# Exact Loop Controllers for ReLU Realization of Homogeneous Curve Refinements

arXiv: https://arxiv.org/abs/2605.01655 · HTML: https://mlsanity.com/paper/2605.01655/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 2 of its 20 references that are in the map.
- First posted on arXiv 3 May 2026.

## Builds on (most cited first, 2 of 2)

- [Neural Network Approximation of Refinable Functions](https://mlsanity.com/paper/2107.13191/) (2021)
- [Expression of Fractals Through Neural Network Functions](https://mlsanity.com/paper/1905.11345/) (2019)

## Built on by (most cited first, 4 of 4)

- [Exact ReLU realization of affine one-dimensional refinement iterates via residual memory and offset frames](https://mlsanity.com/paper/2607.20586/) (2026)
- [Exact ReLU realization of tensor-product refinement iterates](https://mlsanity.com/paper/2605.03917/) (2026)
- [Neural network realization of binary refinement iterates via a two-chart atlas selector](https://mlsanity.com/paper/2608.02624/) (2026)
- [Exact ReLU realization of binary affine refinement iterates via reflection folding and cone switching](https://mlsanity.com/paper/2609.11962/) (2026)

## Abstract (arXiv)

We study homogeneous refinement operators ((Vγ)(t)=sumj∈mathbb ZAjγ(Mt-j)), acting on compactly supported continuous piecewise linear curves (γ:mathbb R→mathbb Rp), where (M≥2) and only finitely many matrices (Aj∈mathbb Rp× p) are nonzero. We prove that the iterates (Vnγ) 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(n2)), 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.