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On Computing Multilinear Polynomials Using Multi-r -ic Depth Four Circuits
Suryajith Chillara
To cite this version:
Suryajith Chillara. On Computing Multilinear Polynomials Using Multi-r -ic Depth Four Circuits.
37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020), LIRMM, CNRS, Mar 2020, Montpellier, France. �hal-02366791�
On Computing Multilinear Polynomials Using Multi- r -ic Depth Four Circuits
Suryajith Chillara∗ University of Haifa November 16, 2019
Abstract
In this paper, we are interested in understanding the complexity of computing multilinear polynomials using depth four circuits in which polynomial computed at every node has a bound on the individual degree ofr (referred to as multi-r-ic circuits). The goal of this study is to make progress towards proving superpolynomial lower bounds for general depth four circuits computing multilinear polynomials, by proving better and better bounds as the value ofr increases.
Recently, Kayal, Saha and Tavenas (Theory of Computing, 2018) showed that any depth four arithmetic circuit of bounded individual degreer computing a multilinear polynomial onnO(1) variables and degreed=o(n), must have size at least rn1.1
Ωq
d r
whenr iso(d)and is striclty less thann1.1. This bound however deteriorates with increasingr. It is a natural question to ask if we can prove a bound that does not deteriorate with increasingror a bound that holds for alarger regime ofr.
We here prove a lower bound which does not deteriorate withr, however for a specific instance ofd =d(n)but for awiderrange ofr. Formally, we show that there exists an explicit polynomial onnO(1) variables and degreeΘ(log2n)such that any depth four circuit of bounded individual degreer<n0.2must have size at least exp Ω log2n
. Thisimprovementis obtained by suitably adapting the complexity measure of Kayal et al. (Theory of Computing, 2018). This adaptation of the measure is inspired by the complexity measure used by Kayal et al. (SIAM J. Computing, 2017).
1 Introduction
One of the major focal points in the area of algebraic complexity theory is to show that certain polynomials are hard to compute syntactically. Here, the hardness of computation is quantified by the number of arithmetic operations that are needed to compute the target polynomial. Instead of the standard turing machine model, we consider arithmetic circuits and formulas as models of computation.
Arithmetic circuits are directed acyclic graphs such that the leaf nodes are labeled by variables or constants from the underlying field, and every non-leaf node is labeled either by a+or×. Every node computes a polynomial by operating on its inputs with the operand given by its label. The flow of computation flows from the leaf to the output node. We refer the readers to the standard resources[SY10,Sap19]for more information on arithmetic formulas and arithmetic circuits.
Valiant conjectured that the Permanent does not have polynomial sized arithmetic circuits[Val79].
Working towards that conjecture, we aim to prove superpolynomial circuit size lower bounds. However,
∗suryajith@cmi.ac.in. This work was done while the author was a post doctoral fellow at IIT Bombay.
the best known circuit size lower bound isΩ(nlogn), for a power symmetric polynomial, due to Baur and Strassen[Str73, BS83], and, the best known formula size lower bound is Ω(n2), due to Kalorkoti[Kal85]. Due to the slow progress towards proving general circuit/formula lower bounds, it is natural to study some restricted class of arithmetic circuits and formulas.
Since most of the polynomials of interest such as Determinant, Permanent, etc., are multilinear polynomials, it is natural to consider the restriction where every intermediate computation is in fact multilinear. Due to the phenomenal work in the last two decades[NW97,Raz06,Raz04,RY08,RY09, HY11,RSY08,AKV18,CLS19,CELS18,CLS18], the complexity of multilinear formulas and circuits is better understood than that of general formulas and circuits.
Backed with this progress it is natural to try to extend these results to a circuit model where the individual degree of every variable in the polynomial computed at every node in the circuit isr. We refer to these circuits as multi-r-ic circuits. Whenr =1, the circuit model is multilinear.
Kayal and Saha[KS17]first studied multi-r-ic circuits of depth three and proved exponential lower bounds. Kayal, Saha and Tavenas[KST18]have extended this and proved exponential lower bounds at depth three and depth four. These circuits that were considered were syntactically multi-r-ic . That is, at any product node, any variable appears in the support of at most r many operands, and the total of the individual degrees is also at most r. Henceforth, all the multi-r-ic depth four circuits that we talk about shall be syntactically multi-r-ic .
Recently, Kumar, Oliviera and Saptharishi[KdOS19]showed that there is a chasm1for multi-r-ic circuits too. Formally, they showed that any polynomial sized (saync for a fixed constantc) multi-r-ic circuit of arbitrary depth computing a polynomial onnvariables can be depth reduced to syntactical multi-r-ic depth four circuits of size exp(O(p
r nlogn)). This provides us a motivation to study multi-r-ic depth four circuits and prove strong lower bounds against them.
