Classification by Separating Hypersurfaces: An Entropic Approach

Argimiro Arratia; Henryk Gzyl; Mahmoud El Daou

REVIEW 3 major objections 4 minor 19 references

Classification by Separating Hypersurfaces: An Entropic Approach

T0 review · 3 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Binary classification can be recast as the minimization of an entropy-like convex function over a box, with the optimal separating hyperplane given by a closed-form dual solution.

desk verdict Correct convex dual for separable data, but the experiments on nonseparable data run outside the theorem's scope and the linear method is essentially the maxent-in-mean approach of [8]. read the letter →

arxiv 2507.02732 v1 pith:A4GMXCX5 submitted 2025-07-03 cs.LG cs.ITmath.ITphysics.data-anstat.ML

Argimiro Arratia , Mahmoud El Daou , Henryk Gzyl This is my paper · ORCID

classification cs.LGcs.ITmath.ITphysics.data-anstat.ML MSC 90C0590C2590C4790C5268T0168T0568T0768T20

keywords ClassificationLineardiscriminationIll-posedinverseproblemsConvexoptimizationEntropyminimizationSeparatinghypersurfacesLagrangedualityPolynomialclassifiers

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

This paper proposes a new way to train a binary classifier: instead of solving the usual separation inequalities, it turns them into equalities by giving each training point its own parallel separating plane with a positive distance. The training problem becomes the minimization of a strictly convex, entropy-like function Ψ over a bounded box, subject to exact linear constraints Ax = 0. The paper proves (Theorem 7) that the optimal weights and distances have an explicit closed form in terms of a Lagrange multiplier that minimizes a dual objective, so training reduces to a smooth convex optimization problem. The same construction is extended to polynomial decision surfaces by lifting the data into monomial features. If the result is correct, classification becomes a parameter-free convex program that also outputs an uncertainty band between the two classes, and numerical experiments show it is competitive with SVM, KNN, logistic regression, and perceptron on several datasets.

What carries the argument

The load-bearing object is the pair of convex functions Ψ(x) and M(τ, ζ) in equations (12) and (13). Ψ is defined on the constraint box: each weight component w_j contributes a binary entropy term and each distance b_j contributes b_j(ln b_j − 1), making the box constraints automatic. M is the logarithm of the Laplace transform of a measure that places unit mass at the corners of the weight box and at all nonnegative integers for the distances, and it is the Legendre–Fenchel dual of Ψ. The key identity ∇M = (∇Ψ)^{-1} lets the authors solve the constrained primal problem through its unconstrained dual: the Lagrange multiplier λ* of the equality constraints minimizes λ ↦ M(A^t λ), and the primal variables are recovered by evaluating ∇M at A^t λ*. This dual substitution is what turns classification into a smooth convex optimization problem that standard spectral gradient methods can handle.

What would settle it

Take a linearly separable dataset with a known separating hyperplane, run the dual minimization until the gradient norm is below ε, and compute the primal residual ‖Ax*‖. If Theorem 7 is right, this residual should tend to zero as ε decreases; if it stalls above zero, the asserted equivalence between the primal and dual minimizers fails. The same test on the Circles dataset with a degree-1 polynomial will show a residual bounded away from zero, confirming that the infeasible-data case does not satisfy the theorem's premise.

Watch

Extended reading notes

Core claim

The central claim is that the separation problem can be restated as finding x* = (w*, b*) that minimizes Ψ(x) over K = [-1,1]^N × R_{++}^M subject to Ax = 0, where the data matrix A encodes the requirement that every good point g_i satisfies ⟨g_i, w⟩ = b_1(i) and every defective point d_i satisfies ⟨d_i, w⟩ = -b_2(i) with all b components positive. The paper proves that this primal solution is given in closed form: the weight components are hyperbolic-tangent-like functions of the first N components of A^t λ*, while the distances are exponentials of the remaining components, and λ* minimizes the dual function M(A^t λ) defined in (13). This is Theorem 7, and it holds because the entropy function Ψ and the log-Laplace transform M are Legendre–Fenchel duals whose gradients are inverses of each other. A byproduct of the solution is the separation measure Δ(w*) = b_+ − b_- and an uncertainty band between the two hyperplanes H_g and H_d, which gives the classifier a natural region of uncertain prediction.

