Sparse Polynomial Regression under Anomalous Data

Maryam Dehghani; Mohammad Reza Hesamzadeh; Roozbeh Abolpour

REVIEW 4 major objections 3 minor 28 references

Sparse Polynomial Regression under Anomalous Data

T0 review · 4 major / 3 minor · reviewed 2026-08-05 · deepseek-v4-flash

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Pith's one-line read This paper sets out to show that sparse polynomial regression with anomalous data filtering can be reformulated as a fractional program with better computational properties than the original nonconvex QCQP, and that a two-step convex relaxa

desk verdict A plausible method-level contribution that could not be audited because the supplied full text is corrupted; referee only after a clean version is available. read the letter →

arxiv 2508.18199 v1 pith:GU4RTILS submitted 2025-08-25 math.OC

Roozbeh Abolpour , Mohammad Reza Hesamzadeh , Maryam Dehghani This is my paper

classification math.OC MSC 90C2090C2690C11

keywords sparsepolynomialregressionanomalousdatafilteringmixed-integerlinearprogramquadraticallyconstrainedquadraticfractionalconicrelaxationoutlier-robustTS-CRR

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper aims to show that sparse polynomial regression can handle anomalous data without separating outlier detection from fitting. It casts the problem as a mixed-integer linear program, rewrites that as a non-convex quadratically constrained quadratic program, then maps the QCQP to a fractional program that it claims has better computational properties. The proposed TS-CRR algorithm solves a convex relaxation of the fractional program and recovers the sparse polynomial coefficients. If the reformulation is as tight as claimed, the approach would give a principled, optimization-based way to fit interpretable polynomial models while discarding corrupted observations. The authors support the claim with experiments on two datasets against several regression and AI baselines.

What carries the argument

The load-bearing object is the Fractional Program obtained from the QCQP by the proposed mapping, together with the TS-CRR algorithm. The FP carries the tractability claim: because the paper argues it has better computational properties, the subsequent convex relaxation starts from a better formulation than relaxing the QCQP directly. TS-CRR's recovery step converts the relaxed solution back into a sparse polynomial coefficient vector while the anomaly-filtering variables remain part of the model.

What would settle it

Construct a synthetic data set with a known sparse polynomial and inject a known set of anomalies; solve TS-CRR and compare its recovered support and objective value with the global optimum obtained by enumerating all plausible anomaly subsets. If the conic relaxation's objective falls far short of the global value, or the recovered anomaly set is wrong at a modest anomaly fraction, the claim that the FP relaxation preserves optimality is refuted.

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Extended reading notes

Core claim

On its own terms, the paper's central claim is a reformulation result plus an algorithm. The Sparse Polynomial Regression Model with anomalous data filtering is first expressed as a MILP; the MILP is converted into a non-convex QCQP; and a proposed mapping turns that QCQP into a Fractional Program. The paper states that the FP has better computational properties than the QCQP. TS-CRR then relaxes the FP to a convex problem, solves it, and applies a recovery step to obtain the final sparse polynomial and anomaly pattern. The reported experiments compare this recovered model with several regression and artificial-intelligence methods on two datasets and find the proposed algorithm competitive.

Load-bearing premise

The load-bearing premise is that the mapping from the QCQP to the fractional program is exact and that the conic relaxation of the fractional program stays close to its global optimum, so the recovered sparse polynomial is actually near-optimal.

