Separated borders: Exponential-gap fanin-hierarchy theorem for approximative depth-3 circuits

Mulmuley and Sohoni (2001) proposed an ambitious program, the Geometric Complexity Theory (GCT), to prove $P\neq NP$ and related conjectures using algebraic geometry and representation theory. Gradually, GCT has introduced new structures and questions in complexity. GCT tries to capture the algebraic/geometric notion of ’approximation’ by defining border classes. Surprisingly, (Kumar ToCT’20) proved the universal power of the border of top-fanin- 2 depth-3 circuits $(\overline{\Sigma^{[2]}\Pi\Sigma})$; which is in complete contrast to its classical model. Recently, (Dutta,Dwivedi,Saxena, FOCS’21) put an upper bound, by showing that bounded-top-fanin border depth-3 circuits $(\overline{\Sigma^{[k]}\Pi\Sigma}$ for constant $k)$ can be computed by a polynomial-size algebraic branching program (ABP). It was left open to show an exponential separation between the class of ABPs and $\overline{\Sigma^{[k]}\Pi\Sigma}$. In this article, we show a strongly-exponential separation between any two consecutive border classes, $\overline{\Sigma^{[k]}\Pi\Sigma}$ and $\Sigma^{[k+1]}\Pi\Sigma$, establishing an optimal hierarchy of constant topfanin border depth- 3 circuits. Put in GCT language: we prove an exponential-hierarchy for padded- k-th-secant-varieties of the Chow variety of $\mathbb{F}^{n+1} $. This positively answers [Open question 2 of Dutta,Dwivedi,Saxena FOCS’21] and [Problem 8.10 with constant r, of Landsberg, Annal.Ferrara’15]. Full version: https://www.cse.iitk.ac.in/users/nitin/papers/exphierarchy.pdf