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A Near-Optimal Depth-Hierarchy Theorem for Small-Depth Multilinear - - PowerPoint PPT Presentation
A Near-Optimal Depth-Hierarchy Theorem for Small-Depth Multilinear - - PowerPoint PPT Presentation
A Near-Optimal Depth-Hierarchy Theorem for Small-Depth Multilinear Circuits Nutan Limaye Compuer Science and Engineering Department, Indian Institute of Technology, Bombay, (IITB) India. Joint work with Suryajith Chillara, Christian Engels,
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More Resources
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More Resources More power?
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Power of resources
Turing machines
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Power of resources
Turing machines With more time, can Turing machines compute more languages?
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Power of resources
Turing machines With more time, can Turing machines compute more languages? With more space, can Turing machines compute more languages?
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Power of resources
Turing machines With more time, can Turing machines compute more languages? With more space, can Turing machines compute more languages?
Theorem (Time Hierarchy Theorem, [Hartmanis and Stearns, 65])
There exists a language L that is computed by a TM in time O(t(n) log t(n)) such that no TM running in time o(t(n)) can compute it.
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Power of resources
Turing machines With more time, can Turing machines compute more languages? With more space, can Turing machines compute more languages?
Theorem (Time Hierarchy Theorem, [Hartmanis and Stearns, 65])
There exists a language L that is computed by a TM in time O(t(n) log t(n)) such that no TM running in time o(t(n)) can compute it.
Theorem (Space Hierarchy Theorem, [Stearns, Hartmanis, Lewis, 65])
There exists a language L that is computed by a TM in space O(s(n)) such that no TM running in space o(s(n)) can compute it.
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Power of resources
Turing machines With more time, can Turing machines compute more languages? With more space, can Turing machines compute more languages?
Theorem (Time Hierarchy Theorem, [Hartmanis and Stearns, 65])
There exists a language L that is computed by a TM in time O(t(n) log t(n)) such that no TM running in time o(t(n)) can compute it.
Theorem (Space Hierarchy Theorem, [Stearns, Hartmanis, Lewis, 65])
There exists a language L that is computed by a TM in space O(s(n)) such that no TM running in space o(s(n)) can compute it. Non-explicit separations.
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Power of resources
We can ask a similar question for any model of computation and a resource.
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Power of resources
We can ask a similar question for any model of computation and a resource. Model of computation Resources of interest
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Power of resources
We can ask a similar question for any model of computation and a resource. Model of computation Resources of interest Turing machines time, space, number of random bits, non-determinism, advice
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Power of resources
We can ask a similar question for any model of computation and a resource. Model of computation Resources of interest Turing machines time, space, number of random bits, non-determinism, advice Boolean circuits size, depth
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Power of resources
We can ask a similar question for any model of computation and a resource. Model of computation Resources of interest Turing machines time, space, number of random bits, non-determinism, advice Boolean circuits size, depth Today we will focus on arithmetic formulas as a model of computation.
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A model of computation for polynomials
Arithmetic formula
+ × × x1 x1 x2 + × x1 x1 −1
Definition: An arithmetic formula
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A model of computation for polynomials
Arithmetic formula
+ × × x1 x1 x2 + × x1 x1 −1
Definition: An arithmetic formula a directed tree
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A model of computation for polynomials
Arithmetic formula
+ × × x1 x1 x2 + × x1 x1 −1
Definition: An arithmetic formula a directed tree with nodes labeled by +, ×, x1, . . . , xn or constants .
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A model of computation for polynomials
Arithmetic formula
+ × × x1 x1 x2 + × x1 x1 −1
Definition: An arithmetic formula a directed tree with nodes labeled by +, ×, x1, . . . , xn or constants . indegree 0 nodes : input gates
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A model of computation for polynomials
Arithmetic formula
+ × × x1 x1 x2 + × x1 x1 −1
Definition: An arithmetic formula a directed tree with nodes labeled by +, ×, x1, . . . , xn or constants . indegree 0 nodes : input gates
- utdegree 0 nodes: output gates
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A model of computation for polynomials
Arithmetic formula
+ × × x1 x1 x2 + × x1 x1 −1
Definition: An arithmetic formula a directed tree with nodes labeled by +, ×, x1, . . . , xn or constants . indegree 0 nodes : input gates
- utdegree 0 nodes: output gates
without loss of generality alternate +, × layers.
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A model of computation for polynomials
Arithmetic formula
+ × × x1 x1 x2 + × x1 x1 −1
Definition: An arithmetic formula a directed tree with nodes labeled by +, ×, x1, . . . , xn or constants (say over F). indegree 0 nodes : input gates
- utdegree 0 nodes: output gates
without loss of generality alternate +, × layers. Size and product-depth The number of nodes in the tree is the size of the formula.
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A model of computation for polynomials
Arithmetic formula
+ × × x1 x1 x2 + × x1 x1 −1
Definition: An arithmetic formula a directed tree with nodes labeled by +, ×, x1, . . . , xn or constants (say over F). indegree 0 nodes : input gates
- utdegree 0 nodes: output gates
without loss of generality alternate +, × layers. Size and product-depth The number of nodes in the tree is the size of the formula. The maximum number of product gates along any leaf to root path is its product-depth.
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A model of computation for polynomials
Arithmetic formula
+ × × x1 x1 x2 + × x1 x1 −1
Definition: An arithmetic formula a directed tree with nodes labeled by +, ×, x1, . . . , xn or constants (say over F). indegree 0 nodes : input gates
- utdegree 0 nodes: output gates
without loss of generality alternate +, × layers. Size and product-depth The number of nodes in the tree is the size of the formula. The maximum number of product gates along any leaf to root path is its product-depth. (Closely related to the depth.)
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Small depth formulas
We will focus on small product-depth formulas.
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Small depth formulas
We will focus on small product-depth formulas. Product-depth ∆ = 1: formulas
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Small depth formulas
We will focus on small product-depth formulas. Product-depth ∆ = 1: formulas The model is complete.
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Small depth formulas
We will focus on small product-depth formulas. Product-depth ∆ = 1: formulas The model is complete. Let X = {x1, . . . , xn}.
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Small depth formulas
We will focus on small product-depth formulas. Product-depth ∆ = 1: formulas The model is complete. Let X = {x1, . . . , xn}. Let p(X) ∈ F[X] be a degree d polynomial.
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Small depth formulas
We will focus on small product-depth formulas. Product-depth ∆ = 1: formulas The model is complete. Let X = {x1, . . . , xn}. Let p(X) ∈ F[X] be a degree d polynomial. p(x) =
- m∈M
cm · m,
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Small depth formulas
We will focus on small product-depth formulas. Product-depth ∆ = 1: formulas The model is complete. Let X = {x1, . . . , xn}. Let p(X) ∈ F[X] be a degree d polynomial. p(x) =
- m∈M
cm · m, where M set of all monomials in n variables of degree at most d.
