Limitations of the Shpilka-Volkovich generator Arpita Korwar joint - - PowerPoint PPT Presentation

Limitations of the Shpilka-Volkovich generator

Arpita Korwar

joint work with Herv´ e Fournier

Universit´ e Denis Diderot - Paris 7

March 16, 2018

Arpita Korwar (Universit´ e Denis Diderot - Paris 7) Limitations of the Shpilka-Volkovich generator March 16, 2018 1 / 23

1 Polynomial Identity Testing 2 Shpilka-Volkovich (SV) Generator 3 Finding the Annihilating polynomial

Arpita Korwar (Universit´ e Denis Diderot - Paris 7) Limitations of the Shpilka-Volkovich generator March 16, 2018 2 / 23

Polynomial Identity Testing

Section 1 Polynomial Identity Testing

Arpita Korwar (Universit´ e Denis Diderot - Paris 7) Limitations of the Shpilka-Volkovich generator March 16, 2018 3 / 23

Polynomial Identity Testing

Polynomial Identity Testing (PIT)

PIT: Is a given input polynomial identically zero?

Arpita Korwar (Universit´ e Denis Diderot - Paris 7) Limitations of the Shpilka-Volkovich generator March 16, 2018 4 / 23

Polynomial Identity Testing

Input model: Arithmetic circuits

x y × × + 5 A natural and succinct representation of a polynomial.

Arpita Korwar (Universit´ e Denis Diderot - Paris 7) Limitations of the Shpilka-Volkovich generator March 16, 2018 5 / 23

Polynomial Identity Testing

Blackbox test (a.k.a. Hitting set)

PIT can be classified according to how the polynomial is given to the algorithm.

Arpita Korwar (Universit´ e Denis Diderot - Paris 7) Limitations of the Shpilka-Volkovich generator March 16, 2018 6 / 23

Polynomial Identity Testing

Blackbox test (a.k.a. Hitting set)

PIT can be classified according to how the polynomial is given to the algorithm. P ∈ P a1, a2, . . . , ah P(a1), P(a2), . . . , P(ah) ai = (ai1, ai2, . . . , ain).

Arpita Korwar (Universit´ e Denis Diderot - Paris 7) Limitations of the Shpilka-Volkovich generator March 16, 2018 6 / 23

Polynomial Identity Testing

Blackbox test (a.k.a. Hitting set)

PIT can be classified according to how the polynomial is given to the algorithm. P ∈ P a1, a2, . . . , ah P(a1), P(a2), . . . , P(ah) ai = (ai1, ai2, . . . , ain). Example: PIT for univariate polynomials of degree bounded by d.

Arpita Korwar (Universit´ e Denis Diderot - Paris 7) Limitations of the Shpilka-Volkovich generator March 16, 2018 6 / 23

Polynomial Identity Testing

Blackbox test (a.k.a. Hitting set)

PIT can be classified according to how the polynomial is given to the algorithm. P ∈ P a1, a2, . . . , ah P(a1), P(a2), . . . , P(ah) ai = (ai1, ai2, . . . , ain). Example: PIT for univariate polynomials of degree bounded by d. For n-variate P, a small-degree univariate substitution is enough.

Arpita Korwar (Universit´ e Denis Diderot - Paris 7) Limitations of the Shpilka-Volkovich generator March 16, 2018 6 / 23

Polynomial Identity Testing

Hitting set generator

For a family P of n-variate, a polynomial map to k-variate polynomials (f1(y), f2(y), . . . , fn(y)) is a hitting set generator if for all polynomials P(x1, x2, . . . , xn) = 0 ∈ P, P(f1(y), f2(y), . . . , fn(y)) = 0. Final time complexity = (δd + 1)k, where d is the degree of fis and the polys in P are of degree δ. Poly when k is constant. Quasipoly when k is log n.

Arpita Korwar (Universit´ e Denis Diderot - Paris 7) Limitations of the Shpilka-Volkovich generator March 16, 2018 7 / 23

Shpilka-Volkovich (SV) Generator

Section 2 Shpilka-Volkovich (SV) Generator

Arpita Korwar (Universit´ e Denis Diderot - Paris 7) Limitations of the Shpilka-Volkovich generator March 16, 2018 8 / 23

Shpilka-Volkovich (SV) Generator

Applications of the SV generator

sO(1)-size hitting set for Read-once formulas [Shpilka and Volkovich, 2009, Minahan and Volkovich, 2016]. sO(1)-size hitting set for Constant-read multilinear formulas [Anderson et al., 2015]. sO(log log s)-size hitting set for Commutative Read-once ABPs [Forbes et al., 2014].