Kayal, Saha and Tavenas[KST18]proved an exponential size lower bound against multi-r-ic depth four circuits computing a variant of the iterated matrix multiplication polynomial. They achieved this bound using a measure that is inspired by the methodShifted Partial Derivatives[Kay12,GKKS14]and the method ofSkew Partial Derivatives[KNS16]. They referred to this new technique as the method ofShifted Skew Partial Derivatives. Hegde and Saha[HS17]improved upon[KST18]and showed a near-optimalsize lower bound. However, the "best known" lower bounds are for polynomials that are multi-r-ic.
Motivation for this work
Raz and Yehudayoff[RY09]showed a lower bound of exp(Ωp
dlogd
)against multilinear depth four circuits which compute a multilinear polynomial overnvariables and degreed n(cf.[KST18, Footnote 9]). Kayal, Saha and Tavenas [KST18] have shown a lower bound of rn1.1
ΩÇ
d r
for a multilinear polynomial overnO(1)variables and degreed that is computed by a multi-r-ic depth four circuit. This lower bound degrades with the increasing value ofr and is superpolynomial only when r iso(d)and is strictly less thann1.1. This raises a natural question if the dependence onr could be improved upon.
In this work, we show that for a certain regime ofd, we can prove a lower bound that does not deteriorate with increasing values ofr.
1Agrawal and Vinay[AV08], Koiran, and Tavenas[Tav15]showed that any general circuit can be depth reduced to a depth four circuit of non-trivial size.
Theorem 1. Let n and r be integers such that r<n0.2. There exists an explicit polynomial Qnof degree Θ(log2n), over nO(1)variables such that any syntactically multi-r -ic depth four circuit computing it must have sizeexp Ω(log2n)
.
The explicit polynomial that we consider is a close variant of the polynomial considered by Kayal et al.[KST18]. By substituting these parameters into the bound from[KST18], their bound evaluates tonΩ
logn pr
. Note that this bound is superpolynomial only when r =o(log2n). Our lower bound is quantitatively better in this regime of parameters.
If we can show superpolynomial size lower bounds against multi-r-ic depth four circuits forr =nc for any constantc, then we indeed have superpolynomial circuit size lower bounds against depth four circuits. We believe that by building on the work of[KST18,HS17],Theorem 1is a step towards that direction.
In a subsequent work to this, we observed that by careful analysis and by building upon this work and work of Kumar and Saraf[KS14], we can in fact prove exponential lower bounds against depth four multi-r-ic circuits computing the Iterated Matrix Multiplication polynomial.
Proof overview:
Analogous to the work of Fournier et al.[FLMS15], and Kumar and Saraf[KS14], we first consider multi-r-ic depth four circuits of low bottom support2and prove lower bounds against them.
LetT1,T2, . . . ,Tsbe the terms corresponding to the product gates feeding into the output sum gate.
The output polynomial is obtained by adding the termsT1,T2, . . . ,Ts. Note that each of theseTi’s is a product of low support polynomialsQi,j, that is, every monomial in theseQi,j’s is supported on a small set of variables (sayµmany). One major observation at this point is to see that there can be at mostN·r many factors in any of theTi’s.
Kayal et al.[KST18]observed that the measure of shifted partial derivates[KSS14,FLMS15]does not yield any non-trivial lower bound if the number of factors is much larger than the number of variables itself. They worked around this obstacle by defining ahybridcomplexity measure (refered to asShifted Skew Partial Derivatives) where they first split the variables into two disjoint and unequal sets Y andZ such that|Y| |Z|, then considered theloworder partial derivatives with respect to only the Y variables and subsequently set all theY variables to zero in the partial derivatives obtained. This effectively reduces the number of factors in a partial derivative to at most|Z| ·r. They then shift these polynomials by monomials inZ variables and look at the dimension of the polynomials thus obtained.
This measure gave them a size lower bound of rn1.1
ΩÇ
d r
against multi-r-ic depth four circuits computing an explicit polynomial onnO(1)variables and degreed=o(n)whenr =o(d). To improve the dependence on r in the lower bound, we consider a variant ofShifted Skew Partial Derivativesthat we callProjected Shifted Skew Partial Derivatives. Here, we project down the space of Shifted Skew Partials and only look at the multilinear terms. Since the polynomial of interest is multilinear, it makes sense to only look at the multilinear terms obtained after the shifts of the skew partial derivatives. This is analogous to the method employed by Kayal et al.[KLSS17]to prove exponential lower bounds for Homogeneous depth four circuits, through the measure ofProjected Shifted Partial Derivatives.
