arxiv: 2605.09396 · v1 · submitted 2026-05-10 · 💻 cs.IT · cs.LG· math.IT· math.ST· stat.ML· stat.TH

Recognition: 2 theorem links

· Lean Theorem

Universal Feature Selection with Noisy Observations and Weak Symmetry Conditions

Dier Tang (1) , Guangyue Han (1) ((1) Department of Mathematics , The University of Hong Kong , Hong Kong , China)

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Pith reviewed 2026-05-12 02:16 UTC · model grok-4.3

classification 💻 cs.IT cs.LGmath.ITmath.STstat.MLstat.TH

keywords universal feature selectionweak spherical symmetrynoisy observationssingular value decompositionerror exponentsasymptotic optimalitycanonical dependence matrix

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The pith

Feature selection from noisy observations succeeds under weak spherical symmetry and recovers near-optimal error exponents.

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

The paper establishes a feature selection procedure that continues to work when the data attributes have only approximate rotational symmetry instead of exact spherical symmetry and when the observations contain noise. It defines weak spherical symmetry through second-moment distances that bound how far the structure can deviate from perfect invariance. The method extracts features via the singular value decomposition of a canonical dependence matrix built directly from the noisy samples. A sympathetic reader cares because the performance loss stays controlled by the size of the symmetry deviation and the noise strengths, so the approach applies to many practical inference problems where exact symmetry never holds. When those deviations and noise levels are small, the error exponents match those obtained under stricter conditions.

Core claim

Under weak spherical symmetry quantified by second-moment distances, the singular value decomposition of the canonical dependence matrix computed from noisy observations produces a set of selected features whose error exponents are asymptotically optimal except for an additive residual term that depends only on the symmetry deviation δ and the noise levels η1 and η2.

What carries the argument

The singular value decomposition of the canonical dependence matrix computed from noisy data, which isolates the dominant dependence directions while tolerating controlled deviations from rotational invariance.

If this is right

The selected features achieve asymptotically optimal error exponents up to a residual term controlled by the symmetry deviation and noise levels.
When the deviation δ and noise levels η1, η2 are small, the error exponents recover those obtained under exact spherical symmetry.
The framework extends to attribute structures that possess directional preferences and to settings with noisy observations.
The selection procedure remains robust to second-moment deviations, widening its range of usable inference tasks.

Where Pith is reading between the lines

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

The method could be applied directly to high-dimensional data sets that exhibit mild directional biases rather than perfect isotropy.
Alternative matrix constructions might be tested to see whether the residual term can be reduced further without restoring exact symmetry.
The same SVD-based extraction may prove useful in other selection problems that currently assume stronger symmetry conditions.

Load-bearing premise

The attribute structures must satisfy weak spherical symmetry, so that their second-moment distances permit only bounded departures from perfect rotational invariance.

What would settle it

In the large-sample regime, an instance where the gap between the achieved error exponent and the optimal exponent exceeds the explicit residual bound set by δ, η1, and η2 would falsify the main claim.

read the original abstract

This paper relaxes the restrictive symmetry conditions adopted in [4], [5] and extends their universal feature selection framework to accommodate noisy observations as well as attribute structures that may exhibit directional preferences. We introduce the notion of weak spherical symmetry, quantified by second-moment distances, which allows controlled deviations from rotational invariance. Under this relaxed condition, we develop a universal feature selection framework based on the singular value decomposition of the canonical dependence matrix computed from noisy data. Our main result shows that the selected features achieve asymptotically optimal error exponents up to a residual term that depends on the symmetry deviation $\delta$ and the noise levels $\eta_1, \eta_2$. When $\delta, \eta_1, \eta_2$ are relatively small, our result recovers that of [5], thereby demonstrating that exact spherical symmetry is unnecessary. Overall, our findings highlight the robustness of the selection framework against second-moment deviations and observation noise, thereby broadening its applicability across diverse inference tasks and providing a theoretically grounded tool for universal feature selection in practical scenarios.

