Exponential lower bounds for hom. depth-5 circuits over finite - - PowerPoint PPT Presentation
Exponential lower bounds for hom. depth-5 circuits over finite fields Mrinal Kumar Ramprasad Saptharishi TIFR, Mumbai CCC 2017 Riga Rutgers Harvard Algebraic Circuits f ( x 1 , x 2 , x 3 ) = 2 x 2 1 + 2 x 1 x 2 + 2 x 1 x 3 + 2 x 2 x 3 =
Mrinal Kumar Ramprasad Saptharishi Rutgers → Harvard TIFR, Mumbai CCC 2017 Riga
x1 x2 x3
+
(x1 + x2)
+
(x1 + x3)
+
x1
+
(x1 + x2 + x3)
+
x2
+
x3
×
(x1 + x2)(x1 + x3)
×
x1(x1 + x2 + x3)
×
x2x3
+
f (x1, x2, x3)
= 2x2
1 + 2x1x2 + 2x1x3 + 2x2x3
= 0 over 2
x1 x2 x3
+
(x1 + x2)
+
(x1 + x3)
+
x1
+
(x1 + x2 + x3)
+
x2
+
x3
×
(x1 + x2)(x1 + x3)
×
x1(x1 + x2 + x3)
×
x2x3
+
f (x1, x2, x3)
= 2x2
1 + 2x1x2 + 2x1x3 + 2x2x3
= 0 over 2 Size = number of gates
x1 x2 x3
+
(x1 + x2)
+
(x1 + x3)
+
x1
+
(x1 + x2 + x3)
+
x2
+
x3
×
(x1 + x2)(x1 + x3)
×
x1(x1 + x2 + x3)
×
x2x3
+
f (x1, x2, x3)
= 2x2
1 + 2x1x2 + 2x1x3 + 2x2x3
= 0 over 2 Depth
x1 x2 x3
+
(x1 + x2)
+
(x1 + x3)
+
x1
+
(x1 + x2 + x3)
+
x2
+
x3
×
(x1 + x2)(x1 + x3)
×
x1(x1 + x2 + x3)
×
x2x3
+
f (x1, x2, x3)
= 2x2
1 + 2x1x2 + 2x1x3 + 2x2x3
= 0 over 2
Σ Π Σ
x1 x2 x3
+
(x1 + x2)
+
(x1 + x3)
+
x1
+
(x1 + x2 + x3)
+
x2
+
x3
×
(x1 + x2)(x1 + x3)
×
x1(x1 + x2 + x3)
×
x2x3
+
f (x1, x2, x3)
= 2x2
1 + 2x1x2 + 2x1x3 + 2x2x3
= 0 over 2
x1 x2 x3
+
(x1 + x2)
+
(x1 + x3)
+
x1
+
(x1 + x2 + x3)
+
x2
+
x3
×
(x1 + x2)(x1 + x3)
×
x1(x1 + x2 + x3)
×
x2x3
+
f (x1, x2, x3)
= 2x2
1 + 2x1x2 + 2x1x3 + 2x2x3
= 0 over 2
x1 x2 x3
+
(x1 + x2)
+
(x1 + x3)
+
x1
+
(x1 + x2 + x3)
+
x2
+
x3
×
(x1 + x2)(x1 + x3)
×
x1(x1 + x2 + x3)
×
x2x3
+
f (x1, x2, x3)
= 2x2
1 + 2x1x2 + 2x1x3 + 2x2x3
= 0 over 2
x1 x2 x3
+
(x1 + x2)
+
(x1 + x3)
+
x1
+
(x1 + x2 + x3)
+
x2
+
x3
×
(x1 + x2)(x1 + x3)
×
x1(x1 + x2 + x3)
×
x2x3
+
f (x1, x2, x3)
= 2x2
1 + 2x1x2 + 2x1x3 + 2x2x3
= 0 over 2
x1 x2 x3
+
(x1 + x2)
+
(x1 + x3)
+
x1
+
(x1 + x2 + x3)
+
x2
+
x3
×
(x1 + x2)(x1 + x3)
×
x1(x1 + x2 + x3)
×
x2x3
+
f (x1, x2, x3)
= 2x2
1 + 2x1x2 + 2x1x3 + 2x2x3
= 0 over 2
x1 x2 x3
+
(x1 + x2)
+
(x1 + x3)
+
x1
+
(x1 + x2 + x3)
+
x2
+
x3
×
(x1 + x2)(x1 + x3)
×
x1(x1 + x2 + x3)
×
x2x3
+
f (x1, x2, x3)
= 2x2
1 + 2x1x2 + 2x1x3 + 2x2x3
= 0 over 2
NP P VP VNP is simpler to prove than P NP. Ultimate goal: Find an explicit
polynomial that requires large arithmetic circuits to compute it.
VNP NP P VP VP VNP is simpler to prove than P NP. Ultimate goal: Find an explicit
polynomial that requires large arithmetic circuits to compute it.
