Lower Bounds for Depth Three Circuits with small bottom fanin Neeraj Kayal Chandan Saha Indian Institute of Science

A lower bound

Remark:  Bad news.  Good news.

Background/Motivation

Arithmetic Circuits …

… … Arithmetic Circuits

… … … Arithmetic Circuits

… … … Arithmetic Circuits

… … … Arithmetic Circuits

… … …

… … … Size = Number of Edges

… … … Depth

… … …

Two Fundamental Questions Can explicit polynomials be efficiently computed? Can computation be efficiently parallelized?

Two Fundamental Questions

Can computation be efficiently parallelized?

Question: Is this optimal?

Can computation be efficiently parallelized? Question: Is this optimal?

Can computation be efficiently parallelized?

A possible way to approach VP vs VNP

Lower Bound in VNPGKKS13+KSS 14 IMMFLMS14 in VNPKLSS14 IMMKS14 IMMThis work in VNPThis work IMMNext talk

A possible way to approach VP vs VNP

A common Proof Strategy and some technical ingredients

Proof Strategy shallow circuit C

Proof Strategy shallow circuit C

Lower Bounding rank of large matrices If a matrix M(f) has a large upper triangular submatrix, then it has large rank (Alon): If the columns of M(f) are almost orthogonal then M(f) has large rank.

shallow circuit C

Finding a geometric property GP of T V(T) is a union of low-degree hypersurfaces V(T) has lots of high-order singularities

Finding a geometric property GP of T

V(T) is a union of low-degree hypersurfaces V(T) has lots of high-order singularities

shallow circuit C

Expressing largeness of a variety in terms of rank

shallow circuit C

Restrictions

Employing restrictions  Yields lower bounds for homogeneous depth four (KLSS14 and KS14).

Employing Restrictions

 Yields lower bounds for homogeneous depth five with low bottom fanin (KS15 and BC15).

A lemma by Shpilka and Wigderson  Yields lower bounds mentioned earlier.

Conclusion