On obtaining a polynomial as a projection of another

polynomial

Neeraj KayalMicrosoft Research

A dream

• Conjecture #1: The determinantal complexity of the permanent is superpolynomial

• Conjecture #2: The arithmetic complexity of matrix multiplication is

• Conjecture #3: The depth three complexity of the permanent is

Projections of polynomials

• Definition (Valiant): An n-variate polynomial is said to be a projection of an m-variate polynomial if

for some matrix A and vector . That is ,

where the ’s are affine functions. (Affine function : ) We will say that such a projection is invertible if is invertible.

Some families of polynomials

• Determinant:

• Permanent:

• Sum of Products:

Some families of polynomials• Trace of Matrix Multiplication

where X, Y and Z are matrices

• Sum of powers:

• Elementary Symmetric Polynomial:

][

dn, :POWni

dix

)(Trace:MMn ZYX

dSnS Si

ixYM||],[

dn, :S

The dream

• Conjecture #1: The determinantal complexity of the permanent is superpolynomial

• Conjecture #2: The arithmetic complexity of matrix multiplication is

• Conjecture #3: The depth three complexity of the permanent is

The dream

• Conjecture #1: If is a projection of then m is superpolynomial in n.

• Conjecture #2: The arithmetic complexity of matrix multiplication is

• Conjecture #3: The depth three complexity of the permanent is

The dream

• Conjecture #1: If is a projection of then m is superpolynomial in n.

• Conjecture #2: can be expressed as a projection of for m =

• Conjecture #3: The depth three complexity of the permanent is

The dream

• Conjecture #1: If is a projection of then m is superpolynomial in n.

• Conjecture #2: can be expressed as a projection of for m =

• Conjecture #3: If is a projection of then .

A computational problem

• POLY_PROJECTION: Given polynomials and , determine if is a projection of and if so find A and b such that

Some conventions

• We consider polynomials over , the field of complex numbers.

• Polynomials are encoded as arithmetic circuits (unless mentioned otherwise).

Unfortunately …

Theorem: POLY_PROJECTION is NP-complete.

A more modest ambition

• Given a polynomial and an integer n determine if is a projection of say .

• Given a polynomial and integers n,d determine if is a projection of .

A conjecture

Conjecture (Scott Aaronson): A random low rank projection of is indistinguishable from a truly random polynomial.

More precisely, if are random m-variate affine functions then is indistuinguishable from a random m-variate polynomial of degree n.

POLY_EQUIVALENCE: Given polynomials and , determine if for some invertible matrix .

Theorem (Agrawal-Saxena): POLY_EQUIVALENCE is at least as hard as graph isomorphism.

(Probably much harder than graph isomorphism.)

Polynomial Equivalence

• Given a polynomial and an integer n determine if is equivalent to .

• Given a polynomial and integers n,d determine if is equivalent to .

Lowering our sights further

Polynomial Equivalence Results

Theorem #1: There is an efficient randomized algorithm that determines if a given polynomial is an invertible projection of .

Theorem #2: There is an efficient randomized algorithm that determines if a given polynomial is an invertible projection of .

Invertible Projections

Theorem #3: There is an efficient randomized algorithm that determines if a given polynomial is an invertible projection of .

Theorem #4: There is an efficient randomized algorithm that determines if a given polynomial is an invertible projection of .

Invertible ProjectionsTheorem #3: There is an efficient randomized algorithm that determines if a given polynomial is an invertible projection of .

Theorem #4: There is an efficient randomized algorithm that determines if a given polynomial is an invertible projection of .

… and so on for the other families of polynomials

PreliminariesFact: Given polynomials in randomized polynomial time we can determine a basis of the vector space

}0...:),...,,{(: 1121 mmm

m ffCV

Fact: Given an n-variate polynomial , in randomized polynomial time we can find an invertible matrix A such that has fewer than n variables, if such an exists.

