1. Lower Bounds using shifted partials Chandan Saha Indian Institute of Science Workshop on Algebraic Complexity Theory 2016 Tel-Aviv University

2. Background

3. Arithmetic Circuit + xxxx ++++ xxxx …. ….. x1x1 x2x2 x n-1 xnxn f(x 1, x 2, …, x n ) x g h gh + g h g+h Product gate Sum gate There are `field constants’ on the wires

4. Arithmetic Circuit + xxxx ++++ xxxx …. ….. x1x1 x2x2 x n-1 xnxn f(x 1, x 2, …, x n ) Depth

5. Arithmetic Circuit + xxxx ++++ xxxx …. ….. x1x1 x2x2 x n-1 xnxn f(x 1, x 2, …, x n ) Size = no. of wires

6. Arithmetic Circuit + xxxx ++++ xxxx …. ….. x1x1 x2x2 x n-1 xnxn f(x 1, x 2, …, x n ) Depth = 4

7. Arithmetic Circuit + xxxx ++++ xxxx …. ….. x1x1 x2x2 x n-1 xnxn f(x 1, x 2, …, x n ) f = ∑ ∏ Q ij i j

8. Arithmetic Circuit + xxxx ++++ xxxx …. ….. x1x1 x2x2 x n-1 xnxn f(x 1, x 2, …, x n ) f = ∑ ∏ Q ij i j sum of monomials

9. Arithmetic Circuit + xxxx ++++ xxxx …. ….. x1x1 x2x2 x n-1 xnxn f(x 1, x 2, …, x n ) f = ∑ ∏ Q ij i j Top fan-in = s number of summands = s

10. Arithmetic Circuit + xxxx ++++ xxxx …. ….. x1x1 x2x2 x n-1 xnxn f(x 1, x 2, …, x n ) f = ∑ ∏ Q ij i j Bottom fan-in ≤ t degree ≤ t

11. Arithmetic Circuit + xxxx ++++ xxxx …. ….. x1x1 x2x2 x n-1 xnxn f(x 1, x 2, …, x n ) f = ∑ ∏ Q ij i j This talk: Lower bounds for restricted depth four circuits

12. Arithmetic Circuit + xxxx ++++ xxxx …. ….. x1x1 x2x2 x n-1 xnxn f(x 1, x 2, …, x n ) f = ∑ ∏ Q ij i j Notation: n = number of variables d = degree of f

13. Recap: A template for proving lower bound Step 1: Define a suitable measure function μ μ : F [x 1, …, x n ] R such that μ is able to exploit some ‘weakness’ of the circuit. Step 2: Show an upper bound for μ (Circuit) in terms of size of the circuit.

14. Recap: A template for proving lower bound Step 1: Define a suitable measure function μ μ : F [x 1, …, x n ] R such that μ is able to exploit some ‘weakness’ of the circuit. Step 2: Show an upper bound for μ (Circuit) in terms of size of the circuit. Step 3: Find a ‘hard’ polynomial f and lower bound μ (f).

15. Recap: A template for proving lower bound Step 1: Define a suitable measure function μ μ : F [x 1, …, x n ] R such that μ is able to exploit some ‘weakness’ of the circuit. Step 2: Show an upper bound for μ (Circuit) in terms of size of the circuit. Step 3: Find a ‘hard’ polynomial f and lower bound μ (f). Step 4: Set the parameters correctly.

16. Recap: Space of partial derivatives Notation: ∂ k f = Set of all k-th order derivatives of f This talk: multilinear derivatives, i.e. we do not derive with respect to the same variable more than once.

17. Recap: Space of partial derivatives Notation: ∂ k f = Set of all k-th order derivatives of f 〈S〉 = The vector space spanned by F -linear combinations of polynomials in S

18. Recap: Space of partial derivatives Notation: ∂ k f = Set of all k-th order derivatives of f 〈S〉 = The vector space spanned by F -linear combinations of polynomials in S Definition: PD k (f) = dim 〈∂ k f〉 Property: (Sub-additive) PD k (f 1 + f 2 ) ≤ PD k (f 1 ) + PD k (f 2 )

19. Recap: Diagonal depth three circuits C = ℓ 1 + … + ℓ s Upper bound for the circuit : PD k (C) ≤ s e 1 eses

20. Recap: Diagonal depth three circuits C = ℓ 1 + … + ℓ s Upper bound for the circuit : PD k (C) ≤ s Lower bound for the ‘hard’ polynomial: f = x 1 x 2 x 3 ···x n PD k (f) = ( ) e 1 eses n k

21. Recap: Diagonal depth three circuits C = ℓ 1 + … + ℓ s Upper bound for the circuit : PD k (C) ≤ s Lower bound for the ‘hard’ polynomial: f = x 1 x 2 x 3 ···x n PD k (f) = ( ) Setting parameter: Choose k = n/2 Top fan-in lower bound: s = 2 Ω(n) e 1 eses n k

22. Extending the circuit model: A motivating example

23. Sum of powers of quadratics C = Q 1 + … + Q s e 1 e s Quadratic polynomials

24. Sum of powers of quadratics C = Q 1 + … + Q s A simple case: Q = x 1 x 2 + x 3 x 4 + + x n-1 x n + 1 Notation: y i = x i+1 if i is odd = x i –1 if i is even e 1 e s

25. Sum of powers of quadratics C = Q 1 + … + Q s A simple case: Q = x 1 x 2 + x 3 x 4 + + x n-1 x n + 1 Derivatives of a term: T = Q e suppose e ≥ n e 1 e s

26. Sum of powers of quadratics C = Q 1 + … + Q s A simple case: Q = x 1 x 2 + x 3 x 4 + + x n-1 x n + 1 Derivatives of a term: T = Q e e 1 e s ∂ x T = e. y. Q e-1 i1i1 i1i1

27. Sum of powers of quadratics C = Q 1 + … + Q s A simple case: Q = x 1 x 2 + x 3 x 4 + + x n-1 x n + 1 Derivatives of a term: T = Q e e 1 e s ∂ x T = e. y. Q e-1 i1i1 i1i1 ∂ x x T = e(e-1). y y. Q e-2 i2i2 i1i1 2 i1i1 i2i2

28. Sum of powers of quadratics C = Q 1 + … + Q s A simple case: Q = x 1 x 2 + x 3 x 4 + + x n-1 x n + 1 Derivatives of a term: T = Q e e 1 e s ∂ x T = e. y. Q e-1 i1i1 i1i1 ∂ x x T = e(e-1). y y. Q e-2 i2i2 i1i1 2 i1i1 i2i2 ∂ x x … x T = e(e-1)···. y y ··· y. Q e-k i2i2 i1i1 ikik i1i1 i 2 k ikik......

29. Sum of powers of quadratics C = Q 1 + … + Q s A simple case: Q = x 1 x 2 + x 3 x 4 + + x n-1 x n + 1 Derivatives of a term: T = Q e e 1 e s ∂ x T = e. y. Q e-1 i1i1 i1i1 ∂ x x T = e(e-1). y y. Q e-2 i2i2 i1i1 2 i1i1 i2i2 ∂ x x … x T = e(e-1)···. y y ··· y. Q e-k i2i2 i1i1 ikik i1i1 i 2 k ikik...... Uniquely determines the derivative

30. Sum of powers of quadratics C = Q 1 + … + Q s A simple case: Q = x 1 x 2 + x 3 x 4 + + x n-1 x n + 1 Observation: The k -th order derivatives of T = Q e are F -linearly independent. e 1 e s

31. Sum of powers of quadratics C = Q 1 + … + Q s A simple case: Q = x 1 x 2 + x 3 x 4 + + x n-1 x n + 1 Observation: PD k (T) = ( ) e 1 e s n k

32. Sum of powers of quadratics C = Q 1 + … + Q s A simple case: Q = x 1 x 2 + x 3 x 4 + + x n-1 x n + 1 Observation: PD k (T) = ( ) Observation: For any f, PD k (f) ≤ ( ) e 1 e s n k n k Note: In this setting, PD k is unable to distinguish between a power of a quadratic and any ‘hard’ polynomial !

33. Sum of powers of quadratics C = Q 1 + … + Q s A simple case: Q = x 1 x 2 + x 3 x 4 + + x n-1 x n + 1 Observation: PD k (T) = ( ) Observation: For any f, PD k (f) ≤ ( ) e 1 e s n k n k … seems like we need something new! [Kayal’12]

34. Shifted partials: An augmentation of partial derivatives

35. Shifted partial derivatives ∂ k f = Set of all k-th order derivatives of f Notation: 〈S〉 = The vector space spanned by F -linear combinations of polynomials in S x ≤ℓ = Set of monomials of degree ≤ ℓ

36. Shifted partial derivatives ∂ k f = Set of all k-th order derivatives of f Notation: 〈S〉 = The vector space spanned by F -linear combinations of polynomials in S Definition: (Kayal’12) SPD k, ℓ (f) = dim 〈x ≤ℓ · ∂ k f〉 x ≤ℓ = Set of monomials of degree ≤ ℓ Set of polynomials formed by multiplying a monomial in x ≤ℓ with a polynomial in ∂ k f.

37. Shifted partial derivatives ∂ k f = Set of all k-th order derivatives of f Notation: 〈S〉 = The vector space spanned by F -linear combinations of polynomials in S Definition: (Kayal’12) SPD k, ℓ (f) = dim 〈x ≤ℓ · ∂ k f〉 Property: (Sub-additive) SPD k, ℓ (f 1 + f 2 ) ≤ SPD k, ℓ (f 1 ) + SPD k, ℓ (f 2 ) x ≤ℓ = Set of monomials of degree ≤ ℓ

38. How large can SPD k, ℓ be? Definition: SPD k, ℓ (f) = dim 〈x ≤ℓ · ∂ k f〉 Observation: SPD k, ℓ (f) ≤ min ( )( ), ( ) n k n + ℓ n n + ℓ + d - k n Proof : The set x ≤ℓ · ∂ k f has at most | x ≤ℓ | · |∂ k f| polynomials each of degree at most ℓ + d – k.

