{"id":"3a3ecde7-7be4-4870-9ac2-c797cb9b635e","arxiv_id":"2509.09380","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":3.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"A deterministic polynomial-regression estimator for the HGR correlation and a fast one-sided version are proposed for use as fair-ML regularizers.","lead":"Polynomial regression is used to estimate the HGR correlation coefficient, a general nonlinear dependence measure. The estimator is deterministic, more stable under noise than neural or randomized alternatives, and can act as a penalty term in fair machine learning.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Eqs. (8)–(9) replace the least-squares minimization needed for HGR by an unbounded maximization of the squared residual; as written HGR-KB is undefined, not a correlation estimator.","rationale":"The central claim depends on HGR-KB being a valid estimator of maximal correlation. The weakest point is the step from Eq. (6) to Eqs. (8)–(9). The reader identified Appendix B and Section C, but the sign error is prior: if arg max is literal, no reformulation can be correct, and the h=k=1 case gives a crisp counterexample. This is an internal inconsistency, not a disagreement with consensus. The empirical results may come from an implementation that uses arg min, but the manuscript as written does not define the method it evaluates. Therefore I recommend keeping the conditional verdict until the authors correct the optimization problem and supply a valid proof of global optimality, or explicitly weaken the claim to a local-optimum heuristic.","tokens_in":12934,"tokens_out":12913,"duration_ms":164206,"concrete_test":"Take h=k=1 and any dataset with non-constant a. In Eq. (9), β reduces to ±1/σ(b); maximizing over α gives infinite objective, so check whether Eqs. (8)–(9) have any finite solution. If the authors agree this is a typo for arg min, test the corrected formulation on synthetic data (e.g., independent Gaussians and the circular relation from Fig. 6) by comparing its optima with a dense grid search over normalized α,β for small degree; the corrected minimizer should match the Eq. (4) maximum. Also re-derive the Lagrangian proof in Section C for the corrected problem, since the current proof does not establish convexity or global optimality.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Equation (4) defines HGR-KB as max_{α,β} ρ(P^h_a α, P^k_b β). By Appendix A, for fixed f,g the optimal r in the least-squares objective is the correlation, so maximizing ρ^2 is equivalent to minimizing the joint squared residual; this is why Eq. (6) is an arg min. The substitution eα = α r / σ(P^h_a α) does not change the optimization direction. Yet Eqs. (8) and (9) state arg max over eα,β of ||eP^h_a eα − eP^k_b β||^2 (after fixing σ(eP^k_b β)^2 = 1). This is the opposite objective. It is also unbounded: in the simplest allowed case h=k=1, the constraint fixes β up to sign while α is free, so the quadratic form ||(a−mean(a))α − β(b−mean(b))||^2 diverges as |α|→∞. No finite global maximizer exists. Hence the object defined by Eqs. (8)–(9) cannot equal the maximum in Eq. (4). The concern is not merely the Appendix B invertibility claim or Section C's invalid convexity argument; even setting those aside, the written objective has the wrong sign and is ill-posed. If the authors intended arg min, the derivation must be corrected and a valid global-optimality/convexity proof supplied; Section C currently proves neither.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes two polynomial-kernel estimators of the Hirschfeld–Gebelein–Rényi (HGR) correlation coefficient: HGR-KB, which maximizes the Pearson correlation between polynomial transformations of the two variables, and HGR-SK, a restricted version in which one of the two transformations is required to be linear. The authors derive a least-squares reformulation for HGR-KB, claim a convex Lagrangian formulation with a globally optimal solution, prove monotonicity of HGR-KB in the polynomial degrees, and report experiments on synthetic correlation detection and fairness-constrained machine learning. The general direction is interesting and the intended estimator is potentially useful, but the formal development as written contains major flaws: the central optimization problem is stated as an arg max of a squared residual, which is both the wrong direction for least-squares correlation and an unbounded problem; and the supporting appendices contain incorrect or unsupported claims.","tokens_in":13359,"tokens_out":15028,"duration_ms":176637,"significance":"If the polynomial HGR estimator were correctly formulated and validated, it would provide a deterministic, interpretable, and differentiable alternative to neural, KDE, and randomized HGR estimators, with plausible value as a fairness regularizer. The paper also contains a correct monotonicity argument (Appendix D) and a reasonable experimental design with oracle baselines. These strengths are real, but the central algorithmic definition is currently not mathematically valid, so the significance of the contribution is not established as written.","major_comments":[{"comment":"The optimization is stated as arg max over (eα,β) of the squared residual ||eP_h^a eα − eP_k^b β / σ(eP_k^b β)||^2. This is the opposite of the least-squares minimization that is needed to maximize correlation, and it is unbounded: in the simplest case h=k=1, the constraint fixes β up to sign, while α is free, so the objective grows without bound as |α|→∞. Thus Eqs. (8)-(9) do not define a finite estimator and cannot be equivalent to Eq. (4) or to Eq. (6), which is already written as an arg min. The same error propagates to HGR-SK in Eqs. (10)-(11), where β is fixed by the variance constraint and the arg max over α is unbounded. Please correct the sign and provide a well-posed formulation with appropriate normalization or constraints.","section":"Section 