arxiv: 2605.01655 · v1 · submitted 2026-05-03 · 🧮 math.CA · cs.LG

Recognition: unknown

Exact Loop Controllers for ReLU Realization of Homogeneous Curve Refinements

Boldsaikhan Bolorkhuu, Tsogtgerel Gantumur

Pith reviewed 2026-05-09 17:08 UTC · model grok-4.3

classification 🧮 math.CA cs.LG

keywords ReLU realizationhomogeneous refinementloop controllerpiecewise linear curvesresidual dynamicscurve iterationpolygonal generators

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The pith

Homogeneous curve refinement operators admit exact ReLU realizations with fixed width and depth linear in the iteration count.

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

The paper establishes that repeated application of a homogeneous refinement operator to a compactly supported piecewise linear curve can be computed exactly by a ReLU network whose width stays constant while depth grows only linearly with the number of iterations. The construction replaces scalar residual tracking with an exact state maintained on a polygonal loop; scalar multipliers and selection digits are then read out from this state by complementary piecewise linear functions. A reader would care because the result supplies an error-free embedding of iterative geometric constructions into neural networks, extending scalar binary refinable-function methods to vector-valued M-ary cases with a geometric rather than surrogate-based controller. The approach also covers affine forcing terms by finite expansion into homogeneous parts and reduces anchored curves to compactly supported defects.

Core claim

The iterates V^n γ of a homogeneous refinement operator (Vγ)(t) = sum A_j γ(Mt - j) admit exact ReLU realizations of fixed width and depth O(n). The central mechanism is an exact loop controller that transports the residual orbit as a forward-exact state on a polygonal loop; scalar factors and digit selectors are recovered from this state by complementary continuous piecewise linear readouts. Any remaining seam ambiguity is isolated to the final readout stage and does not affect the result because the scalar atom is supported away from the seam. Affine forcing is handled by expanding iterates into finite sums of homogeneous terms, yielding exact fixed-width realizations of depth O(n^2). The

What carries the argument

The exact loop controller for residual dynamics, which maintains the residual orbit as a forward-exact state on a polygonal loop and recovers scalars and selectors via complementary continuous piecewise linear readouts.

Load-bearing premise

The refinement operator has finitely many nonzero matrices and acts on compactly supported continuous piecewise linear curves, with loop seam ambiguity confined to the final readout stage where it remains harmless.

What would settle it

For the dragon-curve refinement matrices, construct the explicit ReLU network for n=2 and verify that its output coincides exactly with the twice-refined curve at every point, up to machine precision.

Figures

Figures reproduced from arXiv: 2605.01655 by Boldsaikhan Bolorkhuu, Tsogtgerel Gantumur.

**Figure 1.** Figure 1: First and second ternary digit and residual maps. view at source ↗

**Figure 2.** Figure 2: The exact loop-controller boundary field for view at source ↗

**Figure 3.** Figure 3: Complementary scalar readouts ρ − (left) and ρ + (right). The dashed line is the identity and the solid curve is the corresponding readout. The horizontal shaded strip is the interval [ϱ, 1−ϱ] containing supp h. The vertical shaded strip marks the seam interval modified by the readout: [1 − ε, 1] on the left and [0, ε] on the right. Lemma 3.3 (Scalar readouts). Let h be a special-hat, and assume 0 < ε < ϱ.… view at source ↗

**Figure 4.** Figure 4: Loop-state selector profiles for M = 3. Away from the red transition intervals Jn, the selectors agree with the exact digit indicators; inside Jn, affine ramps produce a continuous partition of unity. The last selector carries the seam value 0 ∼ 1. (i) For every z ∈ Γ, we have 0 ≤ χq(z) ≤ 1 and M X−1 q=0 χq(z) = 1. (ii) If t ∈ [0, 1]\Jn, then exactly one selector is equal to 1 at E(t), and it is the select… view at source ↗

