Lower Bounds using shifted partials - PowerPoint Presentation

1. Lower Bounds using shifted partialsChandan SahaIndian Institute of ScienceWorkshop on Algebraic Complexity Theory 2016 Tel-Aviv University

2. Background

3. Arithmetic Circuit+xxxx++++xxxx….…..x1x2xn-1xnf(x1, x2, …, xn) xghgh+ghg+hProduct gateSum gateThere are `field constants’ on the wires

4. Arithmetic Circuit+xxxx++++xxxx….…..x1x2xn-1xnf(x1, x2, …, xn) Depth

5. Arithmetic Circuit+xxxx++++xxxx….…..x1x2xn-1xnf(x1, x2, …, xn) Size = no. of wires

6. Arithmetic Circuit+xxxx++++xxxx….…..x1x2xn-1xnf(x1, x2, …, xn) Depth = 4

7. Arithmetic Circuit+xxxx++++xxxx….…..x1x2xn-1xnf(x1, x2, …, xn) f = ∑ ∏ Qijij

8. Arithmetic Circuit+xxxx++++xxxx….…..x1x2xn-1xnf(x1, x2, …, xn) f = ∑ ∏ Qijijsum of monomials

9. Arithmetic Circuit+xxxx++++xxxx….…..x1x2xn-1xnf(x1, x2, …, xn) f = ∑ ∏ QijijTop fan-in = s number of summands = s

10. Arithmetic Circuit+xxxx++++xxxx….…..x1x2xn-1xnf(x1, x2, …, xn) f = ∑ ∏ QijijBottom fan-in ≤ t degree ≤ t

11. Arithmetic Circuit+xxxx++++xxxx….…..x1x2xn-1xnf(x1, x2, …, xn) f = ∑ ∏ QijijThis talk: Lower bounds for restricted depth four circuits

12. Arithmetic Circuit+xxxx++++xxxx….…..x1x2xn-1xnf(x1, x2, …, xn) f = ∑ ∏ QijijNotation: n = number of variables d = degree of f

13. Recap: A template for proving lower boundStep 1: Define a suitable measure function μ μ : F[x1, …, xn] 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 boundStep 1: Define a suitable measure function μ μ : F[x1, …, xn] 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 boundStep 1: Define a suitable measure function μ μ : F[x1, …, xn] 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 derivativesNotation: ∂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 derivativesNotation: ∂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 derivativesNotation: ∂k f = Set of all k-th order derivatives of f 〈S〉 = The vector space spanned by F-linear combinations of polynomials in SDefinition: PDk (f) = dim 〈∂k f〉 Property: (Sub-additive) PDk (f1 + f2) ≤ PDk (f1) + PDk (f2)

19. Recap: Diagonal depth three circuits C = ℓ1 + … + ℓs Upper bound for the circuit : PDk (C) ≤ s e1es

20. Recap: Diagonal depth three circuits C = ℓ1 + … + ℓs Upper bound for the circuit : PDk (C) ≤ s Lower bound for the ‘hard’ polynomial: f = x1x2x3···xn PDk (f) = ( ) e1esnk

21. Recap: Diagonal depth three circuits C = ℓ1 + … + ℓs Upper bound for the circuit : PDk (C) ≤ s Lower bound for the ‘hard’ polynomial: f = x1x2x3···xn PDk (f) = ( )Setting parameter: Choose k = n/2 Top fan-in lower bound: s = 2Ω(n) e1esnk

22. Extending the circuit model: A motivating example

23. Sum of powers of quadratics C = Q1 + … + Qs e1 esQuadratic polynomials

24. Sum of powers of quadratics C = Q1 + … + QsA simple case: Q = x1x2 + x3x4 + + xn-1xn + 1Notation: yi = xi+1 if i is odd = xi –1 if i is even e1 es

25. Sum of powers of quadratics C = Q1 + … + QsA simple case: Q = x1x2 + x3x4 + + xn-1xn + 1Derivatives of a term: T = Qe suppose e ≥ n e1 es

26. Sum of powers of quadratics C = Q1 + … + QsA simple case: Q = x1x2 + x3x4 + + xn-1xn + 1Derivatives of a term: T = Qe e1 es∂x T = e . y . Qe-1 i1i1

27. Sum of powers of quadratics C = Q1 + … + QsA simple case: Q = x1x2 + x3x4 + + xn-1xn + 1Derivatives of a term: T = Qe e1 es∂x T = e . y . Qe-1 i1i1∂x x T = e(e-1) . y y . Qe-2 i2i12i1i2

28. Sum of powers of quadratics C = Q1 + … + QsA simple case: Q = x1x2 + x3x4 + + xn-1xn + 1Derivatives of a term: T = Qe e1 es∂x T = e . y . Qe-1 i1i1∂x x T = e(e-1) . y y . Qe-2 i2i12i1i2∂x x … x T = e(e-1)··· . y y ··· y . Qe-k i2i1iki1 i2kik...

29. Sum of powers of quadratics C = Q1 + … + QsA simple case: Q = x1x2 + x3x4 + + xn-1xn + 1Derivatives of a term: T = Qe e1 es∂x T = e . y . Qe-1 i1i1∂x x T = e(e-1) . y y . Qe-2 i2i12i1i2∂x x … x T = e(e-1)··· . y y ··· y . Qe-k i2i1iki1 i2kik...Uniquely determines the derivative

30. Sum of powers of quadratics C = Q1 + … + QsA simple case: Q = x1x2 + x3x4 + + xn-1xn + 1Observation: The k-th order derivatives of T = Qe are F-linearly independent. e1 es

31. Sum of powers of quadratics C = Q1 + … + QsA simple case: Q = x1x2 + x3x4 + + xn-1xn + 1Observation: PDk (T) = ( ) e1 esnk

32. Sum of powers of quadratics C = Q1 + … + QsA simple case: Q = x1x2 + x3x4 + + xn-1xn + 1Observation: PDk (T) = ( )Observation: For any f, PDk (f) ≤ ( ) e1 esnknkNote: In this setting, PDk is unable to distinguish between a power of a quadratic and any ‘hard’ polynomial !

33. Sum of powers of quadratics C = Q1 + … + QsA simple case: Q = x1x2 + x3x4 + + xn-1xn + 1Observation: PDk (T) = ( )Observation: For any f, PDk (f) ≤ ( ) e1 esnknk… 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 SDefinition: (Kayal’12) SPDk, ℓ(f) = dim 〈x ≤ℓ · ∂k f〉 x ≤ℓ = Set of monomials of degree ≤ ℓSet 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 SDefinition: (Kayal’12) SPDk, ℓ(f) = dim 〈x ≤ℓ · ∂k f〉 Property: (Sub-additive) SPDk, ℓ (f1 + f2) ≤ SPDk, ℓ (f1) + SPDk, ℓ (f2)x ≤ℓ = Set of monomials of degree ≤ ℓ

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

40. How large can SPDk, ℓ be?Definition: SPDk, ℓ(f) = dim 〈x ≤ℓ · ∂k f〉 Observation: SPDk, ℓ (f) ≤ min ( )( ) , ( ) nkn + ℓ nn + ℓ + d - knQuestion: Are there explicit polynomials that achieve this or close to this bound?For now, suppose f has SPDk, 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 = Q1 + … + Qs e1 espolynomials with degree ≤ t

