Arithmetic circuits with locally low algebraic rank Mrinal Kumar - - PowerPoint PPT Presentation

Arithmetic circuits with locally low algebraic rank

Mrinal Kumar

Joint work with Shubhangi Saraf

Plan for the talk

Plan for the talk

• Depth four arithmetic circuits.

Plan for the talk

• Depth four arithmetic circuits.
• The problem we do not solve.

Plan for the talk

• Depth four arithmetic circuits.
• The problem we do not solve.
• The problem we do solve.

Plan for the talk

• Depth four arithmetic circuits.
• The problem we do not solve.
• The problem we do solve.
• Structure of algebraically dependent polynomials.

Plan for the talk

• Depth four arithmetic circuits.
• The problem we do not solve.
• The problem we do solve.
• Structure of algebraically dependent polynomials.
• Ramblings about the proof.

Plan for the talk

• Depth four arithmetic circuits.
• The problem we do not solve.
• The problem we do solve.
• Structure of algebraically dependent polynomials.
• Ramblings about the proof.
• Open questions.

Arithmetic circuits

• Field F
• Variables
• Sum and Product operations
• Depth - 2
• Size - 6

X1, X2,..., Xn

Arithmetic circuits

Arithmetic circuits

• Succinctly represent multivariate polynomials

Arithmetic circuits

• Succinctly represent multivariate polynomials
• Have nice algebraic properties - (partial

derivatives, interpolation, factoring)

Arithmetic circuits

• Succinctly represent multivariate polynomials
• Have nice algebraic properties - (partial

derivatives, interpolation, factoring)

• Many upper bounds for evaluating multivariate

polynomials are arithmetic circuits

Resources

Resources

• Size - number of gates/wires in the circuit

Resources

• Size - number of gates/wires in the circuit
• Depth - longest path from output to input

Fundamental questions

Fundamental questions

• Upper bounds - interesting polynomial families can

be computed succinctly

Fundamental questions

• Upper bounds - interesting polynomial families can

be computed succinctly

• Lower bounds - there are explicit polynomial

families which require large circuits

Depth reductions [H, VSBR, AV, K, T, GKKS]

Depth reductions [H, VSBR, AV, K, T, GKKS]

• ‘Suffices’ to study structured arithmetic circuits
• f low depth

Depth reductions [H, VSBR, AV, K, T, GKKS]

• ‘Suffices’ to study structured arithmetic circuits
• f low depth
• For instance - strong enough lower bounds for

homogeneous depth-4 arithmetic circuits imply general arithmetic circuit lower bounds.

Notation

Notation

• n - Number of variables

Notation

• n - Number of variables
• d - Degree of the polynomial (d < poly(n))

….. ….. …..

+ + + +

….. ….. ….. Monomials Sparse poly. Bottom fan-in

∑∏∑∏

Circuits

• Prod. of

sparse poly.

Depth-4 circuits

C = Qi1

i=1 T

∑

⋅Qi2!Qik

Depth-4 circuits

• Q - Sparse polynomials

C = Qi1

i=1 T

∑

⋅Qi2!Qik

Depth-4 circuits

• Q - Sparse polynomials
• C is homogeneous -

C = Qi1

i=1 T

∑

⋅Qi2!Qik

Deg

j=1 k

∑

(Qij) ≤ d

Depth-4 circuits

• Q - Sparse polynomials
• C is homogeneous -
• In particular, k is at most d

C = Qi1

i=1 T

∑

⋅Qi2!Qik

Deg

j=1 k

∑

(Qij) ≤ d

Depth-4 circuits

Depth-4 circuits

• Homogeneous - exponential (and almost optimal*)

lower bounds known [GKKS, FLMS, KS, KLSS…… ]

Depth-4 circuits

• Homogeneous - exponential (and almost optimal*)

lower bounds known [GKKS, FLMS, KS, KLSS…… ]

• Non-homogeneous - best known lower bounds are

superlinear [Raz] - even when bottom fan-in is 2

Depth-4 circuits

• Homogeneous - exponential (and almost optimal*)

lower bounds known [GKKS, FLMS, KS, KLSS…… ]

• Non-homogeneous - best known lower bounds are

superlinear [Raz] - even when bottom fan-in is 2

• Bottom fan-in 1- over small finite fields, exponential

lower bounds are known [GK98, GR00]

Depth-4 circuits

• Homogeneous - exponential (and almost optimal*)

lower bounds known [GKKS, FLMS, KS, KLSS…… ]

• Non-homogeneous - best known lower bounds are

superlinear [Raz] - even when bottom fan-in is 2

• Bottom fan-in 1- over small finite fields, exponential

lower bounds are known [GK98, GR00]

• Bottom fan-in 2- can we get such bounds, over small

fields ?

