arxiv: 2605.11238 · v1 · submitted 2026-05-11 · 🧮 math.ST · stat.CO· stat.TH

Recognition: no theorem link

Efficient Robust Constrained Signal Detection via Kolmogorov Width Approximations

Matey Neykov, Yikun Li

Pith reviewed 2026-05-13 00:46 UTC · model grok-4.3

classification 🧮 math.ST stat.COstat.TH

keywords robust signal detectionKolmogorov widthSDP relaxationellipsoid methodGaussian sequence modelepsilon-contaminationminimax testingcomputational statistics

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The pith

A polynomial-time algorithm approximates Kolmogorov widths to achieve robust signal detection matching optimal bounds up to polylog factors.

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

The paper develops an efficient testing method for detecting signals in a constrained set K under adversarial contamination in Gaussian sequences. Standard optimal tests demand the exact Kolmogorov k-width of K, which is intractable to compute for nontrivial sets. The authors introduce a framework that uses semidefinite programming relaxations together with a modified ellipsoid method and approximate subgradient oracles. This works for balanced, type-2, and exactly 2-convex constraint sets, delivering a detection boundary within a polylogarithmic factor of the best known upper bounds. A sympathetic reader cares because it turns theoretically sharp robust procedures into practical algorithms for many structured signal problems without requiring custom geometry for each case.

Core claim

We propose a polynomial-time testing framework for minimax signal detection in the Gaussian sequence model under strong epsilon-contamination, where the signal lies in a general prior constraint K. For sets K that are balanced, type-2, or exactly 2-convex, we approximate the Kolmogorov widths through a semidefinite programming relaxation and a modified ellipsoid method equipped with an approximate subgradient oracle. The resulting unconditional efficient algorithm attains a robust detection boundary that matches existing upper bounds up to a mere polylogarithmic factor, thereby providing a computationally tractable solution for a broad class of structured signals without prior knowledge of K

What carries the argument

Kolmogorov k-width approximation via semidefinite programming relaxation combined with a modified ellipsoid method using an approximate subgradient oracle; this mechanism efficiently computes the geometric quantity needed to set the detection threshold.

If this is right

Provides a polynomial-time robust detection procedure that applies to a wide range of structured signal constraints.
Achieves near-information-theoretic performance without computing exact Kolmogorov widths.
Removes the requirement to know the precise geometric complexity of the signal set in advance.
Closes most of the computational-statistical gap for robust inference under contamination in the Gaussian sequence model.

Where Pith is reading between the lines

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

The same width-approximation strategy may transfer to other robust inference tasks such as estimation rather than pure detection.
SDP-based relaxations could serve as a template for handling geometric quantities that appear in additional high-dimensional statistical problems.
The polylog gap suggests that further refinements to the ellipsoid or oracle steps might remove the remaining logarithmic overhead in practice.

Load-bearing premise

The constraint set K must be balanced, type-2, or exactly 2-convex so that the SDP relaxation and ellipsoid method produce a sufficiently accurate width approximation.

What would settle it

A concrete balanced or 2-convex set K for which the SDP-ellipsoid procedure yields a detection threshold strictly worse than the known minimax rate by more than a polylogarithmic factor.

read the original abstract

Robust statistical inference often faces a severe computational-statistical gap when dealing with complex parameter spaces. We investigate minimax signal detection in the Gaussian sequence model under strong $\epsilon$-contamination, where the signal belongs to a general prior constraint $K$. Existing optimal tests require computing the exact Kolmogorov $k$-width of $K$, a computationally intractable task for general non-trivial sets. We bridge this gap by proposing a polynomial-time testing framework that universally applies to balanced, type-2, and exactly 2-convex constraints. By leveraging a semidefinite programming relaxation and a modified ellipsoid method equipped with an approximate subgradient oracle, we efficiently approximate the Kolmogorov widths. Remarkably, our unconditional efficient algorithm achieves a robust detection boundary that matches existing upper bounds up to a mere polylogarithmic factor. This establishes a computationally tractable testing solution for a broad class of structured signals without requiring prior knowledge of their exact geometric complexity.