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Small depth formulas
We will focus on small product-depth formulas. Product-depth ∆ = 1: formulas The model is complete. Let X = {x1, . . . , xn}. Let p(X) ∈ F[X] be a degree d polynomial. p(x) =
- m∈M
cm · m, where M set of all monomials in n variables of degree at most d. It may not always be a succinct representation.
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Examples of polynomials
Let p(X) =
- S⊆[n]
- i∈S
xi
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Examples of polynomials
Let p(X) =
- S⊆[n]
- i∈S
xi This has size O(2n).
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Examples of polynomials
Let p(X) =
- S⊆[n]
- i∈S
xi This has size O(2n). However, here is its succinct representation.
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Examples of polynomials
Let p(X) =
- S⊆[n]
- i∈S
xi This has size O(2n). However, here is its succinct representation. p(X) =
i∈[n](1 + xi)
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Examples of polynomials
Let p(X) =
- S⊆[n]
- i∈S
xi This has size O(2n). However, here is its succinct representation. p(X) =
i∈[n](1 + xi) of size O(n).
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Examples of polynomials
Let p(X) =
- S⊆[n]
- i∈S
xi This has size O(2n). However, here is its succinct representation. p(X) =
i∈[n](1 + xi) of size O(n).
This is a formula for the same polynomial.
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Multilinear polynomials
Let X = {x1, . . . , xN}.
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Multilinear polynomials
Let X = {x1, . . . , xN}. Let p(X) ∈ F[X] be a degree d multilinear polynomial.
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Multilinear polynomials
Let X = {x1, . . . , xN}. Let p(X) ∈ F[X] be a degree d multilinear polynomial. p(x) =
- S∈[n]:|S|≤d
cS ·
- i∈S
xi,
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Multilinear polynomials
Let X = {x1, . . . , xN}. Let p(X) ∈ F[X] be a degree d multilinear polynomial. p(x) =
- S∈[n]:|S|≤d
cS ·
- i∈S
xi, Many interesting polynomials are multilinear.
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Multilinear polynomials
Let X = {x1, . . . , xN}. Let p(X) ∈ F[X] be a degree d multilinear polynomial. p(x) =
- S∈[n]:|S|≤d
cS ·
- i∈S
xi, Many interesting polynomials are multilinear. Determinant: Det(X) =
σ∈Sn sgn(σ) n i=1 xiσ(i)
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Multilinear polynomials
Let X = {x1, . . . , xN}. Let p(X) ∈ F[X] be a degree d multilinear polynomial. p(x) =
- S∈[n]:|S|≤d
cS ·
- i∈S
xi, Many interesting polynomials are multilinear. Determinant: Det(X) =
σ∈Sn sgn(σ) n i=1 xiσ(i)
Permanent: Perm(X) =
σ∈Sn
n
i=1 xiσ(i)
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Multilinear polynomials
Let X = {x1, . . . , xN}. Let p(X) ∈ F[X] be a degree d multilinear polynomial. p(x) =
- S∈[n]:|S|≤d
cS ·
- i∈S
xi, Many interesting polynomials are multilinear. Determinant: Det(X) =
σ∈Sn sgn(σ) n i=1 xiσ(i)
Permanent: Perm(X) =
σ∈Sn
n
i=1 xiσ(i)
Matrix Multiplication: (X × Y )i,j = n
k=1 xik × ykj
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Multilinear polynomials
Let X = {x1, . . . , xN}. Let p(X) ∈ F[X] be a degree d multilinear polynomial. p(x) =
- S∈[n]:|S|≤d
cS ·
- i∈S
xi, Many interesting polynomials are multilinear. Determinant: Det(X) =
σ∈Sn sgn(σ) n i=1 xiσ(i)
Permanent: Perm(X) =
σ∈Sn
n
i=1 xiσ(i)
Matrix Multiplication: (X × Y )i,j = n
k=1 xik × ykj
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Multilinear formulas
A formula is multilinear if every gate in it computes a multilinear polynomial.
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Multilinear formulas
A formula is multilinear if every gate in it computes a multilinear polynomial. Many tools and techniques A breakthrough result of Raz [Raz04] gave a strong lower bound.
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Multilinear formulas
A formula is multilinear if every gate in it computes a multilinear polynomial. Many tools and techniques A breakthrough result of Raz [Raz04] gave a strong lower bound. Multilinear formulas for Det/Perm must have superpolynomial size.
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Multilinear formulas
A formula is multilinear if every gate in it computes a multilinear polynomial. Many tools and techniques A breakthrough result of Raz [Raz04] gave a strong lower bound. Multilinear formulas for Det/Perm must have superpolynomial size. A set of tools introduced in [Raz04].
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Multilinear formulas
A formula is multilinear if every gate in it computes a multilinear polynomial. Many tools and techniques A breakthrough result of Raz [Raz04] gave a strong lower bound. Multilinear formulas for Det/Perm must have superpolynomial size. A set of tools introduced in [Raz04]. Extended and appended by a line of work. [Raz06,RSY07,RY09,DMPY12,KV17].
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Small depth formulas
We will focus on small product-depth multilinear formulas.
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Small depth formulas
We will focus on small product-depth multilinear formulas. Product-depth ∆ = 1: or formulas
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Small depth formulas
We will focus on small product-depth multilinear formulas. Product-depth ∆ = 1: or formulas formulas are not succinct.
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Small depth formulas
We will focus on small product-depth multilinear formulas. Product-depth ∆ = 1: or formulas formulas are not succinct. What about formulas?
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Small depth formulas
We will focus on small product-depth multilinear formulas. Product-depth ∆ = 1: or formulas formulas are not succinct. What about formulas? p(x) =
- i∈[s]
- j∈[s′]
Li,j, where, Li,j are linear polynomials in X. The model is surprisingly powerful!
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Small depth formulas
We will focus on small product-depth multilinear formulas. Product-depth ∆ = 1: or formulas formulas are not succinct. What about formulas? p(x) =
- i∈[s]
- j∈[s′]
Li,j, where, Li,j are linear polynomials in X. The model is surprisingly powerful! [AV08,Koi09,Tav10,GKKS12] Any polynomial on n variables of degree d computable by a size s circuit can be computed by
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Small depth formulas
We will focus on small product-depth multilinear formulas. Product-depth ∆ = 1: or formulas formulas are not succinct. What about formulas? p(x) =
- i∈[s]
- j∈[s′]
Li,j, where, Li,j are linear polynomials in X. The model is surprisingly powerful! [AV08,Koi09,Tav10,GKKS12] Any polynomial on n variables of degree d computable by a size s circuit can be computed by formula of size sO(
√ d).