We first show that the dimension of Projected Shifted Skew Partial derivatives is not too large for small multi-r-ic depth four circuits of low bottom support. We then show that there exists an explicit
2That is, all the product gates at the bottom are supported on small set of variables.
polynomial whose dimension of Projected Shifted Skew Partial derivatives is large and thus cannot be computed by small multi-r-ic depth four circuits. We then lift this result to multi-r-ic depth four circuits for suitable set of parameters.
2 Preliminaries
Notation:
• For a polynomial f, we use∂Y=k(f)to refer to the space of partial derivatives of orderk of f with respect to monomials of degreekinY.
• We usez=`andz≤`to refer to the set of all the monomials of degree equal to`and at most`, respectively, inZ variables.
• We usez≤`MLto refer to the set of all the multilinear monomials of degree at most`inZvariables.
• We usez≤`NonMLto refer to the set of all the non-multilinear monomials of degree at most`inZ variables.
• For a monomialmwe use|MonSupp(m)|to refer to the size of the set of variables that appear in it.
• For a polynomial f, we use |MonSupp(f)|to refer to the maximum |MonSupp(m)| over all monomials in f.
Depth four circuits: A depth four circuit (denoted by ΣΠΣΠ) over a field F and variables {x1,x2, . . . ,xn} computes polynomials which can be expressed in the form of sums of products of polynomials. That is,
Xs i=1
di
Y
j=1
Qi,j(x1, . . . ,xn)for some di’s. A depth four circuit is said to have a bottom support oft (denoted byΣΠΣΠ{t}) if it is a depth four circuit and all the monomials in each polynomialQi,j are supported on at mostt variables.
Multi-r-ic arithmetic circuits:
Definition 2(multi-r-ic circuits). Letr= (r1,r2,· · ·,rn). An arithmetic circuitΦis said to be a syntac- tically multi-r-ic circuit if for all product gates u∈Φand u =u1×u2× · · · ×ut, each variable xi can appear in at most rimany of the ui’s (i∈[t]). Further the total degree with respect to every variable over all the polynomials computed at u1,u2,· · ·,ut, is bounded by r , i.e.P
j∈[t]degx
i(fu
j)≤rifor all i∈[n].
Ifr= (r,r,· · ·,r), then we simply refer to them as multi-r -ic circuits.
Complexity Measure: We shall now describe our complexity measure which we shall henceforth refer to as Dimension of Projected Shifted Skew Partial Derivatives. This is a natural extension of the Dimension of Shifted Skew Partial Derivatives as used by[KST18].
This is analogous to the work of[KLSS14]where they study ashifted partials inspiredmeasure calledShifted Projected Partial derivativesand then[KLSS17]where they studyProjected Shifted Partial derivatives.
Since the polynomial of interest is multilinear, it does make sense for us to only look at those shifts of the partial derivatives, that maintain multilinearity. At the same time, since the individual degree of the intermediate computations in the multi-r-ic depth four circuit is large and non-multilinear terms
"cancel" out to generate the multilinear polynomial, we can focus on the multilinear terms generated after the shifts by projecting our linear space of polynomials down to them. We describe this process formally, below.
Let the variable set X be partitioned into two fixed, disjoint setsY andZ such that |Y| is a magnitude larger than|Z|. LetσY:F[YtZ]7→F[Z]be a map such that for any polynomial f(Y,Z), σY(f)∈F[Z]is obtained by setting every variable inY to zero by it and leavingZ variables untouched.
Let mult :F[Z]7→F[Z]be a map such that for any polynomialf(Y,Z), mult(f)∈F[Z]is obtained by setting the coeficients of all the non-multilinear monomials in f to 0 and leaving the rest untouched. We usez≤`·σY(∂Y=kf)to refer to the linear span of polynomials obtained by multiplying each polynomial inσY(∂Y=kf)with monomials of degree at most`inZ variables. We will now define our complexity measure, Dimension of Projected Shifted Skew Partial Derivatives (denoted byΓk,`) as follows.
Γk,`(f) =dim
F-span¦ mult
z≤`·σY
∂Y=kf©
This is a natural generalization of Shifted Skew Partial Derivatives of[KST18]. The following proposition is easy to verify.
Proposition 3(Sub-additivity). Let k and`be integers. Let the polynomials f,f1,f2be such that f =f1+f2. Then,Γk,`(f)≤Γk,`(f1) +Γk,`(f2).
Monomial Distance: We recall the following definition of distance between monomials from [CM19].
Definition 4(Definition 2.7,[CM19]). Let m1,m2be two monomials over a set of variables. Let S1 and S2be the multisets of variables corresponding to the monomials m1and m2respectively. The distance dist(m1,m2)between the monomials m1and m2is themin{|S1| − |S1∩S2|,|S2| − |S1∩S2|}where the cardinalities are the order of the multisets.
