Sufficiency of determinantal generators for two-block shallow ReLU pattern varieties

Prove that, for a shallow ReLU network with two activation pattern blocks, the ideal J^{\mathbf{A}} of the pattern variety is generated by the following families of polynomials: (1) the (r_1+1)-minors of M_1, (2) the (r_2+1)-minors of M_2, (3) the (n_1+1)-minors of [M_1 | M_2] and [M_1^\top | M_2^\top], and (4) the (t+1)-minors of M_1 - M_2, where r_1 and r_2 are the generic ranks of the block matrices, n_1 is the hidden-layer width, and t = r_1 + r_2 - 2s with s the number of shared active neurons.

Background

In Section 6, the authors derive several classes of determinantal invariants for shallow networks partitioned into two activation blocks. These include rank constraints on individual block matrices, on concatenations, and on their differences.

They conjecture that these identified invariants suffice to generate the full ideal of the pattern variety, which would completely characterize the algebraic relations among outputs across the two blocks.