39. How large can SPD k, ℓ be? Definition: SPD k, ℓ (f) = dim 〈x ≤ℓ · ∂ k f〉 Observation: SPD k, ℓ (f) ≤ min ( )( ), ( ) n k n + ℓ n n + ℓ + d - k n Question: Are there explicit polynomials that achieve this or close to this bound? Yes, we will see shortly…

40. How large can SPD k, ℓ be? Definition: SPD k, ℓ (f) = dim 〈x ≤ℓ · ∂ k f〉 Observation: SPD k, ℓ (f) ≤ min ( )( ), ( ) n k n + ℓ n n + ℓ + d - k n Question: Are there explicit polynomials that achieve this or close to this bound? For now, suppose f has SPD k, ℓ as high as possible, and let’s get back to the circuit.

41. Applying shifted partials to the ‘motivating’ circuit model

42. Sum of powers of low degree polynomials C = Q 1 + … + Q s e 1 e s polynomials with degree ≤ t

43. Sum of powers of low degree polynomials C = Q 1 + … + Q s Observation: SPD k, ℓ (C) ≤ ∑ SPD k, ℓ (Q i ) Proof: By subadditivity of the measure SPD k, ℓ. e 1 e s e i i

44. Sum of powers of low degree polynomials C = Q 1 + … + Q s Observation: SPD k, ℓ (C) ≤ ∑ SPD k, ℓ (Q i ) Focus on a term: T = Q e e 1 e s e i i

45. Sum of powers of low degree polynomials C = Q 1 + … + Q s Observation: SPD k, ℓ (C) ≤ ∑ SPD k, ℓ (Q i ) Focus on a term: T = Q e Observation: ∂ x x … x T = “ polynomial of degree ≤ kt”. Q e-k Proof: Chain rule of derivatives. e 1 e s e i i i2i2 i1i1 ikik k

46. Sum of powers of low degree polynomials C = Q 1 + … + Q s Observation: SPD k, ℓ (C) ≤ ∑ SPD k, ℓ (Q i ) Focus on a term: T = Q e Observation: Let m be a monomial in x ≤ℓ. Then m. ∂ x x … x T = “ polynomial of degree ≤ ℓ + kt”. Q e-k e 1 e s e i i i2i2 i1i1 ikik k

47. Sum of powers of low degree polynomials C = Q 1 + … + Q s Observation: SPD k, ℓ (C) ≤ ∑ SPD k, ℓ (Q i ) Lemma: SPD k, ℓ (T) ≤ ( ) e 1 e s e i i n + ℓ + kt n

48. Sum of powers of low degree polynomials C = Q 1 + … + Q s Observation: SPD k, ℓ (C) ≤ ∑ SPD k, ℓ (Q i ) Lemma: SPD k, ℓ (T) ≤ ( ) Upper bound for the circuit: SPD k, ℓ (C) ≤ s · ( ) e 1 e s e i i n + ℓ + kt n n

49. Lower bound on the top fan-in C = Q 1 + … + Q s = f Upper bound for the circuit: SPD k, ℓ (C) ≤ s · ( ) Lower bound for the ‘hard’ polynomial: e 1 e s n + ℓ + kt n SPD k, ℓ (f) = min ( )( ), ( ) n k n + ℓ n n + ℓ + d - k n assume there’s such an f

50. Lower bound on the top fan-in C = Q 1 + … + Q s = f Upper bound for the circuit: SPD k, ℓ (C) ≤ s · ( ) Lower bound for the `hard’ polynomial: e 1 e s n + ℓ + kt n SPD k, ℓ (f) = min ( )( ), ( ) n k n + ℓ n n + ℓ + d - k n Together: s ≥ min ( )( ), ( ) n k n + ℓ n n + ℓ + d - k n ( ) n + ℓ + kt n

51. Lower bound on the top fan-in C = Q 1 + … + Q s = f Upper bound for the circuit: SPD k, ℓ (C) ≤ s · ( ) Lower bound for the `hard’ polynomial: e 1 e s n + ℓ + kt n SPD k, ℓ (f) = min ( )( ), ( ) n k n + ℓ n n + ℓ + d - k n Together: s ≥ min ( )( ), ( ) n k n + ℓ n n + ℓ + d - k n ( ) n + ℓ + kt n Choose the parameters k and ℓ to maximize the ratio

52. Setting the parameters k and ℓ Let R 1 = and R 2 = ( )( ) n k n + ℓ n ( ) n + ℓ + kt n ( ) n + ℓ + kt n ( ) n + ℓ + d - k n Setting k : Observe that kt ≤ d-k for R 2 ≥ 1

53. Setting the parameters k and ℓ Let R 1 = and R 2 = ( )( ) n k n + ℓ n ( ) n + ℓ + kt n ( ) n + ℓ + kt n ( ) n + ℓ + d - k n Setting k : Observe that kt ≤ d-k for R 2 ≥ 1 k = δ · (d/t) ( δ ≤ 1 is a constant )

54. Setting the parameters k and ℓ Let R 1 = and R 2 = ( )( ) n k n + ℓ n ( ) n + ℓ + kt n ( ) n + ℓ + kt n ( ) n + ℓ + d - k n Setting k : Observe that kt ≤ d-k for R 2 ≥ 1 k = δ · (d/t) Setting ℓ : ℓ R2R2 R1R1 ℓ * ≈ nd log ( ) n k

55. Setting the parameters k and ℓ Let R 1 = and R 2 = ( )( ) n k n + ℓ n ( ) n + ℓ + kt n ( ) n + ℓ + kt n ( ) n + ℓ + d - k n Setting k : Observe that kt ≤ d-k for R 2 ≥ 1 k = δ · (d/t) Setting ℓ : ℓ = ℓ * “Best” possible lower bound for top fan-in: s ≥ ( ) = ( ) n k 1-δ nt d Ω (d/t)

56. Setting the parameters k and ℓ Let R 1 = and R 2 = ( )( ) n k n + ℓ n ( ) n + ℓ + kt n ( ) n + ℓ + kt n ( ) n + ℓ + d - k n Setting k : Observe that kt ≤ d-k for R 2 ≥ 1 k = δ · (d/t) Setting ℓ : ℓ = ℓ * “Best” possible lower bound for top fan-in: s ≥ ( ) = ( ) n k 1-δ nt d Ω (d/t) Is the upper bound on SPD of the circuit optimum?

57. Setting the parameters k and ℓ Let R 1 = and R 2 = ( )( ) n k n + ℓ n ( ) n + ℓ + kt n ( ) n + ℓ + kt n ( ) n + ℓ + d - k n Setting k : Observe that kt ≤ d-k for R 2 ≥ 1 k = δ · (d/t) Setting ℓ : ℓ = ℓ * “Best” possible lower bound for top fan-in: s ≥ ( ) = ( ) n k 1-δ nt d Ω (d/t) ‘Yes’ [Fournier, Limaye, Malod, Srinivasan 14]

58. It remains to find polynomials with high SPD

59. It remains to find polynomials with high SPD … but let’s make the circuit model stronger first.

60. Shifted partials: Touching the threshold

61. Sum of products of low degree polynomials C = Q 11 Q 12 ··· Q 1m + … + Q s1 Q s2 ··· Q sm Q ij ’s have degree ≤ t

62. Sum of products of low degree polynomials C = Q 11 Q 12 ··· Q 1m + … + Q s1 Q s2 ··· Q sm C is homogeneous; implying m = O(d/t)

63. Sum of products of low degree polynomials C = Q 11 Q 12 ··· Q 1m + … + Q s1 Q s2 ··· Q sm Notation: h -∑∏∑∏ [t] circuits

64. Sum of products of low degree polynomials C = Q 11 Q 12 ··· Q 1m + … + Q s1 Q s2 ··· Q sm Fischers (1994) C’ = Q 1 + … + Q s·2 m e 1 e s’

65. Sum of products of low degree polynomials C = Q 11 Q 12 ··· Q 1m + … + Q s1 Q s2 ··· Q sm Fischers (1994) C’ = Q 1 + … + Q s·2 m e 1 e s’ Q i ’s have degree ≤ t

66. Sum of products of low degree polynomials C = Q 11 Q 12 ··· Q 1m + … + Q s1 Q s2 ··· Q sm Fischers (1994) C’ = Q 1 + … + Q s·2 m e 1 e s’ … a blow-up by a factor of 2 in the top fan-in where m = O(d/t) m Q i ’s have degree ≤ t

67. Sum of products of low degree polynomials C = Q 11 Q 12 ··· Q 1m + … + Q s1 Q s2 ··· Q sm Fischers (1994) C’ = Q 1 + … + Q s·2 m e 1 e s’ Observation: A lower bound of ( ) on the top fan-in of C’ implies the same lower bound on the top fan-in of C. Ω (d/t) nt d

68. Sum of products of low degree polynomials C = Q 11 Q 12 ··· Q 1m + … + Q s1 Q s2 ··· Q sm Fischers (1994) C’ = Q 1 + … + Q s·2 m e 1 e s’ Recall: An lower bound of n on the top fan-in of h -∑∏∑∏ [t] circuits (for t ≤ √d ) computing a polynomial f in VNP implies VP ≠ VNP. ω(d/t)

69. Sum of products of low degree polynomials C = Q 11 Q 12 ··· Q 1m + … + Q s1 Q s2 ··· Q sm Fischers (1994) C’ = Q 1 + … + Q s·2 m e 1 e s’ Recall: An lower bound of n on the top fan-in of h -∑∏∑∏ [t] circuits (for t ≤ √d ) computing a polynomial f in VNP implies VP ≠ VNP. ω(d/t) threshold

70. Sum of products of low degree polynomials C = Q 11 Q 12 ··· Q 1m + … + Q s1 Q s2 ··· Q sm Fischers (1994) C’ = Q 1 + … + Q s·2 m e 1 e s’ Observation: Compare ( ) with n. Ω (d/t) nt d ω(d/t) Any asymptotic improvement in the exponent implies VP ≠ VNP.