Equivalence to Problem restatement: Given an n-variate polynomial of degree d, determine if there exist n linearly independent affine functions such that

dn

ddxf ...)( 21

Introducing the Hessian

Definition: The hessian of an n-variate polynomial g is the following matrix:

ggg

gggggg

xH

nnnn

n

n

f

21

22

212

1212

1

:)(

Property: If then

(here g is a shorthand for the second order derivative )

Using the HessianProperty: If then

In particular and

Fact:

2

22

21

00

0000

)1()(,

dn

d

d

POW

x

xx

ddxHdn

Using the Hessian

In particular, if

dn

ddxf ...)( 21

In particular, then

222

21))(( d

ndd

f cxHDET

This gives the algorithm for equivalence to

Equivalence to Problem restatement: Given an n-variate polynomial of degree d, determine if there exist n linearly independent affine functions such that

),...,,()( 21, ndnSYMxf

Fact: is a multilinear polynomial, so

].[ allfor 0,2 niSYM dni

Fact: All the other second-order partial derivatives of are linearly independent.

Outline of Algorithm

Input: A polynomial 1. Compute all the second order partial derivatives of .

2. Compute all the linear dependencies between these second order partials of f.

3. This gives us a linear space of second order differential operators which vanish at f.

4. A second order differential operator naturally corresponds to a matrix. Find a basis of the corresponding linear space of matrices consisting of rank one matrices.

Equivalence to Problem restatement: Given an -variate polynomial of degree n, determine if there exist n linearly independent affine functions such that

),...,,()( 221 nnPERMxf

Fact: The group of symmetries of is continuous.

The following approach was suggested by Mulmuley and Sohoni.

Symmetries of

nnnn

n

nnnn

n

n

xxx

xxxxxx

PERM

xxx

xxxxxx

PERM

21

211

211

211

11211

21

22221

11211

Fact: If we take to be aribtarily close to 1 then we get a matrix A arbitarily close to identity such that

)()( xPERMxAPERM nn

Symmetries of

Let be a formal variable with . Then there exist nontrivial matrices A such that

)())1(( xPERMxAPERM nn

Fact: For any polynomial , the set of matrices A such that

forms a vector space. )())1(( xfxAf

Fact: For a given , a basis for this space can be computed in random polynomial time.

Outline of Algorithm

Input: A polynomial 1. Compute a basis for the space of matrices A satisfying .

2. Let the basis be . These matrices act on an dimensional vector space V.

3. Find all subspaces such that for all .

4. Infer the appropriate equivalence of and from the 1-dimensional invariant subspaces U.

That’s all fine, but what about POLY_PROJECTION?

Theorem (Kaltofen): Projections of can be reconstructed efficiently. (This is just polynomial factoring)

Theorem (follows quickly from the work of Kleppe): Given a univariate polynomial of degree d and an integer s, we can efficiently compute ’s and ’s (if they exist) such that

dss

dd bxabxabxaxf )(...)()()( 2211

Scott’s conjecture

Conjecture (Scott Aaronson): If are random m-variate affine functions then is indistuinguishable from a random m-variate polynomial of degree n.

In other words, solving random instances of projections of is conjecturally hard.

Digression: Background of Scott’s conjecture

Observation: All the known bound proofs (for projections of a family of polynomials ) follow the following strategy:

Step 1. Find an “efficiently computable” property P that is satisfied by projections of small polynomials from .

Step 2. (Relatively easy) Find an explicit polynomial not having the property P.

Example: Mignon and Ressayre

Theorem (Mignon and Ressayre): If is a projection of then m is at least .

Lemma: Let be a projection of . Then for any zero of ,

maH f 2))((Rank

Example: Projections of

Lemma (implicit in Grigoriev-Karpinski): Let be a projection of . Then the set of all possible partial derivatives of contain at most linearly independent polynomials.

Consequences of Scott’s conjecture

• If the conjecture is true then no current proof technique will yield superpolynomial formula size lower bound.

• If the conjecture is false then we would have obtained a good handle on the determinant versus permanent conjecture.

Probing a little deeper

• For families of polynomials such as we have lower bounds.

• Given , where the ’s are m-variate affine functions randomly chosen, can we efficiently recover the ’s?

POLY_PROJECTION on the average

Theorem: Projections of can be reconstructed efficiently on the average.

Theorem: For bounded n, projections of can be reconstructed efficiently on the average.

Summary• POLY_PROJECTION and POLY_EQUIVALENCE are

difficult computational problems in general.

• Empirically, for most families of polynomials that we actually care about and/or encounter in practice, we can solve POLY_EQUIVALENCE efficiently.

• Empirically, efficient average-case algorithms for POLY_PROJECTION of some family seem to be closely related to lower bounds for projections of .

Conclusion: An easier(?) open problem

Conjecture (Amir Shpilka): If is a projection of then .

THE END