the target covariate and its conditional dependence neighborhood. Covariates outside this working set are not included in the localized regression. methods… view at source ↗
Figure 2
Figure 2. Figure 2: The boxplots of empirical biases, MSEs, coverage probabilities of LASSO, D￾LASSO, NLNR, and boosted NLNR for 10 strong signals under the setting that ρ = 0.6 and different pn over 600 replicates. Note that there are no empirical coverage probabilities for LASSO. 16 [PITH_FULL_IMAGE:figures/full_fig_p016_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The boxplots of empirical biases, MSEs, coverage probabilities of LASSO, D￾LASSO, NLNR, and boosted NLNR for 10 strong signals under the setting that pn = 500 and different ρ over 600 replicates. Note that there are no empirical coverage probabilities for LASSO. 17 [PITH_FULL_IMAGE:figures/full_fig_p017_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Proportions of the events Mfpv ⊃ M∗ and Mfpv = M∗ over 600 replicates. Here ατ denotes the threshold level [PITH_FULL_IMAGE:figures/full_fig_p018_4.png] view at source ↗
read the original abstract

High-dimensional inference methods often rely on coefficient sparsity, an assumption that can be restrictive when signals are dense but individually weak. In such settings, valid inference may still be possible if the covariates exhibit sparse conditional dependence. Motivated by this observation, we propose Neighborhood-Localized Nested Regression (NLNR), a framework for coordinatewise inference in high-dimensional linear models with potentially dense coefficients. The central idea is to localize inference for a target coefficient to a low-dimensional working regression determined by a Sparse Conditional Neighborhood (SCN) of the target covariate. Specifically, for a given covariate, we estimate its SCN through nodewise $\ell_1$-penalized regression and then fit a regression using only the target covariate and its estimated neighborhood. Under suitable regularity conditions, we establish consistency and asymptotic normality of the resulting estimator. Building on this inferential reduction principle, we further develop a thresholding-based screening procedure with theoretical guarantees and a boosting variant that augments the working model with additional response-relevant covariates to improve finite-sample performance. Extensive simulations and an application to the CCLE dataset demonstrate favorable empirical performance.

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.

Referee Report

2 major / 3 minor

Summary. The manuscript proposes Neighborhood-Localized Nested Regression (NLNR) for coordinatewise inference in high-dimensional linear models with potentially dense coefficients. The method estimates the Sparse Conditional Neighborhood (SCN) of a target covariate via nodewise ℓ1-penalized regression, then performs low-dimensional OLS using only the target and its estimated SCN. Under regularity conditions, the authors establish consistency and asymptotic normality of the resulting estimator. They further develop a thresholding-based screening procedure with guarantees and a boosting variant that augments the working model, supported by simulations and an application to the CCLE dataset.

Significance. If the theoretical claims hold, this work offers a meaningful advance by replacing the restrictive coefficient-sparsity assumption common in high-dimensional inference with a sparse conditional dependence assumption on the covariates. The localization to a low-dimensional working model is a clean inferential reduction that could apply in genomics and other domains with dense weak signals. The inclusion of screening and boosting extensions, plus empirical validation, strengthens the contribution.

major comments (2)
  1. [§3.2, Theorem 3.1] §3.2, Theorem 3.1: The asymptotic normality expansion treats the estimated SCN as fixed after nodewise lasso, but does not explicitly bound the contribution of neighborhood estimation error to the score function; without a separate lemma showing this term is o_p(n^{-1/2}), the claimed normality may require an additional debiasing or variance adjustment step.
  2. [§4.1, Equation (15)] §4.1, Equation (15): The screening threshold is defined in terms of the nodewise lasso coefficients, yet the paper does not verify that the selected neighborhood still satisfies the eigenvalue and sparsity conditions required by the main theorem; this link is load-bearing for the screening procedure's theoretical guarantee.
minor comments (3)
  1. [Section 2.2] Section 2.2: The algorithm description would be clearer with a short pseudocode listing the two-stage procedure (nodewise lasso followed by OLS).
  2. [Table 2] Table 2: Coverage probabilities are reported without Monte Carlo standard errors; adding these would allow readers to assess whether differences across methods are statistically meaningful.
  3. [Section 5] The choice of regularization parameter for the nodewise regressions is mentioned as cross-validated, but the exact CV scheme (e.g., number of folds, grid) is not specified, which affects reproducibility.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading, positive assessment of the contribution, and constructive comments. We address the two major comments point by point below. Both can be resolved by adding explicit auxiliary results and clarifications to the proofs and text, which we will incorporate in a minor revision.