71. Sum of products of low degree polynomials C = Q 11 Q 12 ··· Q 1m + … + Q s1 Q s2 ··· Q sm Fischers (1994) C’ = Q 1 + … + Q s·2 m e 1 e s’ Why does shifted partials work for this model?

72. Geometric insight

73. Ideals and varieties Intuition: Let g 1, …, g m ϵ C [x 1,…, x n ] I = ideal generated by g 1, …, g m V = set of common zeroes of g 1, …, g m ‘larger’ the variety V ‘smaller’ the ideal I

74. Ideals and varieties Intuition: Let g 1, …, g m ϵ C [x 1,…, x n ] I = ideal generated by g 1, …, g m V = V (I) ‘larger’ the variety V ‘smaller’ the ideal I largeness measured in terms of dimension of a variety

75. Ideals and varieties Intuition: Let g 1, …, g m be the k -th derivatives of T = Q e I = ideal generated by g 1, …, g m V = V (I)  Every polynomial in I is divisible by Q e-k  So, V (Q) ⊆ V

76. Ideals and varieties Intuition: Let g 1, …, g m be the k -th derivatives of T = Q e I = ideal generated by g 1, …, g m V = V (I)  Every polynomial in I is divisible by Q e-k  So, V (Q) ⊆ V dim(V) is `large’  So, we expect I to be ‘small’

77. Ideals and varieties Intuition: Let g 1, …, g m be the k -th derivatives of T = Q e I = ideal generated by g 1, …, g m V = V (I)  Every polynomial in I is divisible by Q e-k  So, V (Q) ⊆ V dim(V) is `large’  So, we expect I to be ‘small’ We would like to capture ‘smallness’ of an ideal by a measurable quantity.

78. Hilbert function Notation: C [x 1, …, x n ] ≤L = all degree ≤ L polynomials. I ≤L = I ⋂ C [x 1, …, x n ] ≤L

79. Hilbert function Notation: C [x 1, …, x n ] ≤L = all degree ≤ L polynomials I ≤L = I ⋂ C [x 1, …, x n ] ≤L vector space over C

80. Hilbert function Notation: C [x 1, …, x n ] ≤L = all degree ≤ L polynomials I ≤L = I ⋂ C [x 1, …, x n ] ≤L Hilbert function: H I (L) := ( ) - dim C I ≤L n + L n Hilbert polynomial: H I (L) is a polynomial in L for sufficiently large L.

81. Hilbert function Notation: C [x 1, …, x n ] ≤L = all degree ≤ L polynomials I ≤L = I ⋂ C [x 1, …, x n ] ≤L Hilbert function: H I (L) := ( ) - dim C I ≤L n + L n Hilbert polynomial: H I (L) is a polynomial in L for sufficiently large L Theorem: degree H I (L) = dim V (I) = dim V

82. Hilbert function Notation: C [x 1, …, x n ] ≤L = all degree ≤ L polynomials I ≤L = I ⋂ C [x 1, …, x n ] ≤L H I (L) = ( ) - dim C I ≤L n + L n Theorem: degree H I (L) = dim V (I) = dim V

83. Hilbert function Notation: C [x 1, …, x n ] ≤L = all degree ≤ L polynomials I ≤L = I ⋂ C [x 1, …, x n ] ≤L H I (L) = ( ) - dim C I ≤L n + L n Theorem: degree H I (L) = dim V (I) = dim V dim C I ≤L = ( ) - H I (L) n + L n

84. Hilbert function Notation: C [x 1, …, x n ] ≤L = all degree ≤ L polynomials I ≤L = I ⋂ C [x 1, …, x n ] ≤L H I (L) = ( ) - dim C I ≤L n + L n Theorem: degree H I (L) = dim V (I) = dim V dim C I ≤L = ( ) - L dim V + lower order terms n + L n

85. Hilbert function Notation: C [x 1, …, x n ] ≤L = all degree ≤ L polynomials I ≤L = I ⋂ C [x 1, …, x n ] ≤L H I (L) = ( ) - dim C I ≤L n + L n larger dim V smaller dim C I ≤L dim C I ≤L = ( ) - L dim V + lower order terms n + L n

86. Hilbert function Notation: C [x 1, …, x n ] ≤L = all degree ≤ L polynomials I ≤L = I ⋂ C [x 1, …, x n ] ≤L H I (L) = ( ) - dim C I ≤L n + L n larger dim V smaller dim C I ≤L dim C I ≤L = ( ) - L dim V + lower order terms n + L n smaller dim C (x ≤ℓ · {g 1, …, g m }) … where L = ℓ + deg(T) - k

87. In search of a ‘hard’ polynomial family

88. Candidate ‘hard’ polynomials 1.Permanent ( Perm d ) 2.Determinant ( Det d ) 3.Iterated Matrix Multiplication (IMM w,d ) 4.Elementary symmetric polynomial (ESym n,d ) VNP VP ABP m -∑∏∑

89. Candidate ‘hard’ polynomials 1.Permanent ( Perm d ) 2.Determinant ( Det d ) 3.Iterated Matrix Multiplication (IMM w,d ) 4.Elementary symmetric polynomial (ESym n,d ) Gupta-Kayal-Kamath-Saptharishi (2013)

90. SPD of the determinant Gupta-Kayal-Kamath-Saptharishi (2013): SPD k,ℓ (Det d ) ≥ ( )( ) d+k 2k n + ℓ -2k ℓ Theorem: Top fan-in lower bound of 2 for h -∑∏∑∏ [t] circuits computing Det d. Ω (d/t)

91. SPD of the determinant Gupta-Kayal-Kamath-Saptharishi (2013): SPD k,ℓ (Det d ) ≥ ( )( ) n k n + ℓ n ( )( ) d+k 2k n + ℓ -2k ℓ Theorem: Top fan-in lower bound of 2 for h -∑∏∑∏ [t] circuits computing Det d. Ω (d/t) <<

92. SPD of the determinant Gupta-Kayal-Kamath-Saptharishi (2013): SPD k,ℓ (Det d ) ≥ ( )( ) n k n + ℓ n ( )( ) d+k 2k n + ℓ -2k ℓ SPD k,ℓ (Det d ) ≤ (k+1) 2 ( ) ( ) d - 1 k n + ℓ -2k ℓ 2 <<

93. Candidate ‘hard’ polynomials 1.Permanent ( Perm d ) 2.Determinant ( Det d ) 3.Iterated Matrix Multiplication (IMM w,d ) 4.Elementary symmetric polynomial (ESym n,d )

94. Candidate ‘hard’ polynomials 1.Permanent ( Perm d ) 2.Determinant ( Det d ) 3.Iterated Matrix Multiplication (IMM w,d ) 4.Elementary symmetric polynomial (ESym n,d )

95. Candidate ‘hard’ polynomials 1.Permanent ( Perm d ) 2.Determinant ( Det d ) 3.Iterated Matrix Multiplication (IMM w,d ) 4.Elementary symmetric polynomial (ESym n,d ) Conj: (GKKS’13) SPD k,ℓ (Perm d ) >> SPD k,ℓ (Det d )

96. Candidate ‘hard’ polynomials 1.Permanent ( Perm d ) 2.Determinant ( Det d ) 3.Iterated Matrix Multiplication (IMM w,d ) 4.Elementary symmetric polynomial (ESym n,d ) Conj: (GKKS’13) Open !

97. Candidate ‘hard’ polynomials 1.Nisan-Wigderson ( NW n,d ) 2.Determinant ( Det d ) 3.Iterated Matrix Multiplication (IMM w,d ) 4.Elementary symmetric polynomial (ESym n,d ) Kayal-S.-Saptharishi (2014)

98. SPD of the Nisan-Wigderson polynomial Kayal-S.-Saptharishi (2014): SPD k,ℓ (NW n,d ) ≥ Theorem: Top fan-in lower bound of n for h -∑∏∑∏ [t] circuits computing NW n,d. Ω (d/t) 1/n · ( ) n + ℓ + d - k n

99. SPD of the Nisan-Wigderson polynomial Kayal-S.-Saptharishi (2014): SPD k,ℓ (NW n,d ) ≥ Theorem: Top fan-in lower bound of n for h -∑∏∑∏ [t] circuits computing NW n,d. Ω (d/t) 1/n · ( ) n + ℓ + d - k n … nearly the best possible n + ℓ + d - k n ( )

100. SPD of the Nisan-Wigderson polynomial Kayal-S.-Saptharishi (2014): SPD k,ℓ (NW n,d ) ≥ Theorem: Top fan-in lower bound of n for h -∑∏∑∏ [t] circuits computing NW n,d. Ω (d/t) 1/n · ( ) n + ℓ + d - k n … nearly the best possible n + ℓ + d - k n ( ) … might see a prove in this talk

101. Candidate ‘hard’ polynomials 1.Nisan-Wigderson ( NW n,d ) 2.Determinant ( Det d ) 3.Iterated Matrix Multiplication (IMM w,d ) 4.Elementary symmetric polynomial (ESym n,d )

102. Candidate ‘hard’ polynomials 1.Nisan-Wigderson ( NW n,d ) 2.Determinant ( Det d ) 3.Iterated Matrix Multiplication (IMM w,d ) 4.Elementary symmetric polynomial (ESym n,d ) Fournier-Limaye-Malod-Srinivasan (2014)

103. SPD of Iterated matrix multiplication Fournier-Limaye-Malod-Srinivasan (2014): SPD k,ℓ (IMM w,d ) ≥ Theorem: Top fan-in lower bound of n for h -∑∏∑∏ [t] circuits computing IMM w,d. Ω (d/t) 1/2 · ( ) ( ) w 4 n + ℓ n k n = w 2. d

104. SPD of Iterated matrix multiplication Fournier-Limaye-Malod-Srinivasan (2014): SPD k,ℓ (IMM w,d ) ≥ Theorem: Top fan-in lower bound of n for h -∑∏∑∏ [t] circuits computing IMM w,d. Ω (d/t) 1/2 · ( ) ( ) w 4 n + ℓ n k ( )( ) n k n + ℓ n < …but quite close n = w 2. d

105. SPD of Iterated matrix multiplication Fournier-Limaye-Malod-Srinivasan (2014): SPD k,ℓ (IMM w,d ) ≥ Theorem: Top fan-in lower bound of n for h -∑∏∑∏ [t] circuits computing IMM w,d. Ω (d/t) 1/2 · ( ) ( ) w 4 n + ℓ n k Observation: No significant improvement possible on the upper bound for SPD k, ℓ (C).