Load-bearing premise

The whole construction assumes that the training data admit an exact vector x in the box with positive distances satisfying Ax = 0, meaning a separating surface of the chosen degree that passes through every training point with a positive margin; when the data are not separable (as with a linear classifier on the concentric Circles dataset), the claimed primal–dual solution lies outside the theorem's scope.

Editorial extensions

If this is right

Training the entropic classifier reduces to minimizing a smooth convex function of the Lagrange multiplier λ, so no quadratic programming or nonsmooth subgradient methods are needed.
The construction extends to polynomial decision surfaces by enlarging the feature vector with monomials, and the reported experiments show near-perfect accuracy on the Circles, Spiral, and Moons datasets for degree 3 or 4.
Every solution comes with an explicit uncertainty band between the two boundary hyperplanes, so predictions can be assigned a confident region and an uncertain region.
Because the primal objective and constraints are convex, the method has a unique global minimizer whenever the feasibility assumption holds, avoiding the local-minimum issues of gradient-descent-trained classifiers.

Reading between the lines

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

The dual formulation suggests a probabilistic reading of the distances b_i = e^{-λ_i}, which could be used to turn the uncertainty band into a calibrated probability of class membership; the paper does not develop this.
The same entropy–dual scheme could be applied to other feature maps, such as radial basis functions, by changing the matrix A; only polynomial features are tested, so the comparison to kernel SVMs is not direct.
A natural extension is to relax the hard equality constraints with slack variables or a norm penalty, which would let the method handle non-separable and noisy data while retaining the entropic dual structure; the paper notes the Ax = y case in passing but does not implement it.
The separation measure Δ(w*) could be used as a data-dependent margin for model selection, analogous to the margin in SVMs, but the paper does not explore this connection.

Share X Bluesky LinkedIn Reddit HN

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

Two things to know: the convex-duality derivation in Section 4 is correct for the feasible case, and the method's uncertainty band is a nice touch. But the linear classifier is the maxent-in-the-mean method from reference [8], dressed in new notation, and the experiments on nonseparable data operate outside the theorem's scope because the equality constraint Ax=0 requires strict homogeneous separability.

What is actually new: the polynomial extension (feature expansion followed by the same dual minimization) and the explicit uncertainty band between b+ and b-. The derivation of Theorem 7 is sound when a feasible x exists. The dual objective λ↦M(A^tλ) is convex, and the gradient stopping rule is a sensible way to approach the constraint for separable data.

Where it gets soft. First, the novelty claim: equations (15)-(16) identify Ψ as the Fenchel dual of the log-Laplace transform M, and the solution x*=∇M(A^tλ*) is exactly the maxentropic mean used in [8]. The paper cites [8] but never says the linear algorithm is equivalent to it. That is a disclosure problem.

Second, and more load-bearing: Theorem 7 assumes Ax=0 is satisfiable. For a linear classifier with K=[-1,1]^N×R_{++}^M, feasibility means some w in the box satisfies Dw>0, i.e., the data are strictly separable by a hyperplane through the origin. The Circles dataset is not separable this way, so the Entropic-Linear row in Table 3 is not a valid instance of the theorem. The stress-test note is right: the dual objective is bounded below but its infimum is not attained; along an unbounded λ direction the gradient goes to zero with w and b shrinking to zero. The spectral-gradient stopping rule then reports a degenerate solution, which explains the 0.46 accuracy and zero precision/recall. The paper never checks feasibility before running the method, and Section 5.3.2 treats the result as just another linear classifier's failure.

Third, the experiments use a single train-test split with no error bars, degree selection by grid search, and no polynomial-kernel SVM baseline. The Entropic-Polynomial results are encouraging but not persuasive without repeated splits.