For example, let m1 = x12x2x32x4 and m2 = x1x22x3x5x6. ThenS1 = {x1,x1,x2,x3,x3,x4}, S2 = {x1,x2,x2,x3,x5,x6},|S1|=6,|S2|=6 and dist(m1,m2) =3. It is important to note that two distinct monomials could have distance 0 between them if one of them is a multiple of the other and hence the triangle inequality does not hold.
The following beautiful lemma (from[GKKS14]) is key to the asymptotic estimates required for the lower bound analyses.
Lemma 5 (Lemma 6,[GKKS14]). Let a(n),f(n),g(n):Z≥0→Z≥0be integer valued functions such that(f +g) =o(a). Then,
ln(a+f)!
(a−g)!= (f +g)lna±O
(f +g)2 a
We use the following strengthening of the Principle of Inclusion and Exclusion in our proof.
Lemma 6(Strong Inclusion-Exclusion[KS14]). Let W1,W2,· · ·,Wt be subsets of a finite set W . For a parameterλ≥1, letP
i,j∈[t]
i6=j
Wi∩Wj ≤λP
i∈[t]|Wi|.Then,
∪i∈[t]Wi
≥ 4λ1 P
i∈[t]|Wi|.
Polynomial Families: Let n,α,k be positive integers. We define the polynomial family Pn,α,k n,α,k≥0as follows. Let the variable setX ={x1, . . . ,xN
0}be partitioned into two fixed, and disjoint setsY andZ. We first define the polynomial family fn,α,k(Y,Z)as follows (as it was defined in [KST18]).
fn,α,k=Yk
i=1
gi(Yi,Zi)where gi(Yi,Zi) = X
a,b∈[n]
ya,b(i) Y
c∈[α]
za,c(i,1)zc(i,2+α),b.
It is easy to see that|Y|is n2k and|Z|is 2αnk. We shall henceforth usemto refer to|Z|. Thus, N0=|X|=|Y|+|Z|=k(n2+2αn).
Letc be a fixed constant in(0, 1). We shall now define another polynomial familyPn,α,kbased on the definition of fn,α,k. Let p=N0−c. Let ˆX ={ˆx1,1, ˆx1,2, . . . , ˆx1,t, . . . , ˆxN0,1, ˆxN0,2, . . . , ˆxN0,t}be a variable set distinct fromX such thatt= N0logp N0. Let Linp:X 7→Xˆbe a linear map such thatxi7→Pt
j=1ˆxi,j for alli∈[N0]. Then the polynomialPn,α,k(Xˆ)be defined to be fn,α,k◦Linp(Xˆ). That is,
Pn,α,k=fn,α,k
t
X
j=1
ˆx1,j,
t
X
j=1
ˆx2,j,· · ·,
t
X
j=1
xˆN0,j
!
wheret= N0logN0
p .
Note thatPn,α,kis a polynomial onN=N02+clogN0many variables.
We will now recall the following lemma which in the mentioned form is due to Kumar and Saptharishi[Sap19].
Lemma 7(Analogous to Lemma 20.5,[Sap19]). Letρbe a random restriction on the variable setX thatˆ sets each variable to zero with a probability of at least(1−p)where p=N0−c for some constant c∈(0, 1). Then fn,α,k(X)is a projection ofρ(Pn,α,k(Xˆ))with a probability of at least(1−2−N0).
3 Multi-r -ic Depth Four Circuits of Low Bottom Support
For some carefully chosen parameterskand`, we shall first show that if a multi-r-ic depth four circuit C of bottom supportµissmallthenΓk,`(C)is not too large. We shall then show thatΓk,`(fn,α,k)is large and thus it cannot be computed byC.
3.1 Upper Bound onΓk,`(C)
Recall thatC is a sum of at mosts many products of polynomialsT(1)+· · ·+T(s)where eachTi is a syntactically multi-r-ic product of polynomials of low monomial support.
We shall first prove a bound onΓk,`(Ti)for an arbitraryTi and derive a bound onΓk,`(C)by using sub-additivity of the measure (cf.Theorem 3).
LetT be a syntactic multi-r-ic product of polynomials ˜Q1(Y,Z)·Q˜2(Y,Z)·. . .·Q˜D(Y,Z)·R(Y) such that
MonSupp(Q˜i)
≤µ. We will first argue thatD is not too large sinceT is a syntactically multi-r-ic product. We shall first pre-process the productT by doing the following procedure.
Repeat this process until all but at most one of the factors inT (exceptR) have aZ-support of at leastµ2.