106. Candidate ‘hard’ polynomials 1.Nisan-Wigderson ( NW n,d ) 2.Determinant ( Det d ) 3.Iterated Matrix Multiplication (IMM w,d ) 4.Elementary symmetric polynomial (ESym n,d )

107. Candidate ‘hard’ polynomials 1.Nisan-Wigderson ( NW n,d ) 2.Determinant ( Det d ) 3.Iterated Matrix Multiplication (IMM w,d ) 4.Elementary symmetric polynomial (ESym n,d ) Kamath (2013) & Fournier-Limaye-Mahajan-Srinivasan (2015)

108. SPD of Elementary symmetric polynomial Kamath (2013) & Fournier-Limaye-Mahajan-Srinivasan (2015): SPD k,ℓ (ESym n,d ) = Theorem: (FLMS’15) Top fan-in lower bound of n for h -∑∏∑∏ [t] circuits computing ESym n,d where d ≈ (log n)/(log log n). Ω (d/t) “Fairly large”

109. Candidate ‘hard’ polynomials 1.Nisan-Wigderson ( NW n,d ) 2.Determinant ( Det d ) 3.Iterated Matrix Multiplication (IMM w,d ) 4.Elementary symmetric polynomial (Esym n,d )

110. Shifted partials of Nisan- Wigderson polynomials

111. Nisan-Wigderson polynomials Definition: Nisan-Wigderson family of polynomials: NW n,d := ∑ ∏ x i, h(i) d h(z) in F [z], deg(h) < k i in [d] Identify the elements of F with {1,2, …, d} Total number of monomials = d k Number of variables n = d 2 d

112. Nisan-Wigderson polynomials Definition: Nisan-Wigderson family of polynomials: NW n,d := ∑ ∏ x i, h(i) d h(z) in F [z], deg(h) < k i in [d] Property: (Disjointness) Two distinct monomials of NW n,d share at most k-1 variables, i.e. the ‘distance’ between two monomials is at least d-k.

113. Nisan-Wigderson polynomials Definition: Nisan-Wigderson family of polynomials: NW n,d := ∑ ∏ x i, h(i) d h(z) in F [z], deg(h) < k i in [d] What follows is an analysis by Chillara-Mukhopadhyay (2014) that is also inspired by FLMS (2014).

114. Nisan-Wigderson polynomials Definition: Nisan-Wigderson family of polynomials: NW n,d := ∑ ∏ x i, h(i) d h(z) in F [z], deg(h) < k i in [d] A subset of all derivatives: Consider the following set of derivatives: D := { ∂ x x … x } D NW n,d := { ∂ x x … x NW n,d } 2, * 1, * k k, * 2, * 1, * k, * k

115. Nisan-Wigderson polynomials Definition: Nisan-Wigderson family of polynomials: NW n,d := ∑ ∏ x i, h(i) d h(z) in F [z], deg(h) < k i in [d] A subset of all derivatives: Consider the following set of derivatives: D := { ∂ x x … x } D NW n,d := { ∂ x x … x NW n,d } 2, * 1, * k k, * 2, * 1, * k, * k Observation: D NW n,d is a set of monomials. (disjointness property)

116. Nisan-Wigderson polynomials Definition: Nisan-Wigderson family of polynomials: NW n,d := ∑ ∏ x i, h(i) d h(z) in F [z], deg(h) < k i in [d] A subset of all derivatives: Consider the following set of derivatives: D := { ∂ x x … x } D NW n,d := { ∂ x x … x NW n,d } 2, * 1, * k k, * 2, * 1, * k, * k | D NW n,d | = d k Observation:

117. Nisan-Wigderson polynomials Definition: Nisan-Wigderson family of polynomials: NW n,d := ∑ ∏ x i, h(i) d h(z) in F [z], deg(h) < k i in [d] A subset of all derivatives: Consider the following set of derivatives: D := { ∂ x x … x } D NW n,d := { ∂ x x … x NW n,d } 2, * 1, * k k, * 2, * 1, * k, * k Notation: D NW n,d = { m 1, m 2, …, m } dkdk degree m i = d - k

118. Nisan-Wigderson polynomials Definition: Nisan-Wigderson family of polynomials: NW n,d := ∑ ∏ x i, h(i) d h(z) in F [z], deg(h) < k i in [d] A subset of all derivatives: Consider the following set of derivatives: D := { ∂ x x … x } D NW n,d := { ∂ x x … x NW n,d } 2, * 1, * k k, * 2, * 1, * k, * k Notation: D NW n,d = { m 1, m 2, …, m } dkdk m i, m j share ≤ k variables

119. Nisan-Wigderson polynomials Definition: Nisan-Wigderson family of polynomials: NW n,d := ∑ ∏ x i, h(i) d h(z) in F [z], deg(h) < k i in [d] A subset of shifted derivatives: x ≤ℓ · D NW n,d = x ≤ℓ · { m 1, m 2, …, m } = (x ≤ℓ · m 1 ) U (x ≤ℓ · m 2 ) U … U (x ≤ℓ · m ) dkdk dkdk

120. Nisan-Wigderson polynomials Definition: Nisan-Wigderson family of polynomials: NW n,d := ∑ ∏ x i, h(i) d h(z) in F [z], deg(h) < k i in [d] A subset of shifted derivatives: x ≤ℓ · D NW n,d = x ≤ℓ · { m 1, m 2, …, m } = (x ≤ℓ · m 1 ) U (x ≤ℓ · m 2 ) U … U (x ≤ℓ · m ) dkdk dkdk B1B1 B2B2 dkdk B

121. Nisan-Wigderson polynomials Definition: Nisan-Wigderson family of polynomials: NW n,d := ∑ ∏ x i, h(i) d h(z) in F [z], deg(h) < k i in [d] A subset of shifted derivatives: x ≤ℓ · D NW n,d = x ≤ℓ · { m 1, m 2, …, m } = (x ≤ℓ · m 1 ) U (x ≤ℓ · m 2 ) U … U (x ≤ℓ · m ) dkdk dkdk B1B1 B2B2 dkdk B |B i | = n + ℓ n ( )

122. Nisan-Wigderson polynomials Definition: Nisan-Wigderson family of polynomials: NW n,d := ∑ ∏ x i, h(i) d h(z) in F [z], deg(h) < k i in [d] A subset of shifted derivatives: x ≤ℓ · D NW n,d = B 1 U B 2 U … U B dkdk |B i | = ( ) n + ℓ n | x ≤ℓ · D NW n,d | = ∑ | B i | - ½· ∑ | B i ⋂ B j | ii≠j

123. Nisan-Wigderson polynomials Definition: Nisan-Wigderson family of polynomials: NW n,d := ∑ ∏ x i, h(i) d h(z) in F [z], deg(h) < k i in [d] A subset of shifted derivatives: x ≤ℓ · D NW n,d = B 1 U B 2 U … U B dkdk |B i | = ( ) n + ℓ n | x ≤ℓ · D NW n,d | = ∑ | B i | - ½· ∑ | B i ⋂ B j | ≥ d k · ( ) - ?? ii≠j n + ℓ n

124. Nisan-Wigderson polynomials Definition: Nisan-Wigderson family of polynomials: NW n,d := ∑ ∏ x i, h(i) d h(z) in F [z], deg(h) < k i in [d] A subset of shifted derivatives: x ≤ℓ · D NW n,d = B 1 U B 2 U … U B dkdk | x ≤ℓ · D NW n,d | = ∑ | B i | - ½· ∑ | B i ⋂ B j | ≥ d k · ( ) - ii≠j n + ℓ n Can upper bound this quantity using disjointness property

125. Nisan-Wigderson polynomials Definition: Nisan-Wigderson family of polynomials: NW n,d := ∑ ∏ x i, h(i) d h(z) in F [z], deg(h) < k i in [d] A subset of shifted derivatives: x ≤ℓ · D NW n,d = B 1 U B 2 U … U B dkdk | x ≤ℓ · D NW n,d | = ∑ | B i | - ½· ∑ | B i ⋂ B j | ≥ d k · ( ) - ii≠j n + ℓ n ≤ ½ · d k · ( ) n + ℓ n

126. Nisan-Wigderson polynomials Definition: Nisan-Wigderson family of polynomials: NW n,d := ∑ ∏ x i, h(i) d h(z) in F [z], deg(h) < k i in [d] A subset of shifted derivatives: x ≤ℓ · D NW n,d = B 1 U B 2 U … U B dkdk | x ≤ℓ · D NW n,d | = ∑ | B i | - ½· ∑ | B i ⋂ B j | ≥ ½ · d k · ( ) ii≠j n + ℓ n

127. Nisan-Wigderson polynomials Definition: Nisan-Wigderson family of polynomials: NW n,d := ∑ ∏ x i, h(i) d h(z) in F [z], deg(h) < k i in [d] A subset of shifted derivatives: x ≤ℓ · D NW n,d = B 1 U B 2 U … U B dkdk | x ≤ℓ · D NW n,d | = ∑ | B i | - ½· ∑ | B i ⋂ B j | ≥ ½ · d k · ( ) ii≠j n + ℓ n ( )( ) n k n + ℓ n close

128. Nisan-Wigderson polynomials Definition: Nisan-Wigderson family of polynomials: NW n,d := ∑ ∏ x i, h(i) d h(z) in F [z], deg(h) < k i in [d] Upper bound for | B i ⋂ B j | : Let m ϵ | B i ⋂ B j |  Then m = r i.m i = r j.m j s.t degree r i, r j ≤ ℓ and degree m i, m j ≤ d-k