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Small depth formulas
We will focus on small product-depth multilinear formulas. Product-depth ∆ = 1: or formulas formulas are not succinct. What about formulas? p(x) =
- i∈[s]
- j∈[s′]
Li,j, where, Li,j are linear polynomials in X. The model is surprisingly powerful! [AV08,Koi09,Tav10,GKKS12] Any polynomial on n variables of degree d computable by a size s circuit can be computed by formula of size sO(
√ d).
(Assume characteristic 0.)
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Small depth formulas
We will focus on small product-depth multilinear formulas. Product-depth ∆ = 1: or formulas formulas are not succinct. What about formulas? p(x) =
- i∈[s]
- j∈[s′]
Li,j, where, Li,j are linear polynomials in X. The model is surprisingly powerful! [AV08,Koi09,Tav10,GKKS12] Any polynomial on n variables of degree d computable by a size s circuit can be computed by formula of size sO(
√ d).
(Assume characteristic 0.) This realization is non-multilinear!
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Product-depth ∆ = 1
Non-multilinear to multilinear formula conversion.
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Product-depth ∆ = 1
Non-multilinear to multilinear formula conversion. Let p(X) be a multilinear polynomial computable by a formula of size s.
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Product-depth ∆ = 1
Non-multilinear to multilinear formula conversion. Let p(X) be a multilinear polynomial computable by a formula of size s. Does p(X) have a multilinear formula of size sO(1)?
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Product-depth ∆ = 1
Non-multilinear to multilinear formula conversion. Let p(X) be a multilinear polynomial computable by a formula of size s. Does p(X) have a multilinear formula of size sO(1)? [Chillara, L, Srinivasan, 18] prove that the answer is no.
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Product-depth ∆ = 1
Non-multilinear to multilinear formula conversion. Let p(X) be a multilinear polynomial computable by a formula of size s. Does p(X) have a multilinear formula of size sO(1)? [Chillara, L, Srinivasan, 18] prove that the answer is no. Product-depth ∆ = 2 to ∆ = 1 conversion
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Product-depth ∆ = 1
Non-multilinear to multilinear formula conversion. Let p(X) be a multilinear polynomial computable by a formula of size s. Does p(X) have a multilinear formula of size sO(1)? [Chillara, L, Srinivasan, 18] prove that the answer is no. Product-depth ∆ = 2 to ∆ = 1 conversion Let p(X) be a multilinear polynomial computable by a multilinear formula of size s.
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Product-depth ∆ = 1
Non-multilinear to multilinear formula conversion. Let p(X) be a multilinear polynomial computable by a formula of size s. Does p(X) have a multilinear formula of size sO(1)? [Chillara, L, Srinivasan, 18] prove that the answer is no. Product-depth ∆ = 2 to ∆ = 1 conversion Let p(X) be a multilinear polynomial computable by a multilinear formula of size s. Does p(X) have a multilinear formula of size sO(1)?
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Product-depth ∆ = 1
Non-multilinear to multilinear formula conversion. Let p(X) be a multilinear polynomial computable by a formula of size s. Does p(X) have a multilinear formula of size sO(1)? [Chillara, L, Srinivasan, 18] prove that the answer is no. Product-depth ∆ = 2 to ∆ = 1 conversion Let p(X) be a multilinear polynomial computable by a multilinear formula of size s. Does p(X) have a multilinear formula of size sO(1)? [Kayal, Nair, Saha, 15] show that this is not possible.
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How expensive ∆ = 2 − → ∆ = 1?
Consider formula of size s.
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How expensive ∆ = 2 − → ∆ = 1?
Consider formula of size s. Consider the layer
i∈[t] Qi.
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How expensive ∆ = 2 − → ∆ = 1?
Consider formula of size s. Consider the layer
i∈[t] Qi.
That is, [t] .
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How expensive ∆ = 2 − → ∆ = 1?
Consider formula of size s. Consider the layer
i∈[t] Qi.
That is, [t] . Open up the multiplication of summands as a sum of multiplications.
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How expensive ∆ = 2 − → ∆ = 1?
Consider formula of size s. Consider the layer
i∈[t] Qi.
That is, [t] . Open up the multiplication of summands as a sum of multiplications. [t] − → [exp(t)]
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How expensive ∆ = 2 − → ∆ = 1?
Consider formula of size s. Consider the layer
i∈[t] Qi.
That is, [t] . Open up the multiplication of summands as a sum of multiplications. [t] − → [exp(t)] − → [exp(t)]
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How expensive ∆ = 2 − → ∆ = 1?
Consider formula of size s. Consider the layer
i∈[t] Qi.
That is, [t] . Open up the multiplication of summands as a sum of multiplications. [t] − → [exp(t)] − → [exp(t)] The conversion incurs an exponential blow-up.
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How expensive ∆ = 2 − → ∆ = 1?
Consider formula of size s. Consider the layer
i∈[t] Qi.
That is, [t] . Open up the multiplication of summands as a sum of multiplications. [t] − → [exp(t)] − → [exp(t)] The conversion incurs an exponential blow-up. [Kayal, Nair, Saha, 15] show that this exponential blow-up is essential while going from ∆ = 2 to ∆ = 1.
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Larger product depth ∆ + 1 − → ∆
Consider . . . formula of size s and product depth ∆ + 1.
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Larger product depth ∆ + 1 − → ∆
Consider . . . formula of size s and product depth ∆ + 1. Consider the layer
i∈[t] Qi, such that t ≤ sO(1/∆).
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Larger product depth ∆ + 1 − → ∆
Consider . . . formula of size s and product depth ∆ + 1. Consider the layer
i∈[t] Qi, such that t ≤ sO(1/∆).
That is, . . . [(sO(1/∆))] . . . .
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Larger product depth ∆ + 1 − → ∆
Consider . . . formula of size s and product depth ∆ + 1. Consider the layer
i∈[t] Qi, such that t ≤ sO(1/∆).
That is, . . . [(sO(1/∆))] . . . . Open up the multiplication of summands as a sum of multiplications.
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Larger product depth ∆ + 1 − → ∆
Consider . . . formula of size s and product depth ∆ + 1. Consider the layer
i∈[t] Qi, such that t ≤ sO(1/∆).
That is, . . . [(sO(1/∆))] . . . . Open up the multiplication of summands as a sum of multiplications. . . . [(sO(1/∆))] . . . − → . . . [exp((sO(1/∆)))] . . .
