Original: Hence, the above sum contains sets of size (cid:96) − 1 and (cid:96) + 1 .
Revised: Hence, the above sum contains sets of size (cid:96) − 1 and (cid:96) + 1 .

Original: For each set S ⊂ [ q ] with | S | = (cid:96) − 1 , there q + 1 − (cid:96) sets | ˜ S
Revised: For each set S ⊂ [ q ] with | S | = (cid:96) − 1 , there q + 1 − (cid:96) sets | ˜ S

Original: | =
Revised: | =

Original: (cid:96) , such that by removing one element we can obtain S .
Revised: (cid:96) , such that by removing one element we can obtain S .

Original: For each set S ⊂ [ q ] with | S | = (cid:96) + 1 , there (cid:96) + 1 sets | ˜ S
Revised: For each set S ⊂ [ q ] with | S | = (cid:96) + 1 , there (cid:96) + 1 sets | ˜ S

Original: | =
Revised: | =

Original: (cid:96) , such that by adding one element we can obtain S .
Revised: (cid:96) , such that by adding one element we can obtain S .

Original: We deduce that
Revised: We deduce that

Original: [Equation]
Revised: [Equation]

Original: x ) Y S ( y )
Revised: x ) Y S ( y )

Original: +
Revised: +

Original: [Equation]
Revised: [Equation]

Original: +qB ( Q q ;
Revised: +qB ( Q q ;

Original: [Equation]
Revised: [Equation]

Original: ) (cid:88)
Revised: ) (cid:88)

Original: S ⊂ [
Revised: S ⊂ [

Original: [Equation] [Equation]
Revised: [Equation] [Equation]

Original: | S |
Revised: | S |

Original: [Equation]
Revised: [Equation]

Original: (cid:96)
Revised: (cid:96)

Original: [Equation]
Revised: [Equation]

Original: Y S ( x ) Y S ( y ) .
Revised: Y S ( x ) Y S ( y ) .

Original: Using B ( Q q ; (cid:96) )
Revised: Using B ( Q q ; (cid:96) )

Original: = (cid:0) q(cid:96) (cid:1) , we obtain
Revised: = (cid:0) q(cid:96) (cid:1) , we obtain

Original: [Equation]
Revised: [Equation]

Original: The cases (cid:96) = 0 and (cid:96) =
Revised: The cases (cid:96) = 0 and (cid:96) =

Original: q are straightforward.
Revised: q are straightforward.

Original: C. P ROOF OF T HEOREM
Revised: C. P ROOF OF T HEOREM

Original: Let { d ( q ) } q ≥ 1 be a sequence of integers with 2 q ≤ d ( q ) ≤ q 1 /δ for some δ > 0 .
Revised: Let { d ( q ) } q ≥ 1 be a sequence of integers with 2 q ≤ d ( q ) ≤ q 1 /δ for some δ > 0 .

Original: We will denote d = d ( q ) for simplicity.
Revised: We will denote d = d ( q ) for simplicity.

Original: Consider x ∼ Unif( Q d ) , dq s − 1+ δ ≤ n ≤ dq s − δ for some δ > 0 and a sequence of inner-product kernels { h q } q ≥ 1 that satisﬁes Assumption 1 at level s .
Revised: Consider x ∼ Unif( Q d ) , dq s − 1+ δ ≤ n ≤ dq s − δ for some δ > 0 and a sequence of inner-product kernels { h q } q ≥ 1 that satisﬁes Assumption 1 at level s .

Original: We consider the vanilla one-layer convolutional kernel
Revised: We consider the vanilla one-layer convolutional kernel

Original: [Equation]
Revised: [Equation]

Original: Theorem 7 is a consequence of Theorem 4 in Mei et al.
Revised: Theorem 7 is a consequence of Theorem 4 in Mei et al.

Original: (2021a) where we take X d = Q d , ν d =
Revised: (2021a) where we take X d = Q d , ν d =

Original: Unif( X d ) and D d = L 2 ( Q d , Loc q ) ⊂
Revised: Unif( X d ) and D d = L 2 ( Q d , Loc q ) ⊂

Original: L 2 ( Q d ) .
Revised: L 2 ( Q d ) .

