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 y x 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) x 1 + x 2 + x 3 x 1 − x 2 x 1 − x 2 3 x 2 2 x 1 − x 2 x 3 s x 1 + 2 x 2 t x 1 2 x 3 x 3 5

ABP (Branching programs) x 1 + x 2 + x 3 x 1 − x 2 x 1 − x 2 3 x 2 2 x 1 − x 2 x 3 ( x 1 − x 2 )( 3 x 2 )(− x 2 ) s x 1 + 2 x 2 t x 1 2 x 3 x 3 • 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 . 6

Coefficient matrices • Π = ( Y , Z ) partition of [ d ] monomials of degree ∣ Z ∣ monomials of degree ∣ Y ∣ m 2 d m 1 α m • f = ∑ m α m . m , homogeneous, degree d , n variables ⎧ ⎪ m ∣ Y = m 1 ⎪ ⎨ m t . q ⎪ • Define matrix M Π ( f ) ⎪ m ∣ Z = m 2 ⎩ 7

Coefficient matrices • Π = ( Y , Z ) partition of [ d ] monomials of degree ∣ Z ∣ monomials of degree ∣ Y ∣ m 2 d m 1 α m • f = ∑ m α m . m , homogeneous, degree d , n variables ⎧ ⎪ m ∣ Y = m 1 ⎪ ⎨ m t . q ⎪ • Define matrix M Π ( f ) ⎪ m ∣ Z = m 2 ⎩ • Complexity measure : rank ( M Π ( f )) . 7

Exercise: the palindrome polynomial w = ( w 1 ,..., w d / 2 ) ∈ [ n ] d / 2 � → w R = ( w d / 2 ,..., w 1 ) x w = x w 1 x w 2 ... x w d / 2 ˜ x w ⋅ ˜ ∑ Pal d X = ˜ x w R w ∈[ n ] d / 2 n ∑ x i ⋅ Pal d X ⋅ x i Pal d + 1 X = i = 1 What is the matrix if we cut in the middle? 8

Exercise: the palindrome polynomial m R 1 m 1 1 9

• Π i = ({ 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 ∑ rank ( M k ( f )) k = 0 10

• Π i = ({ 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 ∑ rank ( M k ( f )) k = 0 Corollary Any homogeneous ABP computing the palindrome of degree d over n variables has size ≥ n d / 2 Any homogeneous ABP computing the permanent has size ≥ 2 n 10

Proof (lower bound) v 1 level k v i s t v t L k R k 11

Proof (lower bound) v 1 level k v i s t v t L k R k v i m ′ n k t coef ( m ′ ) btw coef ( m ) btw v i m s and v i v i and t t n d − k 11

Proof (lower bound) v 1 level k v i s t v t L k R k v i m ′ n k t coef ( m ′ ) btw coef ( m ) btw v i m s and v i v i and t t n d − k M k ( f ) = L k R k rank ( M k ( f )) ≤ t and 11

Proof (upper bound) level k level k + 1 v 1 w 1 w j v i s t w p v t • suppose rank ( L k ) < t , then there is a column i which is a linear combination of the others • the polynomial computed between s and v i is a linear combination of the polynomials computed by the other vertices v ′ j s • we could delete v i and update the weights from level k to level k + 1. • so rank ( L k ) = rank ( R k ) = t = rank ( M k ( f )) . 12

Unambiguous circuits

Parse trees a c b × × + + z w × × × + 13

Parse trees a c b × × + + z w × × × + 14

Parse trees a c b × × + + z w × × × + 15

Parse trees a c b × × + + z w × × × + 16

Parse trees a c a b b × × × + + + z w z × × × × + + Figure 1: val ( T ) = zab • Each parse tree computes a monomial. Lemma f = ∑ val ( T ) T 17

Parse trees of ABPs Right-skew Circuits ABP 18

Parse trees of ABPs Right-skew Circuits ABP D´ efinition A circuit is unambiguous if all its parse trees are isomorphic. 18

Unambiguous circuits a c b × × × 1 − 3 1 2 + + + z w × × × × 2 1 − 1 + + 19

Canonical circuits a c a c b b b 1 1 1 1 1 1 1 + + + + + + + × × × × 1 1 1 1 + + + 20

Canonical circuits a c a c b b b 1 1 1 1 1 1 1 + + + + + + + × × × × 1 1 1 1 + + + Any unambiguous circuit can be rendered canonical at a polynomial cost. 20

Type of a gate d − p − i p i deg. i f α + + β 1 + deg. p × + β k + × + 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 rank ( M ( i , p ) ( P )) , ∑ ( i , p )∈ S where S is the set of all existing types of + -gates in the shape T . Corollary Any UC computing the permanent has size 2 Ω ( n ) 22

Proof (lower bound) Parse tree shape p d − p − i i Π ( p , i ) = • rg ( M Π ( 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 ⃗ x and Q = ∑ ⃗ x ⃗ x , the Hadamard x a ⃗ x b ⃗ x ⃗ product of P and Q , written P ⊙ Q , equals ∑ ⃗ x . x a ⃗ x b ⃗ 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 ). 25

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! 29