**Figure 5.** Figure 5: Residual dynamics of the exact loop controller for view at source ↗

**Figure 6.** Figure 6: Gosper curve (Example 4.12) and self-intersecting Hilbert-type curve (Example 4.16). Example 4.17 (Hilbert-type recursion in R p ). Let p ≥ 2, and set M = 2p . We describe an inductive homogeneous prototype based on the reflected Hilbert symmetry. For p = 1, take P (1) 0 = 0, P(1) 1 = 1 2 , P(1) 2 = 1, and let C (1) 0 = [0, 1 2 ], C (1) 1 = [ 1 2 , 1]. Now assume inductively that, in dimension p − 1, we ha… view at source ↗

read the original abstract

We study homogeneous refinement operators \((V\gamma)(t)=\sum_{j\in\mathbb Z}A_j\gamma(Mt-j)\), acting on compactly supported continuous piecewise linear curves \(\gamma:\mathbb R\to\mathbb R^p\), where \(M\ge2\) and only finitely many matrices \(A_j\in\mathbb R^{p\times p}\) are nonzero. We prove that the iterates \(V^n\gamma\) admit exact ReLU realizations of fixed width and depth \(O(n)\). The main new ingredient is an exact loop controller for the residual dynamics. Instead of propagating scalar residual surrogates, the construction transports the residual orbit by a forward-exact state on a polygonal loop. Scalar factors and digit selectors are then recovered from this loop state by complementary CPwL readouts. The loop seam is not removed, but its remaining ambiguity is confined to the final readout/selector stage, where it is harmless because the scalar atom is supported away from the seam. This gives a homogeneous \(M\)-ary vector-valued extension of the scalar binary refinable-function construction with a more geometric controller architecture. We also record crude exponential bounds on the network weights and biases. Affine forcing terms are handled by expanding affine iterates into finite sums of homogeneous iterates, giving exact fixed-width realizations with depth \(O(n^2)\), and anchored open curves reduce to compactly supported defects with affine anchor mismatch. We also describe homogeneous polygonal generators, including dragon-type examples and a self-intersecting Hilbert-type prototype in arbitrary dimension. The extended version includes stage-dependent forcing, finite-state stacking reductions, and further geometric constructions such as Koch-, Gosper-, Morton-, and connector-based Hilbert-type variants.

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.

Referee Report

2 major / 2 minor

Summary. The paper proves that iterates V^n γ of homogeneous refinement operators Vγ(t) = ∑ A_j γ(Mt - j), acting on compactly supported continuous piecewise-linear curves γ : ℝ → ℝ^p with finitely many nonzero matrices A_j, admit exact ReLU realizations of fixed width and depth O(n). The central construction introduces an exact loop controller that transports residual orbits via a forward-exact polygonal state; scalar factors and digit selectors are recovered by complementary CPwL readouts whose seam ambiguity is confined to the final stage and rendered harmless by support separation. The result extends prior scalar binary work to the vector-valued M-ary setting, supplies crude exponential weight bounds, reduces affine forcing to depth O(n^2), and illustrates the method with dragon-type and self-intersecting Hilbert-type polygonal generators.

Significance. If the central construction holds, the work supplies a geometrically explicit, parameter-free route to exact fixed-width ReLU realizations of subdivision iterates, strengthening the theory of neural-network expressivity for self-similar and fractal curves. The loop-controller architecture and the clean separation of seam ambiguity constitute a genuine technical advance over scalar surrogates; the supplied examples (dragon, Hilbert) and the reduction of affine and open-curve cases further increase utility. The absence of circularity and the standard assumptions on the refinement operator are positive features.

major comments (2)

[Abstract / Main theorem] Abstract and the statement of the main theorem: the claim that the loop controller yields depth exactly O(n) with fixed width rests on the forward-exactness of the polygonal state and the support separation that makes seam ambiguity harmless; the provided sketch does not exhibit the explicit state-transition matrices or the readout functions that realize the digit selectors, so the O(n) bound cannot yet be verified from the text alone.
[Affine forcing paragraph] The reduction of affine forcing to a finite sum of homogeneous iterates (yielding depth O(n^2)) is asserted but the precise expansion and the manner in which the homogeneous controllers are stacked without width growth are not displayed; this step is load-bearing for the affine case and requires an explicit formula or inductive argument.

minor comments (2)