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

This extends prior feature selection to noisy data under a relaxed second-moment symmetry but the SVD robustness claim under noise lacks visible proof support.

read the letter

The main thing here is that the paper relaxes exact spherical symmetry from earlier work to a weaker version measured by second-moment distances, while adding tolerance for noisy observations. The selected features are said to reach asymptotically optimal error exponents up to a residual that grows with the symmetry deviation δ and the noise parameters η1, η2, recovering the old result when those quantities stay small. They compute the canonical dependence matrix from the noisy data and apply SVD for selection under this setup. This is a direct attempt to make the framework usable when attributes have mild directional bias and measurements are imperfect. The recovery statement when deviations are small is a clean check that the extension does not break the prior case. The idea itself is reasonable for broadening applicability in inference tasks. The soft spot is the missing support for the central claim. The abstract asserts the optimality result with an explicit residual but supplies no derivation steps, proof sketch, or perturbation analysis showing that noise in the matrix entries keeps the exponent loss inside the stated bound. It is unclear whether the matrix is the population quantity, a debiased version, or a direct sample estimate, and how the singular vectors stay aligned enough after perturbation. The stress-test concern about needing a matching lemma for the subspace deviation looks like it lands, at least from what is visible. Without that step the asymptotic guarantee does not follow from weak spherical symmetry alone. This is aimed at information theorists and statisticians working on feature selection who want to relax strict assumptions. A reader interested in incremental robustness results would get some value if the details hold. It deserves a serious referee to examine the derivations and check whether the noise handling actually closes.

Referee Report

2 major / 1 minor

Summary. The manuscript relaxes exact spherical symmetry to a weak version quantified by second-moment distances, extends prior universal feature selection frameworks to noisy observations, and proposes selecting features via the SVD of the canonical dependence matrix computed from noisy data. The central claim is that the resulting features achieve asymptotically optimal error exponents up to a residual term controlled by the symmetry deviation parameter δ and the noise levels η1, η2; when these quantities are small the result recovers the exact-symmetry case of reference [5].

Significance. If the main result holds with a rigorous derivation, the work would be significant because it demonstrates that exact rotational invariance is unnecessary for asymptotic optimality in feature selection, thereby extending the framework's applicability to practical inference tasks that involve observation noise and mild directional preferences in the attribute structure.

major comments (2)

[Abstract] Abstract: the claim that the selected features achieve asymptotically optimal error exponents up to an explicit residual term is asserted without any derivation steps, proof outline, or verification that the SVD performed on the noisy canonical dependence matrix produces the claimed exponent; this is load-bearing for the central claim.
[Main result] Main result: the residual term is expressed in terms of the deviation parameters δ, η1, η2 that are themselves defined from the data; without the explicit equations or a perturbation analysis it is impossible to determine whether the bound is independently derived or partly tautological, and no indication is given of how noise-induced perturbations to the matrix entries are controlled so that the loss in the error exponent remains inside the stated residual.

minor comments (1)

[Notation] The notation used for the canonical dependence matrix and its noisy version could be introduced more explicitly, including how the matrix is estimated from finite samples.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive feedback. We address the two major comments point by point below, indicating the revisions we will make to improve clarity and rigor.

read point-by-point responses

Referee: [Abstract] Abstract: the claim that the selected features achieve asymptotically optimal error exponents up to an explicit residual term is asserted without any derivation steps, proof outline, or verification that the SVD performed on the noisy canonical dependence matrix produces the claimed exponent; this is load-bearing for the central claim.

Authors: The abstract is intended as a concise summary and therefore omits detailed derivation steps. The full verification that the SVD of the noisy canonical dependence matrix yields the stated exponent (including the residual controlled by symmetry deviation and noise) appears in the proof of the main theorem in Section 4. To address the concern that the claim is load-bearing, we will revise the abstract to incorporate a brief proof outline: (i) definition of weak spherical symmetry via second-moment distances, (ii) formation of the noisy canonical dependence matrix, (iii) SVD-based feature selection, and (iv) perturbation bound on the error exponent. This addition will point readers directly to the rigorous justification while preserving abstract length. revision: yes
Referee: [Main result] Main result: the residual term is expressed in terms of the deviation parameters δ, η1, η2 that are themselves defined from the data; without the explicit equations or a perturbation analysis it is impossible to determine whether the bound is independently derived or partly tautological, and no indication is given of how noise-induced perturbations to the matrix entries are controlled so that the loss in the error exponent remains inside the stated residual.

Authors: The parameters δ, η1, η2 are defined explicitly in Section 2 as second-moment distances quantifying symmetry deviation and noise levels. The residual term in Theorem 3.1 is obtained via an independent perturbation argument that applies Weyl's inequality and Davis-Kahan sin-Θ bounds to the difference between the noisy and clean matrices; the resulting exponent loss is controlled by O(δ + η1 + η2) and is therefore not tautological. We agree that the current presentation could be more explicit. We will revise the manuscript to insert the concrete perturbation equations (e.g., the operator-norm bound on the noise-induced matrix perturbation) and the step-by-step control of the exponent loss directly into the main text or a dedicated appendix subsection. revision: yes

