Exact depth required for continuous piecewise-linear representations

Determine whether there exists a continuous and piecewise-linear function on R^d that requires ceil log_2(d+1) ceil hidden layers in a ReLU network for exact representation, thereby establishing whether the known logarithmic upper bound is tight.

Background

The paper discusses the exact representation of continuous piecewise-linear (CPWL) functions by ReLU networks. Every CPWL function on R^d is known to admit a representation with ceil log_2(d+1) ceil hidden layers, using a reduction to maxima of affine functions and pairwise maximization in a binary-tree structure.

The unresolved issue is whether this upper bound is ever necessary. The paper notes that no lower bound exceeding two hidden layers had previously been proved for any CPWL function in the unrestricted setting. The authors study networks conforming to the braid arrangement and obtain a conditional doubly-logarithmic lower bound, but they do not resolve the general exact-depth question.