# Exact ReLU realization of affine one-dimensional refinement iterates via residual memory and offset frames

arXiv: https://arxiv.org/abs/2607.20586 · HTML: https://mlsanity.com/paper/2607.20586/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 3 of its 10 references that are in the map.
- First posted on arXiv 22 Jul 2026.

## Builds on (most cited first, 3 of 3)

- [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)
- [Exact Loop Controllers for ReLU Realization of Homogeneous Curve Refinements](https://mlsanity.com/paper/2605.01655/) (2026)

## Built on by (most cited first, 1 of 1)

- [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 vector-valued affine refinement operators of the form [ (Wγ)(t)=sum_j∈mathbbZ Ajγ(Mt-j)+B(t), ] with finitely supported matrix mask and compactly supported continuous piecewise linear input and forcing data. Building on the homogeneous realization theorem for (Bequiv 0), we prove that, for (M≥ 3), every finite affine iterate (Wnγ) admits an exact fixed-width ReLU realization whose depth is (O(n)). The main new ingredient is a residual memory controller. It replaces the noninvertible residual dynamics by an injective skew-product and permits exact backward replay of the residual states required by a Horner-type evaluation of the affine forcing sum. Offset frames align the forcing atoms away from residual seams, allowing complementary loop readouts to recover their values exactly. The remaining branch-selection ambiguity occurs only where the accumulated affine state has already vanished. For (M≥ 3), the result applies to arbitrary compactly supported continuous piecewise linear forcing terms. For (M=2), the same construction applies to ordinary-frame seam-separated forcing. We also prove a stage-dependent extension for forcing terms in a fixed finite-dimensional continuous piecewise linear span and record the resulting linear-depth upgrade for open-curve, finite-state, and Hilbert- and Morton-type recursive constructions.