Non-commutative computations: lower bounds and polynomial identity - - PowerPoint PPT Presentation

Non-commutative computations: lower bounds and polynomial identity testing

Guillaume Malod Joint work with G. Lagarde, S. Perifel IMSc Workshop on Arithmetic Complexity March 1st, 2017

Table of contents

• 1. Introduction
• 2. Nisan’s results
• 3. Unambiguous circuits
• 4. Other results

1

Introduction

Arithmetic circuits

x y z × + π + × + × + +

2

Non-commutative circuits

• F commutative field.
• Non-commutative : xy ≠ yx → distinguish left and right arguments

in a computation gate.

• Various motivations

3

Some results

• No better lower bound for NC circuits than for commutative

circuits

4

Some results

• No better lower bound for NC circuits than for commutative

circuits But for ABPs (Algebraic Branching Programs) :

• (Nisan 1991) Exact characterization of complexity
• (Nisan 1991) Exponential lower bounds for the permanent
• (Arvind, Joglekar, Srinivasan 2009) Deterministic poly-time PIT

4

Some results

• No better lower bound for NC circuits than for commutative

circuits But for ABPs (Algebraic Branching Programs) :

• (Nisan 1991) Exact characterization of complexity
• (Nisan 1991) Exponential lower bounds for the permanent
• (Arvind, Joglekar, Srinivasan 2009) Deterministic poly-time PIT

Also (Limaye,Malod,Srinivasan 2016) Exponentiel lower bounds for skew circuits

4

Nisan’s results

ABP (Branching programs)

s t x1 − x2 x3 x1 x1 + x2 + x3 3x2 2x1 x1 + 2x2 x3 x1 − x2 −x2 2x3

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ABP (Branching programs)

s t (x1 − x2)(3x2)(−x2) x1 − x2 x3 x1 x1 + x2 + x3 3x2 2x1 x1 + 2x2 x3 x1 − x2 −x2 2x3

• DAG : source s, sink t, edges with linear forms
• Weight of a path : product of edge weights
• Computed polynomial : sum of path weights from s to t.

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Coefficient matrices

• Π = (Y ,Z) partition of [d]

d

• f = ∑

m αm.m, homogeneous,

degree d, n variables

• Define matrix MΠ(f )

αm m1 m2 monomials of degree ∣Z∣ monomials of degree ∣Y ∣ m t.q ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ m∣Y = m1 m∣Z = m2

7

Coefficient matrices

• Π = (Y ,Z) partition of [d]

d

• f = ∑

m αm.m, homogeneous,

degree d, n variables

• Define matrix MΠ(f )

αm m1 m2 monomials of degree ∣Z∣ monomials of degree ∣Y ∣ m t.q ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ m∣Y = m1 m∣Z = m2

• Complexity measure : rank(MΠ(f )).

7

Exercise: the palindrome polynomial

w = (w1,...,wd/2) ∈ [n]d/2 → w R = (wd/2,...,w1) ˜ xw = xw1xw2 ...xwd/2 Pald X = ∑

w∈[n]d/2

˜ xw ⋅ ˜ xw R Pald+1 X =

n

∑

i=1

xi ⋅ Pald X ⋅ xi What is the matrix if we cut in the middle?

8

Exercise: the palindrome polynomial

1 1 1 m mR

9

• Πi = ({1,2,...,k},{k + 1,k + 2,...,d})

k d − k

Theorem (Nisan, 1991) For any homogeneous polynomial f of degree d, the size of a smallest homogeneous algebraic branching program for f is equal to

d

∑

k=0

rank(Mk(f ))

10

• Πi = ({1,2,...,k},{k + 1,k + 2,...,d})

k d − k

Theorem (Nisan, 1991) For any homogeneous polynomial f of degree d, the size of a smallest homogeneous algebraic branching program for f is equal to

d

∑

k=0

rank(Mk(f )) Corollary Any homogeneous ABP computing the palindrome of degree d over n variables has size ≥ nd/2 Any homogeneous ABP computing the permanent has size ≥ 2n

