Demystifying the border of depth-3 algebraic circuits


Pranjal Dutta November 05, 2021.


Border complexity of polynomials plays an integral role in GCT (Geometric complexity theory) to approach P != NP. It tries to formalize the notion of ‘approximating a polynomial’ via limits (Burgisser FOCS’01). This raises the open question whether VP ?= VP; as the approximation involves exponential precision which may not be efficiently simulable. Recently (Kumar ToCT’20) proved the universal power of the border of top-fanin-2 depth-3 circuits (border-Σ^[2]ΠΣ). In this talk, we will show that the border of bounded top-fanin depth-3 circuits (border-Σ^[k]ΠΣ for constant k) is relatively easy– it can be computed by a polynomial size algebraic branching program (ABP).

Moreover, we will show a quasipolynomial-time blackbox identity test for the same. Prior best was in PSPACE (Forbes,Shpilka STOC’18). Our de-bordering paradigm is a multi-step process; in short we call it DiDIL – divide, derive, induct, with limit. It ‘almost’ reduces border-Σ^[k]ΠΣ to special cases of read-once oblivious algebraic branching programs (ROABPs) in any-order.

This is based on a joint work with Prateek Dwivedi (IIT Kanpur) and Nitin Saxena (IIT Kanpur), which got accepted to FOCS 2021. A preprint is also available.

Enjoy Reading This Article?

Here are some more articles you might like to read next:

  • Deterministic identity testing paradigms for bounded top-fanin depth-4 circuits
  • Are quantum speedups for learning expressive classes possible?
  • On Probabilistic Approximations of Boolean Functions via Polynomials, and Polynomial Closure Statements over the Boolean Cube
  • Interior Point Methods for Nearly Linear Time Algorithms
  • Derandomization from Time-Space Tradeoffs