#P VNP NP P
SAC1
VP VP VNP is simpler to prove than P NP. Ultimate goal: Find an explicit
polynomial that requires large arithmetic circuits to compute it.
#P VNP NP P
SAC1
VP VP ̸= VNP is simpler to prove than P ̸= NP. Ultimate goal: Find an explicit
polynomial that requires large arithmetic circuits to compute it.
#P VNP NP P
SAC1
VP VP ̸= VNP is simpler to prove than P ̸= NP. Ultimate goal: Find an explicit n-variate degree d polynomial that requires large arithmetic circuits to compute it.
Tieorem ([Agrawal-Vinay + Koiran, Tavenas])
Can be computed by algebraic circuits
Can be computed by depth-4 circuits
(Or) Cannot be computed by algebraic circuits
size Cannot be computed by circuits
size
Tieorem ([Agrawal-Vinay + Koiran, Tavenas])
Can be computed by algebraic circuits
Can be computed by ΣΠ[
(Or) Cannot be computed by algebraic circuits
size Cannot be computed by circuits
size
Tieorem ([Agrawal-Vinay + Koiran, Tavenas])
Can be computed by algebraic circuits
Can be computed by ΣΠ[
(Or) Cannot be computed by algebraic circuits
Cannot be computed by ΣΠ[
Goal: T
Tieorem ([Nisan-Wigderson])
A lower bound for circuits.
Tieorem ([Grigoriev-Karpinski, Grigoriev-Razborov])
A lower bound circuits over any fixed finite field
Tieorem ([Gupta-Kamath-Kayal-S])
A lower bound for circuits.
Tieorem ([Kayal-Limaye-Saha-Srinivasan])
A lower bound for homogeneous depth- circuits.
Goal: T
Tieorem ([Nisan-Wigderson])
A 2Ω(d) lower bound for ΣΠ[d]Σ circuits.
Tieorem ([Grigoriev-Karpinski, Grigoriev-Razborov])
A lower bound circuits over any fixed finite field
Tieorem ([Gupta-Kamath-Kayal-S])
A lower bound for circuits.
Tieorem ([Kayal-Limaye-Saha-Srinivasan])
A lower bound for homogeneous depth- circuits.
Goal: T
Tieorem ([Nisan-Wigderson])
A 2Ω(d) lower bound for ΣΠ[d]Σ circuits.
Tieorem ([Grigoriev-Karpinski, Grigoriev-Razborov])
A 2Ωq(d) lower bound ΣΠΣ circuits over any fixed finite field q
Tieorem ([Gupta-Kamath-Kayal-S])
A lower bound for circuits.
Tieorem ([Kayal-Limaye-Saha-Srinivasan])
A lower bound for homogeneous depth- circuits.
Goal: T
Tieorem ([Nisan-Wigderson])
A 2Ω(d) lower bound for ΣΠ[d]Σ circuits.
Tieorem ([Grigoriev-Karpinski, Grigoriev-Razborov])
A 2Ωq(d) lower bound ΣΠΣ circuits over any fixed finite field q
Tieorem ([Gupta-Kamath-Kayal-S])
A 2Ω(
Tieorem ([Kayal-Limaye-Saha-Srinivasan])
A lower bound for homogeneous depth- circuits.
Goal: T
Tieorem ([Nisan-Wigderson])
A 2Ω(d) lower bound for ΣΠ[d]Σ circuits.
Tieorem ([Grigoriev-Karpinski, Grigoriev-Razborov])
A 2Ωq(d) lower bound ΣΠΣ circuits over any fixed finite field q
Tieorem ([Gupta-Kamath-Kayal-S])
A 2Ω(
Tieorem ([Kayal-Limaye-Saha-Srinivasan])
A nΩ(
Tieorem
An explicit polynomial f (x1,..., xn) of degree d with 0/1 coeffjcients such that, for any fixed finite field q, any homogeneous ΣΠΣΠΣ circuit computing f must have size 2Ωq(
Ingredients for the proof: [Kayal-Limaye-Saha-Srinivasan] + [Grigoriev-Karpinski] + a good amount of sweat ... ought to have been easier than this
Tieorem
An explicit polynomial f (x1,..., xn) of degree d with 0/1 coeffjcients such that, for any fixed finite field q, any homogeneous ΣΠΣΠΣ circuit computing f must have size 2Ωq(
Ingredients for the proof: [Kayal-Limaye-Saha-Srinivasan] + [Grigoriev-Karpinski] + a good amount of sweat ... ought to have been easier than this
Tieorem
An explicit polynomial f (x1,..., xn) of degree d with 0/1 coeffjcients such that, for any fixed finite field q, any homogeneous ΣΠΣΠΣ circuit computing f must have size 2Ωq(
Ingredients for the proof: [Kayal-Limaye-Saha-Srinivasan] + [Grigoriev-Karpinski] + a good amount of sweat ... ought to have been easier than this
Natural proof strategies Construct a map Γ : [x1,..., xn] → , that assigns a number to every polynomial such that: Typically is the rank of some associated linear space.