Intuition

Obs 1 [GK’98, GR’00]: If the rank of the linear forms is ‘high’, then A is almost always zero as a function. Obs 2 [GK’98, GR’00]: If the rank of the linear forms is ‘low’, then A is equal to a low degree polynomial as a function.

A = ℓi

i=1 k

∏

Intuition

So, over constant size finite fields, ‘not too large’ depth-3 circuits can be approximated by a low degree polynomial as a function.

Wishful thinking…

Goal : (Non-homogeneous) depth-4 circuit l.b with bottom fan-in 2, over larger (but constant) sized fields. Need an appropriate notion of rank, such that

• low rank products have ‘low complexity’
• high rank products are ‘easy to handle’

Wishful thinking…

Goal : (Non-homogeneous) depth-4 circuit l.b with bottom fan-in 2, over larger (but constant) sized fields. Need an appropriate notion of rank, such that

• low rank products have ‘low complexity’
• high rank products are ‘easy to handle’

Algebraic rank is a candidate.

Wishful thinking…

Goal : (Non-homogeneous) depth-4 circuit l.b with bottom fan-in 2, over larger (but constant) sized fields. Need an appropriate notion of rank, such that

• low rank products have ‘low complexity’
• high rank products are ‘easy to handle’

Algebraic rank is a candidate.

Step 1 is already non-trivial, and the focus of this talk.

Algebraic (in)dependence

Algebraic (in)dependence

• {Q1, Q2,…, Qk} are algebraically dependent if there is

a non-zero polynomial R(y1, y2, …, yk) such that R(Q1, Q2, …, Qk) is identically zero.

Algebraic (in)dependence

• {Q1, Q2,…, Qk} are algebraically dependent if there is

a non-zero polynomial R(y1, y2, …, yk) such that R(Q1, Q2, …, Qk) is identically zero.

• higher degree generalization of linear dependence

Algebraic (in)dependence

• {Q1, Q2,…, Qk} are algebraically dependent if there is

a non-zero polynomial R(y1, y2, …, yk) such that R(Q1, Q2, …, Qk) is identically zero.

• higher degree generalization of linear dependence
• an underlying matroid, so the size of maximal

independent subset is well defined

Algebraic (in)dependence

• {Q1, Q2,…, Qk} are algebraically dependent if there is

a non-zero polynomial R(y1, y2, …, yk) such that R(Q1, Q2, …, Qk) is identically zero.

• higher degree generalization of linear dependence
• an underlying matroid, so the size of maximal

independent subset is well defined

• the size is called the algebraic rank of {Q1, Q2, …,

Qk}

Depth-4 circuits*

C = Qi1

i=1 T

∑

⋅Qi2!Qik

Depth-4 circuits*

• Algebraic rank of

C = Qi1

i=1 T

∑

⋅Qi2!Qik

{Qi1,Qi2,…,Qik} ≤ d

Depth-4 circuits*

• Algebraic rank of
• In particular, k >> d

C = Qi1

i=1 T

∑

⋅Qi2!Qik

{Qi1,Qi2,…,Qik} ≤ d

Depth-4 circuits*

• Algebraic rank of
• In particular, k >> d
• Includes homogeneous depth-4 circuits

C = Qi1

i=1 T

∑

⋅Qi2!Qik

{Qi1,Qi2,…,Qik} ≤ d

Depth-4 circuits*

• Algebraic rank of
• In particular, k >> d
• Includes homogeneous depth-4 circuits
• Only local upper bounds on algebraic rank

C = Qi1

i=1 T

∑

⋅Qi2!Qik

{Qi1,Qi2,…,Qik} ≤ d

Results

• 1. An lower bound for a family of

explicit polynomials of degree d in n variables.

nΩ( d )

Results

• 2. A quasi-polynomial deterministic identity

test, if the bottom fan-in is small, and algebraic rank is poly-logarithmic.

Remarks

Remarks

• When the global algebraic rank is bounded,

then lower bounds are known via Jacobian based methods (Agrawal et al.).

