Hardness vs Randomness for Bounded Depth Arithmetic Circuits

Hardness vs Randomness for

Bounded Depth Arithmetic Circuits

Chi-Ning Chou Mrinal Kumar Noam Solomon

Harvard University

1

Outline

• Arithmetic circuits and algebraic complexity classes

• Polynomial identity testing (PIT)

• Hardness vs Randomness for arithmetic circuits

• Polynomial factorization

• Open problems

2

Outline

• Arithmetic circuits and algebraic complexity classes

• Polynomial identity testing (PIT)

• Hardness vs Randomness for arithmetic circuits

• Polynomial factorization

• Open problems

3

Arithmetic circuits

4

⇥ ⇥ ⇥

+ + + +

+

x1 x2 x3 x4

Arithmetic circuits

4

⇥ ⇥ ⇥

+ + + +

+

Multivariate polynomialP 2 F[x1,x2, . . . ,xn]

x1 x2 x3 x4

Arithmetic circuits

4

⇥ ⇥ ⇥

+ + + +

+

⌃

⌃

Y

Multivariate polynomialP 2 F[x1,x2, . . . ,xn]

x1 x2 x3 x4

Arithmetic circuits

4

⇥ ⇥ ⇥

+ + + +

+

⌃

⌃

Y Depth - 3

Multivariate polynomialP 2 F[x1,x2, . . . ,xn]

x1 x2 x3 x4

Arithmetic circuits

4

⇥ ⇥ ⇥

+ + + +

+

⌃

⌃

Y Depth - 3

Size - 8

Multivariate polynomialP 2 F[x1,x2, . . . ,xn]

x1 x2 x3 x4

Arithmetic circuits

4

⇥ ⇥ ⇥

+ + + +

+

⌃

⌃

Y Depth - 3

Size - 8

*Assume F = Q

Multivariate polynomialP 2 F[x1,x2, . . . ,xn]

x1 x2 x3 x4

5

Algebraic complexity classes

5

Algebraic complexity classes

C =n{f1, f2, . . . }

o

5

Algebraic complexity classes

C =n{f1, f2, . . . }

o

For simplicity, denote .f = fn

5

Algebraic complexity classes

• : Polynomials computed by poly(n) size, poly(n) degree arithmetic circuits (e.g Determinant).

C =n{f1, f2, . . . }

o

VP

For simplicity, denote .f = fn

5

Algebraic complexity classes

• : Polynomials computed by poly(n) size, poly(n) degree arithmetic circuits (e.g Determinant).

• : Polynomials computed by poly(n) size, poly(n) degree, and depth- arithmetic circuits.

C =n{f1, f2, . . . }

o

Depth-�

VP

For simplicity, denote .f = fn

�

5

Algebraic complexity classes

• : Polynomials computed by poly(n) size, poly(n) degree arithmetic circuits (e.g Determinant).

• : Polynomials computed by poly(n) size, poly(n) degree, and depth- arithmetic circuits.

• Many more such as VF, VBP, VNP…

C =n{f1, f2, . . . }

o

Depth-�

VP

For simplicity, denote .f = fn

�

6

Hardness - Lower bounds

6

Hardness - Lower bounds

Goal: Find an explicit such that .{fn} {fn} 62 C

6

Hardness - Lower bounds

Goal: Find an explicit such that .

• [Strassen 73, Baur & Strassen 83] An n log n lower bound for general arithmetic circuits.

{fn} {fn} 62 C

6

Hardness - Lower bounds

Goal: Find an explicit such that .

• [Strassen 73, Baur & Strassen 83] An n log n lower bound for general arithmetic circuits.

• [Kalorkoti 87] A quadratic lower bound for arithmetic formula.

{fn} {fn} 62 C

6

Hardness - Lower bounds

Goal: Find an explicit such that .

• [Strassen 73, Baur & Strassen 83] An n log n lower bound for general arithmetic circuits.

• [Kalorkoti 87] A quadratic lower bound for arithmetic formula.

• [Kumar 17] A quadratic lower bound for homogeneous algebraic branching programs.

{fn} {fn} 62 C

6

Hardness - Lower bounds

Goal: Find an explicit such that .

• [Strassen 73, Baur & Strassen 83] An n log n lower bound for general arithmetic circuits.

