Expected dimension for non-increasing width profiles

Establish that for any non-increasing width vector d=(d0,d1,…,dL) with output width dL>1 and any activation degree r∈N, the neurovariety V_{d,r} attains its expected dimension.

Background

Beyond the asymptotic conjecture for large activation degree, the authors propose a structural conjecture that non-increasing width profiles (with dL>1) should be non-defective for every activation degree. They report empirical verification for small architectures using a backpropagation-based Jacobian rank routine, but no general proof is known.

This conjecture, if true, would give a broad family of deep architectures whose expressivity (as measured by dimension) matches the expected count derived from parameter symmetries.