Γ(f ) is “large”.
Natural proof strategies Construct a map Γ : [x1,..., xn] → , that assigns a number to every polynomial such that: Typically Γ(f ) is the rank of some associated linear space.
Γ(f ) is “large”.
▶ [Nisan-Wigderson-95]: ΣΠdΣ circuits, sums of terms of the form
ℓ1 ···ℓd.
▶ [Nisan-Wigderson-95]: ΣΠdΣ circuits, sums of terms of the form
ℓ1 ···ℓd. Key observation: There are just d
k
partial derivatives of ℓ1 ···ℓd.
▶ [Nisan-Wigderson-95]: ΣΠdΣ circuits, sums of terms of the form
ℓ1 ···ℓd. Key observation: There are just d
k
partial derivatives of ℓ1 ···ℓd. For a generic polynomial, you would all partial derivatives to be linearly independent.
▶ [Nisan-Wigderson-95]: ΣΠdΣ circuits, sums of terms of the form
ℓ1 ···ℓd. Key observation: There are just d
k
partial derivatives of ℓ1 ···ℓd. For a generic polynomial, you would all partial derivatives to be linearly independent. ∂ =k(ℓ1 ···ℓd) ⊆ span ∏
i∈S
ℓi : S ⊆ [d] , |S| = d − k
▶ [Nisan-Wigderson-95]: ΣΠdΣ circuits, sums of terms of the form
ℓ1 ···ℓd. Key observation: There are just d
k
partial derivatives of ℓ1 ···ℓd. For a generic polynomial, you would all partial derivatives to be linearly independent. ∂ =k(ℓ1 ···ℓd) ⊆ span ∏
i∈S
ℓi : S ⊆ [d] , |S| = d − k
d
k
2 linearly independent (d − k) × (d − k) minors.
▶ [Nisan-Wigderson-95]: ΣΠdΣ circuits, sums of terms of the form
ℓ1 ···ℓd. Key observation: There are just d
k
partial derivatives of ℓ1 ···ℓd. For a generic polynomial, you would all partial derivatives to be linearly independent. ∂ =k(ℓ1 ···ℓd) ⊆ span ∏
i∈S
ℓi : S ⊆ [d] , |S| = d − k
d
k
2 linearly independent (d − k) × (d − k) minors. Therefore, if Detd = ∑s
i=1 ℓi1 ···ℓid, then s ≥
d
d/2
▶ [Nisan-Wigderson-95]: ΣΠdΣ circuits, sum of terms of the form
ℓ1 ···ℓd. Key observation: There are “few” linearly independent partial derivatives of ℓ1 ···ℓd.
▶ [Nisan-Wigderson-95]: ΣΠdΣ circuits, sum of terms of the form
ℓ1 ···ℓd. Key observation: There are “few” linearly independent partial derivatives of ℓ1 ···ℓd. ∂ =k Mons of degree d − k m ∂xα
▶ [Nisan-Wigderson-95]: ΣΠdΣ circuits, sum of terms of the form
ℓ1 ···ℓd. Key observation: There are “few” linearly independent partial derivatives of ℓ1 ···ℓd.
▶ [Nisan-Wigderson-95]: ΣΠdΣ circuits, sum of terms of the form
ℓ1 ···ℓd. Key observation: There are “few” linearly independent partial derivatives of ℓ1 ···ℓd.
▶ [Gupta-Kamath-Kayal-S-13]: ΣΠ
form Q1 ···Q
d.
▶ [Nisan-Wigderson-95]: ΣΠdΣ circuits, sum of terms of the form
ℓ1 ···ℓd. Key observation: There are “few” linearly independent partial derivatives of ℓ1 ···ℓd.
▶ [Gupta-Kamath-Kayal-S-13]: ΣΠ
form Q1 ···Q
d.
∂x(Q1 ···Qr) = ∂x(Q1) · Q2 ···Qr + ··· + Q1 ···Qr−1 · ∂x(Qr)
▶ [Nisan-Wigderson-95]: ΣΠdΣ circuits, sum of terms of the form
ℓ1 ···ℓd. Key observation: There are “few” linearly independent partial derivatives of ℓ1 ···ℓd.
▶ [Gupta-Kamath-Kayal-S-13]: ΣΠ
form Q1 ···Q
d.
∂x(Q1 ···Qr) = ∂x(Q1) · Q2 ···Qr + ··· + Q1 ···Qr−1 · ∂x(Qr)
▶ [Nisan-Wigderson-95]: ΣΠdΣ circuits, sum of terms of the form
ℓ1 ···ℓd. Key observation: There are “few” linearly independent partial derivatives of ℓ1 ···ℓd.
▶ [Gupta-Kamath-Kayal-S-13]: ΣΠ
form Q1 ···Q
d.