1. Pick two factors ˜Qi1 and ˜Qi,2such that they have the smallestZ-support amongstQ1,· · ·,QD. 2. If both of them have support strictly less than µ2, merge these factors to obtain a new factor. Else,
stop.
In the afore mentioned procedure, it is important to note that the monomial support post merging will still be at most beµsince the factors being merged are of support strictly less thanµ2. Henceforth, W.L.O.G we shall consider that every product gate at the top, in any multi-r-ic depth four circuit to be in a processed form.
LetT =Q1(Y,Z)·Q2(Y,Z)·. . .·Qt(Y,Z)·R(Y)be the product obtained after the preprocessing.
Each of theQi has aZ-support of at least µ2. The totalZ-support is at most|Z|r =m r sinceT is a syntactically multi-r-ic product. Thust could at most be2m rµ .
Lemma 8. Let n,k,r,`andµbe positive integers such that`+kµ <m2. Let T be a processed syntactic multi-r -ic product of polynomials Q1(Y,Z)·Q2(Y,Z)·. . .·Qt(Y,Z)·R(Y)such that|MonSupp(Qi)| ≤µ.
Then,Γk,`(T)is at most kt
· `+mkµ
·(`+kµ).
Before provingTheorem 8, we shall first use it to show an upper bound on the dimension of the space of Projected Shifted Skew Partial derivatives of a depth four multi-r-ic circuit of low bottom support.
Lemma 9. Let n,k,r,`andµbe positive integers such that`+kµ < m2. Let C be a processed syntactic multi-r -ic depth four circuit of bottom supportµand size s. Then,Γk,`(C)is at most s·
2m r µ
k
· `+mkµ
·(`+kµ).
Proof. W.L.O.G we can assume thatC be expressed asPs
iT(i)such thatT(i)is a processed syntactically multi-r-ic product of bottom support polynomials at mostµ. FromTheorem 3, we get thatΓk,`(C)≤ Ps
i=1Γk,`(T(i)). From the afore mentioned discussion we know that the number of factors inT(i)with a non-zeroZ-support is at most 2m rµ . FromTheorem 8, we get thatΓk,`(T(i))is at most
2m r µ
k
· `+mkµ
· (`+kµ). By putting all of this together, we get that
Γk,`(C)≤s· 2m r
µ
k
· m
`+kµ
·(`+kµ).
Proof ofTheorem 8. We will first show by induction onk, the following.
∂Y=kT ⊆F-span
[
S∈(t−kt )
¨ Y
i∈S
Qi(Y,Z)·z≤kµML ·F[Y]
«
[
F-span
[
S∈(t−kt )
¨ Y
i∈S
Qi(Y,Z)·z≤k rµNonML·F[Y]
«
The base case of induction fork =0 is trivialT is already in the required form. Let us assume the induction hypothesis for all derivatives of order<k. That is,∂Y=k−1T can be expressed as a linear combination of terms of the form
h(X,Y) =Y
i∈S
Qi(Y,Z)·h1(Z)·h2(Y)
whereSis a set of sizet−(k−1),h1(Z)is a polynomial inZvariables of degree at most(k−1)rµ, andh2(Y)is some polynomial inY variables. In fact,h1(Z)can be expressed as a linear combination of multilinear monomials of degree at most(k−1)µ, and non-multilinear monomials of degree at most (k−1)rµ.
For someu∈[|Y|]and some fixedi0inS,
∂h(Y,Z)
∂yu =
X
j∈S
Y
i∈Si6=j
Qi(Y,Z)·∂Qj(Y,Z)
∂yu ·h1(Z)·h2(Y)
+
Q
i∈SQi Qi0 ·Qi
0(Y,Z)·h1(Z)·∂h2(Y)
∂yk
∈F-span
Y
i∈Si6=j
Qi(Y,Z)·∂Qj(Y,Z)
∂yu ·h1(Z)·F[Y]|j ∈[S]
[F-span
¨ Q
i∈SQi
Qi0 ·Qi0(Y,Z)·h1(Z)·F[Y]
«
⊆F-span
[
T∈(|S|−1S )
¨ Y
i∈T
Qi(Y,Z)·zµML·h1(Z)·F[Y]
«
[
[
T∈(|S|−1S )
¨ Y
i∈T
Qi(Y,Z)·zNonML≤rµ ·h1(Z)·F[Y]
«
⊆F-span
[
T∈(t−kt )
¨ Y
i∈T
Qi(Y,Z)·z≤kµML ·F[Y]
«
[
F-span
[
T∈(t−kt )
¨ Y
i∈T
Qi(Y,Z)·z≤k rNonMLµ ·F[Y]
«
The last inclusion follows from the fact thath1(Z)is a linear combination of multilinear monomials of degree at most(k−1)µ, and non-multilinear monomials of degree at most(k−1)rµ. From the discussion above we know that any polynomial in∂Yk(T)can be expressed as a linear combination of polynomials of the form ∂∂yh
u. Further every polynomial of the form ∂∂yh
u belongs to the set W =F-span
[
T∈(t−kt )
¨ Y
i∈T
Qi(Y,Z)·z≤kMLµ·F[Y]
«
[
F-span
[
T∈(t−kt )
¨ Y
i∈T
Qi(Y,Z)·z≤k rNonMLµ ·F[Y]
«
.