• [Kalorkoti 87] A quadratic lower bound for arithmetic formula.

• [Kumar 17] A quadratic lower bound for homogeneous algebraic branching programs.

• [NW’95, GKKS’14, FLMS’14, KS’14] Exponential lower bounds for depth-3 and depth-4 circuits.

{fn} {fn} 62 C

Outline

• Arithmetic circuits and algebraic complexity classes

• Polynomial identity testing (PIT)

• Hardness vs Randomness for arithmetic circuits

• Polynomial factorization

• Open problems

7

8

Randomness - Polynomial identity testing (PIT)

8

Randomness - Polynomial identity testing (PIT)

Goal: Given , determine whether .f 2 C f ⌘ 0

8

Randomness - Polynomial identity testing (PIT)

Goal: Given , determine whether .f 2 C f ⌘ 0

8

Randomness - Polynomial identity testing (PIT)

Goal: Given , determine whether .

• Easy when using randomness: Schwartz-Zippel.

f 2 C f ⌘ 0

8

Randomness - Polynomial identity testing (PIT)

Goal: Given , determine whether .

• Easy when using randomness: Schwartz-Zippel.

• No non-trivial deterministic PIT for and .

f 2 C f ⌘ 0

Depth-�VPsub-exponential time

8

Randomness - Polynomial identity testing (PIT)

Goal: Given , determine whether .

• Easy when using randomness: Schwartz-Zippel.

• No non-trivial deterministic PIT for and .

f 2 C f ⌘ 0

PIT = Hitting Set

Depth-�VPsub-exponential time

8

Randomness - Polynomial identity testing (PIT)

Goal: Given , determine whether .

• Easy when using randomness: Schwartz-Zippel.

• No non-trivial deterministic PIT for and .

f 2 C f ⌘ 0

PIT = Hitting Set

is a hitting set for if for any non-zeroP C f 2 C

Depth-�VP

9a 2 P, f(a) 6= 0.

sub-exponential time

8

Randomness - Polynomial identity testing (PIT)

PIT = Hitting Set

is a hitting set for if for any non-zeroP C f 2 C

9a 2 P, f(a) 6= 0.

Goal: Explicitly construct a hitting set for .CP

8

Randomness - Polynomial identity testing (PIT)

PIT = Hitting Set

is a hitting set for if for any non-zeroP C f 2 C

9a 2 P, f(a) 6= 0.

Goal: Explicitly construct a hitting set for .

• Running time is .

CP

poly(n, |P|)

Outline

• Arithmetic circuits and algebraic complexity classes

• Polynomial identity testing (PIT)

• Hardness vs Randomness for arithmetic circuits

• Polynomial factorization

• Open problems

9

10

Hardness vs Randomness

RandomnessHardness

10

Hardness vs Randomness

LowerBound PIT

10

Hardness vs Randomness

LowerBound PIT

10

Hardness vs Randomness

• [KI’04]: Permanent not in => PIT for

LowerBound PIT

VPVP

10

Hardness vs Randomness

• [KI’04]: Permanent not in => PIT for

• [DSY’09]: for => PIT for

LowerBound PIT

!(poly(n)) Depth-� Depth-�� 5

VPVP

10

Hardness vs Randomness

• [KI’04]: Permanent not in => PIT for

• [DSY’09]: for => PIT for

LowerBound PIT

!(poly(n))

with bounded individual degree

Depth-� Depth-�� 5

VPVP

multilinear

11

Our result

Theorem: For any ,� � 6

11

Our result

Theorem: For any ,� � 6

multilinear and with degree O(log2 n/ log2 log n)

!(poly(n)) lower bound for Depth-�

11

Our result

Theorem: For any ,� � 6

multilinear and with degree O(log2 n/ log2 log n)

!(poly(n)) lower bound for Depth-�

Sub-exponential time PIT for Depth-�� 5

11

Our result

Theorem: For any ,� � 6

multilinear and with degree O(log2 n/ log2 log n)

!(poly(n)) lower bound for Depth-�

Sub-exponential time PIT for Depth-�� 5

�� 2k � 2

O(logkn/ log

k log n)

11

Our result

Theorem: For any ,� � 6

multilinear and with degree O(log2 n/ log2 log n)

!(poly(n)) lower bound for Depth-�

Sub-exponential time PIT for Depth-�� 5

�� 2k � 2

O(logkn/ log

k log n)

Don’t be bothered by the

constant in depth!

