Exact ReLU realization of tensor-product refinement iterates

TL;DR

This paper demonstrates that certain two-dimensional dyadic refinement operators can be exactly realized using ReLU neural networks with fixed width and depth proportional to the iteration count, extending the theory to higher dimensions.

Contribution

It introduces a method to exactly realize tensor-product refinement iterates in two dimensions using ReLU networks, expanding the existing one-dimensional theory.

Findings

Exact ReLU realizations of 2D refinement iterates with fixed width and depth O(n)

Extension of the one-dimensional exact realization theory to two dimensions

Reduction of the multivariate case to a finite decomposition and gluing process.

Abstract

We study scalar dyadic refinement operators on R^2 of the form (Vf)(x,y) = sum_{(j,k) in Z^2} c_{j,k} f(2x-j, 2y-k), where only finitely many mask coefficients c_{j,k} are nonzero. Under a fixed support-window hypothesis, we prove that for every compactly supported continuous piecewise linear seed g:R^2->R, the iterates V^n g admit exact ReLU realizations of fixed width and depth O(n). This gives a first genuinely two-dimensional extension of the exact realization theory for refinement cascades. Using the one-dimensional exact loop-controller framework, the proof transports the tensor-product residual dynamics exactly on the product of two polygonal loops and reduces the remaining seam ambiguity to a final readout and selector step. The matrix cascade is then handled by a fixed-depth recursive block, and general compactly supported continuous piecewise linear seeds are reduced to a finite decomposition together with exact clamped gluing on the support window. This identifies the tensor-product dyadic case as a natural first multivariate instance of the loop-controller method for refinement iterates.