In this talk, we will discuss some advancements in this area. We will discuss an upper bound result which states that the bounded top-fanin border depth-3 circuits (Σ^[k]ΠΣ-bar for constant k) can be computed by a polynomial-size algebraic branching program (ABP). We will also discuss a strong exponential separation between any two consecutive border classes, Σ^[k]ΠΣ-bar and Σ^[k+1]ΠΣ-bar, establishing an optimal hierarchy of constant top-fanin border depth-3 circuits.
This is based on two works -- (i) the upper bound result is a joint work with Prateek Dwivedi and Nitin Saxena, FOCS'21 and (ii) the lower bound result is a joint work with Nitin Saxena, FOCS'22.