12

Compare with [Dvir-Shpilka-Yehudayoff’09]

[DSY’09] This work

Lower bound for With degree

PIT for With bounded individual degree

Depth-�

Depth-�� 5

O(log2 n/ log2 log n)

13

Hardness vs Randomness framework [KI’04, DSY’09]

13

Hardness vs Randomness framework [KI’04, DSY’09]

Nisan-Wigderson generator

Reduce #variables from n ! `

13

Hardness vs Randomness framework [KI’04, DSY’09]

Nisan-Wigderson generator

Schwartz-Zippel lemma

Reduce #variables from n ! `

Brute-force to find hitting set in time dO(`)

13

Hardness vs Randomness framework [KI’04, DSY’09]

Nisan-Wigderson generator

Schwartz-Zippel lemma

Reduce #variables from n ! `

Brute-force to find hitting set in time dO(`)

Reduce to

factoring problem!

14

NW generator - reducing #variables

q 2 C

14

NW generator - reducing #variables

q 2 C

n

14

NW generator - reducing #variables

q 2 C

n

P ✓ FnGoal: Hitting set

14

NW generator - reducing #variables

q 2 C

n

y`

14

NW generator - reducing #variables

q 2 C

n

y

S1SnS2

`Nisan-

Wigderson Design

m

14

NW generator - reducing #variables

q 2 C

n

y

S1

y|S1 y|Sn

Sn

y|S2

S2

`Nisan-

Wigderson Design

m

14

NW generator - reducing #variables

q 2 C

n

y

f

S1

y|S1 y|Sn

Sn

ff

y|S2

S2

`

m

Nisan-Wigderson

Design

…

m

14

NW generator - reducing #variables

q 2 C

n

y`

14

NW generator - reducing #variables

q 2 C

n

y`

Q(y) = q�f(y|S1), f(y|S2), . . . , f(y|Sn)

�

14

NW generator - reducing #variables

q 2 C

n

y`

Q(y) = q�f(y|S1), f(y|S2), . . . , f(y|Sn)

�

Want: If then .q 6⌘ 0 Q 6⌘ 0

15

Key lemma

15

Key lemma q 2 C

y

Q(y) = q�f(y|S1), f(y|S2), . . . , f(y|Sn)

�

Goal: If , then .q 6⌘ 0 Q 6⌘ 0

15

Key lemma

Lemma: Let non-zero and a m-variate multilinear polynomial of degree . If

q 2 Depth-�f

Q(y) = q�f(y|S1), f(y|S2), . . . , f(y|Sn)

�⌘ 0

d

q 2 C

y

Q(y) = q�f(y|S1), f(y|S2), . . . , f(y|Sn)

�

15

Key lemma

Lemma: Let non-zero and a m-variate multilinear polynomial of degree . If

Then, f can be computed by a size and depth circuit.

q 2 Depth-�f

Q(y) = q�f(y|S1), f(y|S2), . . . , f(y|Sn)

�⌘ 0

�+ 5poly(n, d

pd)

d

q 2 C

y

Q(y) = q�f(y|S1), f(y|S2), . . . , f(y|Sn)

�

15

Key lemma

Lemma: Let non-zero and a m-variate multilinear polynomial of degree . If

Then, f can be computed by a size and depth circuit.

q 2 Depth-�f

Q(y) = q�f(y|S1), f(y|S2), . . . , f(y|Sn)

�⌘ 0

�+ 5poly(n, d

pd)

d

f /2 Depth-�+ 5 Q(y) 6⌘ 0, 8q 2 Depth-�

q 2 C

y

Q(y) = q�f(y|S1), f(y|S2), . . . , f(y|Sn)

�

15

Key lemma

Lemma: Let non-zero and a m-variate multilinear polynomial of degree . If

Then, f can be computed by a size and depth circuit.

q 2 Depth-�f

Q(y) = q�f(y|S1), f(y|S2), . . . , f(y|Sn)