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Larger product depth ∆ + 1 − → ∆
Consider . . . formula of size s and product depth ∆ + 1. Consider the layer
i∈[t] Qi, such that t ≤ sO(1/∆).
That is, . . . [(sO(1/∆))] . . . . Open up the multiplication of summands as a sum of multiplications. . . . [(sO(1/∆))] . . . − → . . . [exp((sO(1/∆)))] . . . − → . . . [exp((sO(1/∆)))] . . .
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Larger product depth ∆ + 1 − → ∆
Consider . . . formula of size s and product depth ∆ + 1. Consider the layer
i∈[t] Qi, such that t ≤ sO(1/∆).
That is, . . . [(sO(1/∆))] . . . . Open up the multiplication of summands as a sum of multiplications. . . . [(sO(1/∆))] . . . − → . . . [exp((sO(1/∆)))] . . . − → . . . [exp((sO(1/∆)))] . . . Careful analysis shows a blow-up of exp(s1/∆+o(1)).
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Larger product depth ∆ + 1 − → ∆
Consider . . . formula of size s and product depth ∆ + 1. Consider the layer
i∈[t] Qi, such that t ≤ sO(1/∆).
That is, . . . [(sO(1/∆))] . . . . Open up the multiplication of summands as a sum of multiplications. . . . [(sO(1/∆))] . . . − → . . . [exp((sO(1/∆)))] . . . − → . . . [exp((sO(1/∆)))] . . . Careful analysis shows a blow-up of exp(s1/∆+o(1)). Is the blow-up essential?
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Depth hierarchy theorem
More Resources
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Depth hierarchy theorem
More Resources
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Depth hierarchy theorem
More Resources More power?
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Depth hierarchy theorem
More Resources More power? More Product-depth
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Depth hierarchy theorem
More Resources More power? More Product-depth
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Depth hierarchy theorem
More Resources More power? More Product-depth More power?
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Depth hierarchy theorems
Arithmetic circuit complexity world
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Depth hierarchy theorems
Arithmetic circuit complexity world [Raz and Yehudayoff, 2009] prove quasipolynomial depth-hierarchy theorem
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Depth hierarchy theorems
Arithmetic circuit complexity world [Raz and Yehudayoff, 2009] prove quasipolynomial depth-hierarchy theorem, i.e. they show quasipolynomial size lower bound for converting product-depth ∆ + 1 multilinear formula into product-depth ∆ formula, as long as ∆ is small.
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Depth hierarchy theorems
Arithmetic circuit complexity world [Raz and Yehudayoff, 2009] prove quasipolynomial depth-hierarchy theorem, i.e. they show quasipolynomial size lower bound for converting product-depth ∆ + 1 multilinear formula into product-depth ∆ formula, as long as ∆ is small. [Kayal, Nair, Saha, 2015] show an exponential size lower bound for converting product-depth 2 multilinear formulas into product-depth 1 formulas.
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Depth hierarchy theorems
Arithmetic circuit complexity world [Raz and Yehudayoff, 2009] prove quasipolynomial depth-hierarchy theorem, i.e. they show quasipolynomial size lower bound for converting product-depth ∆ + 1 multilinear formula into product-depth ∆ formula, as long as ∆ is small. [Kayal, Nair, Saha, 2015] show an exponential size lower bound for converting product-depth 2 multilinear formulas into product-depth 1 formulas. Boolean circuit complexity world
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Depth hierarchy theorems
Arithmetic circuit complexity world [Raz and Yehudayoff, 2009] prove quasipolynomial depth-hierarchy theorem, i.e. they show quasipolynomial size lower bound for converting product-depth ∆ + 1 multilinear formula into product-depth ∆ formula, as long as ∆ is small. [Kayal, Nair, Saha, 2015] show an exponential size lower bound for converting product-depth 2 multilinear formulas into product-depth 1 formulas. Boolean circuit complexity world [Ajtai,Frust et al.,Yao, H˚ astad, 1980s] proved quasipolynomial depth-hierarchy theorem.
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Depth hierarchy theorems
Arithmetic circuit complexity world [Raz and Yehudayoff, 2009] prove quasipolynomial depth-hierarchy theorem, i.e. they show quasipolynomial size lower bound for converting product-depth ∆ + 1 multilinear formula into product-depth ∆ formula, as long as ∆ is small. [Kayal, Nair, Saha, 2015] show an exponential size lower bound for converting product-depth 2 multilinear formulas into product-depth 1 formulas. Boolean circuit complexity world [Ajtai,Frust et al.,Yao, H˚ astad, 1980s] proved quasipolynomial depth-hierarchy theorem. [H˚ astad, 1986] proved exponential depth-hierarchy theorem.
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Depth hierarchy theorems
Arithmetic circuit complexity world
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Depth hierarchy theorems
Arithmetic circuit complexity world [Raz and Yehudayoff, 2009] prove quasipolynomial depth-hierarchy theorem
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Depth hierarchy theorems
Arithmetic circuit complexity world [Raz and Yehudayoff, 2009] prove quasipolynomial depth-hierarchy theorem, i.e. they show quasipolynomial size lower bound for converting product-depth ∆ + 1 multilinear formula into product-depth ∆ formula, as long as ∆ is small.
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Depth hierarchy theorems
Arithmetic circuit complexity world [Raz and Yehudayoff, 2009] prove quasipolynomial depth-hierarchy theorem, i.e. they show quasipolynomial size lower bound for converting product-depth ∆ + 1 multilinear formula into product-depth ∆ formula, as long as ∆ is small. [Kayal, Nair, Saha, 2015] show an exponential size lower bound for converting product-depth 2 multilinear formulas into product-depth 1 formulas.
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Depth hierarchy theorems
Arithmetic circuit complexity world [Raz and Yehudayoff, 2009] prove quasipolynomial depth-hierarchy theorem, i.e. they show quasipolynomial size lower bound for converting product-depth ∆ + 1 multilinear formula into product-depth ∆ formula, as long as ∆ is small. [Kayal, Nair, Saha, 2015] show an exponential size lower bound for converting product-depth 2 multilinear formulas into product-depth 1 formulas. Our result: Near-optimal Depth Hierarchy Theorem
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Depth hierarchy theorems
Arithmetic circuit complexity world [Raz and Yehudayoff, 2009] prove quasipolynomial depth-hierarchy theorem, i.e. they show quasipolynomial size lower bound for converting product-depth ∆ + 1 multilinear formula into product-depth ∆ formula, as long as ∆ is small. [Kayal, Nair, Saha, 2015] show an exponential size lower bound for converting product-depth 2 multilinear formulas into product-depth 1 formulas. Our result: Near-optimal Depth Hierarchy Theorem For any constant ∆, there is an explicit polynomial P∆+1(x1, . . . , xn) such that
SLIDE 104
Depth hierarchy theorems
Arithmetic circuit complexity world [Raz and Yehudayoff, 2009] prove quasipolynomial depth-hierarchy theorem, i.e. they show quasipolynomial size lower bound for converting product-depth ∆ + 1 multilinear formula into product-depth ∆ formula, as long as ∆ is small. [Kayal, Nair, Saha, 2015] show an exponential size lower bound for converting product-depth 2 multilinear formulas into product-depth 1 formulas. Our result: Near-optimal Depth Hierarchy Theorem For any constant ∆, there is an explicit polynomial P∆+1(x1, . . . , xn) such that
P∆+1(X) is computed by multilinear formula F∆+1 of product-depth ∆ + 1 and size O(n).
