Parametrizing Convex Sets Using Sublinear Neural Networks

Eloi Martinet

arxiv: 2605.03520 · v1 · submitted 2026-05-05 · 🧮 math.OC · cs.AI· cs.LG· cs.NA· math.NA

Parametrizing Convex Sets Using Sublinear Neural Networks

Eloi Martinet This is my paper

Pith reviewed 2026-05-07 15:28 UTC · model grok-4.3

classification 🧮 math.OC cs.AIcs.LGcs.NAmath.NA

keywords convex setssublinear functionsneural networkssupport functionsgauge functionsuniversal approximationshape optimizationinverse design

0 comments

The pith

Sublinear neural networks parametrize arbitrary convex sets by learning their support and gauge functions.

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

The paper introduces a parametrization of convex sets through neural networks that learn sublinear functions, which are positively homogeneous and convex. These networks implicitly capture both the support function, describing the set's width in every direction, and the gauge function, describing its size relative to the origin. A universal approximation theorem establishes that any convex body can be represented to arbitrary accuracy in this form. This matters because it converts geometric objects into trainable, differentiable models suitable for optimization and design tasks.

Core claim

We propose a neural parameterization of convex sets by learning sublinear functions. Our networks implicitly represent both the support and gauge functions of a convex body. We prove a universal approximation theorem for convex sets under this parametrization. Empirically, we demonstrate the method on shape optimization and inverse design tasks, achieving accurate reconstruction of target shapes.

What carries the argument

Sublinear neural networks that are positively homogeneous and convex, representing the support and gauge functions of convex bodies.

If this is right

Shape optimization can proceed by directly optimizing the weights of the sublinear network instead of explicit geometric variables.
Inverse design tasks gain the ability to reconstruct target convex shapes from partial observations or objectives.
Convex constraints become differentiable components that can be embedded inside larger end-to-end neural pipelines.
The universal approximation result removes the need to hand-craft different representations for different families of convex sets.

Where Pith is reading between the lines

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

The same homogeneity property might extend the technique to parametrizing convex functions rather than just sets.
Hybrid models could combine these networks with standard architectures to enforce convexity in high-dimensional learning problems.
Training dynamics in high dimensions or with noisy data remain open questions that would affect practical deployment beyond the reported tasks.

Load-bearing premise

Sublinear neural networks can be trained to represent the support and gauge functions of any convex set to arbitrary accuracy without restrictions on the class of convex bodies.

What would settle it

A specific convex body whose support function cannot be approximated below a fixed positive error by any sublinear network, regardless of size or training.

Figures

Figures reproduced from arXiv: 2605.03520 by Eloi Martinet.

**Figure 1.** Figure 1: A sublinear network can represent both the gauge and support functions of convex sets, and view at source ↗

**Figure 2.** Figure 2: Reconstruction of convex shapes from noisy point clouds. First column: reference shapes. view at source ↗

**Figure 3.** Figure 3: Optimal shape for the Poisson-based shape optimization problem (in black) along with the view at source ↗

**Figure 4.** Figure 4: Optimal shapes for maximizing the torsion gradient under two constraints: fixed volume view at source ↗

**Figure 5.** Figure 5: Solution to the Minkowski problem. Left: prescribed curvature. Right: view at source ↗

**Figure 6.** Figure 6: Runtime vs accuracy comparison between classical solvers and PINN-based approaches view at source ↗

**Figure 7.** Figure 7: Minimizers of the Mahler volume under n-fold symmetry constraints. The method recovers polygonal structures, highlighting its ability to represent non-smooth convex shapes. E Minimization of the Mahler volume Gauge and support functions are related to each other through the notion of polar body. The polar body of a convex set K ∈ K is defined as K◦ := x ∈ R d : x · y ≤ 1 for all y ∈ K view at source ↗

**Figure 8.** Figure 8: Robustness to noise and network initialization across 100 runs. Median reconstruction view at source ↗

**Figure 9.** Figure 9: Robustness to amounts of samples and network initialization across 100 runs. Median view at source ↗

**Figure 10.** Figure 10: Optimal shapes for the first six Dirichlet eigenvalues view at source ↗

read the original abstract

We propose a neural parameterization of convex sets by learning sublinear (positively homogeneous and convex) functions. Our networks implicitly represent both the support and gauge functions of a convex body. We prove a universal approximation theorem for convex sets under this parametrization. Empirically, we demonstrate the method on shape optimization and inverse design tasks, achieving accurate reconstruction of target shapes.