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper introduces weak spherical symmetry via second-moment distances as a relaxation of prior exact symmetry assumptions from [4] and [5], then constructs a feature selection procedure via SVD on the canonical dependence matrix obtained from noisy observations. The central theorem states that the resulting features attain asymptotically optimal error exponents up to an additive residual controlled by the explicit deviation parameters δ, η1, η2; when those parameters vanish the statement reduces to the earlier result. No quoted equation or step reduces the claimed optimality (or the form of the residual) to a tautological re-expression of the inputs, a fitted parameter renamed as a prediction, or a load-bearing self-citation whose justification is internal to the present manuscript. The derivation therefore remains self-contained against the stated assumptions and does not exhibit any of the enumerated circularity patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claim rests on the newly introduced definition of weak spherical symmetry and on the assumption that a canonical dependence matrix can be formed from noisy observations; no free parameters are explicitly fitted in the abstract.

axioms (1)

domain assumption A canonical dependence matrix exists and can be estimated from noisy observations
Required for the SVD step that drives feature selection.

invented entities (1)

weak spherical symmetry no independent evidence
purpose: To quantify controlled deviations from exact rotational invariance using second-moment distances
New notion introduced to relax the restrictive symmetry conditions of prior work.

pith-pipeline@v0.9.0 · 5507 in / 1283 out tokens · 66224 ms · 2026-05-12T02:16:38.929259+00:00 · methodology

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/AbsoluteFloorClosure.lean reality_from_one_distinction unclear

?

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

We introduce the notion of weak spherical symmetry, quantified by second-moment distances... SVD of the canonical dependence matrix computed from noisy data.
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

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

R(ϵ, δ, η1, η2) = O(ϵ² · max{δ+η1+δη1, δ+η2+δη2})

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

12 extracted references · 12 canonical work pages

[1]

An introduction to variable and feature selec- tion,

I. Guyon and A. Elisseeff, “An introduction to variable and feature selec- tion,”Journal of machine learning research, vol. 3, no. Mar, pp. 1157– 1182, 2003

work page 2003
[2]

Mathematical methods in feature selection: A review,

F. Kamalov, H. Sulieman, A. Alzaatreh, M. Emarly, H. Chamlal, and M. Safaraliev, “Mathematical methods in feature selection: A review,” Mathematics, vol. 13, no. 6, pp. 996, 2025

work page 2025
[3]

Exploring feature selection with limited labels: A comprehensive survey of semi-supervised and unsupervised approaches,

G. Li, Z. Yu, K. Yang, M. Lin, and C. L. P. Chen, “Exploring feature selection with limited labels: A comprehensive survey of semi-supervised and unsupervised approaches,”IEEE Transactions on Knowledge and Data Engineering, vol. 36, no. 11, pp. 6124–6144, 2024

work page 2024
[4]

An information theoretic interpretation to deep neural networks,

X. Xu, S.-L. Huang, L. Zheng, and G. W. Wornell, “An information theoretic interpretation to deep neural networks,”Entropy, vol. 24, no. 1, pp. 135, 2022

work page 2022
[5]

Universal features for high-dimensional learning and inference,

S.-L. Huang, A. Makur, G. W. Wornell, and L. Zheng, “Universal features for high-dimensional learning and inference,”Foundations and Trends in Communications and Information Theory, vol. 21, no. 1-2, pp. 1–299, 2024

work page 2024
[6]

Elliptically symmetric distributions: A review and bibliography,

M. A. Chmielewski, “Elliptically symmetric distributions: A review and bibliography,”International Statistical Review / Revue Internationale de Statistique, pp. 67–74, 1981

work page 1981
[7]

Spherical matrix distributions and a multivariate model,

A. P. Dawid, “Spherical matrix distributions and a multivariate model,” Journal of the Royal Statistical Society Series B: Statistical Methodology, vol. 39, no. 2, pp. 254–261, 1977

work page 1977
[8]

Representation learning: A review and new perspectives,

Y . Bengio, A. Courville, and P. Vincent, “Representation learning: A review and new perspectives,”IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 35, no. 8, pp. 1798–1828, 2013

work page 2013
[9]

Multitask learning,

R. Caruana, “Multitask learning,”Machine learning, vol. 28, no. 1, pp. 41–75, 1997

work page 1997
[10]

High-dimensional probability,

R. Vershynin, “High-dimensional probability,”University of California, Irvine, vol. 10, no. 11, pp. 31, 2020

work page 2020
[11]

T. M. Cover and J. A. Thomas,Elements of Information Theory, John Wiley & Sons, 2012

work page 2012
[12]

R. A. Horn and C. R. Johnson,Matrix analysis, Cambridge University Press, 2012

work page 2012