�⌘ 0

�+ 5poly(n, d

pd)

d

f /2 Depth-�+ 5 Q(y) 6⌘ 0, 8q 2 Depth-�

q 2 C

y

Q(y) = q�f(y|S1), f(y|S2), . . . , f(y|Sn)

�

Schwartz-Zippel

16

Proof sketch of the key lemma9q 2 Depth-�, Q(y) ⌘ 0

f 2 Depth-�+ 5

16

Proof sketch of the key lemma

If Q(y) = q�f(y|S1), f(y|S2), . . . , f(y|Sn)

�⌘ 0

9q 2 Depth-�, Q(y) ⌘ 0

f 2 Depth-�+ 5

16

Proof sketch of the key lemma

If Q(y) = q�f(y|S1), f(y|S2), . . . , f(y|Sn)

�⌘ 0

q 2 C

x1 x2 xn

6⌘ 0

9q 2 Depth-�, Q(y) ⌘ 0

f 2 Depth-�+ 5

16

Proof sketch of the key lemma

If Q(y) = q�f(y|S1), f(y|S2), . . . , f(y|Sn)

�⌘ 0

q 2 C

x1 x2 xn

y

q 2 C

6⌘ 0

⌘ 0

9q 2 Depth-�, Q(y) ⌘ 0

f 2 Depth-�+ 5

17

Proof sketch of the key lemma9q 2 Depth-�, Q(y) ⌘ 0

f 2 Depth-�+ 5

17

By hybrid argument, there exists

f

y|S1

f

y|S2

xn

q 2 C

Proof sketch of the key lemma

xi

9q 2 Depth-�, Q(y) ⌘ 0

f 2 Depth-�+ 5

17

By hybrid argument, there exists

f

y|S1

f

y|S2

xn

q 2 C

Proof sketch of the key lemma

xi

z = {x1, . . . ,xi�1,xi+1, . . . ,xn,y}Q(z,xi)

9q 2 Depth-�, Q(y) ⌘ 0

f 2 Depth-�+ 5

17

By hybrid argument, there exists

f

y|S1

f

y|S2

xn

q 2 C

Proof sketch of the key lemma

xi

z = {x1, . . . ,xi�1,xi+1, . . . ,xn,y}

•

Q(z,xi)

Q(z,xi) 6⌘ 0

9q 2 Depth-�, Q(y) ⌘ 0

f 2 Depth-�+ 5

17

By hybrid argument, there exists

f

y|S1

f

y|S2

xn

q 2 C

Proof sketch of the key lemma

z = {x1, . . . ,xi�1,xi+1, . . . ,xn,y}

• •

f

y|Si

Q(z,xi)

Q(z,xi) 6⌘ 0

Q(z, f(z)) ⌘ 0

9q 2 Depth-�, Q(y) ⌘ 0

f 2 Depth-�+ 5

17

By hybrid argument, there exists

xn

q 2 C

Proof sketch of the key lemma

• •

f

y|Si

Q(z,xi)

Q(z,xi) 6⌘ 0

Q(z, f(z)) ⌘ 0

9q 2 Depth-�, Q(y) ⌘ 0

f 2 Depth-�+ 5

Fixed Fixed

17

By hybrid argument, there exists

xn

q 2 C

Proof sketch of the key lemma

• •

f

y|Si

Q(z,xi)

Q(z,xi) 6⌘ 0

Q(z, f(z)) ⌘ 0

9q 2 Depth-�, Q(y) ⌘ 0

f 2 Depth-�+ 5

Fixed Fixed

* Q(z,xi) 2 Depth-�+ 1

18

Proof sketch of the key lemma

• • Q(z,xi) 6⌘ 0

Q(z, f(z)) ⌘ 0Q(z,xi) 2 Depth-�+ 1

18

Proof sketch of the key lemma

• • Q(z,xi) 6⌘ 0

Q(z, f(z)) ⌘ 0

xi � f(z) Q(z,xi)divides

Q(z,xi) 2 Depth-�+ 1

18

Proof sketch of the key lemma

• • Q(z,xi) 6⌘ 0

Q(z, f(z)) ⌘ 0

xi � f(z) Q(z,xi)divides

Reducing to polynomial factorization!