Who this is for: someone working on entropy-based inverse problems and classification might find the dual formulation useful, but the added value over [8] is modest. The paper deserves a serious referee because the theorem is non-trivial and the infeasibility issue is addressable with an explicit feasibility test or a relaxed formulation. My recommendation: send it to peer review, but flag that the experimental protocol and the novelty disclosure need fixing. If the authors can show feasibility checks, add error bars, and position the contribution honestly against [8], it could be a decent methods note.

Referee Report

3 major / 4 minor

Summary. The paper proposes a classification method based on minimizing a strictly convex, entropy-like function Psi over a box containing the weight vector and a positive orthant containing per-point margins, subject to equality constraints Ax=0. In the linear case the constraints require every training point to satisfy an exact equality with a positive per-point margin; the polynomial extension maps the data through monomial features and places the coefficient vector in [-e,e]^{T(p)}. Section 3 introduces the Legendre-Fenchel dual M, and Section 4 derives closed-form expressions for the primal variables in terms of a dual vector lambda, culminating in Theorem 7, which states that the solution is obtained by minimizing lambda |-> M(A^t lambda). Section 5 compares the resulting Entropic-Linear and Entropic-Polynomial classifiers against SVM, KNN, logistic regression, and perceptron on four synthetic datasets and on the Breast Cancer dataset.

Significance. If Theorem 7 holds under the appropriate hypotheses, the paper offers an attractive convex-duality formulation: minimizing Psi subject to affine equality constraints is equivalent to minimizing a smooth convex function of the dual variable, with simple gradient formulas. The availability of reproducible code and the inclusion of several standard baselines are positive features. However, the central theorem is conditional on the feasible set {x in K : Ax=0} being nonempty, a condition that is never stated, checked, or enforced; the experiments include datasets for which that set is empty, and the reported degenerate result for Entropic-Linear on Circles is consistent with an unbounded dual sequence rather than a separating solution. The polynomial extension also contains an unresolved scaling gap between the coefficient box [-e,e] and the entropy function derived for [-1,1]. As a result, the paper currently establishes a conditional theoretical result plus an illustrative implementation, but not the broad empirical and methodological claim made in the abstract.

major comments (3)

[Section 4, Theorem 7 and Eq. (6)] The theorem presupposes that the feasible set {x in K : Ax=0} is nonempty, but the paper never states or tests this condition. With A=[D,-I], feasibility is equivalent to b=Dw with b_i>0 and w in [-1,1]^N, i.e. strict linear separability by a hyperplane through the origin. For the Circles dataset in Section 5.3.2 this condition fails, so the Entropic-Linear row of Table 3 (accuracy 0.46) is not a valid instance of Theorem 7. The degenerate confusion matrix, with all test points assigned to one class, is what one expects from an unbounded dual sequence lambda=t v with v>0 and D^t v=0, along which grad_lambda M(A^t lambda)=A x(lambda) tends to 0 while x(lambda) tends to (0,0); the stopping criterion in Section 4.1 can therefore be satisfied without approaching a separating solution. The paper should add a feasibility check before applying the method, or explicitly restrict all claims and experiments to data satisfying Problem 3's equality constraint.
[Section 2, Problem 6 and Eq. (12)] The polynomial formulation places coefficients in the box [-e,e]^{T(p)}, but the entropy function Psi in Eq. (12) and the resulting dual formulas in Eqs. (17) and (20)-(21) are derived only for the box [-1,1]^N. The paper does not provide the rescaled entropy or the corresponding dual formulas (after a change of variables one would expect expressions containing e, such as w_j = e tanh(e tau_j)). Unless the implementation silently sets e=1, the Entropic-Polynomial classifier is not justified by the theory in Sections 3 and 4; if it does set e=1, then e is not an adjustable parameter as claimed. This gap needs to be fixed by generalizing the formulas or by explicitly normalizing the coefficients.
[Section 5.2 and Tables 3-10] All numerical comparisons are based on a single 70:30 train-test split, with no standard errors, confidence intervals, or repeated-seed evaluation. Given that the behavior of Entropic-Linear on nonseparable data is the central empirical question raised by Theorem 7's feasibility condition, single-split accuracy values cannot support the paper's comparative claims. In addition, the polynomial degree is selected by grid search on the same split used for reporting accuracy, which can optimistically bias the reported results. Please report repeated splits or cross-validation, and include a feasibility diagnosis for each dataset.