129. Nisan-Wigderson polynomials Definition: Nisan-Wigderson family of polynomials: NW n,d := ∑ ∏ x i, h(i) d h(z) in F [z], deg(h) < k i in [d] Upper bound for | B i ⋂ B j | : Let m ϵ | B i ⋂ B j |  Then m = r i.m i = r j.m j s.t degree r i, r j ≤ ℓ and degree m i, m j ≤ d-k  Since m i and m j share at most k variables, m = r · · m j s.t. degree r ≤ ℓ - (d-2k) mimi gcd (m i, m j )

130. Nisan-Wigderson polynomials Definition: Nisan-Wigderson family of polynomials: NW n,d := ∑ ∏ x i, h(i) d h(z) in F [z], deg(h) < k i in [d] Upper bound for | B i ⋂ B j | : Let m ϵ | B i ⋂ B j |  Then m = r i.m i = r j.m j s.t degree r i, r j ≤ ℓ and degree m i, m j ≤ d-k  Since m i and m j share at most k variables, m = r · · m j s.t. degree r ≤ ℓ - (d-2k) mimi gcd (m i, m j ) has degree ≥ d – 2k

131. Nisan-Wigderson polynomials Definition: Nisan-Wigderson family of polynomials: NW n,d := ∑ ∏ x i, h(i) d h(z) in F [z], deg(h) < k i in [d] Upper bound for | B i ⋂ B j | : Let m ϵ | B i ⋂ B j |  Then m = r i.m i = r j.m j s.t degree r i, r j ≤ ℓ and degree m i, m j ≤ d-k  Since m i and m j share at most k variables, m = r · · m j s.t. degree r ≤ ℓ - (d-2k) mimi gcd (m i, m j )  Hence, | B i ⋂ B j | ≤ ( ) n + ℓ - (d – 2k) n

132. Nisan-Wigderson polynomials Definition: Nisan-Wigderson family of polynomials: NW n,d := ∑ ∏ x i, h(i) d h(z) in F [z], deg(h) < k i in [d] Upper bound for | B i ⋂ B j | :  | B i ⋂ B j | ≤ ( ) n + ℓ - (d – 2k) n

133. Nisan-Wigderson polynomials Definition: Nisan-Wigderson family of polynomials: NW n,d := ∑ ∏ x i, h(i) d h(z) in F [z], deg(h) < k i in [d] Upper bound for | B i ⋂ B j | :  | B i ⋂ B j | ≤ ( )  ½· ∑ | B i ⋂ B j | ≤ d 2k /2 · ( ) n + ℓ - (d – 2k) n i≠j n + ℓ - (d – 2k) n

134. Nisan-Wigderson polynomials Definition: Nisan-Wigderson family of polynomials: NW n,d := ∑ ∏ x i, h(i) d h(z) in F [z], deg(h) < k i in [d] Upper bound for | B i ⋂ B j | :  | B i ⋂ B j | ≤ ( )  ½· ∑ | B i ⋂ B j | ≤ d 2k /2 · ( ) n + ℓ - (d – 2k) n i≠j n + ℓ - (d – 2k) n ≤ d k /2 · ( ) at ℓ ≈ ℓ * n + ℓ n (the optimum choice of ℓ ) Q.E.D

135. A few questions…

136. Questions Can we improve the following lower bounds? 1. (Baur-Strassen’83) Ω(n log d) lower bound for general circuits. 2. (Kalorkoti’85) Ω(n 2 ) lower bound for general formulas. 3. (Mignon-Ressayre’04) Ω(n 2 ) lower bound for the determinantal complexity of Perm n. 4. (Shpilka-Wigderson’01) Ω(n 2 ) lower bound for general depth three circuits.

137. Questions Can we improve the following lower bounds? 1. (Baur-Strassen’83) Ω(n log d) lower bound for general circuits. 2. (Kalorkoti’85) Ω(n 2 ) lower bound for general formulas. 3. (Mignon-Ressayre’04) Ω(n 2 ) lower bound for the determinantal complexity of Perm n. 4. (Shpilka-Wigderson’01) Ω(n 2 ) lower bound for general depth three circuits. Yes for “regular formulas” [Kayal-S-Saptharishi’14] Seems unlikely (??) [Efremenko-Landsberg-Schenck-Weyman’15] Yes [Kayal-S-Tavenas’16] ???

138. Questions Can we improve the following lower bounds? 1. (Baur-Strassen’83) Ω(n log d) lower bound for general circuits. 2. (Kalorkoti’85) Ω(n 2 ) lower bound for general formulas. 3. (Mignon-Ressayre’04) Ω(n 2 ) lower bound for the determinantal complexity of Perm n. 4. (Shpilka-Wigderson’01) Ω(n 2 ) lower bound for general depth three circuits. Yes for “regular formulas” [Kayal-S-Saptharishi’14] Seems unlikely (??) [Efremenko-Landsberg-Schenck-Weyman’15] Yes [Kayal-S-Tavenas’16] ??? Neeraj’s talk later…

139. Limitations of the shifted partials measure

140. We have already seen one Two ways one could have shown VP ≠ VNP: a. improve the upper bound on SPD (C) b. a better depth reduction to h -∑∏∑∏ [t] circuits.

141. We have already seen one Two ways one could have shown VP ≠ VNP: a. improve the upper bound on SPD (C) b. a better depth reduction to h -∑∏∑∏ [t] circuits. FLMS’14: Top fan-in lower bound of n for h -∑∏∑∏ [t] circuits computing IMM w,d. Ω (d/t) n = w 2. d … the target polynomial family is in VP ! … depth reduction to h -∑∏∑∏ [t] circuits is optimal … upper bound on SPD (C) is optimal

142. We have already seen one h -formulas h - ∑∏∑∏ ABP h - ∑∏∑∏ [t] hard for

143. There’s one more… h -formulas h - ∑∏∑∏ ABP h - ∑∏∑∏ [t] hard for Kumar-Saraf (2014): There is a polynomial computed by a poly(n,d) size h - ∑∏∑∏ circuit whose SPD is close to optimum.

144. There’s one more… h -formulas h - ∑∏∑∏ ABP h - ∑∏∑∏ [t] hard for (how to show this?)

145. There’s one more… h -formulas h - ∑∏∑∏ ABP h - ∑∏∑∏ [t] hard for (how to show this?) Time to introduce variants of shifted partials…

146. Variants of shifted partials

147. Kumar-Saraf separation Theorem: There is an explicit family of polynomials {f n } computable by poly(n) size h - ∑∏∑∏ circuits s.t. – for ω(log d) < t < d/50, any h - ΣΠΣΠ [t] circuit computing f n has top fan-in n Ω(d/t). h -formulas h - ∑∏∑∏ ABP h - ∑∏∑∏ [t] hard for

148. Kumar-Saraf separation Theorem: There is an explicit family of polynomials {f n } computable by poly(n) size h - ∑∏∑∏ circuits s.t. – for ω(log d) < t < d/50, any h - ΣΠΣΠ [t] circuit computing f n has top fan-in n Ω(d/t). h -formulas h - ∑∏∑∏ ABP h - ∑∏∑∏ [t] hard for Observation: SPD of h - ∑∏∑∏ circuits `close’ to the highest possible.

149. Kumar-Saraf separation Theorem: There is an explicit family of polynomials {f n } computable by poly(n) size h - ∑∏∑∏ circuits s.t. – for ω(log d) < t < d/50, any h - ΣΠΣΠ [t] circuit computing f n has top fan-in n Ω(d/t). h -formulas h - ∑∏∑∏ ABP h - ∑∏∑∏ [t] hard for Observation: SPD of h - ∑∏∑∏ circuits `close’ to the highest possible. Observation: No efficient ‘direct reduction’ possible from h - ∑∏∑∏ to h - ∑∏∑∏ [t] circuits for a large range of t.

150. Can we avoid a ‘direct reduction’? Let C be a h - ∑∏∑∏ circuit. It is conceivable that – C = C t + C junk s.t. h -ΣΠΣΠ [t] circuit of size comparable to that of C

151. Can we avoid a ‘direct reduction’? Let C be a h - ∑∏∑∏ circuit. It is conceivable that – C = C t + C junk s.t. μ(C) ≤ μ(C t ) + μ(C junk ) and μ(C junk ) = 0 A measure obeying subadditivity

152. Can we avoid a ‘direct reduction’? Let C be a h - ∑∏∑∏ circuit. It is conceivable that – C = C t + C junk s.t. μ(C) ≤ μ(C t ) + μ(C junk ) μ(C) ≤ μ(C t ) An upper bound on μ(C t ) serves as an upper bound for μ(C) 0

153. The idea at work: an example Recall: Classical [NW95] lower bound for h - ∑∏∑ circuits. C = ∑ ℓ i1 ℓ i2 …ℓ id ( ℓ ij are linear forms) i=1 … k=(d-1)/2 alternate layers of variables w.r.t which derivatives are taken  call this set S IMM w,d s

154. The idea at work: an example Recall: Classical [NW95] lower bound for h - ∑∏∑ circuits. C = ∑ ℓ i1 ℓ i2 …ℓ id ( ℓ ij are linear forms) i=1 … k=(d-1)/2 alternate layers of variables w.r.t which derivatives are taken  call this set S s Circuit upper bound: PD k (C) ≤ s · ( ) d k IMM lower bound: PD k (IMM w,d ) ≥ w 2k

155. The idea at work: an example Recall: Classical [NW95] lower bound for h - ∑∏∑ circuits. C = ∑ ℓ i1 ℓ i2 …ℓ id ( ℓ ij are linear forms) i=1 s Circuit upper bound: PD k (C) ≤ s · ( ) d k IMM lower bound: PD k (IMM w,d ) ≥ w 2k Top fan-in lower bound: If C = IMM w,d then s = w Ω(d) provided w ≥ d.

