equizero_neurips23_format

Proof of Thm. 2. We want to show M G(hx)= hM G(x) for all x 2X and h 2 G. From the definition of M G in equation 4, we have M G(hx)= 1P Similar to Yarotsky (2022), we first define Ksym = S g2G gK. Note that Ksym is also a compact set and Ksym X . We want to show that M G,equi(gx)= gM G,equi(x). Hence, ( h(gx) 1gx) is invariant to actions of G. The proof for invariance of M G,inv(x) follows similarly. In addition to properties discussed in section 3.3, here we show that equizero models have autoregressive and invertibility properties. These properties have not been used in the main paper, but we believe they could be of use for future work in this area.