43. Sum of powers of low degree polynomials C = Q1 + … + QsObservation: SPDk, ℓ (C) ≤ ∑ SPDk, ℓ (Qi ) Proof: By subadditivity of the measure SPDk, ℓ. e1 es ei i

44. Sum of powers of low degree polynomials C = Q1 + … + QsObservation: SPDk, ℓ (C) ≤ ∑ SPDk, ℓ (Qi ) Focus on a term: T = Qe e1 es ei i

45. Sum of powers of low degree polynomials C = Q1 + … + QsObservation: SPDk, ℓ (C) ≤ ∑ SPDk, ℓ (Qi ) Focus on a term: T = Qe Observation: ∂x x … x T = “polynomial of degree ≤ kt” . Qe-k Proof: Chain rule of derivatives. e1 es ei ii2i1ikk

46. Sum of powers of low degree polynomials C = Q1 + … + QsObservation: SPDk, ℓ (C) ≤ ∑ SPDk, ℓ (Qi ) Focus on a term: T = Qe Observation: Let m be a monomial in x ≤ℓ. Then m . ∂x x … x T = “polynomial of degree ≤ ℓ + kt” . Qe-k e1 es ei ii2i1ikk

47. Sum of powers of low degree polynomials C = Q1 + … + QsObservation: SPDk, ℓ (C) ≤ ∑ SPDk, ℓ (Qi ) Lemma: SPDk, ℓ (T) ≤ ( ) e1 es ei in + ℓ + ktn

48. Sum of powers of low degree polynomials C = Q1 + … + QsObservation: SPDk, ℓ (C) ≤ ∑ SPDk, ℓ (Qi ) Lemma: SPDk, ℓ (T) ≤ ( ) Upper bound for the circuit: SPDk, ℓ (C) ≤ s · ( ) e1 es ei in + ℓ + ktnn + ℓ + ktn

49. Lower bound on the top fan-in C = Q1 + … + Qs = fUpper bound for the circuit: SPDk, ℓ (C) ≤ s · ( )Lower bound for the ‘hard’ polynomial: e1 esn + ℓ + ktnSPDk, ℓ (f) = min ( )( ) , ( )nkn + ℓ nn + ℓ + d - knassume there’s such an f

50. Lower bound on the top fan-in C = Q1 + … + Qs = fUpper bound for the circuit: SPDk, ℓ (C) ≤ s · ( )Lower bound for the `hard’ polynomial: e1 esn + ℓ + ktnSPDk, ℓ (f) = min ( )( ) , ( )nkn + ℓ nn + ℓ + d - knTogether:s ≥ min ( )( ) , ( )nkn + ℓ nn + ℓ + d - kn( )n + ℓ + ktn

51. Lower bound on the top fan-in C = Q1 + … + Qs = fUpper bound for the circuit: SPDk, ℓ (C) ≤ s · ( )Lower bound for the `hard’ polynomial: e1 esn + ℓ + ktnSPDk, ℓ (f) = min ( )( ) , ( )nkn + ℓ nn + ℓ + d - knTogether:s ≥ min ( )( ) , ( )nkn + ℓ nn + ℓ + d - kn( )n + ℓ + ktnChoose the parameters k and ℓ to maximize the ratio

52. Setting the parameters k and ℓLet R1 = and R2 = ( )( ) nkn + ℓ n( )n + ℓ + ktn( )n + ℓ + ktn( )n + ℓ + d - knSetting k: Observe that kt ≤ d-k for R2 ≥ 1

53. Setting the parameters k and ℓLet R1 = and R2 = ( )( ) nkn + ℓ n( )n + ℓ + ktn( )n + ℓ + ktn( )n + ℓ + d - knSetting k: Observe that kt ≤ d-k for R2 ≥ 1k = δ · (d/t) ( δ ≤ 1 is a constant )

54. Setting the parameters k and ℓLet R1 = and R2 = ( )( ) nkn + ℓ n( )n + ℓ + ktn( )n + ℓ + ktn( )n + ℓ + d - knSetting k: Observe that kt ≤ d-k for R2 ≥ 1k = δ · (d/t) Setting ℓ:ℓR2R1ℓ* ≈ ndlog ( )nk

55. Setting the parameters k and ℓLet R1 = and R2 = ( )( ) nkn + ℓ n( )n + ℓ + ktn( )n + ℓ + ktn( )n + ℓ + d - knSetting k: Observe that kt ≤ d-k for R2 ≥ 1k = δ · (d/t) Setting ℓ: ℓ = ℓ*“Best” possible lower bound for top fan-in: s ≥ ( ) = ( )nk1-δnt d Ω (d/t)

56. Setting the parameters k and ℓLet R1 = and R2 = ( )( ) nkn + ℓ n( )n + ℓ + ktn( )n + ℓ + ktn( )n + ℓ + d - knSetting k: Observe that kt ≤ d-k for R2 ≥ 1k = δ · (d/t) Setting ℓ: ℓ = ℓ*“Best” possible lower bound for top fan-in: s ≥ ( ) = ( )nk1-δnt d Ω (d/t)Is the upper bound on SPD of the circuit optimum?

57. Setting the parameters k and ℓLet R1 = and R2 = ( )( ) nkn + ℓ n( )n + ℓ + ktn( )n + ℓ + ktn( )n + ℓ + d - knSetting k: Observe that kt ≤ d-k for R2 ≥ 1k = δ · (d/t) Setting ℓ: ℓ = ℓ*“Best” possible lower bound for top fan-in: s ≥ ( ) = ( )nk1-δnt 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 = Q11Q12··· Q1m + … + Qs1Qs2··· Qsm Qij ’s have degree ≤ t

62. Sum of products of low degree polynomials C = Q11Q12··· Q1m + … + Qs1Qs2··· Qsm C is homogeneous; implying m = O(d/t)

63. Sum of products of low degree polynomials C = Q11Q12··· Q1m + … + Qs1Qs2··· Qsm Notation: h-∑∏∑∏[t] circuits

64. Sum of products of low degree polynomials C = Q11Q12··· Q1m + … + Qs1Qs2··· QsmFischers (1994) C’ = Q1 + … + Qs·2m e1 es’

65. Sum of products of low degree polynomials C = Q11Q12··· Q1m + … + Qs1Qs2··· QsmFischers (1994) C’ = Q1 + … + Qs·2m e1 es’ Qi ’s have degree ≤ t

66. Sum of products of low degree polynomials C = Q11Q12··· Q1m + … + Qs1Qs2··· QsmFischers (1994) C’ = Q1 + … + Qs·2m e1 es’… a blow-up by a factor of 2 in the top fan-in where m = O(d/t)m Qi ’s have degree ≤ t

67. Sum of products of low degree polynomials C = Q11Q12··· Q1m + … + Qs1Qs2··· QsmFischers (1994) C’ = Q1 + … + Qs·2m e1 es’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 = Q11Q12··· Q1m + … + Qs1Qs2··· QsmFischers (1994) C’ = Q1 + … + Qs·2m e1 es’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 = Q11Q12··· Q1m + … + Qs1Qs2··· QsmFischers (1994) C’ = Q1 + … + Qs·2m e1 es’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 = Q11Q12··· Q1m + … + Qs1Qs2··· QsmFischers (1994) C’ = Q1 + … + Qs·2m e1 es’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 = Q11Q12··· Q1m + … + Qs1Qs2··· QsmFischers (1994) C’ = Q1 + … + Qs·2m e1 es’Why does shifted partials work for this model?