Thus, we get that∂=kT is a subset ofW. This completes the proof by induction.
From the afore mentioned discussion, we can now derive the following expressions.
σY
∂Y=kT
⊆F-span
[
S∈(t−k[t])
¨ Y
i∈S
σY(Qi)·z≤kMLµ
«
[
F-span
[
S∈(t−k[t])
¨ Y
i∈S
σY(Qi)·z≤k rNonMLµ
«
z≤`σY
∂Y=kT
⊆F-span
[
S∈(t−k[t])
¨ Y
i∈S
σY(Qi)·z≤`+kµML
«
[F-span
[
S∈(t−k[t])
¨ Y
i∈S
σY(Qi)·z≤`+NonMLk rµ
«
=⇒ F-span¦ mult
z≤`·σY
∂Y=kT©
⊆F-span
[
S∈(t−k[t])
¨ Y
i∈S
mult(σY(Qi))·z≤kMLµ+`
«
.
Thus we get that dim F-span
mult z≤`·σY(∂Y=kT) is at most
dim
F-span
[
S∈(t−k[t])
¨ Y
i∈S
mult(σY(Qi))·z≤kµ+`ML
«
≤dim
F-span
[
S∈(t−kt )
¨ Y
i∈S
mult(σY(Qi))
«
·dim
F-span¦
z≤kµ+`ML ©
≤ t
t−k
·
kµ+`
X
i=0
m i
≤ t
k
· m
`+kµ
·(`+kµ) (Since`+kµ <m/2).
3.2 Lower Bound onΓk,`(fn,α,k)
First we recall the generalized Hamming bound[GRS19, Section 1.7].
Claim 10. Let the vectorsa= (a1,a2, . . . ,ak)andb = (b1,b2, . . . ,bk)correspond to the indices of the Y -monomial ya(1)
1,b1ya(2)
2,b2· · ·ya(k)
k,bk that is used to derive fn,α,k with. For every ∆0 < k, there is a subset P∆0⊂[n]2kof size n2k−∆0
∆0(∆02k) such that for all(a,b),(a0,b0)∈ P,dist((a,b),(a0,b0))≥∆0.
Observation 11. It is important to note that ∂kfn,α,k
ya(1)
1,b1ya(2)
2,b2···y(k)
ak,bk
for any choice of(a,b)∈[n]2kis a multilinear monomial over just the Z variables.
Claim 12. Let (a,b) and (a0,b0) be such that dist((a,b),(a0,b0)) ≥ ∆0. Then dist
∂(a,b)k fn,α,k,∂(ak0,b0)fn,α,k
≥α∆0.
Let m1,m2, . . . ,mt be the monomials in the set M0(= ∂P=k
∆0
fn,α,k), over Z variables such that dist(mi,mj)≥ ∆ ≥α∆0for alli 6= j. Further,σY(M0) =M0. LetM be the set of mutlilinear monomials of the formmim0overZ-variables for some 1≤i≤t wherem0is a monomial of length`.
It is important to note that the setM now corresponds to the setz=`·σY
∂P=k
∆0
fn,α,k
. We shall now show that the cardinality of the setM is large enough for a suitable setting of parametersα,∆0andk. Lemma 13. Let m,k,d,r,∆0,∆,`andµbe positive integers such that`+kµ < m2,(d−k)2=o(m),∆2= o(m),∆0=δk and`= m2(1−")for some fixed constantsδand". Then,|M | ≥12 2
1−"
δαk
· m−(`d−k) . Proof. For alli ∈[t], LetBi be the set of multilinear monomials of the formmim0where m0 is a multilinear monomial of degree`. From the previous discussion, it follows that|M |=
∪ti=1Bi . Using the principle of Inclusion and Exclusion, we get that
∪ti=1Bi ≥
Xt i=1
|Bi| − Xt i6=j∈[t]
Bi∩Bj . Claim 14. For all i∈[t],|Bi|= m−(d−k)`
.
Proof. Since deg(mi)is equal tod−k, the cardinality ofBi is equal to m−(`d−k) . Claim 15. For all i,j ∈[t]such that i6= j ,
Bi∩Bj
≤ m−(d−k)−∆`−∆
.