156. The idea at work: an example Consider a slightly general model (Srikanth’s model) C = ∑ Q i1 Q i2 …Q id ( Q ij are sums of univariates) i=1 s f 1 (x 1 ) + f 2 (x 2 ) + … + f n (x n )

157. The idea at work: an example Consider a slightly general model (Srikanth’s model) C = ∑ Q i1 Q i2 …Q id ( Q ij are sums of univariates) i=1 s f 1 (x 1 ) + f 2 (x 2 ) + … + f n (x n ) Note: PD k (Q i1 Q i2 …Q id ) can be as high as ( ). n k …think of Q i1 = … = Q id = x 1 + …. + x n 2 2

158. The idea at work: an example Consider a slightly general model C = ∑ Q i1 Q i2 …Q id ( Q ij are sums of univariates) i=1 s Split Q ij = ℓ ij + J ij affine formSum of powers of variables where every power is ≥ 2

159. The idea at work: an example Consider a slightly general model C = ∑ ℓ i1 ℓ i2 … ℓ id + C junk i=1 s Split Q ij = ℓ ij + J ij Every monomial has a variable with degree ≥ 2

160. The idea at work: an example Consider a slightly general model C = ∑ ℓ i1 ℓ i2 … ℓ id + C junk i=1 s Every monomial has a variable with degree ≥ 2 Projection map. π S : f multilinear (f) x = 0 for x ϵ S S ⊆ [n]

161. The idea at work: an example Consider a slightly general model C = ∑ ℓ i1 ℓ i2 … ℓ id + C junk i=1 s Every monomial has a variable with degree ≥ 2 Projection map. π S : f multilinear (f) x = 0 for x ϵ S Projected PD measure map. PPD S, k (f) := dim (π S ( ∂ f )) S k set of multilinear derivatives w.r.t variables in S

162. The idea at work: an example Consider a slightly general model C = ∑ ℓ i1 ℓ i2 … ℓ id + C junk i=1 s Every monomial has a variable with degree ≥ 2 Projection map. π S : f multilinear (f) x = 0 for x ϵ S Projected PD measure map. PPD S, k (f) := dim (π S ( ∂ f )) S k Note. PPD S, k (f) obeys subadditivity as S is a fixed set

163. The idea at work: an example Apply the measure on the circuit (for any S, k ) PPD S, k (C) ≤ PPD S, k ( ∑ ℓ i1 ℓ i2 … ℓ id ) + PPD S, k (C junk ) i=1 s

164. The idea at work: an example Apply the measure on the circuit (for any S, k ) PPD S, k (C) ≤ PPD S, k ( ∑ ℓ i1 ℓ i2 … ℓ id ) + PPD S, k (C junk ) i=1 s 0 Either setting x = 0 for every x ϵ S makes a monomial zero, or, restricting to only multilinear monomials gets rid of the monomial.

165. The idea at work: an example Apply the measure on the circuit (for any S, k ) PPD S, k (C) ≤ PPD S, k ( ∑ ℓ i1 ℓ i2 … ℓ id ) i=1 s … although, C ≠ ∑ ℓ i1 ℓ i2 …ℓ id

166. The idea at work: an example Apply the measure on the circuit (for any S, k ) PPD S, k (C) ≤ PPD S, k ( ∑ ℓ i1 ℓ i2 … ℓ id ) ≤ s · ( ) i=1 s d k

167. The idea at work: an example Apply the measure on the circuit PPD S, k (C) ≤ PPD S, k ( ∑ ℓ i1 ℓ i2 … ℓ id ) ≤ s · ( ) i=1 s … k=(d-1)/2 alternate layers of variables w.r.t which derivatives are taken  call this set S IMM w,d PPD S, k (IMM w,d ) = w 2k d k Take S and k as in the figure

168. The idea at work: an example Apply the measure on the circuit PPD S, k (C) ≤ PPD S, k ( ∑ ℓ i1 ℓ i2 … ℓ id ) ≤ s · ( ) i=1 s … k=(d-1)/2 alternate layers of variables w.r.t which derivatives are taken  call this set S IMM w,d PPD S, k (IMM w,d ) = w 2k d k Top fan-in lower bound: if C = IMM w,d then s = w Ω(d)

169. What we learn from the example  Every monomial in Q ij has support at most 1.  Every monomial in ℓ ij has degree at most 1.  The projection map π low support low degree

170. What we learn from the example  Every monomial in Q ij has support at most 1.  Every monomial in ℓ ij has degree at most 1.  The projection map π  Towards h -∑∏∑∏ circuit. To apply a projection map π we first need to reduce a h -∑∏∑∏ circuit to one with low bottom support. low support low degree

171. Homogeneous depth four circuit lower bound Kayal-Limaye-S.-Srinivasan (2014)

172. From h - ΣΠΣΠ to h - ΣΠΣΠ {t}  Let C = ∑ Q i1 Q i2 …Q im be a h -ΣΠΣΠ circuit. Random restriction. Set x i = 0 i.a.r with probability 1 – n -ε. Call this map σ R. sufficiently small constant s i=1

173. From h - ΣΠΣΠ to h - ΣΠΣΠ {t}  Let C = ∑ Q i1 Q i2 …Q im be a h -ΣΠΣΠ circuit. Random restriction. Set x i = 0 i.a.r with probability 1 – n -ε. Call this map σ R. s i=1  Then w.h.p σ R (C) = ∑ σ R (Q i1 ). σ R (Q i2 ) … σ R (Q im ) is a h -ΣΠΣΠ {t} circuit for a suitable choice of t. sum of product of support- t polynomials i

174. From h - ΣΠΣΠ to h - ΣΠΣΠ {t}  Let C = ∑ Q i1 Q i2 …Q im be a h -ΣΠΣΠ circuit. s i=1  Then w.h.p σ R (C) = ∑ σ R (Q i1 ). σ R (Q i2 ) … σ R (Q im ) is a h -ΣΠΣΠ {t} circuit for a suitable choice of t. S := total no. of monomials in the Q ij ’s

175. From h - ΣΠΣΠ to h - ΣΠΣΠ {t}  Let C = ∑ Q i1 Q i2 …Q im be a h -ΣΠΣΠ circuit. s i=1  Then w.h.p σ R (C) = ∑ σ R (Q i1 ). σ R (Q i2 ) … σ R (Q im ) is a h -ΣΠΣΠ {t} circuit for a suitable choice of t. S := total no. of monomials in circuit C

176. From h - ΣΠΣΠ to h - ΣΠΣΠ {t}  Let C = ∑ Q i1 Q i2 …Q im be a h -ΣΠΣΠ circuit. s i=1  Then w.h.p σ R (C) = ∑ σ R (Q i1 ). σ R (Q i2 ) … σ R (Q im ) is a h -ΣΠΣΠ {t} circuit for a suitable choice of t. S := total “sparsity” of circuit C

177. From h - ΣΠΣΠ to h - ΣΠΣΠ {t}  Let C = ∑ Q i1 Q i2 …Q im be a h -ΣΠΣΠ circuit. s i=1  Then w.h.p σ R (C) = ∑ σ R (Q i1 ). σ R (Q i2 ) … σ R (Q im ) is a h -ΣΠΣΠ {t} circuit for a suitable choice of t. S := total no. of monomials in circuit C Pr [ there’s a monomial in σ R (C) with support ≥ t ] ≤ S · n -εt Observation. Unless S > n ε/2. t, w.h.p σ R (C) is a h -ΣΠΣΠ {t} circuit

178. From h - ΣΠΣΠ to h - ΣΠΣΠ {t}  Let C = ∑ Q i1 Q i2 …Q im be a h -ΣΠΣΠ circuit. s i=1  Then w.h.p σ R (C) = ∑ σ R (Q i1 ). σ R (Q i2 ) … σ R (Q im ) is a h -ΣΠΣΠ {t} circuit for a suitable choice of t. Summary of random restriction: Unless the circuit is large, random restriction weakens a h -ΣΠΣΠ circuit to a h -ΣΠΣΠ {t} circuit. A ‘hard’ polynomial should remain sufficiently ‘hard’ w.h.p under random restriction.

179. Prove lower bound for h -ΣΠΣΠ {t} circuits Observation: A top fan-in lower bound of n Ω(d/t) for h -ΣΠΣΠ {t} A size lower bound of min {n Ω(t), n Ω(d/t) } for h -ΣΠΣΠ

180. Prove lower bound for h -ΣΠΣΠ {t} circuits Observation: A top fan-in lower bound of n Ω(d/t) for h -ΣΠΣΠ {t} A size lower bound of min {n Ω(t), n Ω(d/t) } for h -ΣΠΣΠ “sparsity” lower bound if random restriction fails.

181. Prove lower bound for h -ΣΠΣΠ {t} circuits Observation: A top fan-in lower bound of n Ω(d/t) for h -ΣΠΣΠ {t} A size lower bound of min {n Ω(t), n Ω(d/t) } for h -ΣΠΣΠ top fan-in lower bound if random restriction succeeds.

182. Prove lower bound for h -ΣΠΣΠ {t} circuits Observation: A top fan-in lower bound of n Ω(d/t) for h -ΣΠΣΠ {t} A size lower bound of n Ω(√d) for h -ΣΠΣΠ …setting t = √d

183. Prove lower bound for h -ΣΠΣΠ {t} circuits Observation: A top fan-in lower bound of n Ω(d/t) for h -ΣΠΣΠ {t} A size lower bound of n Ω(√d) for h -ΣΠΣΠ Note: We know how to prove top fan-in lower bound for h -ΣΠΣΠ [t] circuits. …bottom degree (instead of support) bounded by t

184. Prove lower bound for h -ΣΠΣΠ {t} circuits Observation: A top fan-in lower bound of n Ω(d/t) for h -ΣΠΣΠ {t} A size lower bound of n Ω(√d) for h -ΣΠΣΠ Next step: “Reduce” h -ΣΠΣΠ {t} circuits to h -ΣΠΣΠ [t] circuits with the help of projection.