∂ x(Q1 ···Qr) = span
∏
i∈S
Qi : S ⊂ [r] , |S| = r − 1
▶ [Nisan-Wigderson-95]: ΣΠdΣ circuits, sum of terms of the form
ℓ1 ···ℓd. Key observation: There are “few” linearly independent partial derivatives of ℓ1 ···ℓd.
▶ [Gupta-Kamath-Kayal-S-13]: ΣΠ
form Q1 ···Q
d.
∂ =k(Q1 ···Qr) = span
∏
i∈S
Qi : S ⊂ [r] , |S| = r − k
▶ [Nisan-Wigderson-95]: ΣΠdΣ circuits, sum of terms of the form
ℓ1 ···ℓd. Key observation: There are “few” linearly independent partial derivatives of ℓ1 ···ℓd.
▶ [Gupta-Kamath-Kayal-S-13]: ΣΠ
form Q1 ···Q
d.
∂ =k(Q1 ···Qr) = span
∏
i∈S
Qi : S ⊂ [r] , |S| = r − k
derivatives are zero if all Qis have low degree.
▶ [Nisan-Wigderson-95]: ΣΠdΣ circuits, sum of terms of the form
ℓ1 ···ℓd. Key observation: There are “few” linearly independent partial derivatives of ℓ1 ···ℓd.
▶ [Gupta-Kamath-Kayal-S-13]: ΣΠ
form Q1 ···Q
d.
Key observation: Many low-degree combinations of partial derivatives are zero if all Qis have low degree.
▶ [Nisan-Wigderson-95]: ΣΠdΣ circuits, sum of terms of the form
ℓ1 ···ℓd. Key observation: There are “few” linearly independent partial derivatives of ℓ1 ···ℓd.
▶ [Gupta-Kamath-Kayal-S-13]: ΣΠ
form Q1 ···Q
d.
Key observation: Many low-degree combinations of partial derivatives are zero if all Qis have low degree. Γ(f ) = dim
▶ [Nisan-Wigderson-95]: ΣΠdΣ circuits, sum of terms of the form
ℓ1 ···ℓd. Key observation: There are “few” linearly independent partial derivatives of ℓ1 ···ℓd.
▶ [Gupta-Kamath-Kayal-S-13]: ΣΠ
form Q1 ···Q
d.
x=ℓ∂ =k Mons of degree ℓ + d − k m xβ∂xα
▶ [Gupta-Kamath-Kayal-S-13]: ΣΠ
form Q1 ···Q
d.
Key observation: Many low-degree combinations of partial derivatives are zero if all Qis have low degree.
▶ [Gupta-Kamath-Kayal-S-13]: ΣΠ
form Q1 ···Q
d.
Key observation: Many low-degree combinations of partial derivatives are zero if all Qis have low degree.
▶ hom. ΣΠΣΠ circuits: terms like Q1 ···Qb with total degree d.
▶ [Gupta-Kamath-Kayal-S-13]: ΣΠ
form Q1 ···Q
d.
Key observation: Many low-degree combinations of partial derivatives are zero if all Qis have low degree.
▶ hom. ΣΠΣΠ circuits: terms like Q1 ···Qb with total degree d.
▶ [Gupta-Kamath-Kayal-S-13]: ΣΠ
form Q1 ···Q
d.
Key observation: Many low-degree combinations of partial derivatives are zero if all Qis have low degree.
▶ hom. ΣΠΣΠ circuits: terms like Q1 ···Qb with total degree d.
Low degree mons. High degree mons.
▶ [Gupta-Kamath-Kayal-S-13]: ΣΠ
form Q1 ···Q
d.
Key observation: Many low-degree combinations of partial derivatives are zero if all Qis have low degree.
▶ hom. ΣΠΣΠ circuits: terms like Q1 ···Qb with total degree d.
Low degree mons. High degree mons. [GKKS-12] ✓
▶ [Gupta-Kamath-Kayal-S-13]: ΣΠ
form Q1 ···Q
d.
Key observation: Many low-degree combinations of partial derivatives are zero if all Qis have low degree.
▶ hom. ΣΠΣΠ circuits: terms like Q1 ···Qb with total degree d.
Low degree mons. High degree large support mons.
High degree small support mons.
1
xd/2
2
[GKKS-12] ✓
▶ [Gupta-Kamath-Kayal-S-13]: ΣΠ
form Q1 ···Q
d.
Key observation: Many low-degree combinations of partial derivatives are zero if all Qis have low degree.
▶ hom. ΣΠΣΠ circuits: terms like Q1 ···Qb with total degree d.
Low degree mons. High degree large support mons.
High degree small support mons.
1
xd/2
2
[GKKS-12] ✓
▶ Idea 1 - Random restrictions: Randomly set a small number of
variables to zero
▶ [Gupta-Kamath-Kayal-S-13]: ΣΠ
form Q1 ···Q
d.
Key observation: Many low-degree combinations of partial derivatives are zero if all Qis have low degree.