Editorial extensions

If this is right

If the FP reformulation truly has better computational properties, the same relaxation pipeline should scale to larger sparse-polynomial instances than direct QCQP solving.
Anomalous data filtering becomes part of the optimization model itself, so the recovered polynomial does not depend on a separate outlier-removal pre-processing step.
TS-CRR returns a sparse polynomial, giving an interpretable model with anomalous observations down-weighted or excluded.
The reformulation chain provides a template for other mixed-integer polynomial regression problems where a QCQP can be mapped to a fractional program before relaxation.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Beyond the paper, the QCQP-to-fractional mapping may transfer to other cardinality-constrained estimation problems whenever a ratio form relaxes better than the original quadratic form.
The paper's 'better computational properties' claim would be made concrete by comparing relaxation gaps or condition numbers on a common instance set; such measurements are not in the abstract.
A natural testable extension is synthetic experiments with known ground-truth polynomials and injected outliers, checking exact support recovery and coefficient error rather than only predictive fit on two datasets.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The one thing you should know: I could only read the abstract. The supplied full text is mojibake, so every body-level claim — the exact QCQP-to-FP mapping, the theoretical argument, the experimental setup, the benchmark identities — sits outside what I can verify. The reader's UNVERDICTED score is the right posture, and the stress-test note is fair: the central assertion, 'We theoretically show that the reformulated FP has better computational properties than the original QCQP,' appears with no statement of what property is improved or what assumptions hold. No recovery guarantee or optimality gap bound is visible in the abstract. That is a missing support, not an internal contradiction.

Give credit where it is due. The problem is well-motivated: sparse polynomial regression with automatic anomaly filtering is a practical task, and the pipeline they describe — MILP to QCQP to fractional program, then conic relaxation — is a coherent assembly of standard blocks. The named TS-CRR algorithm is a reasonable design choice for this problem class. If the mapping is exact and the relaxation is tight, this would be a useful solver. The abstract does not oversell the contribution as anything beyond a method-level improvement, which I appreciate.

The soft spots are exactly where the reader put them. First, the theoretical claim is a promise, not a theorem: no bound, no statement of which computational property improves (conditioning? tractability? gap?), and no comparison to existing relaxation bounds for such QCQPs. Second, the numerical claims report 'promising performance' without error bars, variance, or statistical tests; two datasets is too thin to establish superiority over the listed AI models unless the differences are large. Third, there is no code or data pointer, so reproducibility is currently zero. None of these are fatal if the body actually contains the missing details — but I cannot confirm that from what I was given.

I agree with the reader that this is not a circularity problem at the abstract level; the reformulation chain is a derivation. The classic risk is the relaxation gap, and the paper has not yet shown it is benign.

For whom is this? Someone working on nonconvex sparse regression solvers, especially with robust/anomaly filtering, would want to look at the body once a readable version exists. The abstract alone is not enough to cite. I would not desk-reject a paper with this topic and this structure if the proof and experiments check out — but I would not send the current corrupted file to referees. My recommendation: ask for a clean manuscript and a statement of the theorem with its assumptions, then send it out. If the body is as thin as the abstract, it will get a hard time; if the proof is real, it is a solid minor contribution.

Referee Report

4 major / 3 minor

Summary. The paper proposes a pipeline for sparse polynomial regression with anomalous-data filtering: the problem is formulated as a MILP, converted to a non-convex QCQP, and then, via a proposed mapping, reformulated as a Fractional Program (FP). The abstract claims a theoretical result that the FP has better computational properties than the original QCQP, and proposes a conic-relaxation-based algorithm (TS-CRR). Numerical results on two datasets are reported to show 'promising performance' compared with several regression and AI baselines. However, the supplied full text is corrupted mojibake, so no derivation, theorem, proof, or experimental table can be read. The abstract itself does not state the theorem, its assumptions, the sense of 'better computational properties,' or any optimality-gap or recovery guarantee. Consequently, the central claims are not auditable from the submitted manuscript.

Significance. If the claimed reformulation and conic relaxation were correct and came with a rigorous guarantee, the work could offer a practical approach to sparse polynomial regression that is robust to anomalies, a topic of genuine interest in optimization and statistics. The proposed pipeline (MILP-to-QCQP-to-FP) is methodologically interesting. However, the significance cannot currently be assessed: no formal theorem statement, proof, gap bound, or reproducible experimental evidence is accessible. The manuscript also provides no verification artifacts (e.g., machine-checked proofs or code) that would compensate for the unreadable body.

major comments (4)