185. A simple projection map Projection map: π (g) := sum of the multilinear monomials in g Observation: If every monomial of g has support ≤ t then every monomial of π (g) has degree ≤ t.

186. Projected Shifted Partials PSPD k,ℓ (f) := dim (π (x =ℓ. ∂ k f) ) multilinear shifts of degree ℓ

187. Projected Shifted Partials PSPD k,ℓ (f) := dim (π (x =ℓ. ∂ k f) ) Subadditivity. PSPD k,ℓ (f 1 + f 2 ) ≤ PSPD k,ℓ (f 1 ) + PSPD k,ℓ (f 2 ) How large can PSPD be? PSPD k,ℓ (f) ≤ min ( )·( ), ( ) n k n ℓ n ℓ + d - k

188. Projected Shifted Partials PSPD k,ℓ (f) := dim (π (x =ℓ. ∂ k f) ) Subadditivity. PSPD k,ℓ (f 1 + f 2 ) ≤ PSPD k,ℓ (f 1 ) + PSPD k,ℓ (f 2 ) How large can PSPD be? PSPD k,ℓ (f) ≤ min ( )·( ), ( ) n k n ℓ n ℓ + d - k …assume that there’s an explicit f with highest possible PSPD

189. “Reducing” h -ΣΠΣΠ {t} to h -ΣΠΣΠ [2t] circuits C = Q 11 Q 12 …Q 1m + … + Q s1 Q s2 …Q sm support of every monomial in every Q ij is bounded by t

190. “Reducing” h -ΣΠΣΠ {t} to h -ΣΠΣΠ [2t] circuits C = Q 11 Q 12 …Q 1m + … + Q s1 Q s2 …Q sm Split: Q ij = Q’ ij + Every variable in every monomial has degree 2 or less Every monomial has a variable with degree 3 or more

191. “Reducing” h -ΣΠΣΠ {t} to h -ΣΠΣΠ [2t] circuits C = Q 11 Q 12 …Q 1m + … + Q s1 Q s2 …Q sm Split: Q ij = Q’ ij + Q i1 Q i2 …Q im = Q’ i1 Q’ i2 …Q’ im + Every monomial has a variable with degree 3 or more

192. “Reducing” h -ΣΠΣΠ {t} to h -ΣΠΣΠ [2t] circuits C = Q 11 Q 12 …Q 1m + … + Q s1 Q s2 …Q sm Split: Q ij = Q’ ij + Q i1 Q i2 …Q im = Q’ i1 Q’ i2 …Q’ im + PSPD k,ℓ (Q i1 Q i2 …Q im ) ≤ PSPD k,ℓ (Q’ i1 Q’ i2 …Q’ im ) + PSPD k,ℓ ( )

193. “Reducing” h -ΣΠΣΠ {t} to h -ΣΠΣΠ [2t] circuits C = Q 11 Q 12 …Q 1m + … + Q s1 Q s2 …Q sm Split: Q ij = Q’ ij + Q i1 Q i2 …Q im = Q’ i1 Q’ i2 …Q’ im + PSPD k,ℓ (Q i1 Q i2 …Q im ) ≤ PSPD k,ℓ (Q’ i1 Q’ i2 …Q’ im ) + PSPD k,ℓ ( ) 0

194. “Reducing” h -ΣΠΣΠ {t} to h -ΣΠΣΠ [2t] circuits C = Q 11 Q 12 …Q 1m + … + Q s1 Q s2 …Q sm Split: Q ij = Q’ ij + Q i1 Q i2 …Q im = Q’ i1 Q’ i2 …Q’ im + PSPD k,ℓ (Q i1 Q i2 …Q im ) ≤ PSPD k,ℓ (Q’ i1 Q’ i2 …Q’ im ) degree of every Q’ ij ≤ 2t

195. Upper bounding PSPD of a h -ΣΠΣΠ [2t] circuit ∂ k Q i1 …Q im = Q i1 Q i2 …Q ik …Q im + Q i1 Q i2 …Q ik Q i(k+1) …Q im + … x...... = Q i(k+1) … Q im + Q i1 Q i(k+2) …Q im + … degree ≤ 2kt Reusing notation: Call Q’ i j as Q ij

196. Upper bounding PSPD of a h -ΣΠΣΠ [2t] circuit ∂ k Q i1 …Q im = Q i1 Q i2 …Q ik …Q im + Q i1 Q i2 …Q ik Q i(k+1) …Q im + … x...... = Q i(k+1) … Q im + Q i1 Q i(k+2) …Q im + … Reusing notation: Call Q’ i j as Q ij u. ∂ k Q i1 …Q im = Q i(k+1) … Q im + Q i1 Q i(k+2) …Q im + … degree = ℓ degree ≤ ℓ + 2kt

197. Upper bounding PSPD of a h -ΣΠΣΠ [2t] circuit ∂ k Q i1 …Q im = Q i1 Q i2 …Q ik …Q im + Q i1 Q i2 …Q ik Q i(k+1) …Q im + … x...... = Q i(k+1) … Q im + Q i1 Q i(k+2) …Q im + … Reusing notation: Call Q’ i j as Q ij u. ∂ k Q i1 …Q im = Q i(k+1) … Q im + Q i1 Q i(k+2) …Q im + … π(u.∂ k Q i1 …Q im )=π( Q i(k+1) … Q im )+π( Q i1 Q i(k+2) …Q im )… multilinear and degree ≤ ℓ + 2kt

198. Upper bounding PSPD of a h -ΣΠΣΠ [2t] circuit ∂ k Q i1 …Q im = Q i1 Q i2 …Q ik …Q im + Q i1 Q i2 …Q ik Q i(k+1) …Q im + … x...... = Q i(k+1) … Q im + Q i1 Q i(k+2) …Q im + … Reusing notation: Call Q’ i j as Q ij u. ∂ k Q i1 …Q im = Q i(k+1) … Q im + Q i1 Q i(k+2) …Q im + … π(u.∂ k Q i1 …Q im )=π( Q i(k+1) … Q im )+π( Q i1 Q i(k+2) …Q im )… PSPD k,ℓ (C) ≤ s. ( ). ( ) m k n ℓ + 2kt

199. Upper bounding PSPD of a h -ΣΠΣΠ [2t] circuit ∂ k Q i1 …Q im = Q i1 Q i2 …Q ik …Q im + Q i1 Q i2 …Q ik Q i(k+1) …Q im + … x...... = Q i(k+1) … Q im + Q i1 Q i(k+2) …Q im + … Reusing notation: Call Q’ i j as Q ij u. ∂ k Q i1 …Q im = Q i(k+1) … Q im + Q i1 Q i(k+2) …Q im + … π(u.∂ k Q i1 …Q im )=π( Q i(k+1) … Q im )+π( Q i1 Q i(k+2) …Q im )… PSPD k,ℓ (C) ≤ s. ( ). ( ) d/2t k n ℓ + 2kt

200. Lower bound for h -ΣΠΣΠ {t} circuits “Best” possible lower bound for top fan-in: s ≥ n k n ℓ n ℓ + d - k min ( ).( ), ( ) ( ).( ) d/2t k n ℓ + 2kt

201. Lower bound for h -ΣΠΣΠ {t} circuits “Best” possible lower bound for top fan-in: s ≥ n k n ℓ n ℓ + d - k min ( ).( ), ( ) ( ).( ) d/2t k n ℓ + 2kt Set parameters:  Choose k = δ · d/t for second ratio ≥ 1  Choose ℓ = ℓ* to make the two ratio equal = n Ω(d/t)

202. NW has near optimal PSPD d2d2 Theorem (KLSS’14) For r = d/3, t = √d, k = δ.√d and ℓ ≈ ℓ* PSPD k,ℓ (NW n,d ) ≥ n -9 · min { ( ).( ), ( )} n k n ℓ n ℓ + d - k Nisan-Wigderson family of polynomials: NW n,d := ∑ ∏ x i, h(i) h(z) in F [z], deg(h) < r i in [d]

203. NW has near optimal PSPD d2d2 Theorem (KLSS’14) Any h -ΣΠΣΠ circuit computing NW n,d has size n Ω(√d) Nisan-Wigderson family of polynomials: NW n,d := ∑ ∏ x i, h(i) h(z) in F [z], deg(h) < r i in [d]

204. Proof idea Issue: k -th order derivatives are not monomials. PSPD k,ℓ ( NW n,d ) = rank (M) n k n ℓ M := ( ).( ) rows π (x = ℓ. ∂ k NW) (0/1)-matrix of coefficients n ℓ + d - k ( ) columns

205. Proof idea Lemma (informal). Because of large pairwise distance of monomials in NW, the columns of M are nearly orthogonal. n k n ℓ M := ( ).( ) rows π (x = ℓ. ∂ k NW) (0/1)-matrix of coefficients n ℓ + d - k ( ) columns

206. Proof idea Corollary. B := M T M is a diagonally dominant symmetric matrix. Also, rank (M) ≥ rank (B). n k n ℓ M := ( ).( ) rows π (x = ℓ. ∂ k NW) (0/1)-matrix of coefficients n ℓ + d - k ( ) columns

207. Proof idea Alon’s bound. rank (B) ≥ n k n ℓ M := ( ).( ) rows π (x = ℓ. ∂ k NW) (0/1)-matrix of coefficients n ℓ + d - k ( ) columns Tr (B) 2 Tr (B 2 )

208. NW has near optimal PSPD h -formulas h - ∑∏∑∏ ABP hard for VNP

209. IMM w,d has high PSPD h -formulas h - ∑∏∑∏ ABP hard for VNP Kumar-Saraf (2014) …over any field

210. IMM w,d hard for h -formulas ? h -formulas h - ∑∏∑∏ ABP hard for ? VNP Open!

211. A similar question h -formulas h - ∑∏∑∏ ABP hard for ? VNP formulas ABP hard for ? Non-commutative Open!

212. IMM w,d hard for m -formulas ? m -formulas m - ∑∏∑∏ ABP hard for IMM w,d formulas ABP hard for Non-commutative m stands for multilinear

213. IMM w,d hard for m -formulas ? m -formulas m - ∑∏∑∏ ABP hard for ? Open! m stands for multilinear IMM w,d

214. IMM w,d hard for m -formulas ? m -formulas m - ∑∏∑∏ ABP hard for ? Open! IMM w,d A related result. Dvir, Malod, Perifel, Yehudayoff (2012) A separation between m -ABP and m -formulas, but the hard polynomial is not IMM w,d

215. IMM w,d hard for m - ∑∏∑∏ circuits ? m -formulas m - ∑∏∑∏ ABP hard for ? IMM w,d

216. IMM w,d hard for m - ∑∏∑∏ circuits m -formulas m - ∑∏∑∏ ABP hard for IMM w,d Kayal-S.-Tavenas (2016)

217. IMM w,d hard for m - ∑∏∑∏ circuits m -formulas m - ∑∏∑∏ ABP hard for IMM w,d Kayal-S.-Tavenas (2016) Issue. Not clear how to use a multilinear projection map π to exploit any weakness of a m -circuit.