72. Geometric insight

73. Ideals and varieties Intuition: Let g1 , … , gm ϵ C[x1 ,…, xn] I = ideal generated by g1 , … , gm V = set of common zeroes of g1 , … , gm ‘larger’ the variety V ‘smaller’ the ideal I

74. Ideals and varieties Intuition: Let g1 , … , gm ϵ C[x1 ,…, xn] I = ideal generated by g1 , … , gm V = V(I) ‘larger’ the variety V ‘smaller’ the ideal Ilargeness measured in terms of dimension of a variety

75. Ideals and varieties Intuition: Let g1 , … , gm be the k-th derivatives of T = Qe I = ideal generated by g1 , … , gm V = V(I)Every polynomial in I is divisible by Qe-k So, V(Q) ⊆ V

76. Ideals and varieties Intuition: Let g1 , … , gm be the k-th derivatives of T = Qe I = ideal generated by g1 , … , gm V = V(I)Every polynomial in I is divisible by Qe-k So, V(Q) ⊆ V dim(V) is `large’So, we expect I to be ‘small’

77. Ideals and varieties Intuition: Let g1 , … , gm be the k-th derivatives of T = Qe I = ideal generated by g1 , … , gm V = V(I)Every polynomial in I is divisible by Qe-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[x1 , … , xn]≤L = all degree ≤ L polynomials. I≤L = I ⋂ C[x1 , … , xn]≤L

79. Hilbert function Notation: C[x1 , … , xn]≤L = all degree ≤ L polynomials I≤L = I ⋂ C[x1 , … , xn]≤Lvector space over C

80. Hilbert function Notation: C[x1 , … , xn]≤L = all degree ≤ L polynomials I≤L = I ⋂ C[x1 , … , xn]≤LHilbert function: HI (L) := ( ) - dimC I≤Ln + L nHilbert polynomial: HI (L) is a polynomial in L for sufficiently large L .

81. Hilbert function Notation: C[x1 , … , xn]≤L = all degree ≤ L polynomials I≤L = I ⋂ C[x1 , … , xn]≤LHilbert function: HI (L) := ( ) - dimC I≤Ln + L nHilbert polynomial: HI (L) is a polynomial in L for sufficiently large LTheorem: degree HI (L) = dim V(I) = dim V

82. Hilbert function Notation: C[x1 , … , xn]≤L = all degree ≤ L polynomials I≤L = I ⋂ C[x1 , … , xn]≤L HI (L) = ( ) - dimC I≤Ln + L nTheorem: degree HI (L) = dim V(I) = dim V

83. Hilbert function Notation: C[x1 , … , xn]≤L = all degree ≤ L polynomials I≤L = I ⋂ C[x1 , … , xn]≤L HI (L) = ( ) - dimC I≤Ln + L nTheorem: degree HI (L) = dim V(I) = dim V dimC I≤L = ( ) - HI (L)n + L n

84. Hilbert function Notation: C[x1 , … , xn]≤L = all degree ≤ L polynomials I≤L = I ⋂ C[x1 , … , xn]≤L HI (L) = ( ) - dimC I≤Ln + L nTheorem: degree HI (L) = dim V(I) = dim V dimC I≤L = ( ) - Ldim V + lower order termsn + L n

85. Hilbert function Notation: C[x1 , … , xn]≤L = all degree ≤ L polynomials I≤L = I ⋂ C[x1 , … , xn]≤L HI (L) = ( ) - dimC I≤Ln + L n larger dim V smaller dimC I≤L dimC I≤L = ( ) - Ldim V + lower order termsn + L n

86. Hilbert function Notation: C[x1 , … , xn]≤L = all degree ≤ L polynomials I≤L = I ⋂ C[x1 , … , xn]≤L HI (L) = ( ) - dimC I≤Ln + L n larger dim V smaller dimC I≤L dimC I≤L = ( ) - Ldim V + lower order termsn + L n smaller dimC (x≤ℓ · {g1 , … , gm})… where L = ℓ + deg(T) - k

87. In search of a ‘hard’ polynomial family

88. Candidate ‘hard’ polynomialsPermanent (Permd)Determinant (Detd)Iterated Matrix Multiplication (IMMw,d)Elementary symmetric polynomial (ESymn,d)VNPVPABPm-∑∏∑

89. Candidate ‘hard’ polynomialsPermanent (Permd)Determinant (Detd)Iterated Matrix Multiplication (IMMw,d)Elementary symmetric polynomial (ESymn,d)Gupta-Kayal-Kamath-Saptharishi (2013)

90. SPD of the determinantGupta-Kayal-Kamath-Saptharishi (2013):SPDk,ℓ (Detd) ≥ ( )( ) d+k2kn + ℓ -2k ℓTheorem: Top fan-in lower bound of 2 for h-∑∏∑∏[t] circuits computing Detd . Ω (d/t)

91. SPD of the determinantGupta-Kayal-Kamath-Saptharishi (2013):SPDk,ℓ (Detd) ≥ ( )( ) nkn + ℓ n( )( ) d+k2kn + ℓ -2k ℓTheorem: Top fan-in lower bound of 2 for h-∑∏∑∏[t] circuits computing Detd . Ω (d/t)<<

92. SPD of the determinantGupta-Kayal-Kamath-Saptharishi (2013):SPDk,ℓ (Detd) ≥ ( )( ) nkn + ℓ n( )( ) d+k2kn + ℓ -2k ℓSPDk,ℓ (Detd) ≤ (k+1)2 ( ) ( ) d - 1 kn + ℓ -2k ℓ2<<

93. Candidate ‘hard’ polynomialsPermanent (Permd)Determinant (Detd)Iterated Matrix Multiplication (IMMw,d)Elementary symmetric polynomial (ESymn,d)

94. Candidate ‘hard’ polynomialsPermanent (Permd)Determinant (Detd)Iterated Matrix Multiplication (IMMw,d)Elementary symmetric polynomial (ESymn,d)

95. Candidate ‘hard’ polynomialsPermanent (Permd)Determinant (Detd)Iterated Matrix Multiplication (IMMw,d)Elementary symmetric polynomial (ESymn,d)Conj: (GKKS’13) SPDk,ℓ (Permd) >> SPDk,ℓ (Detd)

96. Candidate ‘hard’ polynomialsPermanent (Permd)Determinant (Detd)Iterated Matrix Multiplication (IMMw,d)Elementary symmetric polynomial (ESymn,d)Conj: (GKKS’13) Open !