▶ hom. ΣΠΣΠ circuits: terms like Q1 ···Qb with total degree d.
Low degree mons. High degree large support mons.
High degree small support mons.
1
xd/2
2
[GKKS-12] ✓
▶ Idea 1 - Random restrictions: Randomly set a small number of
variables to zero
▶ [Gupta-Kamath-Kayal-S-13]: ΣΠ
form Q1 ···Q
d.
Key observation: Many low-degree combinations of partial derivatives are zero if all Qis have low degree.
▶ hom. ΣΠΣΠ circuits: terms like Q1 ···Qb with total degree d.
Low degree mons. High degree large support mons.
High degree small support mons.
1
xd/2
2
[GKKS-12] ✓
▶ Idea 1 - Random restrictions: Randomly set a small number of
variables to zero
▶ Idea 2 - Multilinear projection: Discard all non-multilinear
monomials
▶ [Gupta-Kamath-Kayal-S-13]: ΣΠ
form Q1 ···Q
d.
Key observation: Many low-degree combinations of partial derivatives are zero if all Qis have low degree.
▶ hom. ΣΠΣΠ circuits: terms like Q1 ···Qb with total degree d.
Low degree mons. High degree large support mons.
High degree small support mons.
1
xd/2
2
[GKKS-12] ✓
▶ Idea 1 - Random restrictions: Randomly set a small number of
variables to zero
▶ Idea 2 - Multilinear projection: Discard all non-multilinear
monomials
▶ [Gupta-Kamath-Kayal-S-13]: ΣΠ
form Q1 ···Q
d.
Key observation: Many low-degree combinations of partial derivatives are zero if all Qis have low degree.
▶ [Kayal-Limaye-Saha-Srinivasan-13], [Kumar-Saraf-13]:
Low degree mons. High degree large support mons.
High degree small support mons.
1
xd/2
2
[GKKS-12] ✓
▶ Idea 1 - Random restrictions: Randomly set a small number of
variables to zero
▶ Idea 2 - Multilinear projection: Discard all non-multilinear
monomials
▶ [Kayal-Limaye-Saha-Srinivasan-13], [Kumar-Saraf-13]:
▶ Idea 1 - Random restrictions: Randomly set a small number of
variables to zero
▶ Idea 2 - Multilinear projection: Discard all non-multilinear
monomials
▶ [Kayal-Limaye-Saha-Srinivasan-13], [Kumar-Saraf-13]:
▶ Idea 1 - Random restrictions: Randomly set a small number of
variables to zero
▶ Idea 2 - Multilinear projection: Discard all non-multilinear
monomials
Γ(f ) = dim(x=ℓ∂ =k(f )) Dimension of shifted partials of f .
▶ [Kayal-Limaye-Saha-Srinivasan-13], [Kumar-Saraf-13]:
▶ Idea 1 - Random restrictions: Randomly set a small number of
variables to zero
▶ Idea 2 - Multilinear projection: Discard all non-multilinear
monomials
Γ(f ) = dim(x=ℓ∂ =k(ρ(f ))) Dimension of shifted partials of a random restriction of f .
▶ [Kayal-Limaye-Saha-Srinivasan-13], [Kumar-Saraf-13]:
▶ Idea 1 - Random restrictions: Randomly set a small number of
variables to zero
▶ Idea 2 - Multilinear projection: Discard all non-multilinear
monomials
Γ(f ) = dim(mult ◦ x=ℓ∂ =k(ρ(f ))) Dimension of projected shifted partials of a random restriction of f .
▶ [Kayal-Limaye-Saha-Srinivasan-13], [Kumar-Saraf-13]:
Γ(f ) = dim(mult ◦ x=ℓ∂ =k(ρ(f ))) Dimension of projected shifted partials of a random restriction of f .
▶ [Kayal-Limaye-Saha-Srinivasan-13], [Kumar-Saraf-13]:
Γ(f ) = dim(mult ◦ x=ℓ∂ =k(ρ(f ))) Dimension of projected shifted partials of a random restriction of f . x=ℓ∂ =k Multilinear mons of degree ℓ + d − k m xβ∂xα
We already have a complexity measure
PSPD for hom. depth- circuits.
How large is
PSPD for a generic depth- circuit?
Small a lower bound against depth- circuits. Large separation between depth- and depth- circuits. ... still don’t know
We already have a complexity measure ΓPSPD for hom. depth-4 circuits.
How large is
PSPD for a generic depth- circuit?
Small a lower bound against depth- circuits. Large separation between depth- and depth- circuits. ... still don’t know
We already have a complexity measure ΓPSPD for hom. depth-4 circuits.
How large is ΓPSPD for a generic depth-5 circuit?
Small a lower bound against depth- circuits. Large separation between depth- and depth- circuits. ... still don’t know
We already have a complexity measure ΓPSPD for hom. depth-4 circuits.
How large is ΓPSPD for a generic depth-5 circuit?
