Sufficiency of listed generators for a three-block shallow network with n0=n1=n2=4

Ascertain whether the specified generating set—comprising 48 cubic minors of M_i and M_i−M_j, 120 quartic minors of [M_i | M_j] and [M_i^\top | M_j^\top], 40 quartic minors of [M_1−M_2 | M_2−M_3] and [(M_1−M_2)^\top | (M_2−M_3)^\top], and 2000 quintic minors of certain block matrices—suffices to generate the ideal J^{\mathbf{A}} for the shallow ReLU network with n_0=n_1=n_2=4 and activation patterns A_1=[1,1,0,0], A_2=[1,0,1,0], and A_3=[1,0,0,1].

Background

In the multi-block shallow setting with three specific activation patterns and equal layer widths, the authors enumerate large families of determinantal generators obtained from individual blocks, differences, concatenations, and structured block matrices.

They report that computational checks did not resolve whether these generators are sufficient—pointing to a concrete unresolved case where algorithmic implicitization proved intractable.