97. Candidate ‘hard’ polynomialsNisan-Wigderson (NWn,d)Determinant (Detd)Iterated Matrix Multiplication (IMMw,d)Elementary symmetric polynomial (ESymn,d) Kayal-S.-Saptharishi (2014)

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

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

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

101. Candidate ‘hard’ polynomialsNisan-Wigderson (NWn,d)Determinant (Detd)Iterated Matrix Multiplication (IMMw,d)Elementary symmetric polynomial (ESymn,d)

102. Candidate ‘hard’ polynomialsNisan-Wigderson (NWn,d)Determinant (Detd)Iterated Matrix Multiplication (IMMw,d)Elementary symmetric polynomial (ESymn,d) Fournier-Limaye-Malod-Srinivasan (2014)

103. SPD of Iterated matrix multiplicationFournier-Limaye-Malod-Srinivasan (2014):SPDk,ℓ (IMMw,d) ≥ Theorem: Top fan-in lower bound of n for h-∑∏∑∏[t] circuits computing IMMw,d . Ω (d/t)1/2 · ( ) ( ) w4n + ℓ nkn = w2 . d

104. SPD of Iterated matrix multiplicationFournier-Limaye-Malod-Srinivasan (2014):SPDk,ℓ (IMMw,d) ≥ Theorem: Top fan-in lower bound of n for h-∑∏∑∏[t] circuits computing IMMw,d . Ω (d/t)1/2 · ( ) ( ) w4n + ℓ nk( )( ) nkn + ℓ n<…but quite closen = w2 . d

105. SPD of Iterated matrix multiplicationFournier-Limaye-Malod-Srinivasan (2014):SPDk,ℓ (IMMw,d) ≥ Theorem: Top fan-in lower bound of n for h-∑∏∑∏[t] circuits computing IMMw,d . Ω (d/t)1/2 · ( ) ( ) w4n + ℓ nkObservation: No significant improvement possible on the upper bound for SPDk, ℓ (C) .

106. Candidate ‘hard’ polynomialsNisan-Wigderson (NWn,d)Determinant (Detd)Iterated Matrix Multiplication (IMMw,d)Elementary symmetric polynomial (ESymn,d)

107. Candidate ‘hard’ polynomialsNisan-Wigderson (NWn,d)Determinant (Detd)Iterated Matrix Multiplication (IMMw,d)Elementary symmetric polynomial (ESymn,d) Kamath (2013) & Fournier-Limaye-Mahajan-Srinivasan (2015)

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

109. Candidate ‘hard’ polynomialsNisan-Wigderson (NWn,d)Determinant (Detd)Iterated Matrix Multiplication (IMMw,d)Elementary symmetric polynomial (Esymn,d)

110. Shifted partials of Nisan-Wigderson polynomials

111. Nisan-Wigderson polynomialsDefinition: Nisan-Wigderson family of polynomials: NWn,d := ∑ ∏ xi, h(i)d h(z) in F [z] , deg(h) < ki in [d] Identify the elements of F with {1,2, … , d} Total number of monomials = dk Number of variables n = d2d

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

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

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

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

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

117. Nisan-Wigderson polynomialsDefinition: Nisan-Wigderson family of polynomials: NWn,d := ∑ ∏ xi, h(i)d h(z) in F [z] , deg(h) < ki in [d] A subset of all derivatives: Consider the following set of derivatives: D := { ∂x x … x } D NWn,d := { ∂x x … x NWn,d }2, *1, *kk, *2, *1, *k, *kNotation: D NWn,d = {m1 , m2 , …, m }dkdegree mi = d - k

118. Nisan-Wigderson polynomialsDefinition: Nisan-Wigderson family of polynomials: NWn,d := ∑ ∏ xi, h(i)d h(z) in F [z] , deg(h) < ki in [d] A subset of all derivatives: Consider the following set of derivatives: D := { ∂x x … x } D NWn,d := { ∂x x … x NWn,d }2, *1, *kk, *2, *1, *k, *kNotation: D NWn,d = {m1 , m2 , …, m }dk mi , mj share ≤ k variables

119. Nisan-Wigderson polynomialsDefinition: Nisan-Wigderson family of polynomials: NWn,d := ∑ ∏ xi, h(i)d h(z) in F [z] , deg(h) < ki in [d] A subset of shifted derivatives:x≤ℓ ·D NWn,d = x≤ℓ · {m1 , m2 , …, m } = (x≤ℓ · m1)U(x≤ℓ · m2)U…U(x≤ℓ · m )dkdk

120. Nisan-Wigderson polynomialsDefinition: Nisan-Wigderson family of polynomials: NWn,d := ∑ ∏ xi, h(i)d h(z) in F [z] , deg(h) < ki in [d] A subset of shifted derivatives:x≤ℓ ·D NWn,d = x≤ℓ · {m1 , m2 , …, m } = (x≤ℓ · m1)U(x≤ℓ · m2)U…U(x≤ℓ · m )dkdkB1B2dk B

121. Nisan-Wigderson polynomialsDefinition: Nisan-Wigderson family of polynomials: NWn,d := ∑ ∏ xi, h(i)d h(z) in F [z] , deg(h) < ki in [d] A subset of shifted derivatives:x≤ℓ ·D NWn,d = x≤ℓ · {m1 , m2 , …, m } = (x≤ℓ · m1)U(x≤ℓ · m2)U…U(x≤ℓ · m )dkdkB1B2dk B|Bi| = n + ℓ n( )

122. Nisan-Wigderson polynomialsDefinition: Nisan-Wigderson family of polynomials: NWn,d := ∑ ∏ xi, h(i)d h(z) in F [z] , deg(h) < ki in [d] A subset of shifted derivatives:x≤ℓ ·D NWn,d = B1 U B2 U … U Bdk|Bi| = ( ) n + ℓ n| x≤ℓ ·D NWn,d | = ∑ |Bi| - ½· ∑ |Bi ⋂ Bj|ii≠j

123. Nisan-Wigderson polynomialsDefinition: Nisan-Wigderson family of polynomials: NWn,d := ∑ ∏ xi, h(i)d h(z) in F [z] , deg(h) < ki in [d] A subset of shifted derivatives:x≤ℓ ·D NWn,d = B1 U B2 U … U Bdk|Bi| = ( ) n + ℓ n| x≤ℓ ·D NWn,d | = ∑ |Bi| - ½· ∑ |Bi ⋂ Bj| ≥ dk · ( ) - ??ii≠jn + ℓ n

124. Nisan-Wigderson polynomialsDefinition: Nisan-Wigderson family of polynomials: NWn,d := ∑ ∏ xi, h(i)d h(z) in F [z] , deg(h) < ki in [d] A subset of shifted derivatives:x≤ℓ ·D NWn,d = B1 U B2 U … U Bdk| x≤ℓ ·D NWn,d | = ∑ |Bi| - ½· ∑ |Bi ⋂ Bj| ≥ dk · ( ) - ii≠jn + ℓ nCan upper bound this quantity using disjointness property

125. Nisan-Wigderson polynomialsDefinition: Nisan-Wigderson family of polynomials: NWn,d := ∑ ∏ xi, h(i)d h(z) in F [z] , deg(h) < ki in [d] A subset of shifted derivatives:x≤ℓ ·D NWn,d = B1 U B2 U … U Bdk| x≤ℓ ·D NWn,d | = ∑ |Bi| - ½· ∑ |Bi ⋂ Bj| ≥ dk · ( ) - ii≠jn + ℓ n≤ ½ · dk · ( ) n + ℓ n

126. Nisan-Wigderson polynomialsDefinition: Nisan-Wigderson family of polynomials: NWn,d := ∑ ∏ xi, h(i)d h(z) in F [z] , deg(h) < ki in [d] A subset of shifted derivatives:x≤ℓ ·D NWn,d = B1 U B2 U … U Bdk| x≤ℓ ·D NWn,d | = ∑ |Bi| - ½· ∑ |Bi ⋂ Bj| ≥ ½ · dk · ( ) ii≠jn + ℓ n