Arpita Korwar (Universit´ e Denis Diderot - Paris 7) Limitations of the Shpilka-Volkovich generator March 16, 2018 9 / 23

Shpilka-Volkovich (SV) Generator

Polynomials for Lagrange interpolation

Building blocks of the SV generator. Choose (a1, a2, . . . , an) such that all ais are unique. Lr(y) :=

j=r (y−aj) (ar−aj).

Lr(b) =

• 1 if b = ar,

0 if b ∈ {a1, a2, . . . , an} , b = ar.

Arpita Korwar (Universit´ e Denis Diderot - Paris 7) Limitations of the Shpilka-Volkovich generator March 16, 2018 10 / 23

Shpilka-Volkovich (SV) Generator

Shpilka-Volkovich map (SVn,k)[Shpilka and Volkovich, 2009]

SVn,1(y, z) : F[x] − → F[y, z], given by SVn,1(y, z) : xr − → zLr(y). SVn,1 is a bivariate map. SVn,k(y1, z1, y2, z2, . . . , yk, zk) : F[x] − → F[y, z], given by SVn,k(y, z) : xr − → k

i=1 ziLr(yi).

SVn,k is a 2k-variate map.

Arpita Korwar (Universit´ e Denis Diderot - Paris 7) Limitations of the Shpilka-Volkovich generator March 16, 2018 11 / 23

Shpilka-Volkovich (SV) Generator

Some properties

SVn,k of each xi is a linear form1 in z. SVn,k is a hitting set generator for 2k-sparse polynomials. SVn,k is a hitting set generator for degree-k polynomials.

1constant part of the linear polynomial is 0 Arpita Korwar (Universit´ e Denis Diderot - Paris 7) Limitations of the Shpilka-Volkovich generator March 16, 2018 12 / 23

Shpilka-Volkovich (SV) Generator

Question

We want f such that SVn,k(f ) = 0. What is the smallest degree polynomial that evaluates to 0 at SVn,k? Conjecture: There exists a degree k + 1 multilinear polynomial on n = 2k + 1 variables that maps to 0 on applying SVn,k. I.e. A multilinear, degree k + 1 annihilating polynomial for SVn,k exists.

Arpita Korwar (Universit´ e Denis Diderot - Paris 7) Limitations of the Shpilka-Volkovich generator March 16, 2018 13 / 23

Finding the Annihilating polynomial

Section 3 Finding the Annihilating polynomial

Arpita Korwar (Universit´ e Denis Diderot - Paris 7) Limitations of the Shpilka-Volkovich generator March 16, 2018 14 / 23

Finding the Annihilating polynomial

Finding a small annihilating polynomial - homogeneity

Let f (x1, x2, . . . , xn) =

• S:|S|≤k+1

γS

• r∈S

xr. Recall that SVn,k(y, z) : xr − → k

i=1 ziLr(yi).

The polynomial SVn,k(f ) can be seen as a polynomial in F[y][z]. After the map is applied, the z-degree of a degree-d monomial is d. So, without loss of generality, f is homogeneous. f (x1, x2, . . . , xn) =

• S:|S|=k+1

γS

• r∈S

xr.

Arpita Korwar (Universit´ e Denis Diderot - Paris 7) Limitations of the Shpilka-Volkovich generator March 16, 2018 15 / 23

Finding the Annihilating polynomial

Coefficients of each monomials

The coefficient of any such monomial after the map should be 0. This gives a set of linear constraints on γSs.

Arpita Korwar (Universit´ e Denis Diderot - Paris 7) Limitations of the Shpilka-Volkovich generator March 16, 2018 16 / 23

Finding the Annihilating polynomial

... After some calculations

After cleaning the conditions on the coefficients, our problem reduces to finding (αR)R:|R|=k such that the following linear constraints are satisfied: ∀S ⊆ [n], |S| = k − 1 :

• R:|R|=k,S⊆R

αR = 0 and

• R:|R|=k,S⊆R

aR\S · αR = 0. E.g. when k = 1, n = 2k + 1 = 3. Then, we want to find (α1, α2, α3) such that αi = 0 and ai · αi = 0.

Arpita Korwar (Universit´ e Denis Diderot - Paris 7) Limitations of the Shpilka-Volkovich generator March 16, 2018 17 / 23

Finding the Annihilating polynomial

The (αR)Rs that satisfy the first set of constraints are nullvectors of the inclusion matrix M with the rows indexed by {S : |S| = k − 1} and the columns indexed by {R : |R| = k} with MS,R =

• 1 if S ⊆ R,

0 otherwise. The (αR)Rs that satisfy the second set of constraints are null vectors

• f N, where

NS,R =

• aR\S if S ⊆ R,

0 otherwise. When k = 1, n = 2k + 1 = 3, M = [1 1 1] and N = [a1 a2 a3].

