Reconstruction of full rank algebraic branching programs Vineet - - PowerPoint PPT Presentation

Reconstruction of full rank algebraic branching programs

Vineet Nair

Joint work with: Neeraj Kayal, Chandan Saha, Sebastien Tavenas

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Arithmetic circuits

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Reconstruction problem

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➢ f(X)  Q[X] is an m-variate degree d polynomial computable by a size s circuit in circuit class C. ➢ Input: α ϵ Fm f(α)

Blackbox access

Reconstruction problem

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➢ Input: ➢ Output: A small arithmetic circuit computing f. ➢ The algorithm should run in time poly(m,s,d,b) where (b is the bit length of the coefficients of f). α ϵ Fm f(α)

Polynomial identity testing (PIT):

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 Input:  Randomized algorithm for PIT follows easily

from Schwartz-Zippel lemma

 Unlike PIT no efficient randomized algorithm is

known for reconstruction.

α ϵ Fm f(α) Is f(X) = 0 ?

Previous works

 Over finite fields [Shp07],[KS09] gave quasi-poly

time deterministic reconstruction algorithm for depth three circuits with constant number of product gates.

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Previous works

 Over characteristic zero fields [Sinha16] gave a

poly time randomized algorithm for depth three circuits with two product gates.

 [GKL12] gave poly time randomized algorithm for

multilinear depth four circuits with two top-level product gates.

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Previous works

 [SV09], [MV16] gave deterministic poly time

reconstruction for read-once formulas

 [KS03], [FS13] gave deterministic quasi-poly time

reconstruction for ROABPs, set-multilinear ABPs and non-commutative ABPs

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Average-case reconstruction

 Progress in reconstruction is slow.  Can we do reconstruction for most circuits in a

circuit class C ?

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C

Efficiently reconstructed

Average-case reconstruction

 Problem definition: The input f is an m variate

degree d polynomial picked according to a distribution D on circuit class C

 Output an efficient reconstruction algorithm for f.  [GKL11], [GKQ13] gave randomized poly time

algorithm for average-case reconstruction of multilinear formulas and formulas.

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Algebraic branching programs (ABP)

 Definition: Consider the product of d matrices as

X1• X2 • … • Xd , where X1 is a row vector of length w, Xd is a column vector of length w and X2, … , Xd-1 are wxw matrices.

 Each entry of Xi, i  [d] is an affine form in X

• variables. |X| = m, example a0 + a1x1+ … + am.

 Polynomial computed by the ABP is the entry in

the 1x1 matrix computed as above. Length and width of the ABP is w and d respectively.

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Distribution on ABPs

 Random ABP: Fix w,d and m. Pick the constants

• f the linear forms independently and uniformly

at random from a large set S ⊆ Q.

 Average-case reconstruction: Design a

reconstruction algorithm for random(m,w,d,S) ABP.

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Average-case reconstruction for ABPs

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➢ Input: Blackbox access to f(X) computable by random(m,w,d,S) ABP. ➢ Output: A small ABP computing f with high probability. ➢ The algorithm should run in time poly(m,w,d,ρ) - (ρ bit length of an element in S). α ϵ Fn f(α)

Pseudo-random family

 A distribution D on m variate degree d

polynomial family with seed length s=(md)O(1) generates a pseudo-random family if

 Every algorithm that distinguishes a polynomial

coming from D and uniformly random m-variate polynomial with a non-negligible bias runs in time exponential in s.

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Candidate family

 [Aar08] conjectures the family Detn(AX) where

every entry of A ϵ Ft x m is chosen uniformly at random from a finite field and m << t=n2 is pseudo-random

 Example

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x1+x2 6x1+x2 x1+3x2 5x1+4x2 8x1+x2 10x1+x2 8x1+3x2 3x1+2x2 8x1+2x2 5x1+4x2 7x1+9x2 11x1+x2 4x1+3x2 9x1+3x2 5x1+6x2 9x1+7x2

m = 2, n = 4

Iterated matrix multiplication

 Definition: Consider the product of d matrices as

X1• X2 • … • Xd , where X1 is a row vector of length

w, Xd is a column vector of length w and X2, … , Xd-1 are wxw matrices.