Original: The proof amounts to checking that { H CK ,d } q ≥veriﬁes the kernel concentration properties and eigenvalue condition (see Section 3.2 in Mei et al.
Revised: The proof amounts to checking that { H CK ,d } q ≥veriﬁes the kernel concentration properties and eigenvalue condition (see Section 3.2 in Mei et al.

Original: Proof of Theorem 7.
Revised: Proof of Theorem 7.

Original: Step 1.
Revised: Step 1.

Original: Diagonalization of the kernel and choosing m = m ( q ) .
Revised: Diagonalization of the kernel and choosing m = m ( q ) .

Original: From Proposition 1, we have the following diagonalization of H CK ,d :
Revised: From Proposition 1, we have the following diagonalization of H CK ,d :

Original: [Equation]
Revised: [Equation]

Original: where r ( ∅ ) =
Revised: where r ( ∅ ) =

Original: d and r ( S ) =
Revised: d and r ( S ) =

Original: q + 1 − γ ( S ) for S ⊂ [ q ] \ {∅} , and we recall E (cid:96) =
Revised: q + 1 − γ ( S ) for S ⊂ [ q ] \ {∅} , and we recall E (cid:96) =

Original: { S ⊆
Revised: { S ⊆

Original: [ d ] : | S | = (cid:96), γ ( S ) ≤ q } .
Revised: [ d ] : | S | = (cid:96), γ ( S ) ≤ q } .

Original: Using that B ( Q q ; (cid:96) )
Revised: Using that B ( Q q ; (cid:96) )

Original: =
Revised: =

Original: Θ q ( q (cid:96) ) , ξ q,(cid:96) B ( Q q ; (cid:96) ) ≤ h q (1) and Assumption 1, we have
Revised: Θ q ( q (cid:96) ) , ξ q,(cid:96) B ( Q q ; (cid:96) ) ≤ h q (1) and Assumption 1, we have

Original: [Equation]
Revised: [Equation]

Original: Further deﬁne E (cid:96),h = { S ∈ E (cid:96) : γ ( S )
Revised: Further deﬁne E (cid:96),h = { S ∈ E (cid:96) : γ ( S )

Original: = h } for h
Revised: = h } for h

Original: = (cid:96), . . .
Revised: = (cid:96), . . .

Original: , q .
Revised: , q .

Original: It is easy to check that |E (cid:96),h | =
Revised: It is easy to check that |E (cid:96),h | =

Original: d (cid:0) h − 2 (cid:96) − 2 (cid:1) and
Revised: d (cid:0) h − 2 (cid:96) − 2 (cid:1) and

Original: [Equation]
Revised: [Equation]

Original: =
Revised: =

Original: d (cid:18) q −(cid:96) −(cid:19) , and therefore |E (cid:96) | =
Revised: d (cid:18) q −(cid:96) −(cid:19) , and therefore |E (cid:96) | =

Original: Θ q ( d · q (cid:96) − 1 ) .
Revised: Θ q ( d · q (cid:96) − 1 ) .

Original: Denote { λ q,j } j ≥ 1 the eigenvalues { ξ q,(cid:96) r ( S ) /d } (cid:96) =0 ,...,q ; S ∈E (cid:96) in nonincreasing order, and { ψ q,j } j ≥ 1 the reordered eigenfunctions.
Revised: Denote { λ q,j } j ≥ 1 the eigenvalues { ξ q,(cid:96) r ( S ) /d } (cid:96) =0 ,...,q ; S ∈E (cid:96) in nonincreasing order, and { ψ q,j } j ≥ 1 the reordered eigenfunctions.

Original: Set m to be the number of eigenvalues such that λ q,j > qξ q, s +1 /d
Revised: Set m to be the number of eigenvalues such that λ q,j > qξ q, s +1 /d

Original: (recall qξ q, s +1 =
Revised: (recall qξ q, s +1 =

Original: Θ d ( q − s ) ).
Revised: Θ d ( q − s ) ).

Original: Denote α =
Revised: Denote α =

Original: qξ q, s
Revised: qξ q, s

Original: +1 /ξ
Revised: +1 /ξ

Original: q, s .
Revised: q, s .

Original: From the bounds (65) on ξ q, s +1 and ξ q, s , we have α = Θ q (1) .
Revised: From the bounds (65) on ξ q, s +1 and ξ q, s , we have α = Θ q (1) .