Q(z,xi) 2 Depth-�+ 1

Outline

• Arithmetic circuits and algebraic complexity classes

• Polynomial identity testing (PIT)

• Hardness vs Randomness for arithmetic circuits

• Polynomial factorization

• Open problems

19

20

Polynomial factorization (Simplified setting)

Goal: For any such that . Show that .

P (z, y) 2 C P (z, f(z)) = 0f 2 C0

20

Polynomial factorization (Simplified setting)

Goal: For any such that . Show that .

P (z, y) 2 C P (z, f(z)) = 0f 2 C0

[Kal89]

C C0

VP VP

20

Polynomial factorization (Simplified setting)

Goal: For any such that . Show that .

P (z, y) 2 C P (z, f(z)) = 0f 2 C0

[Kal89]

[DSY09]with bounded individual degree

C C0

VP VP

Depth-� Depth-�+ 3

20

Polynomial factorization (Simplified setting)

Goal: For any such that . Show that .

P (z, y) 2 C P (z, f(z)) = 0f 2 C0

[Kal89]

[DSY09]with bounded individual degree

[DSS18]

C C0

VP VP

Depth-� Depth-�+ 3

(resp. VBP(nlogn), VNP(nlogn))

VF(nlogn))(resp. VBP(nlogn), VNP(nlogn))

VF(nlogn))

20

Polynomial factorization (Simplified setting)

Goal: For any such that . Show that .

P (z, y) 2 C P (z, f(z)) = 0f 2 C0

[Kal89]

[DSY09]with bounded individual degree

[DSS18]

Our result with degree

C C0

VP VP

Depth-� Depth-�+ 3

(resp. VBP(nlogn), VNP(nlogn))

VF(nlogn))(resp. VBP(nlogn), VNP(nlogn))

VF(nlogn))

Depth-� Depth-�+ 3O(log2 n/ log2 log n)

20

Polynomial factorization (Simplified setting)

Goal: For any such that . Show that .

P (z, y) 2 C P (z, f(z)) = 0f 2 C0

[Kal89]

[DSY09]with bounded individual degree

[DSS18]

Our result with degree

C C0

VP VP

Depth-� Depth-�+ 3

(resp. VBP(nlogn), VNP(nlogn))

VF(nlogn))(resp. VBP(nlogn), VNP(nlogn))

VF(nlogn))

non-deterministic (existential)

Depth-� Depth-�+ 3O(log2 n/ log2 log n)

21

Factorization for bounded depth circuits

Goal: For any s.t. . Show that .

P (z, y) 2 Depth-� P (z, f(z)) = 0f 2 Depth-�+O(1)

21

Factorization for bounded depth circuits

Goal: For any s.t. . Show that .

P (z, y) 2 Depth-� P (z, f(z)) = 0f 2 Depth-�+O(1)

Newton iteration

21

Factorization for bounded depth circuits

Goal: For any s.t. . Show that .

P (z, y) 2 Depth-� P (z, f(z)) = 0f 2 Depth-�+O(1)

Newton iteration

Structure lemma

21

Factorization for bounded depth circuits

Goal: For any s.t. . Show that .

P (z, y) 2 Depth-� P (z, f(z)) = 0f 2 Depth-�+O(1)

Newton iteration

Depth reduction

Structure lemma

22

Newton iteration (Sloppy Hensel Lifting)

Goal: Hi[hi] = Hi[f ].

22

Newton iteration (Sloppy Hensel Lifting)

Goal: Hi[hi] = Hi[f ].

Def: (Homogeneous components)

22

Newton iteration (Sloppy Hensel Lifting)

Goal: Hi[hi] = Hi[f ].

Def: (Homogeneous components)The degree i homogeneous component is the collection of monomials of degree i.

22

Newton iteration (Sloppy Hensel Lifting)

Goal: Hi[hi] = Hi[f ].

Def: (Homogeneous components)The degree i homogeneous component is the collection of monomials of degree i.Example: f(x1, x2, x3) = x3

1x2 + x1x2x3 + x22 + x1x3 + x4

3

22

Newton iteration (Sloppy Hensel Lifting)

Goal: Hi[hi] = Hi[f ].

Def: (Homogeneous components)The degree i homogeneous component is the collection of monomials of degree i.Example:•

f(x1, x2, x3) = x31x2 + x1x2x3 + x2

2 + x1x3 + x43

H0[f ] = 0

22

Newton iteration (Sloppy Hensel Lifting)

Goal: Hi[hi] = Hi[f ].