SLIDE 105
Depth hierarchy theorems
Arithmetic circuit complexity world [Raz and Yehudayoff, 2009] prove quasipolynomial depth-hierarchy theorem, i.e. they show quasipolynomial size lower bound for converting product-depth ∆ + 1 multilinear formula into product-depth ∆ formula, as long as ∆ is small. [Kayal, Nair, Saha, 2015] show an exponential size lower bound for converting product-depth 2 multilinear formulas into product-depth 1 formulas. Our result: Near-optimal Depth Hierarchy Theorem For any constant ∆, there is an explicit polynomial P∆+1(x1, . . . , xn) such that
P∆+1(X) is computed by multilinear formula F∆+1 of product-depth ∆ + 1 and size O(n). However, any multilinear formula of product-depth ≤ ∆ for P∆+1(X) must have size exp(nα∆)
SLIDE 106
Depth hierarchy theorems
Arithmetic circuit complexity world [Raz and Yehudayoff, 2009] prove quasipolynomial depth-hierarchy theorem, i.e. they show quasipolynomial size lower bound for converting product-depth ∆ + 1 multilinear formula into product-depth ∆ formula, as long as ∆ is small. [Kayal, Nair, Saha, 2015] show an exponential size lower bound for converting product-depth 2 multilinear formulas into product-depth 1 formulas. Our result: Near-optimal Depth Hierarchy Theorem For any constant ∆, there is an explicit polynomial P∆+1(x1, . . . , xn) such that
P∆+1(X) is computed by multilinear formula F∆+1 of product-depth ∆ + 1 and size O(n). However, any multilinear formula of product-depth ≤ ∆ for P∆+1(X) must have size exp(nα∆), where α∆ = Ω(1/∆).
SLIDE 107
Depth hierarchy theorems
Arithmetic circuit complexity world [Raz and Yehudayoff, 2009] prove quasipolynomial depth-hierarchy theorem, i.e. they show quasipolynomial size lower bound for converting product-depth ∆ + 1 multilinear formula into product-depth ∆ formula, as long as ∆ is small. [Kayal, Nair, Saha, 2015] show an exponential size lower bound for converting product-depth 2 multilinear formulas into product-depth 1 formulas. Our result: Near-optimal Depth Hierarchy Theorem For any constant ∆(can be slowly growing function of n), there is an explicit polynomial P∆+1(x1, . . . , xn) such that
P∆+1(X) is computed by multilinear formula F∆+1 of product-depth ∆ + 1 and size O(n). However, any multilinear formula of product-depth ≤ ∆ for P∆+1(X) must have size exp(nα∆), where α∆ = Ω(1/∆).
SLIDE 108
Hard Polynomial
Construction of the hard polynomial
SLIDE 109
Hard Polynomial
Construction of the hard polynomial Polynomial P∆ is constructed inductively.
SLIDE 110
Hard Polynomial
Construction of the hard polynomial Polynomial P∆ is constructed inductively. This polynomial is inspired by the construction of [Chen, Oliviera, Servedio, Tan, 2016] who prove near optimal Boolean circuit lower bounds for checking graph connectivity at small depth.
SLIDE 111
Hard Polynomial
Construction of the hard polynomial Polynomial P∆ is constructed inductively. This polynomial is inspired by the construction of [Chen, Oliviera, Servedio, Tan, 2016] who prove near optimal Boolean circuit lower bounds for checking graph connectivity at small depth.
SLIDE 112
Hard Polynomial
P(0) is a 4 layered Algebraic Branching Program defined by G (0).
SLIDE 113
Hard Polynomial
P(0) is a 4 layered Algebraic Branching Program defined by G (0). x1,1 x1,2 x3,1 x3,2 x2,1 x2,2 x4,1 x4,2
Figure: Definition of G (0)
SLIDE 114
Hard Polynomial
P(0) is a 4 layered Algebraic Branching Program defined by G (0). x1,1 x1,2 x3,1 x3,2 x2,1 x2,2 x4,1 x4,2
Figure: Definition of G (0)
P(0) = x1,1x1,2x3,1x3,2 + x1,1x1,2x4,1x4,2 + x2,1x2,2x3,1x3,2 + x2,1x2,2x4,1x4,2.
SLIDE 115
Hard Polynomial
P(0) is a 4 layered Algebraic Branching Program defined by G (0). x1,1 x1,2 x3,1 x3,2 x2,1 x2,2 x4,1 x4,2
Figure: Definition of G (0)
P(0) = x1,1x1,2x3,1x3,2 + x1,1x1,2x4,1x4,2 + x2,1x2,2x3,1x3,2 + x2,1x2,2x4,1x4,2. This is a succint expression of the form ΣΠ, i.e., product-depth 1
SLIDE 116
Hard Polynomial
P(0) is a 4 layered Algebraic Branching Program defined by G (0). x1,1 x1,2 x3,1 x3,2 x2,1 x2,2 x4,1 x4,2
Figure: Definition of G (0)
P(0) = x1,1x1,2x3,1x3,2 + x1,1x1,2x4,1x4,2 + x2,1x2,2x3,1x3,2 + x2,1x2,2x4,1x4,2. This is a succint expression of the form ΣΠ, i.e., product-depth 1 and of size O(1).
SLIDE 117
Hard Polynomial
H(1) is obtained by composing two copies of G (0) in parallel.
SLIDE 118
Hard Polynomial
H(1) is obtained by composing two copies of G (0) in parallel. G (1) is obtained by composing m series of H(1) in series.
SLIDE 119
Hard Polynomial
H(1) is obtained by composing two copies of G (0) in parallel. G (1) is obtained by composing m series of H(1) in series. P(1) is the sum of weights of all the source to sink paths in G (1).
SLIDE 120
Hard Polynomial
H(1) is obtained by composing two copies of G (0) in parallel. G (1) is obtained by composing m series of H(1) in series. P(1) is the sum of weights of all the source to sink paths in G (1).
Figure: H(1).