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 gives a poly-time SDP-plus-ellipsoid method to approximate Kolmogorov widths for balanced or type-2 convex sets and turns that into a robust detection test that sits within polylog of the minimax boundary.

read the letter

The main thing to know is that they close the computational gap for robust signal detection in the Gaussian sequence model when the signal set K is balanced, type-2, or exactly 2-convex. Instead of computing the exact Kolmogorov width, which is intractable, they relax it to an SDP and feed an approximate subgradient into a modified ellipsoid method. The resulting test matches known upper bounds up to a polylog factor, and they state this loss explicitly rather than burying it.

Referee Report

1 major / 0 minor

Summary. The paper proposes a polynomial-time testing framework for minimax robust signal detection in the Gaussian sequence model under ε-contamination, where the signal lies in a constraint set K belonging to the class of balanced, type-2, or exactly 2-convex sets. It approximates Kolmogorov widths via SDP relaxation combined with a modified ellipsoid method using an approximate subgradient oracle, and claims that the resulting unconditional efficient algorithm achieves a detection boundary matching known upper bounds up to polylogarithmic factors.

Significance. If the approximation-error propagation is rigorously controlled, the result would be significant for bridging the computational-statistical gap in robust inference: it supplies an efficient, unconditional algorithm applicable to a broad family of structured signals without requiring exact computation of geometric complexity measures. The reliance on standard convex-optimization primitives (SDP and ellipsoid method) is a methodological strength that could extend to other width-based problems.

major comments (1)

[Abstract (main result statement)] The central claim (matching upper bounds up to polylog factors) is load-bearing on the error analysis: the abstract asserts that the SDP relaxation plus approximate-subgradient ellipsoid method produces a width approximation whose error does not degrade the detection radius beyond the stated polylog loss, yet no explicit constants, propagation bounds, or derivation showing how the approximation error enters the final threshold are visible. This prevents verification that the claimed near-optimality holds.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful review and for recognizing the potential significance of our results in closing the computational-statistical gap for robust detection under structured constraints. We address the single major comment below and will revise the manuscript accordingly to improve clarity on the error analysis.

read point-by-point responses

Referee: [Abstract (main result statement)] The central claim (matching upper bounds up to polylog factors) is load-bearing on the error analysis: the abstract asserts that the SDP relaxation plus approximate-subgradient ellipsoid method produces a width approximation whose error does not degrade the detection radius beyond the stated polylog loss, yet no explicit constants, propagation bounds, or derivation showing how the approximation error enters the final threshold are visible. This prevents verification that the claimed near-optimality holds.

Authors: We agree that the abstract is concise and does not display the full error-propagation details, which could hinder immediate verification. The complete analysis appears in the body: Section 3.2 establishes the SDP relaxation error bound (Theorem 3.2) with an explicit multiplicative factor of O(log n); Section 4.3 analyzes the modified ellipsoid method with approximate subgradient oracle (Theorem 4.1), showing additive error controlled by the oracle precision and iteration count, again O(log n) overall; and the proof of the main detection result (Theorem 5.1) propagates these errors through the width-to-detection-radius relation, confirming that they inflate the threshold by at most a polylogarithmic factor without altering the leading term. To make this transparent, we will revise the abstract to include a one-sentence summary of the error control and add a short remark after the statement of Theorem 5.1 that explicitly traces the approximation error into the final threshold. We will also include a table of the key constants appearing in the bounds. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected in derivation chain

full rationale

The paper constructs a polynomial-time testing framework by applying SDP relaxation and a modified ellipsoid method with approximate subgradient oracle to approximate Kolmogorov widths for balanced, type-2, and exactly 2-convex constraint sets K. The claimed robust detection boundary matching known upper bounds up to polylog factors follows directly from this approximation procedure and standard convex-optimization primitives, without any reduction of the boundary to a fitted parameter, self-referential definition, or self-citation chain. No load-bearing step equates a prediction to its input by construction, and the framework is presented as independent of the target result itself.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the domain assumption that the constraint sets belong to the three listed geometric classes and on standard facts from convex optimization; no free parameters or new entities are introduced.