Def: (Homogeneous components)The degree i homogeneous component is the collection of monomials of degree i.Example:• •

f(x1, x2, x3) = x31x2 + x1x2x3 + x2

2 + x1x3 + x43

H0[f ] = 0H1[f ] = 0

22

Newton iteration (Sloppy Hensel Lifting)

Goal: Hi[hi] = Hi[f ].

Def: (Homogeneous components)The degree i homogeneous component is the collection of monomials of degree i.Example:• • •

f(x1, x2, x3) = x31x2 + x1x2x3 + x2

2 + x1x3 + x43

H2[f ] = x22 + x1x3

H0[f ] = 0H1[f ] = 0

22

Newton iteration (Sloppy Hensel Lifting)

Goal: Hi[hi] = Hi[f ].

Def: (Homogeneous components)The degree i homogeneous component is the collection of monomials of degree i.Example:• • • •

f(x1, x2, x3) = x31x2 + x1x2x3 + x2

2 + x1x3 + x43

H2[f ] = x22 + x1x3

H0[f ] = 0H1[f ] = 0

H3[f ] = x1x2x3

22

Newton iteration (Sloppy Hensel Lifting)

Goal: Hi[hi] = Hi[f ].

Def: (Homogeneous components) The degree i homogeneous component is the collection of monomials of degree i. Example: • • • • •

f(x1, x2, x3) = x31x2 + x1x2x3 + x2

2 + x1x3 + x43

H2[f ] = x22 + x1x3

H0[f ] = 0H1[f ] = 0

H3[f ] = x1x2x3

H4[f ] = x31x2 + x4

3

22

Newton iteration (Sloppy Hensel Lifting)

Goal: Hi[hi] = Hi[f ].

22

Newton iteration (Sloppy Hensel Lifting)

Goal:

Update:

Hi[hi] = Hi[f ].

hi = hi�1 �Hi[P (z, hi�1(z))]

�.

22

Newton iteration (Sloppy Hensel Lifting)

Goal:

Update:

Hi[hi] = Hi[f ].

hi = hi�1 �Hi[P (z, hi�1(z))]

�.

* Homogenization & partial derivative preserve depth

22

Newton iteration (Sloppy Hensel Lifting)

Goal:

Update:

Intuition: Taylor’s expansion.

Hi[hi] = Hi[f ].

hi = hi�1 �Hi[P (z, hi�1(z))]

�.

* Homogenization & partial derivative preserve depth

22

Newton iteration (Sloppy Hensel Lifting)

Goal:

Update:

Intuition: Taylor’s expansion.

Hi[hi] = Hi[f ].

hi = hi�1 �Hi[P (z, hi�1(z))]

�.

Q: How to efficiently update?

* Homogenization & partial derivative preserve depth

23

Structure lemma

23

Structure lemma

Goal: For any s.t. . Show that .

P (z, y) 2 Depth-� P (z, f(z)) = 0f 2 Depth-�+O(1)

23

Structure lemma

Goal: For any s.t. . Show that .

• P as an univariate polynomial:

P (z, y) 2 Depth-� P (z, f(z)) = 0f 2 Depth-�+O(1)

P (z, y) =kX

i=0

Ci(z)yi.

23

Structure lemma

Goal: For any s.t. . Show that .

• P as an univariate polynomial:

Lemma [DSY’09]: For each , there exists polynomial such that

P (z, y) 2 Depth-� P (z, f(z)) = 0f 2 Depth-�+O(1)

P (z, y) =kX

i=0

Ci(z)yi.

i = 1, 2, . . . , d = deg(f)Ai

Hi[f ] = Hi[Ai(C0, C1, . . . , Ck)].

23

Structure lemma

Goal: For any s.t. . Show that .

• P as an univariate polynomial:

Lemma [DSY’09]: For each , there exists polynomial such that

P (z, y) 2 Depth-� P (z, f(z)) = 0f 2 Depth-�+O(1)

P (z, y) =kX

i=0

Ci(z)yi.

i = 1, 2, . . . , d = deg(f)Ai

Hi[f ] = Hi[Ai(C0, C1, . . . , Ck)].

