Sums of products of polynomials of low-arity - lower bounds and PIT

Transcription

1 Sums of products of polynomials of low-arity - lower bounds and PIT Mrinal Kumar Joint work with Shubhangi Saraf

2 Plan for the talk

3 Plan for the talk Model of computation

4 Plan for the talk Model of computation Prior work

5 Plan for the talk Model of computation Prior work Results

6 Plan for the talk Model of computation Prior work Results Main Techniques

7 Plan for the talk Model of computation Prior work Results Main Techniques Results

8 Plan for the talk Model of computation Prior work Results Main Techniques Results Proof sketch

9 Plan for the talk Model of computation Prior work Results Main Techniques Results Proof sketch Open questions

10 Notation n - Number of variables

11 Notation n - Number of variables d - Degree of the polynomial (d < poly(n))

12 Notation n - Number of variables d - Degree of the polynomial (d < poly(n)) r - parameter to be defined shortly, r = n 0.99

13 ( ) r Circuits ΣΠ ΣΠ T C = Q i1 Q i2!q ik i=1

14 ( ) r Circuits ΣΠ ΣΠ T C = Q i1 Q i2!q ik i=1 Q - sparse polynomials*

15 ( ) r Circuits ΣΠ ΣΠ T C = Q i1 Q i2!q ik i=1 Q - sparse polynomials* Each Q depends on at most r variables

16 ( ) r Circuits ΣΠ ΣΠ T C = Q i1 Q i2!q ik i=1 Q - sparse polynomials* Each Q depends on at most r variables Formal degree of C could be >> d (non-homogeneous)

17 ( ) r Circuits ΣΠ ΣΠ T C = Q i1 Q i2!q ik i=1 Q - sparse polynomials* Each Q depends on at most r variables Formal degree of C could be >> d (non-homogeneous) Generalizes the (well studied) sums of products of univariates

18 ( ) r Circuits ΣΠ ΣΠ T C = Q i1 Q i2!q ik i=1 Q - sparse polynomials* Each Q depends on at most r variables Formal degree of C could be >> d (non-homogeneous) Generalizes the (well studied) sums of products of univariates + depth-3 circuits with low bottom fan-in

19 Prior results

20 SOP of univariates (r = 1)

21 SOP of univariates (r = 1) Lower bounds - exponential lower bounds via evaluation dimension [Nisan, Saxena]

22 SOP of univariates (r = 1) Lower bounds - exponential lower bounds via evaluation dimension [Nisan, Saxena] PIT - Poly-time non-blackbox [Raz-Shpilka], Quasipoly time blackbox [Forbes-Shpilka, Agrawal-Saha-Saxena, ]

23 SOP of univariates (r = 1) Lower bounds - exponential lower bounds via evaluation dimension [Nisan, Saxena] PIT - Poly-time non-blackbox [Raz-Shpilka], Quasipoly time blackbox [Forbes-Shpilka, Agrawal-Saha-Saxena, ] Don t seem to extend to even r = 2

24 Evaluation dimension

25 Evaluation dimension Key idea underlying the results for r = 1 case.

26 Evaluation dimension Key idea underlying the results for r = 1 case. View P(X) as F[Z][Y], for some partition of variables into Y, Z.

27 Evaluation dimension Key idea underlying the results for r = 1 case. View P(X) as F[Z][Y], for some partition of variables into Y, Z. Collect all the coefficients of monomials in Y variables.

28 Evaluation dimension Key idea underlying the results for r = 1 case. View P(X) as F[Z][Y], for some partition of variables into Y, Z. Collect all the coefficients of monomials in Y variables. These are polynomials in Z variables.

29 Evaluation dimension Key idea underlying the results for r = 1 case. View P(X) as F[Z][Y], for some partition of variables into Y, Z. Collect all the coefficients of monomials in Y variables. These are polynomials in Z variables. Look at the dimension of their linear span.

30 Evaluation dimension Key idea underlying the results for r = 1 case. View P(X) as F[Z][Y], for some partition of variables into Y, Z. Collect all the coefficients of monomials in Y variables. These are polynomials in Z variables. Look at the dimension of their linear span.

31 Evaluation dimension Key observation : For r = 1, for every partition of variables, eval dimension of a circuits is small. ΣΠ(ΣΠ) 1

32 Evaluation dimension Key observation : For r = 1, for every partition of variables, eval dimension of a circuits is small. ΣΠ(ΣΠ) 1 ΣΠ(ΣΠ) 2 A natural approach for r = 2 case - do have small eval dimension for some partition? circuits

33 Evaluation dimension Key observation : For r = 1, for every partition of variables, eval dimension of a circuits is small. ΣΠ(ΣΠ) 1 ΣΠ(ΣΠ) 2 A natural approach for r = 2 case - do have small eval dimension for some partition? circuits [Forbes 15] There are poly(n) sized circuits which have nearly full eval dimension for all partitions. ΣΠ(ΣΠ) 2

34 r > 1 T C = Q i1 Q i2!q ik i=1 Degree of each Qij = 1

35 r > 1 T C = Q i1 Q i2!q ik i=1 Degree of each Qij = 1 [Kayal, Saha 15] An n Ω( d ) lower bound.