218. IMM w,d hard for m - ∑∏∑∏ circuits m -formulas m - ∑∏∑∏ ABP hard for IMM w,d Kayal-S.-Tavenas (2016) … turns out that another variant of the shifted partials measure helps.

219. IMM w,d is hard for m - ∑∏∑∏ circuits

220. A bit of history m -formulas m - ∑∏∑∏ ABP hard for Det d IMM w,d Raz (2004): Any m -formula computing Det d has size d Ω(log d). Det d and IMM w,d are both complete for ABP.

221. A bit of history m -formulas m - ∑∏∑∏ ABP Det d IMM w,d Det d and IMM w,d are both complete for ABP. Det d lower bound doesn’t readily imply IMM w,d lower bound Reduction doesn’t preserve multilinearity

222. A bit of history m -formulas m - ∑∏∑∏ ABP Det d IMM w,d Det d and IMM w,d are both complete for ABP. Det d lower bound doesn’t readily imply IMM w,d lower bound. The converse is true. Reduction preserves multilinearity

223. A bit of history Raz (2004): Any m -formula computing Det d has size d Ω(log d). Raz-Yehudayoff (2008): Lower bound for constant depth m -formulas computing Det d. The measure used: Partition the set of variables randomly into two sets, x and y, of roughly equal sizes. Look at the rank of the PD matrix w.r.t this partition.

224. A bit of history Raz (2004): Any m -formula computing Det d has size d Ω(log d). Raz-Yehudayoff (2008): Lower bound for constant depth m -formulas computing Det d. The measure used: Partition the set of variables randomly into two sets, x and y, of roughly equal sizes. Look at the rank of the PD matrix w.r.t this partition. Basically, the following measure dim (∂ y f | y=0 ) Take derivatives and then set all the y -variables to zero *

225. A bit of history Raz (2004): Any m -formula computing Det d has size d Ω(log d). Raz-Yehudayoff (2008): Lower bound for constant depth m -formulas computing Det d. The measure used: Partition the set of variables randomly into two sets, x and y, of roughly equal sizes. Look at the rank of the PD matrix w.r.t this partition. Basically, the following measure dim (∂ y f | y=0 ) The next variant of shifted partials is inspired by partition of variables *

226. Example: IMM w,d hard for m - ∑∏∑ circuits ….................. 12k y variables

227. Example: IMM w,d hard for m - ∑∏∑ circuits ….................. 12k x variables

228. Example: IMM w,d hard for m - ∑∏∑ circuits ….................. 12k edges labeled by 1

229. Example: IMM w,d hard for m - ∑∏∑ circuits ….................. 12k Number of variables: |y| = w 2 k |x| = 2wk Note the difference (skew) in the number of x and y variables

230. Example: IMM w,d hard for m - ∑∏∑ circuits ….................. 12k Note: The above polynomial (call it I w,k ) is a simple projection of IMM w,5k.

231. Example: IMM w,d hard for m - ∑∏∑ circuits Skewed partials: [ Kayal-Nair-S. (2016) ] SkP k, y (f) = dim (∂ y f | y=0 ) k Subadditivity: SkP k, y (f 1 + f 2 ) ≤ SkP k, y (f 1 ) + SkP k, y (f 2 )

232. Example: IMM w,d hard for m - ∑∏∑ circuits Skewed partials: [ Kayal-Nair-S. (2016) ] SkP k, y (f) = dim (∂ y f | y=0 ) k Subadditivity: SkP k, y (f 1 + f 2 ) ≤ SkP k, y (f 1 ) + SkP k, y (f 2 ) Observation: SkP k, y (I w,k ) = w 2k

233. Example: IMM w,d hard for m - ∑∏∑ circuits Skewed partials of a m -ΣΠΣ circuit: C = ∑ ℓ i1 ℓ i2 …ℓ in i=1 s

234. Example: IMM w,d hard for m - ∑∏∑ circuits Skewed partials of a m -ΣΠΣ circuit: A term T = ℓ I ℓ 2 … ℓ n = ℓ I (x, y). ℓ 2 (x, y) … ℓ |x| (x, y). Q(y) C = ∑ ℓ i1 ℓ i2 …ℓ in i=1 s x-free

235. Example: IMM w,d hard for m - ∑∏∑ circuits Skewed partials of a m -ΣΠΣ circuit: A term T = ℓ I ℓ 2 … ℓ n = ℓ I (x, y). ℓ 2 (x, y) … ℓ |x| (x, y). Q(y) C = ∑ ℓ i1 ℓ i2 …ℓ in i=1 s Observation: Since the y -variables are set to zero by SkP k, y SkP k, y (T) ≤ ( ) |x| k

236. Example: IMM w,d hard for m - ∑∏∑ circuits Skewed partials of a m -ΣΠΣ circuit: A term T = ℓ I ℓ 2 … ℓ n = ℓ I (x, y). ℓ 2 (x, y) … ℓ |x| (x, y). Q(y) C = ∑ ℓ i1 ℓ i2 …ℓ in i=1 s Observation: Since the y -variables are set to zero by SkP k, y SkP k, y (T) ≤ ( ) |x| k SkP k, y (C) ≤ s. ( ) ≤ s. (2ew) k 2wk k

237. Example: IMM w,d hard for m - ∑∏∑ circuits SkP k, y (I w,k ) = w 2k SkP k, y (C) ≤ s. (2ew) k Observation: Any m - ∑∏∑ circuit computing I w,k has top fan-in w Ω(k).

238. Example: IMM w,d hard for m - ∑∏∑ circuits SkP k, y (I w,k ) = w 2k SkP k, y (C) ≤ s. (2ew) k Observation: Any m - ∑∏∑ circuit computing IMM w,d has top fan-in w Ω(d).

239. Example: IMM w,d hard for m - ∑∏∑ circuits SkP k, y (I w,k ) = w 2k SkP k, y (C) ≤ s. (2ew) k Observation: Any m - ∑∏∑ circuit computing IMM w,d has top fan-in w Ω(d). Summary: What helps is |y| >> |x|.

240. IMM w,d hard for m - ∑∏∑∏ circuits Shifted Skewed Partials: [ Kayal-S.-Tavenas (2016) ] SSP k, y, ℓ (f) = dim (x ≤ℓ. (∂ y f | y=0 )) k Theorem: Any m - ∑∏∑∏ circuit computing IMM w,d has size w Ω(√d).

241. IMM w,d hard for m - ∑∏∑∏ circuits Shifted Skewed Partials: [ Kayal-S.-Tavenas (2016) ] SSP k, y, ℓ (f) = dim (x ≤ℓ. (∂ y f | y=0 )) k Theorem: Any m - ∑∏∑∏ circuit computing IMM w,d has size w Ω(√d). Idea: Since C is multilinear, SSP k, y, ℓ (C) depends only on |x| (and not |y| ).

242. IMM w,d hard for m - ∑∏∑∏ circuits Shifted Skewed Partials: [ Kayal-S.-Tavenas (2016) ] SSP k, y, ℓ (f) = dim (x ≤ℓ. (∂ y f | y=0 )) k Theorem: Any m - ∑∏∑∏ circuit computing IMM w,d has size w Ω(√d). Idea: Since C is multilinear, SSP k, y, ℓ (C) depends only on |x| (and not |y| ). So, if |y| >> |x| then one would expect SSP k, y, ℓ (C) << SSP k, y, ℓ (IMM).

243. A few problems

244. Problem 1 Separation between non-commutative ABP and formulas:  Sufficient to show that IMM w,d is hard for m -formulas.

245. Problem 2 Super-polynomial lower bound for ∑∏∑ circuits : [Kayal-S. (2015)]: An n Ω(√d) lower bound for ∑∏∑ [t] circuits where t ≤ n ε for any ε < 1. … Problem due to Shpilka-Wigderson (1999)

246. Problem 2 Super-polynomial lower bound for ∑∏∑ circuits : [Kayal-S. (2015)]: An n Ω(√d) lower bound for ∑∏∑ [t] circuits where t ≤ n ε for any ε < 1.  Sufficient to show an n Ω(√d) lower bound for h- ∑∏∑∏∑ circuits. … Problem due to Shpilka-Wigderson (1999) … Ankit’s suggestion

247. Problem 3 Symmetric polynomials and lower bounds for h -formulas:  Prove that ESym n,d is hard for h -formulas

248. Problem 3 Symmetric polynomials and lower bounds for h -formulas:  First, prove that ESym n,d is hard for h - ∑∏∑∏ Issue: PSPD k,ℓ (ESym n,d ) is “small”

249. Problem 4 Complexity of Nisan-Wigderson family:  Is the family of Nisan-Wigderson polynomials VNP -complete? … or not?

250. Problem 4 Complexity of Nisan-Wigderson family:  Is the family of Nisan-Wigderson polynomials VNP -complete? … or not? Thank You!