127. Nisan-Wigderson polynomialsDefinition: Nisan-Wigderson family of polynomials: NWn,d := ∑ ∏ xi, h(i)d h(z) in F [z] , deg(h) < ki in [d] A subset of shifted derivatives:x≤ℓ ·D NWn,d = B1 U B2 U … U Bdk| x≤ℓ ·D NWn,d | = ∑ |Bi| - ½· ∑ |Bi ⋂ Bj| ≥ ½ · dk · ( ) ii≠jn + ℓ n( )( ) nkn + ℓ nclose

128. Nisan-Wigderson polynomialsDefinition: Nisan-Wigderson family of polynomials: NWn,d := ∑ ∏ xi, h(i)d h(z) in F [z] , deg(h) < ki in [d] Upper bound for |Bi ⋂ Bj| : Let m ϵ |Bi ⋂ Bj|Then m = ri.mi = rj.mj s.t degree ri, rj ≤ ℓ and degree mi, mj ≤ d-k

129. Nisan-Wigderson polynomialsDefinition: Nisan-Wigderson family of polynomials: NWn,d := ∑ ∏ xi, h(i)d h(z) in F [z] , deg(h) < ki in [d] Upper bound for |Bi ⋂ Bj| : Let m ϵ |Bi ⋂ Bj|Then m = ri.mi = rj.mj s.t degree ri, rj ≤ ℓ and degree mi, mj ≤ d-k Since mi and mj share at most k variables,m = r · · mj s.t. degree r ≤ ℓ - (d-2k) migcd (mi , mj)

130. Nisan-Wigderson polynomialsDefinition: Nisan-Wigderson family of polynomials: NWn,d := ∑ ∏ xi, h(i)d h(z) in F [z] , deg(h) < ki in [d] Upper bound for |Bi ⋂ Bj| : Let m ϵ |Bi ⋂ Bj|Then m = ri.mi = rj.mj s.t degree ri, rj ≤ ℓ and degree mi, mj ≤ d-k Since mi and mj share at most k variables,m = r · · mj s.t. degree r ≤ ℓ - (d-2k) migcd (mi , mj)has degree ≥ d – 2k

131. Nisan-Wigderson polynomialsDefinition: Nisan-Wigderson family of polynomials: NWn,d := ∑ ∏ xi, h(i)d h(z) in F [z] , deg(h) < ki in [d] Upper bound for |Bi ⋂ Bj| : Let m ϵ |Bi ⋂ Bj|Then m = ri.mi = rj.mj s.t degree ri, rj ≤ ℓ and degree mi, mj ≤ d-k Since mi and mj share at most k variables,m = r · · mj s.t. degree r ≤ ℓ - (d-2k) migcd (mi , mj)Hence, |Bi ⋂ Bj| ≤ ( ) n + ℓ - (d – 2k)n

132. Nisan-Wigderson polynomialsDefinition: Nisan-Wigderson family of polynomials: NWn,d := ∑ ∏ xi, h(i)d h(z) in F [z] , deg(h) < ki in [d] Upper bound for |Bi ⋂ Bj| : |Bi ⋂ Bj| ≤ ( ) n + ℓ - (d – 2k)n

133. Nisan-Wigderson polynomialsDefinition: Nisan-Wigderson family of polynomials: NWn,d := ∑ ∏ xi, h(i)d h(z) in F [z] , deg(h) < ki in [d] Upper bound for |Bi ⋂ Bj| : |Bi ⋂ Bj| ≤ ( ) ½· ∑ |Bi ⋂ Bj| ≤ d2k/2 · ( )n + ℓ - (d – 2k)ni≠jn + ℓ - (d – 2k)n

134. Nisan-Wigderson polynomialsDefinition: Nisan-Wigderson family of polynomials: NWn,d := ∑ ∏ xi, h(i)d h(z) in F [z] , deg(h) < ki in [d] Upper bound for |Bi ⋂ Bj| : |Bi ⋂ Bj| ≤ ( ) ½· ∑ |Bi ⋂ Bj| ≤ d2k/2 · ( )n + ℓ - (d – 2k)ni≠jn + ℓ - (d – 2k)n ≤ dk/2 · ( ) at ℓ ≈ ℓ*n + ℓ n(the optimum choice of ℓ)Q.E.D

135. A few questions…

136. QuestionsCan we improve the following lower bounds?1. (Baur-Strassen’83) Ω(n log d) lower bound for general circuits.2. (Kalorkoti’85) Ω(n2) lower bound for general formulas.3. (Mignon-Ressayre’04) Ω(n2) lower bound for the determinantal complexity of Permn .4. (Shpilka-Wigderson’01) Ω(n2) lower bound for general depth three circuits.

137. QuestionsCan we improve the following lower bounds?1. (Baur-Strassen’83) Ω(n log d) lower bound for general circuits.2. (Kalorkoti’85) Ω(n2) lower bound for general formulas.3. (Mignon-Ressayre’04) Ω(n2) lower bound for the determinantal complexity of Permn .4. (Shpilka-Wigderson’01) Ω(n2) 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. QuestionsCan we improve the following lower bounds?1. (Baur-Strassen’83) Ω(n log d) lower bound for general circuits.2. (Kalorkoti’85) Ω(n2) lower bound for general formulas.3. (Mignon-Ressayre’04) Ω(n2) lower bound for the determinantal complexity of Permn .4. (Shpilka-Wigderson’01) Ω(n2) 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 oneTwo 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 oneTwo 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 IMMw,d . Ω (d/t)n = w2 . 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 oneh-formulash-∑∏∑∏ABPh-∑∏∑∏[t]hard for

143. There’s one more…h-formulash-∑∏∑∏ABPh-∑∏∑∏[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-formulash-∑∏∑∏ABPh-∑∏∑∏[t]hard for (how to show this?)

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

146. Variants of shifted partials

147. Kumar-Saraf separationTheorem: There is an explicit family of polynomials {fn} computable by poly(n) size h-∑∏∑∏ circuits s.t.for ω(log d) < t < d/50, any h-ΣΠΣΠ[t] circuit computing fn has top fan-in nΩ(d/t) . h-formulash-∑∏∑∏ABPh-∑∏∑∏[t]hard for

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

149. Kumar-Saraf separationTheorem: There is an explicit family of polynomials {fn} computable by poly(n) size h-∑∏∑∏ circuits s.t.for ω(log d) < t < d/50, any h-ΣΠΣΠ[t] circuit computing fn has top fan-in nΩ(d/t) . h-formulash-∑∏∑∏ABPh-∑∏∑∏[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 = Ct + Cjunk 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 = Ct + Cjunk s.t.μ(C) ≤ μ(Ct ) + μ(Cjunk) and μ(Cjunk ) = 0A measure obeying subadditivity

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

153. The idea at work: an exampleRecall: Classical [NW95] lower bound for h-∑∏∑ circuits. C = ∑ ℓi1ℓi2…ℓid (ℓij are linear forms)i=1…k=(d-1)/2 alternate layers of variables w.r.t which derivatives are taken  call this set SIMMw,ds

154. The idea at work: an exampleRecall: Classical [NW95] lower bound for h-∑∏∑ circuits. C = ∑ ℓi1ℓi2…ℓid (ℓij are linear forms)i=1…k=(d-1)/2 alternate layers of variables w.r.t which derivatives are taken  call this set SsCircuit upper bound: PDk(C) ≤ s · ( )dkIMM lower bound: PDk(IMMw,d) ≥ w2k

155. The idea at work: an exampleRecall: Classical [NW95] lower bound for h-∑∏∑ circuits. C = ∑ ℓi1ℓi2…ℓid (ℓij are linear forms)i=1sCircuit upper bound: PDk(C) ≤ s · ( )dkIMM lower bound: PDk(IMMw,d) ≥ w2k Top fan-in lower bound: If C = IMMw,d then s = wΩ(d) provided w ≥ d .