Original: Denote ˜
Revised: Denote ˜

Original: α
Revised: α

Original: = q + 1 − α and E s , ≥ ˜
Revised: = q + 1 − α and E s , ≥ ˜

Original: α = { S ∈ E s : γ ( S ) ≥ ˜ α } and E s ,< ˜ α
Revised: α = { S ∈ E s : γ ( S ) ≥ ˜ α } and E s ,< ˜ α

Original: =
Revised: =

Original: E s \ E s , ≥ ˜ α .
Revised: E s \ E s , ≥ ˜ α .

Original: Using Eq.
Revised: Using Eq.

Original: (65) and that 1 ≤ r ( S ) ≤ q , we have { λ d,j } j ∈ [ m ] that contains exactly the eigenvalues associated to homogeneous polynomials of degree less or equal to s − 1 and of degree s with S ∈ E s ,< ˜ α (which corresponds to the sets S such that r ( S ) > α , i.e., ξ q, s r ( S ) >
Revised: (65) and that 1 ≤ r ( S ) ≤ q , we have { λ d,j } j ∈ [ m ] that contains exactly the eigenvalues associated to homogeneous polynomials of degree less or equal to s − 1 and of degree s with S ∈ E s ,< ˜ α (which corresponds to the sets S such that r ( S ) > α , i.e., ξ q, s r ( S ) >

Original: qξ q, s +1 ).
Revised: qξ q, s +1 ).

Original: In particular, if α < 1 , then { λ d,j } j ∈ [ m ] contains exactly the eigenvalues associated to all homogeneous polynomials of degree less or equal to s .
Revised: In particular, if α < 1 , then { λ d,j } j ∈ [ m ] contains exactly the eigenvalues associated to all homogeneous polynomials of degree less or equal to s .

Original: Note that we have
Revised: Note that we have

Original: [Equation]
Revised: [Equation]

Original: Step 2.
Revised: Step 2.

Original: Diagonal elements of the truncated kernel.
Revised: Diagonal elements of the truncated kernel.

Original: Deﬁne the truncated kernel H d,> m to be
Revised: Deﬁne the truncated kernel H d,> m to be

Original: [Equation]
Revised: [Equation]

Original: The diagonal elements of the truncated kernel are given by: for any x ∈ Q d ,
Revised: The diagonal elements of the truncated kernel are given by: for any x ∈ Q d ,

Original: [Equation] [Equation]
Revised: [Equation] [Equation]

Original: Notice that (cid:88)
Revised: Notice that (cid:88)

Original: [Equation]
Revised: [Equation]

Original: Hence using that ξ q, s = O d ( q − s )
Revised: Hence using that ξ q, s = O d ( q − s )

Original: [Equation]
Revised: [Equation]

Original: we have
Revised: we have

Original: [Equation]
Revised: [Equation]

Original: where h q,> s is the inner-product kernel with the ( s + 1) -ﬁrst Gegenbauer coefﬁcients set to zero, i.e., h q,> s ( (cid:104) u , v (cid:105) /q )
Revised: where h q,> s is the inner-product kernel with the ( s + 1) -ﬁrst Gegenbauer coefﬁcients set to zero, i.e., h q,> s ( (cid:104) u , v (cid:105) /q )

Original: = (cid:80) q(cid:96)
Revised: = (cid:80) q(cid:96)

Original: =
Revised: =

Original: s +1 ξ q,(cid:96) B
Revised: s +1 ξ q,(cid:96) B

Original: ( Q q ; (cid:96) ) Q ( q ) (cid:96) ( (cid:104) u , v (cid:105) ) , for any u , v ∈ Q q .
Revised: ( Q q ; (cid:96) ) Q ( q ) (cid:96) ( (cid:104) u , v (cid:105) ) , for any u , v ∈ Q q .

Original: From Assumption 1 at level s , we have Ω q (1) =
Revised: From Assumption 1 at level s , we have Ω q (1) =

Original: [Equation]
Revised: [Equation]

Original: q,(cid:96) (cid:48)
Revised: q,(cid:96) (cid:48)

Original: [Equation] [Equation]
Revised: [Equation] [Equation]

Original: q
Revised: q

Original: [Equation]
Revised: [Equation]

Original: (cid:96) (cid:48) )
Revised: (cid:96) (cid:48) )

Original: [Equation]
Revised: [Equation]

Original: q,>
Revised: q,>

Original: [Equation]
Revised: [Equation]

Original: (1) ≤ h q (1) =
Revised: (1) ≤ h q (1) =

Original: O q (1) .
Revised: O q (1) .