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

This paper parametrizes convex sets with sublinear networks that jointly represent support and gauge functions and proves a universal approximation result that holds up on the stated construction.

read the letter

The core contribution is a parametrization of convex bodies via sublinear neural networks. These networks are built to stay positively homogeneous and convex, so they can stand in for both the support function and the gauge function of the same set. The authors prove a universal approximation theorem showing that such networks can get arbitrarily close to any compact convex body in the Hausdorff metric. That theorem follows from the fact that finite maxima of linear forms are dense among support functions and that sublinear networks are closed under the needed operations. The empirical section applies the method to shape optimization and inverse design tasks, where it reconstructs target shapes with low error. The approach is straightforward and keeps the convexity properties exactly, which is useful when the downstream task requires feasible convex sets. The main limitation is that the experiments are still narrow; they do not include many baselines or a detailed study of how approximation error behaves with network depth or width on more varied families of convex bodies. The proofs appear to rest on standard density arguments rather than new technical machinery, so the novelty is mainly in the clean application to joint support/gauge representation rather than a deep theoretical advance. This work is aimed at people who already work on convex geometry inside optimization or on learning-based inverse problems that need to output convex shapes. It is coherent on its own terms and the central claim does not contain obvious gaps, so it is worth sending to referees who can check the details of the approximation proof and the training procedure.

Referee Report

0 major / 2 minor

Summary. The manuscript proposes a neural parameterization of convex sets by learning sublinear functions (positively homogeneous and convex) that implicitly represent both the support and gauge functions of a convex body. It establishes a universal approximation theorem for convex sets under this parametrization and demonstrates the approach empirically on shape optimization and inverse design tasks, reporting accurate reconstruction of target shapes.

Significance. If the universal approximation result holds, the work provides a theoretically grounded neural representation of convex bodies that exactly preserves convexity and positive homogeneity, enabling differentiable parametrizations suitable for optimization. The empirical demonstrations on shape optimization and inverse design illustrate practical utility, and the explicit proof of density among support functions (via finite maxima of linear forms) strengthens the contribution relative to heuristic convex representations.

minor comments (2)

The abstract and introduction would benefit from a brief statement of the precise function space (e.g., continuous support functions on the unit sphere) and the norm in which approximation is proved, to make the theorem statement immediately self-contained.
In the empirical section, explicit reporting of the number of training samples, the precise loss formulation (e.g., Hausdorff distance or integrated support-function error), and any regularization used to enforce sublinearity would improve reproducibility.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive evaluation of the manuscript, including recognition of the universal approximation theorem for sublinear neural networks and the empirical results on shape optimization and inverse design. We appreciate the recommendation to accept.

Circularity Check

0 steps flagged

No significant circularity; universal approximation theorem is independent

full rationale

The paper establishes a parametrization of convex sets via sublinear neural networks that preserve positive homogeneity and convexity, thereby representing support and gauge functions. The central load-bearing step is the universal approximation theorem, which is stated as proven from first principles using the fact that sublinear networks are closed under the required operations and that finite maxima of linear forms are dense in the uniform norm on the unit sphere (equivalent to Hausdorff distance on compact convex sets). This density argument is a standard result in convex analysis and does not reduce to any fitted parameter, self-citation chain, or redefinition within the paper. Empirical tasks on shape optimization are presented separately as validation and do not feed back into the theorem. No self-definitional, fitted-input, or ansatz-smuggling patterns appear in the derivation chain.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The approach rests on standard definitions from convex analysis; no free parameters or invented entities are indicated in the abstract.

axioms (1)

standard math Sublinear functions are positively homogeneous and convex.
This is the core property defining the neural network outputs for representing support and gauge functions.

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Works this paper leans on

47 extracted references · 4 canonical work pages · 1 internal anchor

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