SLIDE 121
Hard Polynomial
H(1) is obtained by composing two copies of G (0) in parallel. G (1) is obtained by composing m series of H(1) in series. P(1) is the sum of weights of all the source to sink paths in G (1).
Figure: H(1).
. . .
m copies
Figure: G (1)
SLIDE 122
Hard Polynomial
H(1) is obtained by composing two copies of G (0) in parallel. G (1) is obtained by composing m series of H(1) in series. P(1) is the sum of weights of all the source to sink paths in G (1).
Figure: H(1).
. . .
m copies
Figure: G (1)
P(1) =
m
- i=1
(P(0)
(i,1) + P(0) (i,2)).
SLIDE 123
Hard Polynomial
H(1) is obtained by composing two copies of G (0) in parallel. G (1) is obtained by composing m series of H(1) in series. P(1) is the sum of weights of all the source to sink paths in G (1). P(1) is a polynomial over n1 := 8(2m) many variables and has product-depth 2, size O(m) = O(n1) formula.
Figure: H(1).
. . .
m copies
Figure: G (1)
P(1) =
m
- i=1
(P(0)
(i,1) + P(0) (i,2)).
SLIDE 124
Hard Polynomial
H(∆) is obtained by composing two copies of G (∆−1) in parallel.
SLIDE 125
Hard Polynomial
H(∆) is obtained by composing two copies of G (∆−1) in parallel. G (∆) is obtained by composing m series of H(∆) in series.
SLIDE 126
Hard Polynomial
H(∆) is obtained by composing two copies of G (∆−1) in parallel. G (∆) is obtained by composing m series of H(∆) in series. P(∆) is the sum of weights of all the source to sink paths in G (∆).
SLIDE 127
Hard Polynomial
H(∆) is obtained by composing two copies of G (∆−1) in parallel. G (∆) is obtained by composing m series of H(∆) in series. P(∆) is the sum of weights of all the source to sink paths in G (∆).
SLIDE 128
Hard Polynomial
H(∆) is obtained by composing two copies of G (∆−1) in parallel. G (∆) is obtained by composing m series of H(∆) in series. P(∆) is the sum of weights of all the source to sink paths in G (∆). G (∆−1) G (∆−1) H(∆) H(∆) . . . H(∆)
Figure: H(∆) (above) and G (∆)(below).
SLIDE 129
Hard Polynomial
H(∆) is obtained by composing two copies of G (∆−1) in parallel. G (∆) is obtained by composing m series of H(∆) in series. P(∆) is the sum of weights of all the source to sink paths in G (∆). P(∆) is a polynomial over n∆ := 2m · n∆−1 = 8(2m)∆ many variables and has a product-depth of ∆ + 1. G (∆−1) G (∆−1) H(∆) H(∆) . . . H(∆)
Figure: H(∆) (above) and G (∆)(below).
P(∆) =
m
- i=1
(P(∆−1)
(i,1)
+ P(∆−1)
(i,1)
).
SLIDE 130
Hard Polynomial
H(∆) is obtained by composing two copies of G (∆−1) in parallel. G (∆) is obtained by composing m series of H(∆) in series. P(∆) is the sum of weights of all the source to sink paths in G (∆). P(∆) is a polynomial over n∆ := 2m · n∆−1 = 8(2m)∆ many variables and has a product-depth of ∆ + 1. G (∆−1) G (∆−1) H(∆) H(∆) . . . H(∆)
Figure: H(∆) (above) and G (∆)(below).
P(∆) =
m
- i=1
(P(∆−1)
(i,1)
+ P(∆−1)
(i,1)
). Our result restated Any product-depth ∆ formula for P∆ has size exp(Ω(mΩ(1)))
SLIDE 131
Hard Polynomial
H(∆) is obtained by composing two copies of G (∆−1) in parallel. G (∆) is obtained by composing m series of H(∆) in series. P(∆) is the sum of weights of all the source to sink paths in G (∆). P(∆) is a polynomial over n∆ := 2m · n∆−1 = 8(2m)∆ many variables and has a product-depth of ∆ + 1. G (∆−1) G (∆−1) H(∆) H(∆) . . . H(∆)
Figure: H(∆) (above) and G (∆)(below).
P(∆) =
m
- i=1
(P(∆−1)
(i,1)
+ P(∆−1)
(i,1)
). Our result restated Any product-depth ∆ formula for P∆ has size exp(Ω(mΩ(1))) = exp(nΩ(1/∆)
∆+1
).
SLIDE 132
Hard Polynomial
H(∆) is obtained by composing two copies of G (∆−1) in parallel. G (∆) is obtained by composing m series of H(∆) in series. P(∆) is the sum of weights of all the source to sink paths in G (∆). P(∆) is a polynomial over n∆ := 2m · n∆−1 = 8(2m)∆ many variables and has a product-depth of ∆ + 1. G (∆−1) G (∆−1) H(∆) H(∆) . . . H(∆)
Figure: H(∆) (above) and G (∆)(below).
P(∆) =
m
- i=1
(P(∆−1)
(i,1)
+ P(∆−1)
(i,1)
). Our result restated Any product-depth ∆ formula for P∆ has size exp(Ω(mΩ(1))) = exp(nΩ(1/∆)
∆+1
).
SLIDE 133
Conclusion
SLIDE 134
Conclusion
In the multilinear world there is a strict depth-hierarchy.
SLIDE 135
Conclusion
In the multilinear world there is a strict depth-hierarchy.
( )∆ ( )∆+1
SLIDE 136
Conclusion
In the multilinear world there is a strict depth-hierarchy.
( )∆ ( )∆+1 , while ∆ = O(1).
SLIDE 137
Conclusion
In the multilinear world there is a strict depth-hierarchy.
( )∆ ( )∆+1 , while ∆ = O(1). The classes are exponentially separated.
SLIDE 138
Conclusion
In the multilinear world there is a strict depth-hierarchy.
( )∆ ( )∆+1 , while ∆ = O(1). The classes are exponentially separated. The lower bound we prove is near-optimal.
SLIDE 139
Conclusion
In the multilinear world there is a strict depth-hierarchy.
( )∆ ( )∆+1 , while ∆ = O(1). The classes are exponentially separated. The lower bound we prove is near-optimal.
What about general constant depth formuals?
SLIDE 140
Conclusion
In the multilinear world there is a strict depth-hierarchy.
( )∆ ( )∆+1 , while ∆ = O(1). The classes are exponentially separated. The lower bound we prove is near-optimal.
What about general constant depth formuals? Open!
SLIDE 141
Conclusion
In the multilinear world there is a strict depth-hierarchy.
( )∆ ( )∆+1 , while ∆ = O(1). The classes are exponentially separated. The lower bound we prove is near-optimal.