minor comments (4)

[Section 1, Problem 2] The expression 'G S D' should read 'G union D'; the union symbol appears to have been lost in typesetting.
[Eq. (5) and Eq. (13)] The symbol M is overloaded: M denotes the number of samples K1+K2 in Eq. (5), and also denotes the dual function M(tau,zeta) in Eq. (13). This makes Section 4 harder to read; consider using a different symbol, such as m, for the sample size.
[Introduction and reference [8]] The relationship to reference [8] is not made precise. If the dual minimizer coincides with the maxent-in-the-mean solution of [8], the paper should explicitly state the novel contribution, such as the dual optimization algorithm or the polynomial extension, so that the reader can assess the advance over that prior work.
[Theorem 7] Theorem 7 is stated without the convex-analysis hypotheses needed to guarantee that the dual minimizer exists and that strong duality holds, such as feasibility together with a Slater-type condition. A short remark stating these hypotheses would make the theorem rigorous rather than a formal first-order derivation.

Circularity Check

0 steps flagged · score 2.0 of 10

No load-bearing circularity: the entropic classifier's dual solution is derived by convex duality, not by fitting the target metric; the only significant overlap is a minor self-citation to the authors' earlier maxent-in-mean approach, which is not load-bearing.

full rationale

Theorem 7 is a standard primal-dual statement: the Lagrangian stationarity conditions in equations (18)-(21) characterize the constrained minimizer of Psi, and the minimizer lambda* of the dual objective M(A^t lambda) yields x* = grad M(A^t lambda*). This derivation is self-contained and does not assume the classification result it claims to prove. Test-set accuracy is computed on held-out data, so no fitted parameter is renamed as a prediction. The self-citation to reference [8] (Gzyl, ter Horst, and Molina) is real but minor: the function Psi in equation (12) is the Fenchel conjugate of the log-Laplace transform M in equation (13), which is the same measure dQ used in the authors' prior maxent-in-mean classification work, so the present method is mathematically a repackaging of that prior solution. That is a novelty overlap, not a circular derivation, because the proof of Theorem 7 does not rely on [8]. The one genuine caveat is a scope gap rather than circularity: Theorem 7 presupposes feasibility of Ax = 0, and on the nonseparable Circles dataset the equality system has no feasible x, so the Entropic-Linear result in Table 3 is outside the theorem's assumptions. This affects correctness and reporting, not the derivation chain.

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

The central derivation rests on the convex-duality framework, on a specific entropy-like objective, and on the feasibility of the equality constraints. The feasibility assumption is violated in the non-separable experiments, making it the most fragile part of the theoretical setup.

free parameters (3)

Per-point positive distances b_i = Data-dependent, learned during optimization
The M-dimensional vector b converts strict inequalities into equalities (Problem 2). These are decision variables, not constants, but they add M degrees of freedom that are essential to the method.
Polynomial degree p = 2, 3, or 4 per dataset by grid search
Section 5.2 states a grid search over degrees two, three, and four, with higher degrees avoided to prevent overfitting. This is a model selection choice that affects the reported results.
Polynomial coefficient box size e = Not reported
Section 2 defines K=[-e,e]^T(p) with e>0 'to be determined by the model builder.' The value used in the experiments is not stated.

