arxiv: 2604.03871 · v1 · submitted 2026-04-04 · 🧮 math.OC

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· Lean Theorem

Efficient Convexification of Kolmogorov-Arnold Networks with Polynomial Functional Forms Via a Continuous Graham Scan Approach

Barnabas Poczos, Carl Laird, Daniel Ovalle, Ignacio Grossmann, Javier Pena, Tianwei Li

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Pith reviewed 2026-05-13 16:47 UTC · model grok-4.3

classification 🧮 math.OC

keywords Kolmogorov-Arnold networksconvex envelopesGraham scanpolynomial functionsglobal optimizationconvex relaxationsbitangents

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The pith

A continuous Graham scan computes exact convex envelopes for univariate polynomials in KANs by locating bitangents without discretization.

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

The paper develops a continuous convexification method for Kolmogorov-Arnold Networks whose univariate components are polynomials. It reduces the relaxation task to computing exact convex envelopes of those univariate polynomials by means of a continuous Graham scan that identifies the supporting bitangents of each polynomial hull. The envelopes are then combined across the additive structure of the network to form the overall relaxation. A reader would care because tighter, efficiently computable relaxations directly improve the speed and quality of deterministic global optimization for models that employ polynomial KANs.

Core claim

By exploiting the additive separable structure of polynomial KANs, the relaxation problem reduces to computing tight convex envelopes of univariate polynomials. We propose a continuous variant of the classical Graham Scan that constructs these envelopes exactly by identifying the bitangents of the polynomial convex hull without discretization or factorable reformulations. We establish the correctness of the algorithm and characterize its computational complexity, and show how these envelopes can be combined to construct strong convex relaxations for polynomial KANs.

What carries the argument

Continuous Graham scan that identifies bitangents of the convex hull of a univariate polynomial to build its exact envelope.

If this is right

The univariate envelopes combine to yield strong convex relaxations for the full polynomial KAN.
These relaxations produce bounds that are comparable to or orders of magnitude tighter than those from standard global optimization solvers.
The method remains computationally efficient while avoiding discretization.
The algorithm's correctness is proved and its complexity is characterized.

Where Pith is reading between the lines

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

The same bitangent-identification idea could be tested on other additive separable polynomial models outside the KAN architecture.
Avoiding discretization may reduce numerical instability in relaxation-based branch-and-bound procedures for network optimization.
The approach invites direct comparison experiments on benchmark problems containing polynomial KAN substructures to measure bound tightness versus runtime.

Load-bearing premise

The univariate components of the KAN are polynomial functions, allowing the overall relaxation to reduce exactly to univariate convex-envelope computation.

What would settle it

Applying the continuous Graham scan to a specific quartic polynomial and verifying that the resulting piecewise-linear envelope does not coincide with the true convex hull obtained from solving the dual or using exact symbolic methods.

Figures

Figures reproduced from arXiv: 2604.03871 by Barnabas Poczos, Carl Laird, Daniel Ovalle, Ignacio Grossmann, Javier Pena, Tianwei Li.

**Figure 2.** Figure 2: Our Continuous Graham Scan Algorithm (Algorithm [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗

**Figure 3.** Figure 3: Illustration of using Convex Conjugate to binary search for the correct slope of bitangents: [PITH_FULL_IMAGE:figures/full_fig_p016_3.png] view at source ↗

**Figure 4.** Figure 4: Performance of convex relaxations for PKAN models with six input variables and poly [PITH_FULL_IMAGE:figures/full_fig_p031_4.png] view at source ↗

**Figure 5.** Figure 5: Effect of polynomial degree and input dimensionality on the quality and computational [PITH_FULL_IMAGE:figures/full_fig_p032_5.png] view at source ↗

read the original abstract

Deterministic global optimization of nonlinear models is important in many scientific and engineering applications. This framework typically involves repeatedly solving convex relaxations of the nonconvex problem, meaning that the strength of the relaxations and the cost of computing them directly determine overall efficiency and solution quality. In this work, we develop a tailored continuous convexification framework for Kolmogorov-Arnold Networks in which the univariate components are polynomial functions. By exploiting the additive separable structure of this architecture, the relaxation problem reduces to computing tight convex envelopes of univariate polynomials. We propose a continuous variant of the classical Graham Scan that constructs these envelopes exactly by identifying the bitangents of the polynomial convex hull without discretization or factorable reformulations. We establish the correctness of the algorithm and characterize its computational complexity, and show how these envelopes can be combined to construct strong convex relaxations for polynomial KANs. Computational results demonstrate that the proposed relaxations are both strong and robust, often producing bounds that are comparable, or even orders of magnitude tighter than relaxations of state-of-the-art global optimization solvers while remaining computationally efficient.