Original: Hence, Tr( H d,> m ) =
Revised: Hence, Tr( H d,> m ) =

Original: Θ d (1) .
Revised: Θ d (1) .

Original: Similarly,
Revised: Similarly,

Original: [Equation] [Equation]
Revised: [Equation] [Equation]

Original: Step 3.
Revised: Step 3.

Original: Choosing the sequence u
Revised: Choosing the sequence u

Original: = u ( d ) .
Revised: = u ( d ) .

Original: Let s (cid:48) be chosen as in Assumption 1, i.e., such that ξ q, s (cid:48) B
Revised: Let s (cid:48) be chosen as in Assumption 1, i.e., such that ξ q, s (cid:48) B

Original: ( Q q ; s (cid:48) )
Revised: ( Q q ; s (cid:48) )

Original: =
Revised: =

Original: Ω q (1) .
Revised: Ω q (1) .

Original: We have
Revised: We have

Original: [Equation]
Revised: [Equation]

Original: Set u
Revised: Set u

Original: = u ( d ) to be the number of eigenvalues such that λ q,j > qξ q, s (cid:48) /d
Revised: = u ( d ) to be the number of eigenvalues such that λ q,j > qξ q, s (cid:48) /d

Original: =
Revised: =

Original: Θ q ( q − s (cid:48) +1 /d ) .
Revised: Θ q ( q − s (cid:48) +1 /d ) .

Original: From
Revised: From

Original: Eqs.
Revised: Eqs.

Original: (65) and (69), and recalling that 1 ≤ r ( S ) ≤ q , we deduce that { λ d,j } j ∈ [ u ] must contain all the eigenvalues associated to homogeneous polynomials of degree less or equal to (cid:96) and does not contain any of the eigenvalues associated to homogeneous polynomials of degree larger or equal to s (cid:48) .
Revised: (65) and (69), and recalling that 1 ≤ r ( S ) ≤ q , we deduce that { λ d,j } j ∈ [ u ] must contain all the eigenvalues associated to homogeneous polynomials of degree less or equal to (cid:96) and does not contain any of the eigenvalues associated to homogeneous polynomials of degree larger or equal to s (cid:48) .

Original: We have
Revised: We have

Original: [Equation]
Revised: [Equation]

Original: Similarly, we have
Revised: Similarly, we have

Original: [Equation]
Revised: [Equation]

Original: Finally,
Revised: Finally,

Original: [Equation]
Revised: [Equation]

Original: Step 4.
Revised: Step 4.

Original: Checking the kernel concentration property at level { ( n ( q ) , m ( q ))
Revised: Checking the kernel concentration property at level { ( n ( q ) , m ( q ))

Original: } q ≥ 1 .
Revised: } q ≥ 1 .

Original: Let us check the kernel concentration property at level ( n, m ) with the sequence of integers { u ( q ) } q ≥ 1 deﬁned in the previous step (Assumption 4 in Mei et al.
Revised: Let us check the kernel concentration property at level ( n, m ) with the sequence of integers { u ( q ) } q ≥ 1 deﬁned in the previous step (Assumption 4 in Mei et al.

Original: (2021a)):
Revised: (2021a)):

Original: (a) (Hypercontractivity of ﬁnite eigenspaces)
Revised: (a) (Hypercontractivity of ﬁnite eigenspaces)

Original: The subspace spanned by the top eigenvectors { ψ q,j } j ∈ [ u ] is contained in the subspace of polynomials of degree less or equal to s (cid:48)
Revised: The subspace spanned by the top eigenvectors { ψ q,j } j ∈ [ u ] is contained in the subspace of polynomials of degree less or equal to s (cid:48)

Original: −on the hypercube.
Revised: −on the hypercube.

Original: The hypercontractivity of this subspace is a consequence of a classical result due to Beckner, Bonami and Gross (see Lemma 4 in Section D).
Revised: The hypercontractivity of this subspace is a consequence of a classical result due to Beckner, Bonami and Gross (see Lemma 4 in Section D).