[Introduction] The notation for the refinement operator and the loop state could be introduced with a small concrete example (e.g., the dragon curve) immediately after the abstract to improve readability.
[Weight bounds] Crude exponential bounds on weights and biases are mentioned; a short table or explicit dependence on M, p and the spectral radius of the A_j would make the statement more precise.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive assessment of the paper's significance and for identifying the points where the presentation of the constructions requires greater explicitness. We address both major comments below and will incorporate the requested details into the revised manuscript.

read point-by-point responses

Referee: [Abstract / Main theorem] Abstract and the statement of the main theorem: the claim that the loop controller yields depth exactly O(n) with fixed width rests on the forward-exactness of the polygonal state and the support separation that makes seam ambiguity harmless; the provided sketch does not exhibit the explicit state-transition matrices or the readout functions that realize the digit selectors, so the O(n) bound cannot yet be verified from the text alone.

Authors: We agree that the current manuscript presents the loop controller primarily through a descriptive sketch. In the revision we will supply the explicit state-transition matrices that realize the forward-exact polygonal state on the residual orbit, together with the complementary CPwL readout functions that recover the scalar factors and digit selectors. These formulas will make the fixed-width property (constant across all n) and the O(n) depth bound directly verifiable from the text, while preserving the geometric separation of seam ambiguity at the final readout stage. revision: yes
Referee: [Affine forcing paragraph] The reduction of affine forcing to a finite sum of homogeneous iterates (yielding depth O(n^2)) is asserted but the precise expansion and the manner in which the homogeneous controllers are stacked without width growth are not displayed; this step is load-bearing for the affine case and requires an explicit formula or inductive argument.

Authors: We acknowledge that the affine reduction is stated at a high level. The revised manuscript will include the explicit finite-sum expansion of each affine iterate in terms of homogeneous refinement operators, followed by an inductive construction showing how the corresponding loop controllers are applied sequentially. The induction will demonstrate that width remains fixed (reusing the same controller architecture at each summand) while depth accumulates to O(n^2). revision: yes

Circularity Check

0 steps flagged

No significant circularity; construction is self-contained

full rationale

The paper's central result is a direct mathematical construction proving that homogeneous refinement iterates admit exact fixed-width ReLU networks of depth O(n) via a polygonal loop controller for residual dynamics. This is built from the given operator matrices A_j, the compact support and CPwL assumptions on the input curves, and explicit CPwL readouts for scalar factors and selectors. The loop seam ambiguity is confined and neutralized by support separation, with no reduction to fitted parameters, self-referential definitions, or unverified self-citations. The mention of extending prior scalar binary work is a contextual reference rather than a load-bearing premise; the vector-valued M-ary geometric controller and its depth bounds are derived independently and illustrated with concrete examples. Affine and open-curve cases reduce to the homogeneous compact case by finite expansions, preserving independence from the target result.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The proof rests on standard facts about ReLU realizing CPwL functions and the finite-support homogeneous operator definition; the loop controller is a new constructive device rather than an external axiom.

axioms (2)

standard math ReLU networks realize continuous piecewise linear functions
Invoked implicitly when claiming exact realizations of the refined curves.
domain assumption Homogeneous refinement operators with finite nonzero matrices preserve compact support and continuity of CPwL curves
Stated in the setup for the operator V and the curve class γ.

invented entities (1)

exact loop controller no independent evidence
purpose: Transports the residual orbit by a forward-exact state on a polygonal loop so that scalar factors and digit selectors can be recovered by complementary CPwL readouts
New geometric construction introduced to handle residual dynamics without increasing width.

pith-pipeline@v0.9.0 · 5617 in / 1321 out tokens · 60124 ms · 2026-05-09T17:08:22.238917+00:00 · methodology

discussion (0)

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

Exact ReLU realization of tensor-product refinement iterates
math.CA 2026-05 unverdicted novelty 7.0

Tensor-product dyadic refinement iterates on compactly supported continuous piecewise linear functions in R^2 admit exact ReLU realizations with fixed width and depth O(n).