Small ⇒ a lower bound against depth-5 circuits. Large separation between depth- and depth- circuits. ... still don’t know
We already have a complexity measure ΓPSPD for hom. depth-4 circuits.
How large is ΓPSPD for a generic depth-5 circuit?
Small ⇒ a lower bound against depth-5 circuits. Large ⇒ separation between depth-5 and depth-4 circuits. ... still don’t know
We already have a complexity measure ΓPSPD for hom. depth-4 circuits.
How large is ΓPSPD for a generic depth-5 circuit?
Small ⇒ a lower bound against depth-5 circuits. Large ⇒ separation between depth-5 and depth-4 circuits. ... still don’t know
Γk(f ) = dim
∂ =k Monomials of degree d − k m ∂xα
in ∂xα(f )
∂ =k Monomials of degree d − k m ∂xα
in ∂xα(f ) ∂ =k Points in n
q
¯ a ∂xα
∂xα(ρ(f )) at ¯ a
∂ =k Monomials of degree d − k m ∂xα
in ∂xα(f ) ∂ =k Points in n
q
¯ a ∂xα
∂xα(ρ(f )) at ¯ a
Small rank Small rank
f = ℓ11 ···ℓ1d1 + ··· + ℓs1 ···ℓsds
f = ℓ11 ···ℓ1d1 + ··· + ℓs1 ···ℓsds
Low degree terms. High degree terms.
f = ℓ11 ···ℓ1d1 + ··· + ℓs1 ···ℓsds
Low degree terms. High degree high rank terms High degree low rank terms
1
ℓd/3
2
(ℓ1 + 3ℓ2)d/3
f = ℓ11 ···ℓ1d1 + ··· + ℓs1 ···ℓsds
Low degree terms. High degree high rank terms High degree low rank terms
1
ℓd/3
2
(ℓ1 + 3ℓ2)d/3 [NW-95] ✓
f = ℓ11 ···ℓ1d1 + ··· + ℓs1 ···ℓsds
Low degree terms. High degree high rank terms High degree low rank terms
1
ℓd/3
2
(ℓ1 + 3ℓ2)d/3 [NW-95] ✓ [NW-95] ✓
f = ℓ11 ···ℓ1d1 + ··· + ℓs1 ···ℓsds
Low degree terms. High degree high rank terms High degree low rank terms
1
ℓd/3
2
(ℓ1 + 3ℓ2)d/3 [NW-95] ✓ [NW-95] ✓
Observation
If dim{ℓ1,··· ,ℓr} is large, then almost all evaluations of it on n
q are
zero.
f = ℓ11 ···ℓ1d1 + ··· + ℓs1 ···ℓsds
Low degree terms. High degree high rank terms High degree low rank terms
1
ℓd/3
2
(ℓ1 + 3ℓ2)d/3 [NW-95] ✓ [NW-95] ✓
Observation
If dim{ℓ1,··· ,ℓr} is large, then almost all evaluations of it on n
q are
zero.
∂ =k
m ∂xα
∂ =k n
q
¯ a ∂xα
a
∂ =k n
q
¯ a ∂xα
a
∂ =k n
q
¯ a ∂xα
a
Lemma
If f is computable by a small ΣΠΣ circuit over q, then there the above matrix has small rank when a certain small set of columns are removed.
∂ =k n
q
¯ a ∂xα
a
Lemma
If f is computable by a small ΣΠΣ circuit over q, then there the above matrix has small rank when a certain small set of columns are removed.
Lemma
For Detn or Permn the above matrix remains full rank, as long as we removed only few columns.
ΣΠΣΠΣ Types of products of linear polynomials: Low degree products. High degree products. [GKKS]
ΣΠΣΠΣ Types of products of linear polynomials: Low degree products. High degree products. [GKKS] ✓
ΣΠΣΠΣ Types of products of linear polynomials: Low degree products. High degree, large rank products.
High degree, small rank products.
1
ℓd/2
2
[GKKS] ✓
ΣΠΣΠΣ Types of products of linear polynomials: Low degree products. High degree, large rank products.
High degree, small rank products.
1
ℓd/2
2
✓ [GKKS] ✓
ΣΠΣΠΣ Types of products of linear polynomials: Low degree products. High degree, large rank products.
High degree, small rank products.
1
ℓd/2
2
✓ [GKKS] ✓
Observation
If dim{ℓ1,··· ,ℓr} is large, then almost all evaluations of it on n
q are
zero.
ΣΠΣΠΣ Types of products of linear polynomials: Low degree products. High degree, large rank products.
High degree, small rank products.
1
ℓd/2
2
✓ [GKKS] ✓
Observation
If dim{ℓ1,··· ,ℓr} is large, then almost all evaluations of it on n
q are
zero.
