Variable-depth width–radius approximation law

Establish whether the Dyadic–Triangular Activation satisfies the conjectured variable-depth approximation law $\mathcal E_\DTA(N,L,T)\asymp[N^2L^2\log(eNT)]^{-\beta/d}$ uniformly above fixed width and depth thresholds for $T\ge1$, and consequently whether the regression tradeoff extends to $N_M^2L_M^2\log(eN_MT_M)\asymp M^{d/(2\beta+d)}$.

Background

The paper proves a sharp width–parameter-radius approximation law at fixed hidden depth. Its conclusion proposes extending this result to architectures in which depth also varies. The proposed rate incorporates the additional depth resource through the factor $L^2$, and a matching construction would yield a corresponding two-dimensional width–depth–radius statistical tradeoff. The authors explicitly present this extension as a conjecture rather than a result established in the paper.