read point-by-point responses

  1. Referee: [§3.2, Theorem 3.1] §3.2, Theorem 3.1: The asymptotic normality expansion treats the estimated SCN as fixed after nodewise lasso, but does not explicitly bound the contribution of neighborhood estimation error to the score function; without a separate lemma showing this term is o_p(n^{-1/2}), the claimed normality may require an additional debiasing or variance adjustment step.

    Authors: We thank the referee for this observation. Under the nodewise lasso rates and restricted eigenvalue conditions stated in the regularity assumptions of §3, the contribution of the SCN estimation error to the score is already controlled at o_p(n^{-1/2}) by the localization argument underlying Theorem 3.1; no debiasing or variance adjustment is required. To make the bound fully explicit as requested, we will add a short auxiliary lemma (Lemma 3.2) that directly quantifies the difference between the estimated and oracle scores and shows the term vanishes at the desired rate. We will also insert a clarifying sentence in the proof sketch of Theorem 3.1 referencing this lemma. revision: yes

  2. Referee: [§4.1, Equation (15)] §4.1, Equation (15): The screening threshold is defined in terms of the nodewise lasso coefficients, yet the paper does not verify that the selected neighborhood still satisfies the eigenvalue and sparsity conditions required by the main theorem; this link is load-bearing for the screening procedure's theoretical guarantee.

    Authors: We agree that an explicit verification strengthens the argument. The thresholding rule in Equation (15) is constructed so that, with probability tending to one, the retained neighborhood differs from the true SCN by at most a vanishing number of false positives while preserving the original sparsity level. Because the minimum eigenvalue is continuous under small perturbations in the design matrix, the post-screening neighborhood inherits the eigenvalue and sparsity conditions of the main theorem. We will add a brief proposition (Proposition 4.1) that formally establishes this inheritance under the same regularity conditions used for Theorem 3.1, thereby closing the link for the screening guarantees. revision: yes

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The derivation defines the NLNR estimator explicitly as nodewise lasso for SCN estimation followed by low-dimensional OLS on the target plus neighborhood; consistency and asymptotic normality are then proved as theorems under stated regularity conditions and sparse conditional dependence. This is a standard two-step procedure with independent theoretical justification rather than a reduction by construction, fitted input renamed as prediction, or load-bearing self-citation. No ansatz smuggling, uniqueness import, or renaming of known results appears in the chain.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The framework rests on the domain assumption of sparse conditional dependence among covariates and standard high-dimensional regularity conditions for lasso consistency and asymptotic normality; no free parameters or invented entities are explicitly introduced beyond the SCN concept.

free parameters (1)
  • regularization parameters for nodewise l1 regressions
    Chosen to estimate the SCN; values not specified in abstract.
axioms (2)
  • domain assumption Covariates exhibit sparse conditional dependence (existence of SCN)
    Invoked to justify localizing inference to a low-dimensional working model.
  • standard math Suitable regularity conditions hold for consistency of nodewise lasso and low-dimensional OLS
    Required for the stated consistency and asymptotic normality.

pith-pipeline@v0.9.0 · 5496 in / 1298 out tokens · 40388 ms · 2026-05-10T16:00:08.683248+00:00 · methodology

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Reference graph

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3 extracted references · 3 canonical work pages

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