assumptions (3)

standard math Ψ and M are Legendre-Fenchel duals with inverse gradients, as stated in Eq. (16).
The proof of Theorem 7 relies on convex-analysis duality in the style of Borwein and Lewis [2].
domain assumption A feasible point x with Ax=0 exists in K for each dataset used in the experiments.
Problem 4 and Theorem 7 assume an exact solution to the equality constraints. For non-separable data such as Circles with a linear classifier, no such feasible point exists, and the experiments run outside the theorem's scope.
ad hoc to paper The entropy-like function Ψ is an appropriate objective for classification.
Section 3 motivates Ψ by automatic satisfaction of box constraints and dual tractability, but no generalization bound or margin-based justification is provided.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Classification by Separating Hypersurfaces: An Entropic Approach." pith.science (2026). https://pith.science/paper/A4GMXCX5

@misc{pith2026250702732,
  author       = {Pith},
  title        = {Pith review of: Classification by Separating Hypersurfaces: An Entropic Approach},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/A4GMXCX5}},
  note         = {Machine review of arXiv:2507.02732}
}

abstract

We consider the following classification problem: Given a population of individuals characterized by a set of attributes represented as a vector in ${\mathbb R}^N$, the goal is to find a hyperplane in ${\mathbb R}^N$ that separates two sets of points corresponding to two distinct classes. This problem, with a history dating back to the perceptron model, remains central to machine learning. In this paper we propose a novel approach by searching for a vector of parameters in a bounded $N$-dimensional hypercube centered at the origin and a positive vector in ${\mathbb R}^M$, obtained through the minimization of an entropy-based function defined over the space of unknown variables. The method extends to polynomial surfaces, allowing the separation of data points by more complex decision boundaries. This provides a robust alternative to traditional linear or quadratic optimization techniques, such as support vector machines and gradient descent. Numerical experiments demonstrate the efficiency and versatility of the method in handling diverse classification tasks, including linear and non-linear separability.

Figures

Figures reproduced from arXiv: 2507.02732 by the authors.

**Figure 2.** Decision Boundaries on Circles Dataset with Uncertainty Bands [PITH_FULL_IMAGE:figures/full_fig_p013_2.png] view at source ↗

**Figure 3.** Decision Boundaries on Spiral Dataset with Uncertainty Bands [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: Decision Boundaries on Moons Dataset with Uncertainty Bands [PITH_FULL_IMAGE:figures/full_fig_p015_4.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

19 extracted references · 19 canonical work pages

[8]

H. Gzyl, E. ter Horst, and G. Molina. Application of the method of maximum entropy in the mean to classification problems.Physica A, 437:101–108, 2015

work page 2015
[1]

H. D. Block. The Perceptron: A Model for Brain Functioning. I.Reviews of Modern Physics, 34:123–135, 1962. 9 Table 7: Performance Comparison on Moons Dataset Method Accuracy Precision Recall F1-Score Entropic-Linear 0.80 1.00 0.60 0.75 SVM (GridSearch) 0.88 0.88 0.88 0.88 KNN (GridSearch) 1.00 1.00 1.00 1.00 Logistic Regression 0.88 0.88 0.88 0.88 Percept...

work page 1962
[2]

J. M. Borwein and A. S. Lewis.Convex Analysis and Nonlinear Optimization, 2nd Edition. CMS-Springer, Berlin, 2006

work page 2006
[3]

Boyd and L

S. Boyd and L. Vanderberghe.Convex Optimization. Cambridge University Press, 2004. Avail- able at https://stanford.edu/~boyd/cvxbook/

work page 2004
[4]

Cucker and D.-X

F. Cucker and D.-X. Zhou.Learning Theory: An Approximation Theory Viewpoint. Cambridge University Press, Cambridge, 2007

work page 2007
[5]

M. P. Deisenroth, A. A. Faisal, and C. S. Ong.Mathematics for Machine Learning. Cambridge University Press, Cambridge, 2020

work page 2020
[6]

R. O. Duda, P. E. Hart, and D. G. Stork.Pattern Classification. Wiley-Interscience, 2000

work page 2000
[7]

Golan and H

A. Golan and H. Gzyl. A generalized maxentropic inversion procedure for noisy data.Applied Mathematics and Computation, 127:249–260, 2002

work page 2002

Show all 19 references

[9]