Remarks

• When the global algebraic rank is bounded,

then lower bounds are known via Jacobian based methods (Agrawal et al.).

• These results are over fields of

characteristic zero.

Remarks

• When the global algebraic rank is bounded,

then lower bounds are known via Jacobian based methods (Agrawal et al.).

• These results are over fields of

characteristic zero.

• Recently extended to other fields by Pandey,

Saxena and Sinhababu.

Proof sketch

A toy example

Algebraic rank -> linear rank, {Q1, Q2, …, Qk} is in linear span of {Q1, Q2,…, Qd}

A toy example

Algebraic rank -> linear rank, {Q1, Q2, …, Qk} is in linear span of {Q1, Q2,…, Qd}

• Q1.Q2.Q3……Qk = f(Q1.Q2……Qd)

A toy example

Algebraic rank -> linear rank, {Q1, Q2, …, Qk} is in linear span of {Q1, Q2,…, Qd}

• Q1.Q2.Q3……Qk = f(Q1.Q2……Qd)

Function of Qi’ s in basis!

A toy example

Algebraic rank -> linear rank, {Q1, Q2, …, Qk} is in linear span of {Q1, Q2,…, Qd}

• Q1.Q2.Q3……Qk = f(Q1.Q2……Qd)

reduces to

Function of Qi’ s in basis!

C = Qi1

i=1 T

∑

⋅Qi2!Qik

C = F

i(Qi1 i=1 T

∑

⋅Qi2!Qid)

A toy example

Algebraic rank -> linear rank, {Q1, Q2, …, Qk} is in linear span of {Q1, Q2,…, Qd}

• Q1.Q2.Q3……Qk = f(Q1.Q2……Qd)

reduces to Prove lower bounds for the new model.

Function of Qi’ s in basis!

C = Qi1

i=1 T

∑

⋅Qi2!Qik

C = F

i(Qi1 i=1 T

∑

⋅Qi2!Qid)

A toy example

Algebraic rank -> linear rank, {Q1, Q2, …, Qk} is in linear span of {Q1, Q2,…, Qd}

• Q1.Q2.Q3……Qk = f(Q1.Q2……Qd)

reduces to Prove lower bounds for the new model. Partial derivative based methods prove useful!

Function of Qi’ s in basis!

C = Qi1

i=1 T

∑

⋅Qi2!Qik

C = F

i(Qi1 i=1 T

∑

⋅Qi2!Qid)

Key idea

Key idea

• Product of polynomials of linear rank d, is a

function of the polynomials in the basis.

Key idea

• Product of polynomials of linear rank d, is a

function of the polynomials in the basis.

• Use this to reduce express each product

gate as a polynomial in few variables (polynomials in the basis).

Key idea

• Product of polynomials of linear rank d, is a

function of the polynomials in the basis.

• Use this to reduce express each product

gate as a polynomial in few variables (polynomials in the basis).

• Such functions are ‘simple’ even though they

could be of high degree.

Key question

Key question

• Q = {Q1, Q2, …, Qr} - algebraically

independent

Key question

• Q = {Q1, Q2, …, Qr} - algebraically

independent

• F such that {F

, Q1, Q2, …, Qr} - algebraically dependent

Key question

• Q = {Q1, Q2, …, Qr} - algebraically

independent

• F such that {F

, Q1, Q2, …, Qr} - algebraically dependent

• Can we infer something about the structure
• f F

, from Q?

Key question

• Q = {Q1, Q2, …, Qr} - algebraically

independent

• F such that {F

, Q1, Q2, …, Qr} - algebraically dependent

• Can we infer something about the structure
• f F

, from Q?

If this is linear dependence, then yes!

Key question

• Q = {Q1, Q2, …, Qr} - algebraically

independent

• F such that {F

, Q1, Q2, …, Qr} - algebraically dependent

• Can F be expressed a polynomial function of

Q?

Key question

• Q = {Q1, Q2, …, Qr} - algebraically

independent

• F such that {F

, Q1, Q2, …, Qr} - algebraically dependent

• Can F be expressed a polynomial function of

Q?

No! Q = {x^2}, F = x

Key lemma

• For almost all , there is a polynomial

H(y1, y2, …, yr), such that F(x+b) equals the low degree component of H(Q1(x+b), Q2(x+b), …, Qr(x+b)).

b ∈Fn

Key lemma

• For almost all , there is a polynomial

H(y1, y2, …, yr), such that F(x+b) equals the low degree component of H(Q1(x+b), Q2(x+b), …, Qr(x+b)).

b ∈Fn

Algebraic dependence implies functional dependence!