10

Proof (lower bound)

level k s t v1 vi vt Lk Rk

11

Proof (lower bound)

level k s t v1 vi vt Lk Rk vi m

coef(m) btw s and vi

t nk m′ vi

coef(m′) btw vi and t

nd−k t

11

Proof (lower bound)

level k s t v1 vi vt Lk Rk vi m

coef(m) btw s and vi

t nk m′ vi

coef(m′) btw vi and t

nd−k t

Mk(f ) = LkRk and rank(Mk(f )) ≤ t

11

Proof (upper bound)

level k s t v1 vi vt level k + 1 w1 wj wp

• suppose rank(Lk) < t, then there is a column i which is a linear

combination of the others

• the polynomial computed between s and vi is a linear combination of

the polynomials computed by the other vertices v ′

j s

• we could delete vi and update the weights from level k to level k + 1.
• so rank(Lk) = rank(Rk) = t = rank(Mk(f )).

12

Unambiguous circuits

Parse trees

+ × × × z w + + × × a b c

13

Parse trees

+ × × × z w + + × × a b c

14

Parse trees

+ × × × z w + + × × a b c

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Parse trees

+ × × × z w + + × × a b c

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Parse trees

+ × × × z w + + × × a b c + × z + × a b

Figure 1: val(T) = zab

• Each parse tree computes a monomial.

Lemma f = ∑

T

val(T)

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Parse trees of ABPs

ABP Right-skew Circuits

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Parse trees of ABPs

ABP Right-skew Circuits D´ efinition A circuit is unambiguous if all its parse trees are isomorphic.

18

Unambiguous circuits

+ × × × z w + + × × a b c

−3 1

1 2

2 1 −1

+ × + ×

19

Canonical circuits

+ × × + + + a b c 1 1 1 1 1 + × + + a b 1 1 1 + × + + b c 1 1 1

20

Canonical circuits

+ × × + + + a b c 1 1 1 1 1 + × + + a b 1 1 1 + × + + b c 1 1 1 Any unambiguous circuit can be rendered canonical at a polynomial cost.

20

Type of a gate

• deg. i

+ α × × + +

β1

+

βk

• deg. p

+ + p i f d − p − i

21

Results

Theorem Let P be a homogeneous polynomial of degree d and T a shape with d

• leaves. Then the minimal number of addition gates needed to compute P

by a canonical unambiguous circuit with shape T is exactly equal to ∑

(i,p)∈S

rank(M(i,p)(P)), where S is the set of all existing types of +-gates in the shape T . Corollary Any UC computing the permanent has size 2Ω(n)

22

Proof (lower bound)

Parse tree shape Π(p,i) =

p i d − p − i

• rg(MΠ(p,i)(f )) ≤ number of gates of type (p,i)

23

Proof (upper bound)

24

Proof (upper bound)

Clearly it works

24

Other results

PIT

Hadamard product (Arvind, Joglekar, Srinivasan) Given two polynomials P = ∑⃗

x a⃗ x ⃗

x and Q = ∑⃗

x b⃗ x ⃗

x, the Hadamard product of P and Q, written P ⊙ Q, equals ∑⃗

x a⃗ xb⃗ x ⃗

x. Hadamard product of two unambiguous circuits Let C and D be two unambiguous circuits in canonical form, of the same shape, and of size s and s′, that compute two polynomials P and Q. Then P ⊙ Q is computed by an unambiguous circuit of size at most ss′; moreover, this circuit can be constructed in polynomial time. Theorem There is a deterministic polynomial-time algorithm for PIT for polynomials computed by non-commutative unambiguous circuits over R (or C).

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Relationship to other classes

ABP ⊊ UC There are polynomials computed by polynomial-size UC that need exponential-size ABPs. UC and skew are incomparable There are polynomials computed by polynomial-size UC that need exponential-size skew circuits. There are polynomials computed by polynomial-size skew circuits that need exponential-size UC.

26

The future

• Lower bounds for circuits with “similar” shapes.
• Lower bounds for circuits with not too many shapes.
• Poly-time PIT for a sum of UC circuits.

27

The future (much later?)

Lower bounds for general non-commutative circuits?

28

The future (much later?)

Lower bounds for general non-commutative circuits? Theorem (Limaye, Malod, Srinivasan 2016) There exists a polynomial computed by a small non-commutative circuit which is full rank for any partition.

28

Thank you!

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