36 r > 1 T C = Q i1 Q i2!q ik i=1 Degree of each Qij = 1 [Kayal, Saha 15] An n Ω( d ) lower bound. Based on shifted partial derivative based methods.

37 r > 1 T C = Q i1 Q i2!q ik i=1 Degree of each Qij = 1 [Kayal, Saha 15] An n Ω( d ) lower bound. Based on shifted partial derivative based methods. This talk..both degrees and r are larger than 1.

38 Our results

39 Lower bound for r > 1, bottom degree > 1 For every constant, there exists an explicit family { } of polynomials, where is of degree P n ε > 0 d in n variables, such that any which computes has size at least. P n ΣΠ(ΣΠ) n1 ε P n n Ω( d ) circuit

40 Lower bound for r > 1, bottom degree > 1 For every constant, there exists an explicit family { } of polynomials, where is of degree P n ε > 0 d in n variables, such that any which computes has size at least. P n ΣΠ(ΣΠ) n1 ε P n n Ω( d ) circuit n vs d depends on the value of r.

41 Lower bound for r > 1, bottom degree > 1 No bound on the bottom degree. For every constant, there exists an explicit family { } of polynomials, where is of degree P n ε > 0 d in n variables, such that any which computes has size at least. P n ΣΠ(ΣΠ) n1 ε P n n Ω( d ) circuit n vs d depends on the value of r.

42 PIT via hardness vs randomness If top fan-in, individual degree, and r are at most poly(log n), then there is a quasi polynomial time deterministic identity test.

43 Field of play

44 Field of play The proofs in this paper require the characteristic of the field to be large, or zero.

45 Field of play The proofs in this paper require the characteristic of the field to be large, or zero. Forbes shows that the result holds over all fields.

46 Proof sketch

47 Depth-3 circuits of low arity T C = Q i1 Q i2!q ik i=1

48 Depth-3 circuits of low arity T C = Q i1 Q i2!q ik i=1 1. Each Q = L + 1, for a linear form L -> C is a nonhomogeneous depth-3 circuit with low bottom fan-in.

49 Depth-3 circuits of low arity T C = Q i1 Q i2!q ik i=1 1. Each Q = L + 1, for a linear form L -> C is a nonhomogeneous depth-3 circuit with low bottom fan-in. 2. Works over a field of characteristic zero.

50 Depth-3 circuits of low arity T C = Q i1 Q i2!q ik i=1 1. Each Q = L + 1, for a linear form L -> C is a nonhomogeneous depth-3 circuit with low bottom fan-in. 2. Works over a field of characteristic zero. 3. The hard polynomial is homogeneous of degree d.

51 Depth-3 circuits of low arity T C = Q i1 Q i2!q ik i=1 1. Each Q = L + 1, for a linear form L -> C is a nonhomogeneous depth-3 circuit with low bottom fan-in. 2. Works over a field of characteristic zero. 3. The hard polynomial is homogeneous of degree d. 4. Degree d is at most log n (simplifies the proof idea).

52 Plan (Kayal-Saha)

53 Plan (Kayal-Saha) 1. Reduce from non-homogeneous depth-3 circuits with low bottom fan-in to homogeneous depth-5 circuits with low bottom fan-in.

54 Plan (Kayal-Saha) 1. Reduce from non-homogeneous depth-3 circuits with low bottom fan-in to homogeneous depth-5 circuits with low bottom fan-in. 2. Use random restrictions to reduce from homogeneous depth-5 circuits with low bottom fan-in to homogeneous depth-4 circuits with low bottom support.

55 Plan (Kayal-Saha) 1. Reduce from non-homogeneous depth-3 circuits with low bottom fan-in to homogeneous depth-5 circuits with low bottom fan-in. 2. Use random restrictions to reduce from homogeneous depth-5 circuits with low bottom fan-in to homogeneous depth-4 circuits with low bottom support. 3. Prove lower bounds for homogeneous depth-4 circuits with low bottom support. (Known*!)

56 Depth-3 to homogeneous depth-5 Q = (l 1 +1) (l 2 +1)"(l t +1)

57 Depth-3 to homogeneous depth-5 Q = (l 1 +1) (l 2 +1)"(l t +1) Homogeneous degree d component of Q : Hom d [Q] = ( S [t ] ) d i S l i

58 Depth-3 to homogeneous depth-5 Q = (l 1 +1) (l 2 +1)"(l t +1) Homogeneous degree d component of Q : In other words, Hom d [Q] = ( S [t ] ) d i S l i Hom d [Q] = Sym d (l 1,l 2,,l t )

59 Elementary symmetric polynomials of degree d. Depth-3 to homogeneous depth-5 Q = (l 1 +1) (l 2 +1)"(l t +1) Homogeneous degree d component of Q : In other words, Hom d [Q] = ( S [t ] ) d i S l i Hom d [Q] = Sym d (l 1,l 2,,l t )

60 Newton identities [Shpilka, Wigderson] Elementary symmetric polynomials of degree d in n variables can be computed by homogeneous! circuits of size at most poly(n) 2 O( d ) over fields of characteristic zero.