What about general constant depth formuals? Open! Do similar techniques yield a non-commutative formula depth-hierarchy theorem?
SLIDE 142
Proof details
SLIDE 143
Designing a measure
SLIDE 144
Designing a measure
The measure must satisfy
SLIDE 145
Designing a measure
The measure must satisfy If f (X) is computable by a product-depth ∆ = 1 multilinear formula
- f size s then µ(f ) ≤ s × U.
SLIDE 146
Designing a measure
The measure must satisfy If f (X) is computable by a product-depth ∆ = 1 multilinear formula
- f size s then µ(f ) ≤ s × U.
There is a polynomial P(X) computable by product-depth ∆ = 2 multilinear formula such that µ(P) ≥ L.
SLIDE 147
Designing a measure
The measure must satisfy If f (X) is computable by a product-depth ∆ = 1 multilinear formula
- f size s then µ(f ) ≤ s × U.
There is a polynomial P(X) computable by product-depth ∆ = 2 multilinear formula such that µ(P) ≥ L. Conclude that s ≥ L/U.
SLIDE 148
Designing a measure
The measure must satisfy If f (X) is computable by a product-depth ∆ = 1 multilinear formula
- f size s then µ(f ) ≤ s × U.
There is a polynomial P(X) computable by product-depth ∆ = 2 multilinear formula such that µ(P) ≥ L. Conclude that s ≥ L/U.
SLIDE 149
Partial Derivative Matrix & Complexity Measure
Rank measure defined by [Raz 2004] Let ρ : X → Y ⊔ Z be a partitioning function such that |Y | = |Z|.
SLIDE 150
Partial Derivative Matrix & Complexity Measure
Rank measure defined by [Raz 2004] Let ρ : X → Y ⊔ Z be a partitioning function such that |Y | = |Z|. f =
2n
- i=1
ci · mi → f |ρ =
2n
- i=1
ci · mi,Y · mi,Z
SLIDE 151
Partial Derivative Matrix & Complexity Measure
Rank measure defined by [Raz 2004] Let ρ : X → Y ⊔ Z be a partitioning function such that |Y | = |Z|. f =
2n
- i=1
ci · mi → f |ρ =
2n
- i=1
ci · mi,Y · mi,Z M(Y ,Z)(f |ρ) : Monomials in Y Monomials in Z mZ mY coefff |ρ(mY · mZ) Complexity measure: µ(f ) w.r.t. ρ is rank(M(Y ,Z)(f |ρ)).
SLIDE 152
Partial Derivative Matrix & Complexity Measure
Rank measure defined by [Raz 2004] Let ρ : X → Y ⊔ Z be a partitioning function such that |Y | = |Z|. f =
2n
- i=1
ci · mi → f |ρ =
2n
- i=1
ci · mi,Y · mi,Z M(Y ,Z)(f |ρ) : Monomials in Y Monomials in Z mZ mY coefff |ρ(mY · mZ) Complexity measure: µ(f ) w.r.t. ρ is rank(M(Y ,Z)(f |ρ)).
SLIDE 153
Understanding the measure
Example
SLIDE 154
Understanding the measure
Example Let f (x1, . . . x2n) = n
i=1(xi + xn+i).
SLIDE 155
Understanding the measure
Example Let f (x1, . . . x2n) = n
i=1(xi + xn+i).
Let ρ(xi) = yi if 1 ≤ i ≤ n
SLIDE 156
Understanding the measure
Example Let f (x1, . . . x2n) = n
i=1(xi + xn+i).
Let ρ(xi) = yi if 1 ≤ i ≤ n zi−n if n + 1 ≤ i ≤ 2n
SLIDE 157
Understanding the measure
Example Let f (x1, . . . x2n) = n
i=1(xi + xn+i).
Let ρ(xi) = yi if 1 ≤ i ≤ n zi−n if n + 1 ≤ i ≤ 2n Therefore, f |ρ(Y , Z) =
S⊆[n] YSZ[n]\S
SLIDE 158
Understanding the measure
Example Let f (x1, . . . x2n) = n
i=1(xi + xn+i).
Let ρ(xi) = yi if 1 ≤ i ≤ n zi−n if n + 1 ≤ i ≤ 2n Therefore, f |ρ(Y , Z) =
S⊆[n] YSZ[n]\S
That is, M(Y ,Z)(f |ρ) is a disjointness matrix.
SLIDE 159
Understanding the measure
Example Let f (x1, . . . x2n) = n
i=1(xi + xn+i). (∆ = 1)
Let ρ(xi) = yi if 1 ≤ i ≤ n zi−n if n + 1 ≤ i ≤ 2n Therefore, f |ρ(Y , Z) =
S⊆[n] YSZ[n]\S
That is, M(Y ,Z)(f |ρ) is a disjointness matrix. Therefore, µ(f ) w.r.t. the above ρ is 2n.
SLIDE 160
Understanding the measure
Example Let f (x1, . . . x2n) = n
i=1(xi + xn+i).
Let ρ(xi) = yi if 1 ≤ i ≤ n zi−n if n + 1 ≤ i ≤ 2n Therefore, f |ρ(Y , Z) =
S⊆[n] YSZ[n]\S
That is, M(Y ,Z)(f |ρ) is a disjointness matrix. Therefore, µ(f ) w.r.t. the above ρ is 2n. Let ρ′(xi) = yi if 1 ≤ i ≤ n/2 or n + 1 ≤ i ≤ 3n/2
SLIDE 161
Understanding the measure
Example Let f (x1, . . . x2n) = n
i=1(xi + xn+i).
Let ρ(xi) = yi if 1 ≤ i ≤ n zi−n if n + 1 ≤ i ≤ 2n Therefore, f |ρ(Y , Z) =
S⊆[n] YSZ[n]\S
That is, M(Y ,Z)(f |ρ) is a disjointness matrix. Therefore, µ(f ) w.r.t. the above ρ is 2n. Let ρ′(xi) = yi if 1 ≤ i ≤ n/2 or n + 1 ≤ i ≤ 3n/2 zi−n if n/2 + 1 ≤ i ≤ n or 3n/2 + 2 ≤ i ≤ 2n
SLIDE 162
Understanding the measure
Example Let f (x1, . . . x2n) = n
i=1(xi + xn+i).
Let ρ(xi) = yi if 1 ≤ i ≤ n zi−n if n + 1 ≤ i ≤ 2n Therefore, f |ρ(Y , Z) =
S⊆[n] YSZ[n]\S
That is, M(Y ,Z)(f |ρ) is a disjointness matrix. Therefore, µ(f ) w.r.t. the above ρ is 2n. Let ρ′(xi) = yi if 1 ≤ i ≤ n/2 or n + 1 ≤ i ≤ 3n/2 zi−n if n/2 + 1 ≤ i ≤ n or 3n/2 + 2 ≤ i ≤ 2n µ(f ) w.r.t. ρ′ ≪ 2n.