We know this rank is large: x=ℓ∂ =k Mons of degree ℓ + d − k m xβ∂xα
We know this rank is large: x=ℓ∂ =k Mons of degree ℓ + d − k m xβ∂xα
Need to show this rank is large: x=ℓ∂ =k {0,1}n a xβ∂xα
Mons of degree ℓ + d − k n
q
a xβ∂xα Large rank [KLSS,KS] Large rank Vandermonde = x=ℓ∂ =k n
q
a xβ∂xα
Mons of degree ℓ + d − k n
q
a xβ∂xα Large rank ∵ [KLSS,KS] Large rank Vandermonde = x=ℓ∂ =k n
q
a xβ∂xα
Mons of degree ℓ + d − k n
q
a xβ∂xα Large rank ∵ [KLSS,KS] Large rank ∵ Vandermonde = x=ℓ∂ =k n
q
a xβ∂xα
Mons of degree ℓ + d − k n
q
a xβ∂xα Large rank ∵ [KLSS,KS] Large rank ∵ Vandermonde = x=ℓ∂ =k n
q
a xβ∂xα
Issue 1: [Fat matrix] [T all matrix] could be zero, even if both are full rank. Fix: Make the matrix slimmer by only considering evaluations on . Issue 2: But then , over , is never zero over . Fix: Ok fine. Work with for some random . Issue 3: Even with the matrix is still slightly fat and the Vandermonde is slightly tall. Fix: Prove a really good rank lower bound on the left matrix. (Barely manages to work for a specific explicit polynomial. Phew!)
Issue 1: [Fat matrix] × [T all matrix] could be zero, even if both are full rank. Fix: Make the matrix slimmer by only considering evaluations on . Issue 2: But then , over , is never zero over . Fix: Ok fine. Work with for some random . Issue 3: Even with the matrix is still slightly fat and the Vandermonde is slightly tall. Fix: Prove a really good rank lower bound on the left matrix. (Barely manages to work for a specific explicit polynomial. Phew!)
Issue 1: [Fat matrix] × [T all matrix] could be zero, even if both are full rank. Fix: Make the matrix slimmer by only considering evaluations on {0,1}n. Issue 2: But then , over , is never zero over . Fix: Ok fine. Work with for some random . Issue 3: Even with the matrix is still slightly fat and the Vandermonde is slightly tall. Fix: Prove a really good rank lower bound on the left matrix. (Barely manages to work for a specific explicit polynomial. Phew!)
Issue 1: [Fat matrix] × [T all matrix] could be zero, even if both are full rank. Fix: Make the matrix slimmer by only considering evaluations on {0,1}n. Issue 2: But then (x1 + 1)···(xn + 1), over 3, is never zero over {0,1}n. Fix: Ok fine. Work with for some random . Issue 3: Even with the matrix is still slightly fat and the Vandermonde is slightly tall. Fix: Prove a really good rank lower bound on the left matrix. (Barely manages to work for a specific explicit polynomial. Phew!)
Issue 1: [Fat matrix] × [T all matrix] could be zero, even if both are full rank. Fix: Make the matrix slimmer by only considering evaluations on {0,1}n. Issue 2: But then (x1 + 1)···(xn + 1), over 3, is never zero over {0,1}n. Fix: Ok fine. Work with ¯ c + {0,1}n for some random ¯ c ∈ n
q.
Issue 3: Even with the matrix is still slightly fat and the Vandermonde is slightly tall. Fix: Prove a really good rank lower bound on the left matrix. (Barely manages to work for a specific explicit polynomial. Phew!)
Issue 1: [Fat matrix] × [T all matrix] could be zero, even if both are full rank. Fix: Make the matrix slimmer by only considering evaluations on {0,1}n. Issue 2: But then (x1 + 1)···(xn + 1), over 3, is never zero over {0,1}n. Fix: Ok fine. Work with ¯ c + {0,1}n for some random ¯ c ∈ n
q.
Issue 3: Even with ¯ c + {0,1}n the matrix is still slightly fat and the Vandermonde is slightly tall. Fix: Prove a really good rank lower bound on the left matrix. (Barely manages to work for a specific explicit polynomial. Phew!)
Issue 1: [Fat matrix] × [T all matrix] could be zero, even if both are full rank. Fix: Make the matrix slimmer by only considering evaluations on {0,1}n. Issue 2: But then (x1 + 1)···(xn + 1), over 3, is never zero over {0,1}n. Fix: Ok fine. Work with ¯ c + {0,1}n for some random ¯ c ∈ n
q.
Issue 3: Even with ¯ c + {0,1}n the matrix is still slightly fat and the Vandermonde is slightly tall. Fix: Prove a really good rank lower bound on the left matrix. (Barely manages to work for a specific explicit polynomial. Phew!)
Issue 1: [Fat matrix] × [T all matrix] could be zero, even if both are full rank. Fix: Make the matrix slimmer by only considering evaluations on {0,1}n. Issue 2: But then (x1 + 1)···(xn + 1), over 3, is never zero over {0,1}n. Fix: Ok fine. Work with ¯ c + {0,1}n for some random ¯ c ∈ n
q.