3.1, Eqs. (8)-(9)"},{"comment":"The single-level equivalence is justified by the statement 'Given that all transformations are invertible,' but polynomial maps of degree greater than one are not invertible. This invalidates the proof as written. The algebraic step from Eq. (22) to Eq. (23) does not actually require invertibility, so the equivalence may be true, but it is not established by the supplied argument. A correct derivation that does not rely on this false assumption is needed.","section":"Appendix B, Eq. (23)"},{"comment":"The claimed convex Lagrangian formulation does not hold. Equation (9) is an arg max of a convex quadratic c(α,β); the equality constraint σ(eP_k^b β)^2 = 1 defines a nonconvex set, and the proposed penalty p(α,β)=|σ(eP_k^b β)^2−1| is not convex. The inequalities in Eqs. (26)-(28) are only monotonicity relations for minimizers of c+μp and do not imply convexity, existence of a global optimum, or convergence. Consequently, the statement that 'a globally optimal solution exists' and the use of a trust-region solver as a global optimizer are unsupported. For the arg-min version, a correct treatment would be needed (e.g., a generalized-eigenvalue reformulation rather than a Lagrangian convexity argument).","section":"Appendix C"},{"comment":"The claim that HGR-KB yields a 'valid subgradient' through Eq. (7) is not derived. The solution (eα*,β*) of a constrained optimization problem is generally nonsmooth as a function of the data, and no chain-rule, envelope-theorem, or automatic-differentiation argument is supplied for HGR-KB. The fairness experiments in Section 4.2 rely on this subgradient for back-propagation, so this is a load-bearing gap that needs a precise derivation or a reference.","section":"Section 3.3, Differentiability paragraph"}],"minor_comments":[{"comment":"The statement that β is 'completely determined by the constraint and equal to 1/σ(b)' omits the sign; the constraint only fixes β = ±1/σ(b).","section":"Eq. (10)"},{"comment":"The phrase 'inherent uncomputability of HGR' is imprecise. The population HGR coefficient is a well-defined supremum; it is not computable from finite samples in general, but this is a sample-complexity issue rather than an inherent uncomputability of the coefficient itself.","section":"Abstract / Introduction"}],"recommendation":"major_revision","confidential_remarks":"The intended polynomial HGR formulation is salvageable if the authors correct the arg-max/arg-min error, replace the invalid convexity claims with a sound optimization analysis, and rerun or clarify the experiments accordingly. However, the current manuscript is not publishable in this form: the main estimator is undefined as written. I did not inspect the linked code; given the mismatch between the equations and the likely implementation, the authors should also clarify exactly which optimization objective the reported numbers come from."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Hi [Colleague],\n\nQuick take on arXiv:2509.09380. The paper is not ready for serious engagement as written. The central estimator HGR-KB is defined in Eqs. (8)–(9) as the arg max of a squared residual subject to a variance constraint. That is the wrong direction: the least-squares formulation of Pearson correlation is a minimization, and maximizing that same objective is unbounded (α is free) and has no connection to HGR. This is not a minor typo; it means the main methodological contribution is mathematically undefined. The stress-test note is right.\n\nWhat's good: the underlying idea of using polynomial kernels for HGR is sound, and the single-kernel subcase (HGR-SK) is, once corrected to arg min, simply the square root of R² from polynomial regression — fast, deterministic, interpretable. The experiments are thorough in comparing against HGR-NN, HGR-KDE, and RDC, and the determinism advantage is real. The monotonicity result in Appendix D is correct.\n\nWhere it falls apart: (1) the sign error in Eqs. (8)–(9) kills the estimator; (2) Appendix C's convexity claim is wrong — the penalty |σ(β)^2−1| is not convex, and Lagrange multipliers don't fix the max/min mismatch; (3) Appendix B's invertibility assumption is irrelevant, and the equivalence doesn't need it; (4) the differentiation mechanism is hand-waved — you can't just say Eq. (7) is differentiable without a subgradient derivation through the arg max (or arg min); (5) the novelty is overstated — polynomial kernel CCA is in the cited literature.\n\nThe paper needs major revision before it can be taken seriously. I would not cite it or bring it to our reading group until the authors fix the optimization direction and provide a coherent derivation. If they do, the HGR-SK variant could be a useful practical tool. But as it stands, I'd desk reject it in its current form.","headline":"The polynomial-kernel HGR estimator is built on a sign error: Eqs. (8)–(9) maximize a squared residual, making HGR-KB unbounded and unrelated to the HGR objective.","tokens_in":13748,"tokens_out":8252,"would_cite":false,"duration_ms":86362,"reading_group":"no","serious_thinker":"no","would_accept_peer_review":false},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["62H20","62-08"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper introduces HGR-KB and HGR-SK, polynomial-kernel estimators for the HGR maximal correlation that are deterministic, differentiable, and robust, and argues they outperform neural, kernel-density, and randomized baselines.","keywords":["Hirschfeld-Gebelein-Renyi coefficient","polynomial regression","maximal correlation","copula transformation","dependence measure","fairness regularizer","differentiable estimator","least squares"],"falsifier":"On synthetic data with a known non-invertible relationship such as y = x^2, run HGR-KB with degree 2 and compare against the squared Pearson correlation of the true optimal features; if the result differs, the claimed equivalence to maximal correlation fails. A second