 Each entry of Xi, i  [d] is a distinct variable. The

variables are disjoint across matrices.

 IMMw,d is the entry in 1x1 matrix computed as

above.

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Consequence

 Detn and IMMw,d are affine projections of each

• ther [Mahajan, Vinay 97].

 Hence, it makes sense to ask whether

IMMw,d(AX) where A ϵ Ft x m is chosen uniformly at random from a finite S ⊆ Q and m << t = w2(d-2) + 2w is pseudorandom.

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Our Contribution

Main result



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Remarks

 Does not resolve Aaronson’s conjecture  Our result works even if the matrices are not of

uniform width.

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• For IMMw,d the conjecture holds

when m << w2d

• Our result holds when m  w2d

Full rank ABPs

 If m  w2d then the affine forms in the ABP are

Q-linearly independent with high probability.

 Full rank ABPs: the set of linear forms in X1, X2,

…, Xd are Q-linearly independent.

 Example:

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x4+ x5 x5+ x6 x6+ x7 x7+ x8 x8+ x9 x9+ x10 x10+ x11 x11+ x12 x12+ x13 x1+ x2 x2+ x3 x3+ x4 x13+ x14 x14+ x15 x15+ x16

Full rank ABPs

 If m  w2d then the affine forms in the ABP are

Q-linearly independent with high probability.

 Full rank ABPs: the set of linear forms in X1, X2,

…, Xd are Q-linearly independent.

 Main result: We design an efficient randomized

algorithm to reconstruct full rank ABPs.

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Equivalent polynomials

 An n-variate polynomial f is equivalent to an n-

variate polynomial g if there exists an invertible A ϵ Fnxn such that f(X) = g(AX)

 Equivalence test:

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Is there an invertible A in Fnxn such that f(X) = g(AX)

g(X) f(X)

Equivalent polynomials

 Equivalence test:

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Is there an invertible A in Fnxn such that f(X) = IMM(AX)

IMM(X) f(X)

Remark: Computing a full rank ABP for f is the same as designing an efficient randomized equivalence test for IMM

Group of symmetries of IMM

 Group of symmetries: For an n variate

polynomial g(X) it is the set of all invertible A  Fnxn such that g(AX) = g(X).

 Characterization by symmetries: g(X) is

characterized by its group of symmetries then

 The group of symmetries of f(X) and g(X) are equal if

and only if f(X) is a constant multiple of g(X)

 Main theorem 2: IMMw,d is characterized by its

group of symmetries.

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Proof Ideas

Template of the reconstruction algorithm

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Assume the input polynomial f is computable by a full rank ABP Compute a full rank ABP

• 1. Find the layer spaces
• 2. Glue them together

Do a polynomial identity test to check if the polynomial computed by the ABP is f Output the full rank ABP computing f Output `f is not computable by a full rank ABP’ yes no

Pre-processing

 Let an m variate polynomial f be computed by a

width w and length d full rank ABP.

 The number of edges is n = w2(d-2) +2w

 Two steps of pre-processing:

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• Variable reduction: At the end of this step we get an n

variate f computable by a full rank ABP

• Translation equivalence test: The entries in the

matrices of the full rank ABP computing f are linear forms (constant term is 0).

m  n

Multiple full rank ABPs for f

 Suppose f is computable by a full rank ABP

X1• X2 • … • Xd

 Then this full rank ABP for f is not unique  The following transformations still compute f

 Transposition  Left-right multiplication  Corner translations

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Transposition

 Recall X1 and Xd are row and column vectors  Since the eventual product is a 1x1 matrix the

transpose of the product still computes f

 Hence f is also computed by

TXd• TX2 • … • TX1

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Left-right multiplication

 Let A be a wxw invertible matrix with entries

from Q

 Replace X2 with X’2 = X2• A and X’3 = A-1 • X3  f is computed by the product

X1• X’2 • X’3 • … • Xd

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Corner translations

 Let B be an anti-symmetric wxw matrix, then

X1 • B • TX1 = 0

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4 5

• 4

8

• 5
• 8

x1+ x2 x2+ x3 x3+ x4 x1+ x2 x2+ x3 x3+ x4

= 0

Corner translations

 Let B1, B2, … , Bw be anti-symmetric wxw

matrices.