Reference graph

Works this paper leans on

15 extracted references · 5 canonical work pages · cited by 1 Pith paper

[1]

Daubechies, R

I. Daubechies, R. DeVore, S. Foucart, B. Hanin, and G. Petrova,Nonlinear Approximation and (Deep) ReLU Networks, Constr. Approx.55(2022), no. 1, 127–172

2022
[2]

DeVore, B

R. DeVore, B. Hanin, and G. Petrova,Neural Network Approximation, Acta Numerica30 (2021), 327–444. DOI: 10.1017/S0962492921000052

work page doi:10.1017/s0962492921000052 2021
[3]

Daubechies, R

I. Daubechies, R. DeVore, N. Dym, S. Faigenbaum-Golovin, S. Z. Kovalsky, K.-C. Lin, J. Park, G. Petrova, and B. Sober,Neural Network Approximation of Refinable Functions, IEEE Trans. Inform. Theory69(2023), no. 1, 482–495. DOI: 10.1109/TIT.2022.3199601

work page doi:10.1109/tit.2022.3199601 2023
[4]

N. Dym, B. Sober, and I. Daubechies,Expression of Fractals Through Neural Network Functions, IEEE J. Sel. Areas Inf. Theory1(2020), no. 1, 57–66. DOI: 10.1109/JSAIT.2020.2991422

work page doi:10.1109/jsait.2020.2991422 2020
[5]

J. He, L. Li, J. Xu, and C. Zheng,ReLU deep neural networks and linear finite elements, J. Comput. Math.38(2020), no. 3, 502–527. DOI: 10.4208/jcm.1901-m2018-0160

work page doi:10.4208/jcm.1901-m2018-0160 2020
[6]

Hilbert,Ueber die stetige Abbildung einer Linie auf ein Fl¨ achenst¨ uck, Mathematische Annalen38(1891), 459–460

D. Hilbert,Ueber die stetige Abbildung einer Linie auf ein Fl¨ achenst¨ uck, Mathematische Annalen38(1891), 459–460. 38
[7]

Takagi,A simple example of the continuous function without derivative, Proc

T. Takagi,A simple example of the continuous function without derivative, Proc. Phys.-Math. Soc. Japan1(1903), 176–177

1903
[8]

von Koch,Sur une courbe continue sans tangente, obtenue par une construction g´ eom´ etrique ´ el´ ementaire, Arkiv f¨ or Matematik1(1904), 681–704

H. von Koch,Sur une courbe continue sans tangente, obtenue par une construction g´ eom´ etrique ´ el´ ementaire, Arkiv f¨ or Matematik1(1904), 681–704

1904
[9]

Lebesgue,Le¸ cons sur l’int´ egration et la recherche des fonctions primitives, Gauthier-Villars, Paris, 1904

H. Lebesgue,Le¸ cons sur l’int´ egration et la recherche des fonctions primitives, Gauthier-Villars, Paris, 1904

1904
[10]

Knopp,Ein einfaches Verfahren zur Bildung stetiger nirgends differenzierbarer Funktionen, Math

K. Knopp,Ein einfaches Verfahren zur Bildung stetiger nirgends differenzierbarer Funktionen, Math. Z.2(1918), 1–26

1918
[11]

L´ evy,Les courbes planes ou gauches et les surfaces compos´ ees de parties semblables au tout, Journal de l’ ´Ecole Polytechnique (1938), 227–247, 249–291

P. L´ evy,Les courbes planes ou gauches et les surfaces compos´ ees de parties semblables au tout, Journal de l’ ´Ecole Polytechnique (1938), 227–247, 249–291

1938
[12]

G. M. Morton,A Computer-Oriented Geodetic Data Base and a New Technique in File Sequencing, Technical report, IBM Ltd., Ottawa, Canada, 1966

1966
[13]

Davis and D

C. Davis and D. E. Knuth,Number Representations and Dragon Curves, Journal of Recreational Mathematics3(1970), 66–81, 133–149

1970
[14]

B. B. Mandelbrot,The Fractal Geometry of Nature, W. H. Freeman, New York, 1983

1983
[15]

M. F. Barnsley,Fractal functions and interpolation, Constr. Approx.2(1986), 303–329. DOI: 10.1007/BF01893434 39

work page doi:10.1007/bf01893434 1986