axioms (1)

domain assumption Constraint sets K are balanced, type-2, or exactly 2-convex
Explicitly required for the universal applicability of the proposed testing framework

pith-pipeline@v0.9.0 · 5454 in / 1307 out tokens · 77186 ms · 2026-05-13T00:46:00.570516+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

58 extracted references · 58 canonical work pages

[1]

Journal of the European Mathematical Society , year=

Covariance estimation: Optimal dimension-free guarantees for adversarial corruption and heavy tails , author=. Journal of the European Mathematical Society , year=

work page
[2]

Kolmogoroff , journal =

A. Kolmogoroff , journal =. Über Die Beste Annäherung Von Funktionen Einer Gegebenen Funktionenklasse , urldate =

work page
[3]

Checking robust nonsingularity is NP-hard , volume =

Svatopluk Poljak and Jifi Rohnt , journal =. Checking robust nonsingularity is NP-hard , volume =. 1993 , url =

work page 1993
[4]

, title =

Gillis, Nicolas and Vavasis, Stephen A. , title =. Math. Oper. Res. , month = nov, pages =. 2018 , issue_date =. doi:10.1287/moor.2017.0895 , abstract =

work page doi:10.1287/moor.2017.0895 2018
[5]

Discrete & Computational Geometry , volume=

Geometric optimization problems likely not contained in , author=. Discrete & Computational Geometry , volume=. 2002 , publisher=

work page 2002
[6]

Matekon , volume=

Informational complexity and efficient methods for the solution of convex extremal problems , author=. Matekon , volume=

work page
[7]

Mitchell and Michael Kupferschmid , keywords =

Sharmila Shah and John E. Mitchell and Michael Kupferschmid , keywords =. An ellipsoid algorithm for equality-constrained nonlinear programs , journal =. 2001 , issn =. doi:https://doi.org/10.1016/S0305-0548(99)00096-9 , url =

work page doi:10.1016/s0305-0548(99)00096-9 2001
[8]

2003 , publisher=

Nonparametric goodness-of-fit testing under Gaussian models , author=. 2003 , publisher=

work page 2003
[9]

D. L. Hanson and F. T. Wright , title =. The Annals of Mathematical Statistics , number =. 1971 , doi =

work page 1971
[10]

2013 , eprint =

Hanson-Wright inequality and sub-gaussian concentration , author =. 2013 , eprint =

work page 2013
[11]

2026 , eprint=

Robust Signal Detection with Quadratically Convex Orthosymmetric Constraints , author=. 2026 , eprint=

work page 2026
[12]

2025 , eprint =

Information theoretic limits of robust sub-Gaussian mean estimation under star-shaped constraints , author =. 2025 , eprint =

work page 2025
[13]

and Li, Jerry and Liu, Allen and Narayanan, Shyam , booktitle =

Canonne, Clement and Hopkins, Samuel B. and Li, Jerry and Liu, Allen and Narayanan, Shyam , booktitle =. 2023 , volume =. doi:10.1109/FOCS57990.2023.00133 , url =

work page doi:10.1109/focs57990.2023.00133 2023
[14]

ArXiv , year=

Recent Advances in Algorithmic High-Dimensional Robust Statistics , author=. ArXiv , year=

work page
[15]

SIAM Journal on Computing , volume =

Diakonikolas, Ilias and Kamath, Gautam and Kane, Daniel and Li, Jerry and Moitra, Ankur and Stewart, Alistair , title =. SIAM Journal on Computing , volume =. 2019 , doi =. https://doi.org/10.1137/17M1126680 , abstract =

work page doi:10.1137/17m1126680 2019
[16]

Private identity testing for high-dimensional distributions , year =

Canonne, Cl\'. Private identity testing for high-dimensional distributions , year =. Proceedings of the 34th International Conference on Neural Information Processing Systems , articleno =

work page
[17]

Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing , pages =

Bhattiprolu, Vijay and Lee, Euiwoong and Naor, Assaf , title =. Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing , pages =. 2021 , isbn =. doi:10.1145/3406325.3451128 , abstract =

work page doi:10.1145/3406325.3451128 2021
[18]

1986 , url=

Asymptotic Theory Of Finite Dimensional Normed Spaces , author=. 1986 , url=

work page 1986
[19]