Individual degree

24

Structure lemma

24

Structure lemma

Lemma (This work): For each , there exists polynomial such thatAi

i = 1, 2, . . . , d = deg(f)

Hi[f ] = Hi[Ai(g0, g1, . . . , gd)]

24

Structure lemma

Lemma (This work): For each , there exists polynomial such that

where

Ai

i = 1, 2, . . . , d = deg(f)

Hi[f ] = Hi[Ai(g0, g1, . . . , gd)]

gi = Hd

@i

@yiP (z,H[f ])

��H0

@i

@yiP (z,H[f ])

�.

24

Structure lemma

Lemma (This work): For each , there exists polynomial such that

where size degree at mostO(d6) d

Ai

i = 1, 2, . . . , d = deg(f)

Hi[f ] = Hi[Ai(g0, g1, . . . , gd)]

gi = Hd

@i

@yiP (z,H[f ])

��H0

@i

@yiP (z,H[f ])

�.

24

Structure lemma

Lemma (This work): For each , there exists polynomial such that

where size degree at mostO(d6) d

Ai

i = 1, 2, . . . , d = deg(f)

Hi[f ] = Hi[Ai(g0, g1, . . . , gd)]

gi = Hd

@i

@yiP (z,H[f ])

��H0

@i

@yiP (z,H[f ])

�.

Depth with top layer⌃�+ 1

24

Structure lemma

Lemma (This work): For each , there exists polynomial such that

where size degree at mostO(d6) d

Ai

i = 1, 2, . . . , d = deg(f)

Hi[f ] = Hi[Ai(g0, g1, . . . , gd)]

gi = Hd

@i

@yiP (z,H[f ])

��H0

@i

@yiP (z,H[f ])

�.

Depth with top layer⌃�+ 1

* Homogenization & partial derivative preserve depth

25

Structure lemma

25

Structure lemma P (z, y) 2 Depth-�

P (z, f(z)) = 0

25

Structure lemma

f = hd

z

P (z, y) 2 Depth-�

P (z, f(z)) = 0

25

Structure lemma

f = hd

g0 g1 gd…

z

P (z, y) 2 Depth-�

P (z, f(z)) = 0

25

Structure lemma

f = hd

A0 A1 Ad

g0 g1 gd

…

…

z

P (z, y) 2 Depth-�

P (z, f(z)) = 0

25

Structure lemma

f = hd

+

A0 A1 Ad

g0 g1 gd

…

…

z

P (z, y) 2 Depth-�

P (z, f(z)) = 0

25

Structure lemma

f = hd

+

A0 A1 Ad

g0 g1 gd

…

…

z

Depth �+ 1

P (z, y) 2 Depth-�

P (z, f(z)) = 0

25

Structure lemma

f = hd

+

A0 A1 Ad

g0 g1 gd

…

…

z

Depth �+ 1

Depth?

P (z, y) 2 Depth-�

P (z, f(z)) = 0

26

Depth reduction [Gupta-Kamath-Kayal-Saptharishi’13]

26

Depth reduction [Gupta-Kamath-Kayal-Saptharishi’13]

x1 x2 xn

• size • degree d

s

x1 x2 xn

• size • depth 3, i.e.,

(snd)O(pd)

⌃⇧⌃

26

x1 x2 xn

• size • degree d

s

x1 x2 xn

• size • depth

(snd)O(d)1/k

Depth reduction [Agrawal-Vinay’08, Koiran’12, Tavenas’13]

2k

27

Factorization for bounded depth circuits (Wrap up)

Goal: For any s.t. . Show that .

P (z, y) 2 Depth-� P (z, f(z)) = 0f 2 Depth-�+O(1)

Newton iteration

Depth reduction

Structure lemma

27

Factorization for bounded depth circuits (Wrap up)

Goal: For any s.t. . Show that .

P (z, y) 2 Depth-� P (z, f(z)) = 0f 2 Depth-�+O(1)

Newton iteration

Depth reduction

Structure lemma

Hi[hi] = Hi[f ].

27

Factorization for bounded depth circuits (Wrap up)

Goal: For any s.t. . Show that .

