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

2. A lower bound

4. Remark:  Bad news.  Good news.

5. Background/Motivation

6. Arithmetic Circuits …

7. … … Arithmetic Circuits

8. … … … Arithmetic Circuits

9. … … … Arithmetic Circuits

10. … … … Arithmetic Circuits

11. … … …

12. … … … Size = Number of Edges

13. … … … Depth

14. … … …

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

16. Two Fundamental Questions

17. Can computation be efficiently parallelized?

20. Question: Is this optimal?

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

22. Can computation be efficiently parallelized?

23. A possible way to approach VP vs VNP

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

26. A possible way to approach VP vs VNP

27. A common Proof Strategy and some technical ingredients

28. Proof Strategy shallow circuit C

29. Proof Strategy shallow circuit C

30. 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.

31. shallow circuit C

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

33. Finding a geometric property GP of T

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

35. shallow circuit C

37. Expressing largeness of a variety in terms of rank

40. shallow circuit C

41. Restrictions

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

43. Employing Restrictions

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

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

46. Conclusion