Toy example

Toy example

• Q(x) = x^2, F(x) = x

Toy example

• Q(x) = x^2, F(x) = x
• Pick a non zero b.

Toy example

• Q(x) = x^2, F(x) = x
• Pick a non zero b.
• Take H(y) = 1/2b (y^2 + b^2)

Toy example

• Q(x) = x^2, F(x) = x
• Pick a non zero b.
• Take H(y) = 1/2b (y^2 + b^2)
• H(Q) = (x + b) + degree 2 terms!

Subsequently, Pandey, Saxena, Sinhababu showed that the converse of the lemma is also true. In summary, functional dependence is (almost) captured by algebraic dependence and vice versa.

Proof of key lemma

Proof sketch

Proof sketch

• A(x), B(x) are algebraically dependent. Goal is to express B

as a polynomial in A; upto a translation, taking homog. components.

Proof sketch

• A(x), B(x) are algebraically dependent. Goal is to express B

as a polynomial in A; upto a translation, taking homog. components.

• Let G(y, z) be a non-zero polynomial such that G(A, B) = 0

Proof sketch

• A(x), B(x) are algebraically dependent. Goal is to express B

as a polynomial in A; upto a translation, taking homog. components.

• Let G(y, z) be a non-zero polynomial such that G(A, B) = 0
• Let G be the minimum degree annihilator of {A, B}

Proof sketch

• A(x), B(x) are algebraically dependent. Goal is to express B

as a polynomial in A; upto a translation, taking homog. components.

• Let G(y, z) be a non-zero polynomial such that G(A, B) = 0
• Let G be the minimum degree annihilator of {A, B}
• Consider the univariate polynomial in z, G(A(x), z)

Proof sketch

• A(x), B(x) are algebraically dependent. Goal is to express B

as a polynomial in A; upto a translation, taking homog. components.

• Let G(y, z) be a non-zero polynomial such that G(A, B) = 0
• Let G be the minimum degree annihilator of {A, B}
• Consider the univariate polynomial in z, G(A(x), z)
• z-B(x) is a root of G(A(x), z)

Proof sketch

Proof sketch

• Can we say something interesting about the structure of

roots of a polynomial?

Proof sketch

• Can we say something interesting about the structure of

roots of a polynomial?

• Yes!

Proof sketch

• Can we say something interesting about the structure of

roots of a polynomial?

• Yes!
• [Dvir et al.] Under mild constraints* the roots of G(A(x), z)

can be written as the low deg. component of a polynomial in A(x)

Proof sketch

• Can we say something interesting about the structure of

roots of a polynomial?

• Yes!
• [Dvir et al.] Under mild constraints* the roots of G(A(x), z)

can be written as the low deg. component of a polynomial in A(x)

• (*) is where a translation is needed

• Over fields of characteristic zero, the converse is also easy

to show. Relies on Jacobian criterion for algebraic independence

• Over fields of characteristic zero, the converse is also easy

to show. Relies on Jacobian criterion for algebraic independence

• Proofs over positive characteristic are much more technical

Questions

Questions

• Depth 4 circuits over small finite fields:

Questions

• Depth 4 circuits over small finite fields:
• What can we say about product of low

degree polynomials of large algebraic rank

• ver small fields ?

Questions

• Depth 4 circuits over small finite fields:
• What can we say about product of low

degree polynomials of large algebraic rank

• ver small fields ?
• Can they be ‘approximated’ by simple

functions?

Questions

• Depth 4 circuits over small finite fields:
• What can we say about product of low

degree polynomials of large algebraic rank

• ver small fields ?
• Can they be ‘approximated’ by simple

functions?

• True for polynomials of degree 1.

Questions

Questions

• Depth 4 circuits over small finite fields:

Questions

• Depth 4 circuits over small finite fields:
• How is the tuple (Q1(x), Q2(x), …,Qk(x))

distributed when Qi’ s are algebraically independent?

Questions

• Depth 4 circuits over small finite fields:
• How is the tuple (Q1(x), Q2(x), …,Qk(x))

distributed when Qi’ s are algebraically independent?

• Something known when the field is large

enough (Wooley’ s/Bezout’ s theorem).

Thanks!