Arpita Korwar (Universit´ e Denis Diderot - Paris 7) Limitations of the Shpilka-Volkovich generator March 16, 2018 18 / 23

Finding the Annihilating polynomial

N = D′MD, where

D′ is a [n]

k−1

• diagonal matrix. D′

S,S = i∈S 1/ai = 1/aS.

D is a [n]

k

• diagonal matrix. DR,R =

i∈R ai = aR.

D′ does not affect the nullvector of N. Hence, we need to show that N(M) ∩ N(MD) = ∅. N(M) has dimension n

k

• −

n

k−1

• and has been described by

[Graham et al., 1980] and others.

Arpita Korwar (Universit´ e Denis Diderot - Paris 7) Limitations of the Shpilka-Volkovich generator March 16, 2018 19 / 23

Finding the Annihilating polynomial

[Graham et al., 1980]

Some notation [Graham et al., 1980]:

View the nullvector as a multilinear homogeneous polynomial of degree k. Take n variables {x1, x2, . . . , xn}. With a vector (αR)R, associate

• R αRxR.

Define g(x1, x2, . . . , xn) = (x1 − x2)(x3 − x4) · · · (x2k−1 − x2k) .

Lemma[Graham et al., 1980] N(M) = span {gσ|σ ∈ Sn} where, hσ(x1, x2, . . . , xn) = h(xσ(1), xσ(2), . . . , xσ(n)). Thus, N([1 1 1]) = span {x1 − x2, x1 − x3, x3 − x2, . . .}.

Arpita Korwar (Universit´ e Denis Diderot - Paris 7) Limitations of the Shpilka-Volkovich generator March 16, 2018 20 / 23

Finding the Annihilating polynomial

Mv = MDD−1v. Thus, N(MD) = span {ψ(gσ)|σ ∈ Sn}, where ψ : xi − → 1

ai xi.

Let bi = 1

ai .

Thus, N([a1 a2 a3]) = span {b1x1 − b2x2, b1x1 − b3x3, b3x3 − b2x2, . . .}.

Arpita Korwar (Universit´ e Denis Diderot - Paris 7) Limitations of the Shpilka-Volkovich generator March 16, 2018 21 / 23

Finding the Annihilating polynomial

Conjecture: dim(N(M) ∩ N(MD)) = 1. When k = 1, n = 3, this common nullvector is a1a2(x1 − x2) + a2a3(x2 − x3) + a3a1(x3 − x1) = −a3(a1x1 − a2x2) − a1(a2x2 − a3x3) − a2(a3x3 − a1x1).

Arpita Korwar (Universit´ e Denis Diderot - Paris 7) Limitations of the Shpilka-Volkovich generator March 16, 2018 22 / 23

Finding the Annihilating polynomial

Thank you

Arpita Korwar (Universit´ e Denis Diderot - Paris 7) Limitations of the Shpilka-Volkovich generator March 16, 2018 23 / 23

Finding the Annihilating polynomial

Anderson, M., van Melkebeek, D., and Volkovich, I. (2015). Deterministic polynomial identity tests for multilinear bounded-read formulae. computational complexity, 24(4):695–776. Forbes, M. A., Saptharishi, R., and Shpilka, A. (2014). Hitting sets for multilinear read-once algebraic branching programs, in any order. In Symposium on Theory of Computing, STOC 2014, New York, NY, USA, May 31 - June 03, 2014, pages 867–875. Graham, R. L., y. R. Li:i, S., c, W., and Li, W. (1980). On the structure of t-designs.

• SIAM. J. on Algebraic and Discrete Methods, 1:8–14.

Minahan, D. and Volkovich, I. (2016). Complete derandomization of identity testing and reconstruction of read-once formulas. Electronic Colloquium on Computational Complexity (ECCC), 23:171.

Arpita Korwar (Universit´ e Denis Diderot - Paris 7) Limitations of the Shpilka-Volkovich generator March 16, 2018 23 / 23

Finding the Annihilating polynomial

Shpilka, A. and Volkovich, I. (2009). Improved polynomial identity testing of read-once formulas. In Approximation, Randomization and Combinatorial Optimization. Algorithms and Techniques, volume 5687 of LNCS, pages 700–713.

Arpita Korwar (Universit´ e Denis Diderot - Paris 7) Limitations of the Shpilka-Volkovich generator March 16, 2018 23 / 23