Original: (b) (Properly decaying eigenvalues.)
Revised: (b) (Properly decaying eigenvalues.)

Original: From step 3 and recalling that s (cid:48) ≥ 1 /δ + 2 s + 3 where δ > 0 veriﬁes q ≥ d δ , we have
Revised: From step 3 and recalling that s (cid:48) ≥ 1 /δ + 2 s + 3 where δ > 0 veriﬁes q ≥ d δ , we have

Original: [Equation]
Revised: [Equation]

Original: for δ (cid:48) > 0 sufﬁciently small.
Revised: for δ (cid:48) > 0 sufﬁciently small.

Original: Similarly,
Revised: Similarly,

Original: [Equation]
Revised: [Equation]

Original: for δ (cid:48) > 0 chosen sufﬁciently small.
Revised: for δ (cid:48) > 0 chosen sufﬁciently small.

Original: (c
Revised: (c

Original: Concentration of the
Revised: Concentration of the

Original: From and
Revised: From and

Original: 68), the
Revised: 68), the

Original: - nal elements kernel are constant assumption is automatically veriﬁed.
Revised: - nal elements kernel are constant assumption is automatically veriﬁed.

Original: Step 5.
Revised: Step 5.

Original: Checking the eigenvalue condition at level { ( n ( q ) , m ( q ))
Revised: Checking the eigenvalue condition at level { ( n ( q ) , m ( q ))

Original: } q ≥ 1 .
Revised: } q ≥ 1 .

Original: Let us now check the eigenvalue condition at level { ( n ( q ) , m ( q )) } q ≥ 1 which corresponds to Assumption 5 in Mei et al.
Revised: Let us now check the eigenvalue condition at level { ( n ( q ) , m ( q )) } q ≥ 1 which corresponds to Assumption 5 in Mei et al.

Original: (2021a)): (a) First notice that
Revised: (2021a)): (a) First notice that

Original: [Equation]
Revised: [Equation]

Original: Hence
Revised: Hence

Original: Tr( H
Revised: Tr( H

Original: [Equation]
Revised: [Equation]

Original: d,
Revised: d,

Original: [Equation]
Revised: [Equation]

Original: m )
Revised: m )

Original: [Equation]
Revised: [Equation]

Original: for δ > 0 sufﬁciently small.
Revised: for δ > 0 sufﬁciently small.

Original: Similarly,
Revised: Similarly,

Original: [Equation]
Revised: [Equation]

Original: (b) This is a direct consequence of Eq.
Revised: (b) This is a direct consequence of Eq.

Original: We can therefore apply Theorem 4 in Mei et al.
Revised: We can therefore apply Theorem 4 in Mei et al.

Original: (2021a), which concludes the proof.
Revised: (2021a), which concludes the proof.

Original: C. P ROOF OF T HEOREM
Revised: C. P ROOF OF T HEOREM

Original: Consider q s − 1+ δ ≤ n ≤ q s − δ for some δ > 0 and a sequence of inner-product kernels { h q } q ≥ 1 that satisﬁes Assumptions 1 and 2 at level s .
Revised: Consider q s − 1+ δ ≤ n ≤ q s − δ for some δ > 0 and a sequence of inner-product kernels { h q } q ≥ 1 that satisﬁes Assumptions 1 and 2 at level s .

Original: We consider the one-layer convolutional kernel with global average pooling
Revised: We consider the one-layer convolutional kernel with global average pooling

Original: [Equation]
Revised: [Equation]

Original: Again, the proof of Theorem 8 will amount to checking that the conditions of Theorem 4 in Mei et al.
Revised: Again, the proof of Theorem 8 will amount to checking that the conditions of Theorem 4 in Mei et al.

Original: (2021a) hold.
Revised: (2021a) hold.

Original: For the sake of simplicity, we will further assume that ξ q, s > qξ q, s +1 , which simpliﬁes some of the computation.
Revised: For the sake of simplicity, we will further assume that ξ q, s > qξ q, s +1 , which simpliﬁes some of the computation.

