Hardness vs Randomness for Bounded Depth Arithmetic Circuits Chi-Ning Chou Mrinal Kumar Noam Solomon Harvard University 1
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!