156. The idea at work: an exampleConsider a slightly general model (Srikanth’s model) C = ∑ Qi1Qi2…Qid (Qij are sums of univariates) i=1sf1(x1) + f2(x2) + … + fn(xn)

157. The idea at work: an exampleConsider a slightly general model (Srikanth’s model) C = ∑ Qi1Qi2…Qid (Qij are sums of univariates) i=1sf1(x1) + f2(x2) + … + fn(xn) Note: PDk(Qi1Qi2…Qid) can be as high as ( ).nk…think of Qi1 = … = Qid = x1 + …. + xn22

158. The idea at work: an exampleConsider a slightly general model C = ∑ Qi1Qi2…Qid (Qij are sums of univariates) i=1sSplit Qij = ℓij + Jijaffine formSum of powers of variables where every power is ≥ 2

159. The idea at work: an exampleConsider a slightly general model C = ∑ ℓi1ℓi2 …ℓid + Cjunk i=1sSplit Qij = ℓij + JijEvery monomial has a variable with degree ≥ 2

160. The idea at work: an exampleConsider a slightly general model C = ∑ ℓi1ℓi2 …ℓid + Cjunk i=1sEvery monomial has a variable with degree ≥ 2Projection map. πS : f multilinear (f) x = 0 for x ϵ S S ⊆ [n]

161. The idea at work: an exampleConsider a slightly general model C = ∑ ℓi1ℓi2 …ℓid + Cjunk i=1sEvery monomial has a variable with degree ≥ 2Projection map. πS : f multilinear (f) x = 0 for x ϵ S Projected PD measure map. PPDS, k (f) := dim (πS ( ∂ f )) Skset of multilinear derivatives w.r.t variables in S

162. The idea at work: an exampleConsider a slightly general model C = ∑ ℓi1ℓi2 …ℓid + Cjunk i=1sEvery monomial has a variable with degree ≥ 2Projection map. πS : f multilinear (f) x = 0 for x ϵ S Projected PD measure map. PPDS, k (f) := dim (πS ( ∂ f )) SkNote. PPDS, k (f) obeys subadditivity as S is a fixed set

163. The idea at work: an exampleApply the measure on the circuit (for any S, k) PPDS, k (C) ≤ PPDS, k ( ∑ ℓi1ℓi2 …ℓid ) + PPDS, k (Cjunk) i=1s

164. The idea at work: an exampleApply the measure on the circuit (for any S, k) PPDS, k (C) ≤ PPDS, k ( ∑ ℓi1ℓi2 …ℓid ) + PPDS, k (Cjunk) i=1s0Either 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 exampleApply the measure on the circuit (for any S, k) PPDS, k (C) ≤ PPDS, k ( ∑ ℓi1ℓi2 …ℓid ) i=1s… although, C ≠ ∑ ℓi1ℓi2 …ℓid

166. The idea at work: an exampleApply the measure on the circuit (for any S, k) PPDS, k (C) ≤ PPDS, k ( ∑ ℓi1ℓi2 …ℓid ) ≤ s · ( )i=1sdk

167. The idea at work: an exampleApply the measure on the circuit PPDS, k (C) ≤ PPDS, k ( ∑ ℓi1ℓi2 …ℓid ) ≤ s · ( ) i=1s…k=(d-1)/2 alternate layers of variables w.r.t which derivatives are taken  call this set SIMMw,d PPDS, k (IMMw,d) = w2k dkTake S and k as in the figure

168. The idea at work: an exampleApply the measure on the circuit PPDS, k (C) ≤ PPDS, k ( ∑ ℓi1ℓi2 …ℓid ) ≤ s · ( ) i=1s…k=(d-1)/2 alternate layers of variables w.r.t which derivatives are taken  call this set SIMMw,d PPDS, k (IMMw,d) = w2k dk Top fan-in lower bound: if C = IMMw,d then s = wΩ(d)

169. What we learn from the exampleEvery monomial in Qij has support at most 1 .Every monomial in ℓij has degree at most 1 .The projection map πlow support low degree

170. What we learn from the exampleEvery monomial in Qij has support at most 1 .Every monomial in ℓij 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 boundKayal-Limaye-S.-Srinivasan (2014)

172. From h- ΣΠΣΠ to h- ΣΠΣΠ{t}Let C = ∑ Qi1Qi2…Qim be a h-ΣΠΣΠ circuit.Random restriction. Set xi = 0 i.a.r with probability 1 – n-ε. Call this map σR. sufficiently small constantsi=1

173. From h- ΣΠΣΠ to h- ΣΠΣΠ{t}Let C = ∑ Qi1Qi2…Qim be a h-ΣΠΣΠ circuit.Random restriction. Set xi = 0 i.a.r with probability 1 – n-ε. Call this map σR. si=1 Then w.h.p σR (C) = ∑ σR (Qi1) . σR (Qi2) … σR(Qim) is a h-ΣΠΣΠ{t} circuit for a suitable choice of t.sum of product of support-t polynomialsi

174. From h- ΣΠΣΠ to h- ΣΠΣΠ{t}Let C = ∑ Qi1Qi2…Qim be a h-ΣΠΣΠ circuit. si=1 Then w.h.p σR (C) = ∑ σR (Qi1) . σR (Qi2) … σR(Qim) is a h-ΣΠΣΠ{t} circuit for a suitable choice of t. S := total no. of monomials in the Qij’s

175. From h- ΣΠΣΠ to h- ΣΠΣΠ{t}Let C = ∑ Qi1Qi2…Qim be a h-ΣΠΣΠ circuit. si=1 Then w.h.p σR (C) = ∑ σR (Qi1) . σR (Qi2) … σR(Qim) 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 = ∑ Qi1Qi2…Qim be a h-ΣΠΣΠ circuit. si=1 Then w.h.p σR (C) = ∑ σR (Qi1) . σR (Qi2) … σR(Qim) 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 = ∑ Qi1Qi2…Qim be a h-ΣΠΣΠ circuit. si=1 Then w.h.p σR (C) = ∑ σR (Qi1) . σR (Qi2) … σR(Qim) 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 σR (C) with support ≥ t ] ≤ S · n-εtObservation. Unless S > nε/2 . t , w.h.p σR (C) is a h-ΣΠΣΠ{t} circuit

178. From h- ΣΠΣΠ to h- ΣΠΣΠ{t}Let C = ∑ Qi1Qi2…Qim be a h-ΣΠΣΠ circuit. si=1 Then w.h.p σR (C) = ∑ σR (Qi1) . σR (Qi2) … σR(Qim) 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 nΩ(d/t) for h-ΣΠΣΠ{t} A size lower bound of min {nΩ(t) , nΩ(d/t)} for h-ΣΠΣΠ