Proof. Consider any two monomials ˆmi and ˆmj fromBi andBj respectively. For ˆmi and ˆmj to be identical, ˆmi should contain at least∆variables from ˆmj\mˆiand similarly ˆmj should contain at least
∆variables from ˆmi\mˆj. The rest of the at most(`−∆)many variables should be the same both in ˆ
mi and ˆmj. The number of such multilinear monomials overZ variables is at most m−(`−∆d−k)−∆
. PuttingTheorem 14andTheorem 15together, we get the following.
|M |=
∪ti=1Bi ≥t
m−(d−k)
`
− t2 2
m−(d−k)−∆
`−∆
.
LetT1=t m−(d−k)`
andT2= t22 m−(d−k)−∆`−∆
. Let us consider the case whereT2=λT1whereλ≥1 for some setting of the parameters∆,α,`andk.
λ= T2 T1=
t2
2 m−(d−k)−∆
`−∆
t m−(`d−k)
= t
2· (m−(d−k)−∆)!
(`−∆)!(m−`−(d−k))!·(m−`−(d−k))!`!
(m−(d−k))!
= t
2·(m−(d−k)−∆)!
(m−(d−k))! · `!
(`−∆)!
= t
2·(m−(d−k)−∆)!
m! · m!
(m−(d−k))!· `!
(`−∆)!
= t
2· m(d−k)·`∆
m(d−k)+∆ (UsingTheorem 5)
= t 2·
` m
∆
.
The math block above crucially uses the fact that∆2=o(`)and(d−k)2=o(m)while invoking Theorem 5. Sinceλ≥1, we get that t2·`
m
∆
≥1. For some suitably fixed constantsδand", let∆0be set toδk and`be set to m2(1−"). Recall that for a fixed∆0,t = n2k−∆0
∆0(∆2k0) and∆=α∆0=δαk. Thus, n2k−∆0
2 ∆2k
0
·
` m
∆
≥1 n2k−∆0≥2∆0m
` ∆
2k
∆0
n2k−∆0≥
2 1−"
∆ 2k
∆0 ∆0
n(2−δ)k≥
2 1−"
αδk 2k
∆0 δk
and hence,
α≤(2−δ)logn−δlog2
δ
δlog 2
1−"
. InvokingTheorem 6with the previous discussion, we get that
|M | ≥T1
4λ = t m−(`d−k)
4λ = 1
2 m
` ∆
·
m−(d−k)
`
=1 2
2 1−"
δαk
·
m−(d−k)
`
.
Lemma 16. Let m,k,d,r,`andµbe positive integers such that`+kµ < m2,∆0=δk and`= m2(1−") for some fixed constantsδand". Then,Γk,`(fn,α,k)≥ |M | ≥12 2
1−"
δαk
· m−(d`−k) . Proof. Recall thatM corresponds to the setz=`·σY
∂P=k
∆0
fn,α,k
. SinceM is a bag of multilinear monomials over just theZ variables
|M |=dim
F-spann mult
z=`·σY
∂P=k
∆0
fn,α,ko
Since∂P=k
∆0
fn,α,k⊆∂=kfn,α,k andz=`⊆z≤`, we get that dim
F-span
n mult
z=`·σY
∂P=k
∆0fn,α,k o
≤dim
F-span¦ mult
z≤`·σY
∂Y=kfn,α,k©
=Γk,`(fn,α,k). Thus,Γk,`(fn,α,k)≥ |M | ≥12 2
1−"
δαk
· m−(`d−k) .
We shall now a size lower bound on the depth four multi-r-ic circuits of low bottom support that compute fn,α,k.
Theorem 17. Letδ =0.25 and"=0.8. Let n,r,α,k andµbe positive integers such that r ≤ n0.2, µ≤log50n andα= (2−δ)logn
0.9δlog1−"2 .Let C be a depth four multi-r -ic circuit of low bottom supportµand size s.
If C computes the polynomial fn,α,k then s must beexp(Ω(klogn)).
Proof. Recall that the polynomialfn,α,kis defined on the variable setsYandZsuch that|Z|=m=2αnk. Let`be an integer such that`= m2(1−")and`+kµ < m2. Let∆0=δk. Let us assume that the polynomial fn,α,k is computed by a depth four multi-r-ic circuitC of low bottom supportµand sizes. Then it must be the case thatΓk,`(fn,α,k) =Γk,`(C).
s≥
1 2
2
1−"
δαk
· m−(`d−k)
2m rµ
k
· `+kµm
·(`+kµ)
≥ 1
2(`+kµ)·
2 1−"
δαk
· kµ
2e m r k
·
m−(d−k)
`
`+mkµ
= 1
2(`+kµ)·
2 1−"
δαk
· kµ
2e m r k
·(m−(d−k))!
m! · (m−`−kµ)!