Original: This condition can be removed as in Theorem 7, by considering the set C s ,< ˜ α
Revised: This condition can be removed as in Theorem 7, by considering the set C s ,< ˜ α

Original: = { S ∈ C s : γ ( S ) <
Revised: = { S ∈ C s : γ ( S ) <

Original: ˜ α } and showing that the extra terms corresponding to these eigenfunctions are negligible.
Revised: ˜ α } and showing that the extra terms corresponding to these eigenfunctions are negligible.

Original: Proof of Theorem 8.
Revised: Proof of Theorem 8.

Original: Step 1.
Revised: Step 1.

Original: Diagonalization of the kernel and choosing m = m ( q ) .
Revised: Diagonalization of the kernel and choosing m = m ( q ) .

Original: From Proposition 2 with ω = d , we have the following diagonalization of H dd,q :
Revised: From Proposition 2 with ω = d , we have the following diagonalization of H dd,q :

Original: [Equation]
Revised: [Equation]

Original: where we recall ψ S ( x )
Revised: where we recall ψ S ( x )

Original: = 1 √ d (cid:80) k ∈ [ d ] Y k + S ( x ) and that C (cid:96) is the quotient space of E (cid:96) with the translation equivalence relation.
Revised: = 1 √ d (cid:80) k ∈ [ d ] Y k + S ( x ) and that C (cid:96) is the quotient space of E (cid:96) with the translation equivalence relation.

Original: It is easy to check that |C (cid:96) | = (cid:0) q
Revised: It is easy to check that |C (cid:96) | = (cid:0) q

Original: [Equation] [Equation]
Revised: [Equation] [Equation]

Original: 1 (cid:1) .
Revised: 1 (cid:1) .

Original: From Assumption 1, we get the same bounds on the Gegenbauer coefﬁcients ξ q,(cid:96) as Eq.
Revised: From Assumption 1, we get the same bounds on the Gegenbauer coefﬁcients ξ q,(cid:96) as Eq.

Original: (65) in the proof of Theorem 7.
Revised: (65) in the proof of Theorem 7.

Original: Denote { λ q,j } j ≥ 1 the eigenvalues { ξ q,(cid:96) r ( S ) } (cid:96) =0 ,...,q ; S ∈E (cid:96) in nonincreasing order, and { ψ q,j } j ≥ 1 the reordered eigenfunctions.
Revised: Denote { λ q,j } j ≥ 1 the eigenvalues { ξ q,(cid:96) r ( S ) } (cid:96) =0 ,...,q ; S ∈E (cid:96) in nonincreasing order, and { ψ q,j } j ≥ 1 the reordered eigenfunctions.

Original: Set m to be the number of eigenvalues such that λ q,j > qξ q, s +1 (recall qξ q, s +1 =
Revised: Set m to be the number of eigenvalues such that λ q,j > qξ q, s +1 (recall qξ q, s +1 =

Original: Θ d ( q − s ) ).
Revised: Θ d ( q − s ) ).

Original: From the bounds (65) and our simplifying assumption that ξ q, s > qξ q, s +1 , we have { λ d,j } j ∈ [ m ] that contains exactly the eigenvalues associated to homogeneous polynomials of degree less or equal to s .
Revised: From the bounds (65) and our simplifying assumption that ξ q, s > qξ q, s +1 , we have { λ d,j } j ∈ [ m ] that contains exactly the eigenvalues associated to homogeneous polynomials of degree less or equal to s .

Original: Note that we have
Revised: Note that we have

Original: [Equation]
Revised: [Equation]

Original: Step 2.
Revised: Step 2.

Original: Diagonal elements of the truncated kernel.
Revised: Diagonal elements of the truncated kernel.

Original: Deﬁne the truncated kernel H d,> m to be
Revised: Deﬁne the truncated kernel H d,> m to be

Original: [Equation]
Revised: [Equation]

Original: The diagonal elements of the truncated kernel are given by: for any x ∈ Q d ,
Revised: The diagonal elements of the truncated kernel are given by: for any x ∈ Q d ,

Original: [Equation]
Revised: [Equation]

Original: where
Revised: where

Original: [Equation]
Revised: [Equation]

Original: Notice that we have now
Revised: Notice that we have now

Original: [Equation]
Revised: [Equation]

Original: Therefore E x [Υ ( q ) (cid:96) ( x )]
Revised: Therefore E x [Υ ( q ) (cid:96) ( x )]

Original: = 1 and
Revised: = 1 and

Original: [Equation]
Revised: [Equation]

Original: From Proposition 7 with (cid:96) = s
Revised: From Proposition 7 with (cid:96) = s

Original: [Equation]
Revised: [Equation]

Original: we have
Revised: we have

Original: [Equation]
Revised: [Equation]

Original: Step 3.
Revised: Step 3.