SLIDE 163
Designing µ and ρ
SLIDE 164
Designing µ and ρ
The measure must satisfy
SLIDE 165
Designing µ and ρ
The measure must satisfy If f (X) is computable by a product-depth ∆ = 1 multilinear formula
- f size s then µ(f ) ≤ s × U w.r.t ρ.
SLIDE 166
Designing µ and ρ
The measure must satisfy If f (X) is computable by a product-depth ∆ = 1 multilinear formula
- f size s then µ(f ) ≤ s × U w.r.t ρ.
There is a polynomial P(X) computable by product-depth ∆ = 2 multilinear formula such that µ(P) ≥ L w.r.t the same ρ.
SLIDE 167
Designing µ and ρ
The measure must satisfy If f (X) is computable by a product-depth ∆ = 1 multilinear formula
- f size s then µ(f ) ≤ s × U w.r.t ρ.
There is a polynomial P(X) computable by product-depth ∆ = 2 multilinear formula such that µ(P) ≥ L w.r.t the same ρ. Conclude that s ≥ L/U.
SLIDE 168
Designing µ and ρ
The measure must satisfy If f (X) is computable by a product-depth ∆ = 1 multilinear formula
- f size s then µ(f ) ≤ s × U w.r.t ρ.
There is a polynomial P(X) computable by product-depth ∆ = 2 multilinear formula such that µ(P) ≥ L w.r.t the same ρ. Conclude that s ≥ L/U.
SLIDE 169
A random ρ : X → Y ∪ Z ∪ F
Map every copy of H(1) uniformly at random to one of the three possibilities.
SLIDE 170
A random ρ : X → Y ∪ Z ∪ F
Map every copy of H(1) uniformly at random to one of the three possibilities. y1 z1 y2 z2 y1 z1 y2 z2 y1 z1 (y1 + z1)(y2 + z2) (y1 + z1)(y2 + z2) (y1 + z1)
Figure: Map ρ applied to each copy of H(1). Edges that are not labelled have their variables set to 1. Dotted edges have their variables set to 0.
SLIDE 171
Random map ρ
Recall that G (1) is m copies of H(1).
. . .
Under the above choice of random ρ, the resulting polynomial will be P(1)|ρ =
i∈[t](yi + zi)
SLIDE 172
Random map ρ
Recall that G (1) is m copies of H(1).
. . .
Under the above choice of random ρ, the resulting polynomial will be P(1)|ρ =
i∈[t](yi + zi), where t = Ω(m) in expectation.
Therefore, µ(P(1)) = 2Ω(m)
SLIDE 173
Random map ρ
Recall that G (1) is m copies of H(1).
. . .
Under the above choice of random ρ, the resulting polynomial will be P(1)|ρ =
i∈[t](yi + zi), where t = Ω(m) in expectation.
Therefore, µ(P(1)) = 2Ω(m) w.h.p. over the distribution defined by these random partitions.
SLIDE 174
Effect of ρ on
SLIDE 175
Effect of ρ on
P(x) =
s
- i=1
s′
- j=1
Li,j
SLIDE 176
Effect of ρ on
P(x) =
s
- i=1
s′
- j=1
Li,j Easy to see that µ of linear polynomials is small with constant probability.
SLIDE 177
Effect of ρ on
P(x) =
s
- i=1
s′
- j=1
Li,j Easy to see that µ of linear polynomials is small with constant probability. Also, µ of each product term is low, say U, w.h.p.
SLIDE 178
Effect of ρ on
P(x) =
s
- i=1
s′
- j=1
Li,j Easy to see that µ of linear polynomials is small with constant probability. Also, µ of each product term is low, say U, w.h.p. By subadditivity of ranks, µ(P) is at most s · U
SLIDE 179
Effect of ρ on
P(x) =
s
- i=1
s′
- j=1
Li,j Easy to see that µ of linear polynomials is small with constant probability. Also, µ of each product term is low, say U, w.h.p. By subadditivity of ranks, µ(P) is at most s · U Hence, s ≥ 2Ω(m)/U.
SLIDE 180
Effect of ρ on
P(x) =
s
- i=1
s′
- j=1
Li,j Easy to see that µ of linear polynomials is small with constant probability. Also, µ of each product term is low, say U, w.h.p. By subadditivity of ranks, µ(P) is at most s · U Hence, s ≥ 2Ω(m)/U. At larger depths ...
SLIDE 181
Effect of ρ on
P(x) =
s
- i=1
s′
- j=1
Li,j Easy to see that µ of linear polynomials is small with constant probability. Also, µ of each product term is low, say U, w.h.p. By subadditivity of ranks, µ(P) is at most s · U Hence, s ≥ 2Ω(m)/U. At larger depths ... A carefully chosen ρ at each level of the polynomial.
SLIDE 182
Conclusion
SLIDE 183
Conclusion
In the multilinear world there is a strict depth-hierarchy.
SLIDE 184
Conclusion
In the multilinear world there is a strict depth-hierarchy.
( )∆ ( )∆+1
SLIDE 185
Conclusion
In the multilinear world there is a strict depth-hierarchy.
( )∆ ( )∆+1 , while ∆ = O(1).
SLIDE 186
Conclusion
In the multilinear world there is a strict depth-hierarchy.
( )∆ ( )∆+1 , while ∆ = O(1). The classes are exponentially separated.
SLIDE 187
Conclusion
In the multilinear world there is a strict depth-hierarchy.
( )∆ ( )∆+1 , while ∆ = O(1). The classes are exponentially separated. The lower bound we prove is near-optimal.
SLIDE 188
Conclusion
In the multilinear world there is a strict depth-hierarchy.
( )∆ ( )∆+1 , while ∆ = O(1). The classes are exponentially separated. The lower bound we prove is near-optimal.
What about general constant depth formuals?
SLIDE 189
Conclusion
In the multilinear world there is a strict depth-hierarchy.
( )∆ ( )∆+1 , while ∆ = O(1). The classes are exponentially separated. The lower bound we prove is near-optimal.
What about general constant depth formuals? Open!
SLIDE 190
Conclusion
In the multilinear world there is a strict depth-hierarchy.
( )∆ ( )∆+1 , while ∆ = O(1). The classes are exponentially separated. The lower bound we prove is near-optimal.
What about general constant depth formuals? Open! Do similar techniques yield a non-commutative formula depth-hierarchy theorem?
SLIDE 191