180. Prove lower bound for h-ΣΠΣΠ{t} circuits Observation: A top fan-in lower bound of nΩ(d/t) for h-ΣΠΣΠ{t} A size lower bound of min {nΩ(t) , nΩ(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 nΩ(d/t) for h-ΣΠΣΠ{t} A size lower bound of min {nΩ(t) , nΩ(d/t)} for h-ΣΠΣΠtop 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 nΩ(d/t) for h-ΣΠΣΠ{t} A size lower bound of nΩ(√d) for h-ΣΠΣΠ…setting t = √d

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

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

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

187. Projected Shifted Partials PSPDk,ℓ (f) := dim (π (x=ℓ. ∂k f) )Subadditivity.PSPDk,ℓ (f1 + f2) ≤ PSPDk,ℓ (f1) + PSPDk,ℓ (f2)How large can PSPD be? PSPDk,ℓ (f) ≤ min ( )·( ) , ( ) nknℓnℓ + d - k

188. Projected Shifted Partials PSPDk,ℓ (f) := dim (π (x=ℓ. ∂k f) )Subadditivity.PSPDk,ℓ (f1 + f2) ≤ PSPDk,ℓ (f1) + PSPDk,ℓ (f2)How large can PSPD be? PSPDk,ℓ (f) ≤ min ( )·( ) , ( ) nknℓnℓ + d - k…assume that there’s an explicit f with highest possible PSPD

189. “Reducing” h-ΣΠΣΠ{t} to h-ΣΠΣΠ[2t] circuitsC = Q11Q12…Q1m + … + Qs1Qs2…Qsmsupport of every monomial in every Qij is bounded by t

190. “Reducing” h-ΣΠΣΠ{t} to h-ΣΠΣΠ[2t] circuitsC = Q11Q12…Q1m + … + Qs1Qs2…QsmSplit: Qij = Q’ij + Every variable in every monomial has degree 2 or lessEvery monomial has a variable with degree 3 or more

191. “Reducing” h-ΣΠΣΠ{t} to h-ΣΠΣΠ[2t] circuitsC = Q11Q12…Q1m + … + Qs1Qs2…QsmSplit: Qij = Q’ij + Qi1Qi2…Qim = Q’i1Q’i2…Q’im + Every monomial has a variable with degree 3 or more

192. “Reducing” h-ΣΠΣΠ{t} to h-ΣΠΣΠ[2t] circuitsC = Q11Q12…Q1m + … + Qs1Qs2…QsmSplit: Qij = Q’ij + Qi1Qi2…Qim = Q’i1Q’i2…Q’im + PSPDk,ℓ (Qi1Qi2…Qim) ≤ PSPDk,ℓ (Q’i1Q’i2…Q’im) +PSPDk,ℓ ( )

193. “Reducing” h-ΣΠΣΠ{t} to h-ΣΠΣΠ[2t] circuitsC = Q11Q12…Q1m + … + Qs1Qs2…QsmSplit: Qij = Q’ij + Qi1Qi2…Qim = Q’i1Q’i2…Q’im + PSPDk,ℓ (Qi1Qi2…Qim) ≤ PSPDk,ℓ (Q’i1Q’i2…Q’im) +PSPDk,ℓ ( )0

194. “Reducing” h-ΣΠΣΠ{t} to h-ΣΠΣΠ[2t] circuitsC = Q11Q12…Q1m + … + Qs1Qs2…QsmSplit: Qij = Q’ij + Qi1Qi2…Qim = Q’i1Q’i2…Q’im + PSPDk,ℓ (Qi1Qi2…Qim) ≤ PSPDk,ℓ (Q’i1Q’i2…Q’im)degree of every Q’ij ≤ 2t

195. Upper bounding PSPD of a h-ΣΠΣΠ[2t] circuit∂k Qi1…Qim = Qi1Qi2…Q ik …Qim + Qi1Qi2…QikQ i(k+1)…Qim + … x......= Qi(k+1)… Qim + Qi1Qi(k+2)…Qim + …degree ≤ 2ktReusing notation: Call Q’ij as Qij

196. Upper bounding PSPD of a h-ΣΠΣΠ[2t] circuit∂k Qi1…Qim = Qi1Qi2…Q ik …Qim + Qi1Qi2…QikQ i(k+1)…Qim + … x......= Qi(k+1)… Qim + Qi1Qi(k+2)…Qim + …Reusing notation: Call Q’ij as Qij u . ∂k Qi1…Qim = Qi(k+1)… Qim + Qi1Qi(k+2)…Qim + …degree = ℓ degree ≤ ℓ + 2kt

197. Upper bounding PSPD of a h-ΣΠΣΠ[2t] circuit∂k Qi1…Qim = Qi1Qi2…Q ik …Qim + Qi1Qi2…QikQ i(k+1)…Qim + … x......= Qi(k+1)… Qim + Qi1Qi(k+2)…Qim + …Reusing notation: Call Q’ij as Qij u . ∂k Qi1…Qim = Qi(k+1)… Qim + Qi1Qi(k+2)…Qim + …π(u.∂k Qi1…Qim)=π( Qi(k+1)… Qim)+π( Qi1Qi(k+2)…Qim)…multilinear and degree ≤ ℓ + 2kt

198. Upper bounding PSPD of a h-ΣΠΣΠ[2t] circuit∂k Qi1…Qim = Qi1Qi2…Q ik …Qim + Qi1Qi2…QikQ i(k+1)…Qim + … x......= Qi(k+1)… Qim + Qi1Qi(k+2)…Qim + …Reusing notation: Call Q’ij as Qij u . ∂k Qi1…Qim = Qi(k+1)… Qim + Qi1Qi(k+2)…Qim + …π(u.∂k Qi1…Qim)=π( Qi(k+1)… Qim)+π( Qi1Qi(k+2)…Qim)…PSPDk,ℓ (C) ≤ s . ( ) . ( )mknℓ + 2kt

199. Upper bounding PSPD of a h-ΣΠΣΠ[2t] circuit∂k Qi1…Qim = Qi1Qi2…Q ik …Qim + Qi1Qi2…QikQ i(k+1)…Qim + … x......= Qi(k+1)… Qim + Qi1Qi(k+2)…Qim + …Reusing notation: Call Q’ij as Qij u . ∂k Qi1…Qim = Qi(k+1)… Qim + Qi1Qi(k+2)…Qim + …π(u.∂k Qi1…Qim)=π( Qi(k+1)… Qim)+π( Qi1Qi(k+2)…Qim)…PSPDk,ℓ (C) ≤ s . ( ) . ( )d/2tknℓ + 2kt

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

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

202. NW has near optimal PSPDd2Theorem (KLSS’14) For r = d/3, t = √d, k = δ.√d and ℓ ≈ ℓ* PSPDk,ℓ(NWn,d) ≥ n-9 · min { ( ).( ) , ( )}nknℓnℓ + d - kNisan-Wigderson family of polynomials: NWn,d := ∑ ∏ xi, h(i) h(z) in F [z] , deg(h) < ri in [d]

203. NW has near optimal PSPDd2Theorem (KLSS’14) Any h-ΣΠΣΠ circuit computing NWn,d has size nΩ(√d) Nisan-Wigderson family of polynomials: NWn,d := ∑ ∏ xi, h(i) h(z) in F [z] , deg(h) < ri in [d]

204. Proof ideaIssue: k-th order derivatives are not monomials. PSPDk,ℓ (NWn,d) = rank (M)nknℓM := ( ).( )rows π (x=ℓ. ∂k NW)(0/1)-matrix of coefficients nℓ + d - k ( )columns