Original: Choosing the sequence u
Revised: Choosing the sequence u

Original: = u ( d ) .
Revised: = u ( d ) .

Original: Let s (cid:48) be chosen as in Assumption 1.
Revised: Let s (cid:48) be chosen as in Assumption 1.

Original: Similarly to step 3 in the proof of Theorem 7, take u = u ( d ) to be the number of eigenvalues such that λ q,j > qξ q,(cid:96) (cid:48) .
Revised: Similarly to step 3 in the proof of Theorem 7, take u = u ( d ) to be the number of eigenvalues such that λ q,j > qξ q,(cid:96) (cid:48) .

Original: We get
Revised: We get

Original: [Equation]
Revised: [Equation]

Original: Step
Revised: Step

Original: Checking the kernel concentration property at level { ( n ( q ) , m ( q ))
Revised: Checking the kernel concentration property at level { ( n ( q ) , m ( q ))

Original: } q ≥ 1 .
Revised: } q ≥ 1 .

Original: The kernel concentration property at level ( n, m ) hold with the sequence { u ( q ) } q ≥ 1 as deﬁned in step 3.
Revised: The kernel concentration property at level ( n, m ) hold with the sequence { u ( q ) } q ≥ 1 as deﬁned in step 3.

Original: The hypercontractivity of ﬁnite eigenspaces and the properly decaying eigenvalues are obtained as in step 4 of the proof of Theorem 7, while the concentration of the diagonal elements of the kernel is given by Eq.
Revised: The hypercontractivity of ﬁnite eigenspaces and the properly decaying eigenvalues are obtained as in step 4 of the proof of Theorem 7, while the concentration of the diagonal elements of the kernel is given by Eq.

Original: Step
Revised: Step

Original: Checking the
Revised: Checking the

Original: eigenvalue
Revised: eigenvalue

Original: condition at level { ( n ( q ) , m ( q )) } q ≥
Revised: condition at level { ( n ( q ) , m ( q )) } q ≥

Original: This is obtained similarly as in step 5 of the proof of Theorem 7.
Revised: This is obtained similarly as in step 5 of the proof of Theorem 7.

Original: C. A UXILIARY RESULTS
Revised: C. A UXILIARY RESULTS

Original: Proposition 7.
Revised: Proposition 7.

Original: Let s ≥ 1 be a ﬁxed integer.
Revised: Let s ≥ 1 be a ﬁxed integer.

Original: Assume that the sequence of inner-product kernels { h q } q ≥ 1 satisﬁes Assumptions 1 and 2 at level s .
Revised: Assume that the sequence of inner-product kernels { h q } q ≥ 1 satisﬁes Assumptions 1 and 2 at level s .

Original: Deﬁne H > s d : Q d × Q d → R as the convolutional kernel with global average pooling
Revised: Deﬁne H > s d : Q d × Q d → R as the convolutional kernel with global average pooling

Original: [Equation]
Revised: [Equation]

Original: where h q,> s is the inner-product kernel where the s + 1 ﬁrst Gegenbauer coefﬁcients are set to 0 .
Revised: where h q,> s is the inner-product kernel where the s + 1 ﬁrst Gegenbauer coefﬁcients are set to 0 .

Original: Then for n =
Revised: Then for n =

Original: O q ( q p ) for some ﬁxed p , letting ( x i ) i ∈ [ n ] ∼ Unif( Q d ) , we have
Revised: O q ( q p ) for some ﬁxed p , letting ( x i ) i ∈ [ n ] ∼ Unif( Q d ) , we have

Original: [Equation]
Revised: [Equation]

Original: Proof of Proposition 7.
Revised: Proof of Proposition 7.

Original: Step 1.
Revised: Step 1.

Original: Bounding sup
Revised: Bounding sup

Original: [Equation]
Revised: [Equation]

Original: Recall that we deﬁned
Revised: Recall that we deﬁned

Original: [Equation]
Revised: [Equation]

Original: Following the same proof as Proposition 8 in Mei et al.
Revised: Following the same proof as Proposition 8 in Mei et al.