61 Newton identities [Shpilka, Wigderson] Elementary symmetric polynomials of degree d in n variables can be computed by homogeneous! circuits of size at most poly(n) 2 O( d ) over fields of characteristic zero. Corollary : Elementary symmetric polynomials of linear forms have homogeneous! circuits of size at most poly(n) 2 O( d ).

62 Newton identities For d = O(log n) Corollary : Elementary symmetric polynomials of degree d of t linear forms have homogeneous! circuits of size at most poly(t, n).

63 Newton identities For d = O(log n) Corollary : Elementary symmetric polynomials of degree d of t linear forms have homogeneous! circuits of size at most poly(t, n). From non-homogeneous to homogeneous, at the cost of increasing the depth.

64 Depth-3 to homogeneous depth-5 Q = (l 1 +1) (l 2 +1)"(l t +1) Homogeneous degree d component of Q - can be computed by a homogeneous depth-5 circuit with powering gates at second last level.

65 Depth-3 to homogeneous depth-5 Q = (l 1 +1) (l 2 +1)"(l t +1) Homogeneous degree d component of Q - can be computed by a homogeneous depth-5 circuit with powering gates at second last level. Apply this to every product gate in the depth-3 circuit to get a poly sized homogeneous depth-5 circuit.

66 Depth-3 to homogeneous depth-5 Q = (l 1 +1) (l 2 +1)"(l t +1) Homogeneous degree d component of Q - can be computed by a homogeneous depth-5 circuit with powering gates at second last level. Apply this to every product gate in the depth-3 circuit to get a poly sized homogeneous depth-5 circuit. Bottom fan-in is preserved in the transformation.

67 Depth-5 to depth-4 with small support r ( x i ) k = i=1 S [r] d X S P S + (low support stuff)

68 Depth-5 to depth-4 with small support r ( x i ) k = i=1 S [r] d X S P S + (low support stuff) Set every variable to 0, independently with prob. 1-p.

69 Depth-5 to depth-4 with small support r ( x i ) k = i=1 S [r] d X S P S + (low support stuff) Set every variable to 0, independently with prob. 1-p. Pr[X S is alive] = p d

70 Depth-5 to depth-4 with small support r ( x i ) k = i=1 S [r] d X S P S + (low support stuff) Set every variable to 0, independently with prob. 1-p. Pr[X S is alive] = p d Pr[ S s.t X S is alive] = r d p d

71 Depth-5 to depth-4 with small support r ( x i ) k = i=1 S [r] d X S P S + (low support stuff) Set every variable to 0, independently with prob. 1-p. Pr[X S is alive] = p d Pr[ S s.t X S is alive] = r d p d p = 1 n δ r

72 Depth-5 to depth-4 with small support r ( x i ) k = i=1 S [r] d X S P S + (low support stuff) Set every variable to 0, independently with prob. 1-p. Pr[X S is alive] = p d r Pr[ S s.t X S is alive] = d p d W.h.p all large support monomials are killed. p = 1 n δ r

73 H-depth-5 to H-depth-4 with low bottom support Apply the restrictions to all powers of linear forms in the! circuit.

74 H-depth-5 to H-depth-4 with low bottom support Apply the restrictions to all powers of linear forms in the! circuit.! circuit reduces to homogeneous circuit with bottom support at most d.

75 Depth-4 lower bounds Need to show super polynomial lower bounds for homogeneous depth-4 circuits with small bottom support.

76 Depth-4 lower bounds Need to show super polynomial lower bounds for homogeneous depth-4 circuits with small bottom support. Known* from the results of Kayal, Limaye, Saha, Srinivasan and K., Saraf.

77 Depth-4 lower bounds Need to show super polynomial lower bounds for homogeneous depth-4 circuits with small bottom support. Known* from the results of Kayal, Limaye, Saha, Srinivasan and K., Saraf. (*) Need to show such results under very strong random restrictions [ p = 1 ]. n 0.99

78 Bottom degree > 1

79 Bottom degree > 1 Homogenization -> Low formal degree + some more structure (but not homogeneous).

80 Bottom degree > 1 Homogenization -> Low formal degree + some more structure (but not homogeneous). Random restrictions -> Almost identical analysis.

81 Bottom degree > 1 Homogenization -> Low formal degree + some more structure (but not homogeneous). Random restrictions -> Almost identical analysis. Depth-4 lower bounds -> Lower bound for depth-4 circuits with not too high formal degree + some structure.

82 Questions

83 Questions PIT for sums of products of bivariates without restrictions on the top fan-in and individual degree.

84 Questions PIT for sums of products of bivariates without restrictions on the top fan-in and individual degree. Strengthening the lower bounds in the paper to show that the all the multiples of a hard polynomial are also hard, could be one approach.

85 Thanks!