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

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

207. Proof ideaAlon’s bound. rank (B) ≥ nknℓM := ( ).( )rows π (x=ℓ. ∂k NW)(0/1)-matrix of coefficients nℓ + d - k ( )columnsTr (B)2Tr (B2)

208. NW has near optimal PSPDh-formulash-∑∏∑∏ABPhard for VNP

209. IMMw,d has high PSPDh-formulash-∑∏∑∏ABPhard for VNPKumar-Saraf (2014)…over any field

210. IMMw,d hard for h-formulas ?h-formulash-∑∏∑∏ABPhard for ?VNPOpen!

211. A similar questionh-formulash-∑∏∑∏ABPhard for ?VNPformulasABPhard for ?Non-commutativeOpen!

212. IMMw,d hard for m-formulas ?m-formulasm-∑∏∑∏ABPhard forIMMw,dformulasABPhard forNon-commutativem stands for multilinear

213. IMMw,d hard for m-formulas ?m-formulasm-∑∏∑∏ABPhard for ?Open!m stands for multilinearIMMw,d

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

215. IMMw,d hard for m-∑∏∑∏ circuits ?m-formulasm-∑∏∑∏ABPhard for ?IMMw,d

216. IMMw,d hard for m-∑∏∑∏ circuitsm-formulasm-∑∏∑∏ABPhard forIMMw,dKayal-S.-Tavenas (2016)

217. IMMw,d hard for m-∑∏∑∏ circuitsm-formulasm-∑∏∑∏ABPhard forIMMw,dKayal-S.-Tavenas (2016)Issue. Not clear how to use a multilinear projection map π to exploit any weakness of a m-circuit.

218. IMMw,d hard for m-∑∏∑∏ circuitsm-formulasm-∑∏∑∏ABPhard forIMMw,dKayal-S.-Tavenas (2016)… turns out that another variant of the shifted partials measure helps.

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

220. A bit of historym-formulasm-∑∏∑∏ABPhard forDetdIMMw,dRaz (2004): Any m-formula computing Detd has size dΩ(log d).Detd and IMMw,d are both complete for ABP.

221. A bit of historym-formulasm-∑∏∑∏ABPDetdIMMw,dDetd and IMMw,d are both complete for ABP.Detd lower bound doesn’t readily imply IMMw,d lower boundReduction doesn’t preserve multilinearity

222. A bit of historym-formulasm-∑∏∑∏ABPDetdIMMw,dDetd and IMMw,d are both complete for ABP.Detd lower bound doesn’t readily imply IMMw,d lower bound.The converse is true.Reduction preserves multilinearity

223. A bit of historyRaz (2004): Any m-formula computing Detd has size dΩ(log d).Raz-Yehudayoff (2008): Lower bound for constant depth m-formulas computing Detd .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 historyRaz (2004): Any m-formula computing Detd has size dΩ(log d).Raz-Yehudayoff (2008): Lower bound for constant depth m-formulas computing Detd .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 historyRaz (2004): Any m-formula computing Detd has size dΩ(log d).Raz-Yehudayoff (2008): Lower bound for constant depth m-formulas computing Detd .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: IMMw,d hard for m-∑∏∑ circuits…..................12ky variables

227. Example: IMMw,d hard for m-∑∏∑ circuits…..................12kx variables

228. Example: IMMw,d hard for m-∑∏∑ circuits…..................12kedges labeled by 1

229. Example: IMMw,d hard for m-∑∏∑ circuits…..................12kNumber of variables: |y| = w2 k |x| = 2wkNote the difference (skew) in the number of x and y variables

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

231. Example: IMMw,d hard for m-∑∏∑ circuitsSkewed partials: [ Kayal-Nair-S. (2016) ] SkPk , y(f) = dim (∂y f |y=0 )kSubadditivity: SkPk , y(f1 + f2) ≤ SkPk , y(f1) + SkPk , y(f2)

232. Example: IMMw,d hard for m-∑∏∑ circuitsSkewed partials: [ Kayal-Nair-S. (2016) ] SkPk , y(f) = dim (∂y f |y=0 )kSubadditivity: SkPk , y(f1 + f2) ≤ SkPk , y(f1) + SkPk , y(f2)Observation: SkPk , y(Iw,k) = w2k

233. Example: IMMw,d hard for m-∑∏∑ circuitsSkewed partials of a m-ΣΠΣ circuit: C = ∑ ℓi1ℓi2…ℓin i=1s

234. Example: IMMw,d hard for m-∑∏∑ circuitsSkewed partials of a m-ΣΠΣ circuit: A term T = ℓI ℓ2 … ℓn = ℓI(x , y) . ℓ2(x , y) … ℓ|x|(x , y) . Q(y) C = ∑ ℓi1ℓi2…ℓin i=1sx-free

235. Example: IMMw,d hard for m-∑∏∑ circuitsSkewed partials of a m-ΣΠΣ circuit: A term T = ℓI ℓ2 … ℓn = ℓI(x , y) . ℓ2(x , y) … ℓ|x|(x , y) . Q(y) C = ∑ ℓi1ℓi2…ℓin i=1sObservation: Since the y-variables are set to zero by SkPk , ySkPk , y(T) ≤ ( )|x|k

236. Example: IMMw,d hard for m-∑∏∑ circuitsSkewed partials of a m-ΣΠΣ circuit: A term T = ℓI ℓ2 … ℓn = ℓI(x , y) . ℓ2(x , y) … ℓ|x|(x , y) . Q(y) C = ∑ ℓi1ℓi2…ℓin i=1sObservation: Since the y-variables are set to zero by SkPk , ySkPk , y(T) ≤ ( )|x|kSkPk , y(C) ≤ s . ( ) ≤ s . (2ew)k2wkk

237. Example: IMMw,d hard for m-∑∏∑ circuitsSkPk , y(Iw,k) = w2kSkPk , y(C) ≤ s . (2ew)kObservation: Any m-∑∏∑ circuit computing Iw,k has top fan-in wΩ(k) .

238. Example: IMMw,d hard for m-∑∏∑ circuitsSkPk , y(Iw,k) = w2kSkPk , y(C) ≤ s . (2ew)kObservation: Any m-∑∏∑ circuit computing IMMw,d has top fan-in wΩ(d) .

239. Example: IMMw,d hard for m-∑∏∑ circuitsSkPk , y(Iw,k) = w2kSkPk , y(C) ≤ s . (2ew)kObservation: Any m-∑∏∑ circuit computing IMMw,d has top fan-in wΩ(d) .Summary: What helps is |y| >> |x| .

240. IMMw,d hard for m-∑∏∑∏ circuitsShifted Skewed Partials: [ Kayal-S.-Tavenas (2016) ] SSPk, y, ℓ(f) = dim (x≤ℓ . (∂y f |y=0 ))kTheorem: Any m-∑∏∑∏ circuit computing IMMw,d has size wΩ(√d) .

241. IMMw,d hard for m-∑∏∑∏ circuitsShifted Skewed Partials: [ Kayal-S.-Tavenas (2016) ] SSPk, y, ℓ(f) = dim (x≤ℓ . (∂y f |y=0 ))kTheorem: Any m-∑∏∑∏ circuit computing IMMw,d has size wΩ(√d) .Idea: Since C is multilinear, SSPk, y, ℓ(C) depends only on |x| (and not |y|).

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

243. A few problems

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

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

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

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

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

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

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