[Abstract and Full text] The central theoretical claim—'We theoretically show that the reformulated FP has better computational properties than the original QCQP'—is asserted without any visible theorem statement, assumption set, or proof. The full text is provided as mojibake and cannot be decoded. This is a load-bearing missing support: the contribution hinges on this comparison, yet the reader cannot verify it.
[Abstract] The phrase 'better computational properties' is undefined. It must specify which property is improved (e.g., condition number, tractability, tightness of relaxation, complexity) and in what sense. Moreover, the chain MILP→QCQP→FP must be shown to preserve the optimum or provide an approximation guarantee; no exactness or approximation statement appears in the abstract.
[TS-CRR algorithm (conic relaxation)] The proposed TS-CRR algorithm solves a conic relaxation of the FP and then recovers a sparse polynomial. For relaxation-based methods, the relaxed solution may be far from the true optimum unless a gap bound or recovery guarantee is provided. No such bound or guarantee is stated in the abstract or accessible in the full text. This is a standard failure point and must be addressed for the method's claims.
[Numerical experiments (abstract)] The abstract reports 'promising performance' against 'several regression and artificial intelligent models' on 'two different datasets,' but no numerical tables, error bars, variance estimates, statistical tests, dataset descriptions, or parameter settings are visible. The full text is unreadable, so the experimental claims cannot be checked; 'promising performance' is not a statistically grounded claim.

minor comments (3)

[Full text] The manuscript file appears to be encoded in a non-printable character set, producing mojibake. The authors should resubmit a readable PDF and source file.
[Abstract] The abstract should state the main theorem in a precise form, including assumptions and a bound or property comparison, so that the contribution is appraisal-able even before reading the full body.
[Abstract and experiments] The phrase 'several regression and artificial intelligent models' should be made concrete: list the benchmark models, datasets, and evaluation metrics.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity found: the readable abstract describes a reformulation chain and external benchmark comparison, and no equation-level reduction to the paper's own inputs is visible in the supplied text.

full rationale

The only readable portion of the manuscript is the abstract; the body is corrupted mojibake, so no equations, theorem statements, or numerical tables can be inspected. The claimed chain—SPRM → MILP → QCQP → FP → conic relaxation and TS-CRR recovery—is a sequence of reformulations and a relaxation/algorithm, not a fitted-input loop. The abstract's assertion, 'We theoretically show that the reformulated FP has better computational properties than the original QCQP,' is an unverified claim in the supplied text, but an unprovided proof is a verifiability/correctness concern, not circularity: nothing in the text defines 'better computational properties' in terms of the method's own fitted outputs, and no fitted parameter is renamed as a prediction. The empirical comparison against 'several regression and artificial intelligence models' on two datasets is an external benchmark comparison; it would be circular only if the benchmark outcomes were constructed from the proposed method's solutions, which no readable passage suggests. No self-citations are visible in the supplied text, so no self-citation chain can be load-bearing. Per the hard rule requiring a quote and a specific reduction to establish circularity, I find no significant circularity in the accessible material.

Assumptions & free parameters 2 free parameters · 3 assumptions · 0 invented entities

Because only the abstract is legible, the ledger reflects standard hyperparameters and modeling premises implied by the abstract rather than audited quantities from the body.

free parameters (2)

sparsity regularization weight
Controls the trade-off between fit quality and polynomial sparsity in the SPRM; a standard hyperparameter, value not given in the abstract.
anomaly budget or threshold
Limits how many data points can be flagged as anomalous in the MILP; required to make the filtering problem well-posed, value not given in the abstract.

assumptions (3)

domain assumption The MILP formulation and the non-convex QCQP formulation are equivalent descriptions of the same sparse polynomial regression with anomaly filtering.
Central to the reformulation chain; asserted in the abstract, unverifiable from the garbled body.
domain assumption The mapping from the QCQP to the Fractional Program preserves the optimum, and the conic relaxation has a controlled suboptimality gap.
This carries the 'better computational properties' claim; without a bound, relaxation-based recovery can be arbitrarily loose.
domain assumption Anomalies are correctly modeled by binary indicator variables that may discard points under a fixed budget.
Modeling choice for what counts as anomalous; standard in robust regression, but it assumes the true clean data are a subset of the observed points.