Han-Veiga and F

M. Han-Veiga and F. G. Ged.The Mathematics of Machine Learning: Lectures on Supervised Methods and Beyond. De Gruyter, Berlin, 2024

2024
[10]

E. T. Jaynes. Information theory and statistical mechanics. Physical Review, 106:620–630, 1957

1957
[11]

J. N. Kapur.Maximum Entropy Models in Science and Engineering. Wiley, New York, 1998

1998
[12]

W. Kinzel. Statistical mechanics of generalization: new results for perceptrons. Physica A, 200:613–618, 1993. 10 Table 9: Performance Comparison on Breast Cancer Dataset Method Accuracy Precision Recall F1-Score Entropic-Linear 0.94 0.99 0.91 0.95 SVM 0.98 0.98 0.99 0.99 KNN ...

1993
[13]

McCulloch and W

W. McCulloch and W. Pitts. A logical calculus of ideas immanent in nervous activity.Bulletin of Mathematical Biophysics, 5:115–133, 1943

1943
[14]

Mohri, A

M. Mohri, A. Rostamizadeh, and A. Talwalkar.Foundations of Machine Learning. MIT Press, 2012

2012
[15]

Nierotti and D

J.P. Nierotti and D. Saad. Efficient bayesian inference for learning in the ising linear perceptron and signal detection in cdma.Physica A, 365:203–210, 2006

2006
[16]

Rau and J

A. Rau and J. P. Nadal. A model for a multi-class classification machine.Physica A, 185:428– 432, 1992

1992
[17]

Rosenblatt

F. Rosenblatt. The perceptron: A probabilistic model for information storage and organization in the brain.Psychological Review, 65:386–408, 1958

1958
[18]

Soheili and J

N. Soheili and J. F. Peña. A smooth perceptron algorithm.SIAM Journal on Optimization, 106:728–737, 2012

2012
[19]

T. L. H. Watkin and A. Rau. Selecting examples for the optimal perceptron. Physica A, 185:444–448, 1993. 11 Figure 1: Decision Boundaries on Blobs Dataset with Uncertainty Bands 12 Figure 2: Decision Boundaries on Circles Dataset with Uncertainty Bands 13 Figure 3: Decision Bo...

1993

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [8]

H. Gzyl, E. ter Horst, and G. Molina. Application of the method of maximum entropy in the mean to classification problems.Physica A, 437:101–108, 2015

work page 2015

[2] [1]

H. D. Block. The Perceptron: A Model for Brain Functioning. I.Reviews of Modern Physics, 34:123–135, 1962. 9 Table 7: Performance Comparison on Moons Dataset Method Accuracy Precision Recall F1-Score Entropic-Linear 0.80 1.00 0.60 0.75 SVM (GridSearch) 0.88 0.88 0.88 0.88 KNN (GridSearch) 1.00 1.00 1.00 1.00 Logistic Regression 0.88 0.88 0.88 0.88 Percept...

work page 1962

[3] [2]

J. M. Borwein and A. S. Lewis.Convex Analysis and Nonlinear Optimization, 2nd Edition. CMS-Springer, Berlin, 2006

work page 2006

[4] [3]

Boyd and L

S. Boyd and L. Vanderberghe.Convex Optimization. Cambridge University Press, 2004. Avail- able at https://stanford.edu/~boyd/cvxbook/

work page 2004

[5] [4]

Cucker and D.-X

F. Cucker and D.-X. Zhou.Learning Theory: An Approximation Theory Viewpoint. Cambridge University Press, Cambridge, 2007

work page 2007

[6] [5]

M. P. Deisenroth, A. A. Faisal, and C. S. Ong.Mathematics for Machine Learning. Cambridge University Press, Cambridge, 2020

work page 2020

[7] [6]

R. O. Duda, P. E. Hart, and D. G. Stork.Pattern Classification. Wiley-Interscience, 2000

work page 2000

[8] [7]

Golan and H

A. Golan and H. Gzyl. A generalized maxentropic inversion procedure for noisy data.Applied Mathematics and Computation, 127:249–260, 2002

work page 2002