probe: solve the same instance from multiple trust-region starting points; any dependence on initialization would indicate local optima.","tokens_in":12869,"feed_emoji":"📈","tokens_out":4148,"duration_ms":46095,"temperature":0.7,"pith_summary":"The paper is trying to make the HGR correlation coefficient practical. HGR measures arbitrary nonlinear dependence between two variables, but its exact value is uncomputable and existing differentiable estimators are stochastic or opaque. The authors claim that representing the two copula transformations as polynomial kernels converts HGR estimation into a least-squares problem with a single variance constraint, solvable exactly and deterministically. If true, practitioners get a fast, interpretable correlation score that also supplies a subgradient usable for fairness constraints in neural-network training.","feed_headline":"Polynomial fits deliver a stable nonlinear-correlation measure","feed_subtitle":"Kernel-degree HGR estimator is deterministic, fast, and differentiable enough to enforce fairness constraints.","key_machinery":"The Pearson-as-least-squares reformulation: maximizing a sample correlation is equivalent to minimizing the squared error of the best linear fit after standardizing, and replacing unknown copula maps with degree-d polynomial kernels turns HGR into a constrained quadratic optimization. The paper uses the scale symmetry of correlation to merge one coefficient vector with the regression slope, leaving a single nonconvex variance constraint, and solves the full problem with a trust-region method; the single-kernel variant (degree one in one variable) becomes an ordinary least-squares solve.","core_discovery":"The central claim is that finite-degree polynomial expansions of the two copula functions are enough to approximate HGR reliably. By rewriting Pearson correlation as the optimal slope of a least-squares fit between standardized variables, the maximization over functions becomes a quadratic problem in polynomial coefficients with one variance constraint; a single-kernel restriction reduces further to unconstrained least squares. The paper argues the resulting estimators, HGR-KB and HGR-SK, are deterministic, interpretable, differentiable, and more stable than neural, kernel-density, and randomized baselines, with the degrees acting as a transparent bias-variance dial.","pith_inferences":["Because Appendix B's single-level equivalence assumes all transformations are invertible, and polynomials of degree greater than one are not globally invertible, the equality between the computed optimum and the true HGR maximum is not established for the general case; the paper's experimental miss on y = sin(x^2) hints at this gap.","The same least-squares machinery would extend naturally to other basis families (splines, trigonometric polynomials) preserving determinism; the paper does not explore this.","Monotonicity with kernel degree suggests a calibration heuristic: increase degree until the correlation stops improving, though Figure 4 shows the plateau is dataset-dependent.","In the fairness experiments, constraint satisfaction is measured only with the penalizer's own HGR type; the paper states no oracle exists, so cross-method constraint comparisons are semantic rather than direct."],"forward_implications":["HGR-KB yields fully deterministic correlation values, removing seed-dependence of neural and randomized estimators.","Kernel coefficients can be inspected directly, so users can see which polynomial degrees carry the dependence.","The computed value rises monotonically with kernel degrees, giving a principled way to balance underfitting against overfitting.","A subgradient of HGR-KB, and an exact gradient of HGR-SK, can penalize dependence in fair-machine-learning training.","HGR-SK is orders of magnitude faster but only detects functional dependencies, missing relations such as circles."],"fun_headline_variants":["Polynomial kernels make HGR robust and deterministic","Finite-degree polynomials tame HGR's bias-variance trade-off","A deterministic, differentiable HGR via polynomial regression","User-set polynomial degrees dial HGR robustness and speed","Polynomial regression yields a stable HGR subgradient for fairness"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The load-bearing premise is that the bi-level HGR problem can be collapsed to a single convex optimization; Appendix B's collapse requires all transformations be invertible, a condition polynomial kernels of degree greater than one violate, and Section C's convex Lagrangian argument applies to a nonconvex objective—so if these fail, HGR-KB computes only a local polynomial optimum.","fun_headline_variants_meta":{"raw":{"variants":["Polynomial kernels make HGR robust and deterministic","Finite-degree polynomials tame HGR's bias-variance trade-off","A deterministic, differentiable HGR via polynomial regression","User-set polynomial degrees dial HGR robustness and speed","Polynomial regression yields a stable HGR subgradient for fairness"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000167,"raw_usage":{"total_tokens":1066,"prompt_tokens":687,"completion_tokens":379,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":431,"completion_tokens_details":{"reasoning_tokens":309}},"tokens_in":431,"tokens_out":379,"duration_ms":4689,"temperature":1.0,"reasoning_tokens":309,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-04T19:10:34.644026+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"On synthetic data with a known non-invertible relationship such as y = x^2, run HGR-KB with degree 2 and compare against the squared Pearson correlation of the true optimal features; if the result differs, the claimed equivalence to maximal correlation fails. A second probe: solve the same instance from multiple trust-region starting points; any dependence on initialization would indicate local optima.","supporting_citations":[],"review_version":1}