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Cite this review

Pith. "Pith review of Sparse Polynomial Regression under Anomalous Data." pith.science (2026). https://pith.science/paper/GU4RTILS

@misc{pith2026250818199,
  author       = {Pith},
  title        = {Pith review of: Sparse Polynomial Regression under Anomalous Data},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GU4RTILS}},
  note         = {Machine review of arXiv:2508.18199}
}

read the original abstract

This paper starts with the general form of the polynomial regression model. We reformulate the Sparse Polynomial Regression Model (SPRM) with anomalous data filtering as Mixed-Integer Linear Program (MILP). This MILP is then converted to a non-convex Quadratically Constrained Quadratic Program (QCQP). Through a proposed mapping, the derived QCQP is reformulated as a Fractional Program (FP). We theoretically show that the reformulated FP has better computational properties than the original QCQP. We then suggest a conic-relaxation-based algorithm to solve the proposed FP. A Two-Step Convex Relaxation and Recovery (TS-CRR) algorithm is proposed for sparse polynomial regression with anomalous data filtering. Through a series of comprehensive computational experiments (using two different datasets), we have compared the results of our proposed TS-CRR algorithm with the results from several regression and artificial intelligent models. The numerical results show the promising performance of our proposed TS-CRR algorithm as compared to those studied benchmark models.

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Reference graph

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Reviewed August 5, 2026 · model on record in the stance chip above.

[1] [1]

Learning smooth functions in high dimensions: from sparse polynomials to deep neural networks

Adcock, B., Brugiapaglia, S., Dexter, N., Moraga, S.: Learning smooth functions in high dimensions: from sparse polynomials to deep neural networks. arXiv preprint arXiv:2404.03761 (2024)

work page Pith review arXiv 2024

[2] [2]

In: Proceedings of the 32nd Annual European Symposium on Algorithms (E.S.A 2024), Leibniz International Proceedings in Informatics (LIPIcs), vol

Arora, S., Dey, P., Li, Y., Weinberg, S.M.: Outlier robust multivariate polynomial regression. In: Proceedings of the 32nd Annual European Symposium on Algorithms (E.S.A 2024), Leibniz International Proceedings in Informatics (LIPIcs), vol. 308, pp. 12:1-12:17. Schloss Dagstuhl–Leibniz-Zentrum für Informatik (2024)

work page 2024

[3] [3]

Operations Research

Bertsimas, D., Digalakis Jr, V., Li, M.L., Lami, O.S.: Slowly varying regression under sparsity. Operations Research. 73(3), 1581-1597 (2025)

work page 2025

[4] [4]

Analytica Chim

Bertsimas, D., Kitane, D.L., Azami, N., Doucet, F.R.: Novel mixed integer optimization sparse regression approach in chemometrics. Analytica Chim. Acta. 1137, 115-124 (2020)

work page 2020

[5] [5]

Statistical Science

Bertsimas, D., Pauphilet, J., Van Parys, B.: Sparse regression. Statistical Science. 35(4), 555-578 (2020)

work page 2020

[6] [6]

Machine Learning

Bertsimas, D., Van Parys, B.: Sparse hierarchical regression with polynomials. Machine Learning. 109(5), 973-997 (2020)

work page 2020

[7] [7]

Bottmer, L., Croux, C., Wilms, I.: Sparse regression for large data sets with outliers. Eur. J. Oper. Res. 297(2), 782-794 (2022)

work page 2022

[8] [8]

Bridgman, W., Balakrishnan, U., Jones, R.E., Chen, J., Wu, X., Safta, C., Huang, Y., Khalil, M.: A novel probabilistic transfer learning strategy for polynomial regression. Int. J. Uncertainty Quantif. 15(4) (2025)

work page 2025