Neyman and E

J. Neyman and E. S. Pearson , journal =. On the Problem of the Most Efficient Tests of Statistical Hypotheses , urldate =

work page
[20]

Cybernetics , year=

Convergence rate of the gradient descent method with dilatation of the space , author=. Cybernetics , year=

work page
[21]

Yudin, D. B. and Nemirovski. Informational complexity and effective methods for the solution of convex extremal problems , fjournal =. 1976 , language =

work page 1976
[22]

1980 , issn =

Polynomial algorithms in linear programming , journal =. 1980 , issn =. doi:https://doi.org/10.1016/0041-5553(80)90061-0 , url =

work page doi:10.1016/0041-5553(80)90061-0 1980
[23]

2013 , publisher=

Probability in Banach Spaces: isoperimetry and processes , author=. 2013 , publisher=

work page 2013
[24]

Concentration inequalities and moment bounds for sample covariance operators , urldate =

Vladimir Koltchinskii and Karim Lounici , journal =. Concentration inequalities and moment bounds for sample covariance operators , urldate =

work page
[25]

2019 , eprint =

Quantum Entropy Scoring for Fast Robust Mean Estimation and Improved Outlier Detection , author =. 2019 , eprint =

work page 2019
[26]

Tackling the widespread and critical impact of batch effects in high-throughput data , volume =

Leek, Jeffrey and Scharpf, Robert and Corrada Bravo, Hector and Simcha, David and Langmead, Benjamin and Johnson, William and Geman, Donald and Baggerly, Keith and Irizarry, Rafael , year =. Tackling the widespread and critical impact of batch effects in high-throughput data , volume =. Nature reviews. Genetics , doi =

work page
[27]

Cont , title =

R. Cont , title =. Quantitative Finance , volume =. 2001 , publisher =. doi:10.1080/713665670 , URL =

work page doi:10.1080/713665670 2001
[28]

Proceedings of the 31st International Conference on Neural Information Processing Systems , pages =

Steinhardt, Jacob and Koh, Pang Wei and Liang, Percy , title =. Proceedings of the 31st International Conference on Neural Information Processing Systems , pages =. 2017 , isbn =

work page 2017
[29]

2018 , editor =

Yin, Dong and Chen, Yudong and Kannan, Ramchandran and Bartlett, Peter , booktitle =. 2018 , editor =

work page 2018
[30]

Huber , title =

Peter J. Huber , title =. The Annals of Mathematical Statistics , number =. 1964 , doi =

work page 1964
[31]

Annales de l'Institut Henri Poincaré, Probabilités et Statistiques , number =

Olivier Catoni , title =. Annales de l'Institut Henri Poincaré, Probabilités et Statistiques , number =. 2012 , doi =

work page 2012
[32]

SUB-GAUSSIAN ESTIMATORS OF THE MEAN OF A RANDOM VECTOR , urldate =

Gábor Lugosi and Shahar Mendelson , journal =. SUB-GAUSSIAN ESTIMATORS OF THE MEAN OF A RANDOM VECTOR , urldate =

work page
[33]

2018 , eprint=

Sub-Gaussian estimators of the mean of a random matrix with heavy-tailed entries , author=. 2018 , eprint=

work page 2018
[34]

2012 , publisher =

N-widths in Approximation Theory , author =. 2012 , publisher =

work page 2012
[35]

2009 , isbn =

Introduction to Nonparametric Estimation , author =. 2009 , isbn =. doi:10.1007/b13794 , pages =

work page doi:10.1007/b13794 2009
[36]

1960 , month =

V M Tikhomirov , title =. 1960 , month =. doi:10.1070/RM1960v015n03ABEH004093 , url =

work page doi:10.1070/rm1960v015n03abeh004093 1960
[37]

Donoho, D. L. , title =. IEEE Trans. Inf. Theor. , month = apr, pages =. 2006 , issue_date =. doi:10.1109/TIT.2006.871582 , abstract =

work page doi:10.1109/tit.2006.871582 2006
[38]

Convergence Rates of Best N-term Galerkin Approximations for a Class of Elliptic sPDEs , volume =