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 continuous Graham scan for exact univariate polynomial envelopes in KAN convexification.

read the letter

The key point from this paper is their continuous Graham scan algorithm that finds the exact convex envelope of a univariate polynomial by locating the bitangents on the hull. This then feeds into convex relaxations for Kolmogorov-Arnold Networks where the univariate functions are polynomials. They do a good job exploiting the additive separability in KANs. That structure lets them break the relaxation down to independent univariate problems, which they solve exactly instead of using looser factorable approximations or grids. The method avoids discretization entirely, which should preserve tightness. They also claim to have established correctness and given a complexity characterization that looks reasonable for the degrees that show up in practice. The computational results are presented as showing stronger bounds than state-of-the-art solvers in many cases, with good efficiency. If those hold, it would be a useful tool for global optimization in engineering applications that use these networks. One soft spot is the restriction to polynomial univariate components. The paper is clear about this scope, but it means the technique won't apply directly to KANs with other function types without further work. Another thing to check in the full paper is how the complexity behaves as the polynomial degree increases, since KANs can have varying degrees. The abstract suggests it's efficient, but real-world scaling would need confirmation. Overall, this is for researchers focused on deterministic global optimization and convex relaxations for neural network models in optimization problems. Someone working on branch-and-bound methods for nonconvex problems with separable structure could find the algorithmic construction and the bound quality useful. I would recommend sending it for peer review. The claims are concrete, the approach builds on standard convex geometry in a novel way for this setting, and there are no obvious internal contradictions.

Referee Report

0 major / 3 minor

Summary. The paper introduces a continuous variant of the Graham Scan algorithm to compute exact convex envelopes of univariate polynomials by identifying bitangents of the convex hull. Exploiting the additive separability of Kolmogorov-Arnold Networks (KANs) with polynomial univariate components, these envelopes are assembled into strong convex relaxations for deterministic global optimization. The manuscript establishes correctness of the procedure, characterizes its computational complexity, and reports numerical experiments showing bounds that are often comparable or orders of magnitude tighter than those from state-of-the-art solvers while remaining efficient.

Significance. If the central claims hold, the work provides a practical advance for global optimization of KAN-based models by delivering exact, non-discretized convex envelopes that preserve relaxation strength without factorable reformulations. The complexity analysis and focus on polynomial degrees typical in KANs indicate the method scales well for relevant instances, potentially improving bound quality and convergence in branch-and-bound frameworks. The explicit use of convex geometry tools on separable structures is a clear strength.

minor comments (3)

[§3] The description of the continuous Graham scan procedure would benefit from an explicit pseudocode listing or step-by-step walk-through with a low-degree polynomial example to illustrate bitangent identification.
[§5] In the computational experiments, the comparison tables would be strengthened by reporting the fraction of instances solved to global optimality and the average gap closed, rather than only selected bound values.
[§4] Notation for the assembled KAN relaxation (e.g., how univariate envelopes combine across layers) could be clarified with a small schematic diagram.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript on the continuous Graham Scan for convex envelopes of univariate polynomials in KANs. We appreciate the recommendation for minor revision and note that no specific major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper's central derivation introduces a continuous Graham-scan variant that directly computes bitangents to obtain exact univariate polynomial convex envelopes, then assembles them via additive separability into KAN relaxations. This construction is presented as an algorithmic procedure grounded in classical convex geometry (Graham scan) with an independent correctness proof and complexity analysis; no step reduces by definition to a fitted parameter, self-citation chain, or renamed input. The reduction to univariate envelopes follows from the stated polynomial KAN structure without circularity. The method is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The approach rests on standard properties of univariate polynomials and their convex hulls; no free parameters, ad-hoc axioms, or new invented entities are introduced in the abstract.

axioms (2)

standard math Univariate polynomials admit a convex hull that can be exactly described by a finite number of bitangents
Basic fact from convex analysis for smooth univariate functions.
domain assumption The additive separable structure of KANs allows the network relaxation to be assembled from independent univariate envelopes
Follows from the definition of Kolmogorov-Arnold Networks.

pith-pipeline@v0.9.0 · 5501 in / 1218 out tokens · 48886 ms · 2026-05-13T16:47:03.153062+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/BranchSelection.lean branch_selection unclear
the convex envelope of a polynomial over a bounded interval can be computed analytically

Reference graph

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