(m−`−(d−k))!·(`+kµ)!
`!
≥ 1
2(`+kµ)·
2 1−"
δαk
· kµ
2e m r k
· `kµ
m(d−k)·(m−`)(d−k)−kµ
= 1
2(`+kµ)·
2 1−"
δαk
· kµ
2e m r k
·
` m−`
kµ
·
m−` m
d−k
= 1
2(`+kµ)·
2 1−"
δαk
· µ
4eαn r k
·
1−"
1+"
kµ
·
1+"
2
d−k
= 1
2(`+kµ)·
2 1−"
δαk
· µ
4eαn r k
·
1−"
1+"
kµ
·
1+"
2
2αk
= 1
2(`+kµ)·
2 1−"
δ
·
1+"
2
2αk
· µ 4eαn r
k
·
1−"
1+"
kµ
= exp
αklog 2
1−"
δ
·1+"
2
2
−klogn−klogr−klog4eµα+kµlog1−"1+"
2(`+kµ) .
In the above math block, we use Theorem 5 to simplify the terms along with the fact that k2µ2= o(m−`), (d−k)2 =o(m)andk2µ2= o(`). To get a meaningful lower bound, we need
αlog 2
1−"
δ
·1+"
2
2
to be strictly greater than(logn+logr+log4eµα). Let us setαto (2−δ)logn
0.9δlog1−"2 . This reduces to showing that there exist constantsδ,"andνsuch that
(2−δ)·log 2
1−"
δ
·1+"
2
2
0.9δlog1−"2 −1≥ν. (1)
Let us fix the constants as follows: "=0.8,δ=0.25 andν=0.23. Through some calculations, it can be verified thatEquation 1gets satisfied. Thus,
s≥exp k
0.23 logn−logr−log4eαµ +µlog1−"1+"
2(`+kµ) Ifµ≤log50n and r ≤n0.2, we get thats≥exp(Ω(klogn)).
4 Multi-r -ic Depth Four Circuits
To prove the main theorem, we also need the following lemma.
Lemma 18(Analogous to Lemma 20.4,[Sap19]). Let P be a multi-r -ic polynomial that is computed by a syntactically multi-r -ic depth 4 circuit C of size s≤Nγµfor someγ >0. Letρbe a random restriction that sets each variable to zero independently with probability(1−N−2γ). Then with probability at least (1−N−γµ)the polynomialρ(P)is computed by a multi-r -ic depth four circuit C0of bottom support at mostµand size s.
Proof ofTheorem 1. Letnbe a large positive integer. Let us set some relevant parameters in terms ofn or otherwise as follows.
µ=logn
50 , "=0.8,
δ=0.25, α=(2−δ)logn
0.9δlog1−"2 , k= c
100logn N0=k(n2+2αn),
N =N02+clogN0, γis a small constant such thatN2γuN0c.
Let ˆX ={ˆx1,1, ˆx1,2, . . . , ˆx1,t, . . . , ˆxN0,1, ˆxN0,2, . . . , ˆxN0,t}be a set of variables over which the polynomial Pn,α,kis defined. Letρ: ˆX 7→ {0,∗}be a random restriction such that a variable is set to zero with a probability of(1−N−2γ), and is left untouched otherwise. LetC be a syntactically multi-r-ic depth four circuit of sizes≤Nγµthat computesPn,α,k.Theorem 18tells us thatC0=ρ(C)is a multi-r-ic depth four circuit of sizes and bottom support at mostµwith a probability of at least(1−N−γµ).
Conditioned on this probability,ρ(Pn,α,k)has a multi-r-icΣΠΣΠ{µ}size at mosts.
Theorem 7tells us that fn,α,kis a p-projection ofρ(Pn,α,k)with a probability of(1−2−N0)and hence fn,α,kalso has a multi-r-icΣΠΣΠ{µ}of size at mostswith a probability of at least(1−N−γµ−2−N0). In other words, there is a random restrictionσsuch that fn,α,kis a p-projection ofρ(Pn,α,k)andC0=ρ(C) is a multi-r-icΣΠΣΠ{µ}circuit.
On the other hand, fromTheorem 17we know that any multi-r-icΣΠΣΠ{µ}circuit that computes fn,α,kmust be of size exp(Ω(klogn)). Thus, it must be that exp(Ω(klogn))≤s≤Nγµ. We can choose cto be a small enough constant such that the aforementioned expression is satisfied. Thus,smust at least be exp Ω(log2n)
. The explicit polynomialQnisPn,α,kwhereα= (2−δ)logn
0.9δlog1−"2 andk= 100c logn.
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