P (z, y) 2 Depth-� P (z, f(z)) = 0f 2 Depth-�+O(1)

Newton iteration

Depth reduction

Structure lemma

Hi[hi] = Hi[f ].

hi � hi�1 = Ai(g0, g1, . . . , gd)

27

Factorization for bounded depth circuits (Wrap up)

Goal: For any s.t. . Show that .

P (z, y) 2 Depth-� P (z, f(z)) = 0f 2 Depth-�+O(1)

Newton iteration

Depth reduction

Structure lemma

Hi[hi] = Hi[f ].

hi � hi�1 = Ai(g0, g1, . . . , gd)

Depth with top layer⌃�+ 1

size degree at mostO(d6) d

27

Factorization for bounded depth circuits (Wrap up)

Goal: For any s.t. . Show that .

P (z, y) 2 Depth-� P (z, f(z)) = 0f 2 Depth-�+O(1)

Newton iteration

Depth reduction

Structure lemma

Hi[hi] = Hi[f ].

hi � hi�1 = Ai(g0, g1, . . . , gd)

Depth with top layer⌃�+ 1

size degree at mostO(d6) d

hi � hi�1 = Ai(g0, g1, . . . , gd)

27

Factorization for bounded depth circuits (Wrap up)

Goal: For any s.t. . Show that .

P (z, y) 2 Depth-� P (z, f(z)) = 0f 2 Depth-�+O(1)

Newton iteration

Depth reduction

Structure lemma

Hi[hi] = Hi[f ].

hi � hi�1 = Ai(g0, g1, . . . , gd)

Depth with top layer⌃�+ 1

size degree at mostO(d6) d

hi � hi�1 = Ai(g0, g1, . . . , gd)

Depth size with top layer⌃�+ 3 poly(n, dpd)

28

Conclusion

28

Conclusion

Theorem: For any s.t. . If , then .

P (z, y) 2 Depth-� P (z, f(z)) = 0dpd = poly(n) f 2 Depth-�+ 3

28

Conclusion

Theorem: For any s.t. . If , then .

Theorem: For any . If there’s a lower bound for with degree , then there’s a sub-exponential time PIT for .

P (z, y) 2 Depth-� P (z, f(z)) = 0dpd = poly(n) f 2 Depth-�+ 3

� � 6 !(poly(n))Depth-�

Depth-�� 5O(log2 n/ log2 log n)

28

Conclusion

Theorem: For any s.t. . If , then .

Theorem: For any . If there’s a lower bound for with degree , then there’s a sub-exponential time PIT for .

P (z, y) 2 Depth-� P (z, f(z)) = 0dpd = poly(n) f 2 Depth-�+ 3

� � 6 !(poly(n))Depth-�

Depth-�� 5O(log2 n/ log2 log n)

[DSY’09] This work

Lower bound for With degree

PIT for With bounded individual degree

Depth-�

Depth-�� 5

O(log2 n/ log2 log n)

Outline

• Arithmetic circuits and algebraic complexity classes

• Polynomial identity testing (PIT)

• Hardness vs Randomness for arithmetic circuits

• Polynomial factorization

• Open problems

29

30

Open problems

30

Open problems

• Hardness vs Randomness

30

Open problems

• Hardness vs Randomness✦ Remove the degree condition(s)?

30

Open problems

• Hardness vs Randomness✦ Remove the degree condition(s)?✦ More circuit classes?

30

Open problems

• Hardness vs Randomness✦ Remove the degree condition(s)?✦ More circuit classes?

• Polynomial factorization

30

Open problems

• Hardness vs Randomness✦ Remove the degree condition(s)?✦ More circuit classes?

• Polynomial factorization✦ Remove the degree condition(s)?

30

Open problems

• Hardness vs Randomness✦ Remove the degree condition(s)?✦ More circuit classes?

• Polynomial factorization✦ Remove the degree condition(s)?✦ Sparse polynomials?

30

Open problems

• Hardness vs Randomness✦ Remove the degree condition(s)?✦ More circuit classes?

• Polynomial factorization✦ Remove the degree condition(s)?✦ Sparse polynomials?✦ Closure results for VF, VBP?

30

Open problems

• Hardness vs Randomness✦ Remove the degree condition(s)?✦ More circuit classes?

• Polynomial factorization✦ Remove the degree condition(s)?✦ Sparse polynomials?✦ Closure results for VF, VBP? Than

k you!