Issue 3: Even with ¯ c + {0,1}n the matrix is still slightly fat and the Vandermonde is slightly tall. Fix: Prove a really good rank lower bound on the left matrix. (Barely manages to work for a specific explicit polynomial. Phew!)
Tieorem
There is a polynomial f ∈ VNP such that, for every finite field q, any hom. ΣΠΣΠΣ circuit computing f over q must have size exp(Ωq(
Remarks and open problems: [Grigoriev-Karpinski] meets [Kayal-Limaye-Saha-Srinivasan]. Delicate analysis. The proof ought to work for also but we don’t have a tight enough analysis (yet). After this, [Kumar-S] did manage to separate depth- and depth- in the low-degree regime, but via a difgerent complexity measure. Other fields?
Tieorem
There is a polynomial f ∈ VNP such that, for every finite field q, any hom. ΣΠΣΠΣ circuit computing f over q must have size exp(Ωq(
Remarks and open problems:
▶ [Grigoriev-Karpinski] meets [Kayal-Limaye-Saha-Srinivasan].
Delicate analysis. The proof ought to work for also but we don’t have a tight enough analysis (yet). After this, [Kumar-S] did manage to separate depth- and depth- in the low-degree regime, but via a difgerent complexity measure. Other fields?
Tieorem
There is a polynomial f ∈ VNP such that, for every finite field q, any hom. ΣΠΣΠΣ circuit computing f over q must have size exp(Ωq(
Remarks and open problems:
▶ [Grigoriev-Karpinski] meets [Kayal-Limaye-Saha-Srinivasan].
Delicate analysis.
▶ The proof ought to work for IMM also but we don’t have a tight
enough analysis (yet). After this, [Kumar-S] did manage to separate depth- and depth- in the low-degree regime, but via a difgerent complexity measure. Other fields?
Tieorem
There is a polynomial f ∈ VNP such that, for every finite field q, any hom. ΣΠΣΠΣ circuit computing f over q must have size exp(Ωq(
Remarks and open problems:
▶ [Grigoriev-Karpinski] meets [Kayal-Limaye-Saha-Srinivasan].
Delicate analysis.
▶ The proof ought to work for IMM also but we don’t have a tight
enough analysis (yet).
▶ After this, [Kumar-S] did manage to separate depth-4 and depth-5
in the low-degree regime, but via a difgerent complexity measure. Other fields?
Tieorem
There is a polynomial f ∈ VNP such that, for every finite field q, any hom. ΣΠΣΠΣ circuit computing f over q must have size exp(Ωq(
Remarks and open problems:
▶ [Grigoriev-Karpinski] meets [Kayal-Limaye-Saha-Srinivasan].
Delicate analysis.
▶ The proof ought to work for IMM also but we don’t have a tight
enough analysis (yet).
▶ After this, [Kumar-S] did manage to separate depth-4 and depth-5
in the low-degree regime, but via a difgerent complexity measure.
▶ Other fields?
Tieorem
There is a polynomial f ∈ VNP such that, for every finite field q, any hom. ΣΠΣΠΣ circuit computing f over q must have size exp(Ωq(
Remarks and open problems:
▶ [Grigoriev-Karpinski] meets [Kayal-Limaye-Saha-Srinivasan].
Delicate analysis.
▶ The proof ought to work for IMM also but we don’t have a tight
enough analysis (yet).
▶ After this, [Kumar-S] did manage to separate depth-4 and depth-5
in the low-degree regime, but via a difgerent complexity measure.
▶ Other fields?
▶ [Agrawal-Vinay]:
“Arithmetic Circuits: A Chasm at Depth Four” Foundations of Computer Science, 2008
▶ [Koiran]:
“Arithmetic circuits: The chasm at depth four gets wider” Theoretical Computer Science, 2012
▶ [T
avenas]: “Improved bounds for reduction to depth 4 and depth 3” Information and Computation, 2015
▶ [Nisan-Wigderson]:
“Lower Bounds on Arithmetic Circuits Via Partial Derivatives” Computational Complexity, 1997
▶ [Grigoriev-Karpinski]:
“An Exponential Lower Bound for Depth 3 Arithmetic Circuits” Symposium on Theory of Computing, 1998
▶ [Gupta-Kamath-Kayal-S]:
“Approaching the Chasm at Depth Four” Journal of the ACM, 2014
▶ [Kayal-Limaye-Saha-Srinivasan]:
“An Exponential Lower Bound for Homogeneous Depth Four Arithmetic Formulas” SIAM Journal of Computing, 2017
▶ [Kumar-Saraf]:
“On the Power of Homogeneous Depth 4 Arithmetic Circuits” SIAM Journal of Computing, 2017
▶ [Grigoriev-Razborov]:
“Exponential Lower Bounds for Depth 3 Arithmetic Circuits in Algebras of Functions over Finite Fields”
▶ [Kumar-S]:
“Finer Separations Between Shallow Arithmetic Circuits” Foundations of Software T echnology and Theoretical Computer Science, 2016