 Let Y be the matrix such that the i-th column of

Y is Bi • TX1

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Bi • TX1 i-th column of matrix Y

Corner translations

 Replace X2 with X’2 = X2 + Y  Observe that X1 • X’2 = X1 • X2 as X1Y = 0wxw  f is computed by the product

X1• X’2 • X3 • … • Xd = X1• (X2 + Y) • X3 • … • Xd

 Similarly we can define corner translations for

Xd-1

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Uniqueness of the layer spaces

 Suppose f is computable by a full rank ABP

X1• X2 • … • Xd

 Let Xi denote the Q-linear space spanned by the

linear forms in Xi

 X1,2 and Xd-1,d denote the the Q-linear space

spanned by the linear forms in X1,X2 and Xd-1, Xd respectively

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Uniqueness of the layer spaces

 If X’1• X’2 • … • X’d computes f then  either

 X’i = Xi for i ϵ [d]\{2,d-1}  X’1,2 = X1,2 and X’d-1,d = Xd-1,d

 or

 X’i = Xd-i for i ϵ [d]\{2,d-1}  X’1,2 = Xd-1,dand X’d-1,d = X1,2

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Uniqueness of the layer spaces

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• X1

X3 X4 Xd-2 Xd X1,2 Xd-1,d

Uniqueness of the layer spaces

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• Xd

Xd-2 X3 X1 Xd-1,d Xd-3 X1,2

Group of symmetries of IMM

 The set of all invertible A ϵ Fnxn such that

IMMw,d(AX) = IMMw,d.

 We show that the group of symmetries are

generated by the following subgroups

 T denotes the group corresponding to transpositions  M denotes the group corresponding to left-right

multilpications

 C denotes the group corresponding to corner

translations

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Group of symmetries of IMM

 Main Theorem:

GIMM = C ⋊ H , where H = M ⋊ T

 C is a normal subgroup in GIMM and M is a

normal subgroup in H

 We also show that IMM is characterized by its

group of symmetries

 That is any polynomial with the same symmetry

group is a constant multiple of IMM

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Computing the full rank ABP: Step 1

 Computing the layer spaces:

 Study the Lie algebra of the group of symmetries of

IMMw,d

 [Kay12] Lie algebra of the group of symmetries of

f is the set of matrices A = (ai,j) i,j  [n]

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 We just use the vector space property of the

algebra

Invariant spaces

 Invariant space: Let M: Qn  Qn be a linear

• perator. U⊆ Qn is an invariant space if M(U) ⊆

U

 The definition can be extended to a set of linear

• perators {M1, M2, … , Mn}

 The layer spaces of an f computed by a full rank

ABP are intimately connected to the invariant spaces of Lie algebra of f

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Computing the layer spaces

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Compute a basis of the Lie algebra of f Compute the irreducible invariant spaces of the Lie algebra of f Compute the layer spaces from the irreducible invariant spaces Easy: Involves solving a set of linear dependencies Since f and IMM are equivalent their Lie algebras are conjugates of each other We show that the layer spaces are in fact the irreducible invariant spaces in some sense

Computing the full rank ABP: Step 2

 Ordering the layer spaces: We use evaluation

dimension to order the layer spaces.

 Definition:

 Evaluation Dimension for a polynomial H(X) is defined

with respect to a set of variables S ⊆ X

 EvaldimS[H(X) ] is equal to

dim (span{H(X) | xj =ɑj for xj∊S,where ɑj∊F})

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Ordering the layer spaces

 We make the variables in distinct layers are

disjoint by mapping the basis vectors of the layer spaces to distinct variables.

 Then we find the ordering inductively.

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Ordering the layer spaces

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•  Base Case
• Evaluation dimension = w

Evaluation dimension = w2

Ordering the layer spaces

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 Inductive Step

• Evaluation dimension = w

Evaluation dimension = w2

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Thank You