New proof shows ReLU networks exactly simulate binary refinement iterates

A new paper from Tsogtgerel Gantumur, posted on arXiv (2608.02624), tackles a foundational question at the intersection of numerical analysis and deep learning: can refinement operators — the iterative processes behind wavelets, subdivision curves, and geometric modeling — be exactly computed by ReLU neural networks? Earlier work showed that for scalar binary refinement with a finitely supported mask, every compactly supported continuous piecewise linear seed produces finite iterates that admit exact ReLU realizations. The network width stays fixed while depth grows only linearly with the number of refinement steps. The challenge: refinement cascades are driven by discontinuous binary digit choices, yet ReLU networks are continuous piecewise linear functions.

Gantumur's new construction elegantly solves this by representing the residual dynamics on a polygonal model of the circle. Instead of forcing a single coordinate system to handle discontinuities, he introduces two overlapping charts — one ordinary, one shifted by half — whose discontinuities occur at different points. The network switches between them only where both are valid and the underlying linear cascade updates agree, making the switch exact without any variable selector multiplication. This also gives exact readout of every continuous piecewise linear circle function satisfying the natural endpoint compatibility condition. Localized seeds are processed with a two-pass network, while translation covariance, finite decomposition, and gluing extend the result to arbitrary compactly supported continuous piecewise linear seeds within a preserved support window. The 30-page proof offers a cleaner, more direct construction than previous approaches, potentially informing how to design ReLU networks for subdivision-based geometry tasks.