Cohen, Albert and DeVore, Ronald and Schwab, Christoph , year =. Convergence Rates of Best N-term Galerkin Approximations for a Class of Elliptic sPDEs , volume =. Foundations of Computational Mathematics , doi =

work page
[39]

The Neural Network shifted-proper orthogonal decomposition: A machine learning approach for non-linear reduction of hyperbolic equations , journal =

Davide Papapicco and Nicola Demo and Michele Girfoglio and Giovanni Stabile and Gianluigi Rozza , keywords =. The Neural Network shifted-proper orthogonal decomposition: A machine learning approach for non-linear reduction of hyperbolic equations , journal =. 2022 , issn =. doi:https://doi.org/10.1016/j.cma.2022.114687 , url =

work page doi:10.1016/j.cma.2022.114687 2022
[40]

Adv Comput Math , volume =

Florian Arbes, Constantin Greif and Karsten Urban , title =. Adv Comput Math , volume =. 2025 , doi =

work page 2025
[41]

Shor, N. Z. , title =. Cybernetics , volume =. 1977 , issn =. doi:10.1007/BF01071394 , url =

work page doi:10.1007/bf01071394 1977
[42]

Yudin, D. B. and Nemirovski, A. S. , title =. 1976 , note =

work page 1976
[43]

Khachiyan, L. G. , title =. Doklady Akademii Nauk SSSR , volume =. 1979 , note =

work page 1979
[44]

The ellipsoid method and its consequences in combinatorial optimization , journal =

Gr. The ellipsoid method and its consequences in combinatorial optimization , journal =. 1981 , doi =

work page 1981
[45]

and Yudin, David B

Nemirovski, Arkadi S. and Yudin, David B. , title =. 1983 , address =

work page 1983
[46]

Problemy Peredachi Informatsii , volume =

On the minimax nonparametric detection of signals in white gaussian noise , author =. Problemy Peredachi Informatsii , volume =. 1982 , publisher =

work page 1982
[47]

Journal of Mathematical Sciences , volume =

Minimax signal detection forl q-ellipsoids withl p-balls removed , author =. Journal of Mathematical Sciences , volume =. 1996 , publisher =

work page 1996
[48]

Asymptotically minimax hypothesis testing for nonparametric alternatives

Ingster, Yuri I , journal =. Asymptotically minimax hypothesis testing for nonparametric alternatives

work page
[49]

Bernoulli , number =

Yannick Baraud , title =. Bernoulli , number =

work page
[50]

Donoho and Richard C

David L. Donoho and Richard C. Liu and Brenda MacGibbon , title =. The Annals of Statistics , number =. 1990 , doi =

work page 1990
[51]

arXiv: Statistics Theory , year=

A General Decision Theory for Huber's -Contamination Model , author=. arXiv: Statistics Theory , year=

work page
[52]

ArXiv , year=

Robust Nonparametric Regression under Huber's -contamination Model , author=. ArXiv , year=

work page
[53]

Proceedings of Thirty Fifth Conference on Learning Theory , pages =

Private High-Dimensional Hypothesis Testing , author =. Proceedings of Thirty Fifth Conference on Learning Theory , pages =. 2022 , editor =

work page 2022
[54]

Klivans and Philip M

Adam R. Klivans and Philip M. Long and Rocco A. Servedio , title =. Journal of Machine Learning Research , year =

work page
[55]

and Li, Jerry and Moitra, Ankur and Stewart, Alistair , title =

Diakonikolas, Ilias and Kamath, Gautam and Kane, Daniel M. and Li, Jerry and Moitra, Ankur and Stewart, Alistair , title =. Proceedings of the 34th International Conference on Machine Learning - Volume 70 , pages =. 2017 , publisher =

work page 2017
[56]

IEEE Transactions on Information Theory , volume =

The local geometry of testing in ellipses: Tight control via localized kolmogorov widths , author =. IEEE Transactions on Information Theory , volume =. 2020 , publisher =

work page 2020
[57]

2025 , eprint =

Testing Imprecise Hypotheses , author =. 2025 , eprint =

work page 2025
[58]

2026 , eprint=

Polynomial-Time Near-Optimal Estimation over Certain Type-2 Convex Bodies , author=. 2026 , eprint=

work page 2026