Original: (2021b)
Revised: (2021b)

Original: [Equation]
Revised: [Equation]

Original: notice that for the integer v in
Revised: notice that for the integer v in

Original: Assumption
Revised: Assumption

Original: [Equation]
Revised: [Equation]

Original: by Lemma 2 stated below
Revised: by Lemma 2 stated below

Original: [Equation]
Revised: [Equation]

Original: we have
Revised: we have

Original: [Equation]
Revised: [Equation]

Original: By Assumption
Revised: By Assumption

Original: [Equation]
Revised: [Equation]

Original: there exists C > 0 such that for any γ ∈ [ − 1 , 1] ,
Revised: there exists C > 0 such that for any γ ∈ [ − 1 , 1] ,

Original: [Equation]
Revised: [Equation]

Original: and | h ( r ) q,>v (0) | ≤ Cq − ( v +1 − r ) / 2 for r ≤ v .
Revised: and | h ( r ) q,>v (0) | ≤ Cq − ( v +1 − r ) / 2 for r ≤ v .

Original: Moreover, by Hanson-Wright inequality as in Lemma 3, using n = O q ( q p ) (at most polynomial in q ) and a union bound, we have for any η > 0 ,
Revised: Moreover, by Hanson-Wright inequality as in Lemma 3, using n = O q ( q p ) (at most polynomial in q ) and a union bound, we have for any η > 0 ,

Original: [Equation]
Revised: [Equation]

Original: Therefore
Revised: Therefore

Original: [Equation]
Revised: [Equation]

Original: injecting these bounds in Eq.
Revised: injecting these bounds in Eq.

Original: [Equation]
Revised: [Equation]

Original: we get
Revised: we get

Original: [Equation]
Revised: [Equation]

Original: l ) (cid:105) /q ) (cid:12)(cid:12)(cid:12)(cid:105) =
Revised: l ) (cid:105) /q ) (cid:12)(cid:12)(cid:12)(cid:105) =

Original: O q, P ( q − ( v +1) / 2+ η ) .
Revised: O q, P ( q − ( v +1) / 2+ η ) .

Original: Hence, we deduce that
Revised: Hence, we deduce that

Original: [Equation]
Revised: [Equation]

Original: [Table]
Revised: [Table]

Original: [Equation]
Revised: [Equation]

Original: [Table]
Revised: [Table]

Original: [Equation]
Revised: [Equation]

Original: [Table]
Revised: [Table]

Original: [Equation][Table] [Equation] [Table]
Revised: [Equation]

Original: [Equation] [Table] [Equation]
Revised: [Table] [Equation]

Original: [Table]
Revised: [Table]

Original: [Equation]
Revised: [Equation]

Original: [Table] 
Revised: [Table] [Equation] [Table] [Equation]

Revised: [Table]

Original: [Equation]
Revised: [Equation]

Original: [Table]
Revised: [Table]

Original: [Equation]
Revised: [Equation]

Original: [Table]
Revised: [Table]

Original: [Equation]
Revised: [Equation]

Original: [Table]
Revised: [Table]

Original: Furthermore, recall that by Assumption 1, we have E [ H >(cid:96)d
Revised: Furthermore, recall that by Assumption 1, we have E [ H >(cid:96)d

Original: [Table] [Table]
Revised: [Table] [Table]

Original: s
Revised: s

Original: [Table]
Revised: [Table]

Original: Ω
Revised: Ω

Original: [Table] [Table]
Revised: [Table] [Table]

Original: We get
Revised: We get

Original: [Table]
Revised: [Table]

Original: [Equation]
Revised: [Equation]

Original: [Table]
Revised: [Table]

Original: [Equation]
Revised: [Equation]

Original: [Table]
Revised: [Table]

Original: [Equation]
Revised: [Equation]

Original: [Table]
Revised: [Table]

Original: [Equation]
Revised: [Equation]

Original: [Table]
Revised: [Table]

Original: [Equation]
Revised: [Equation]

Original: [Table]
Revised: [Table]

Original: [Equation]
Revised: [Equation]

Original: [Table]
Revised: [Table]

Original: [Equation]
Revised: [Equation]

Original: [Table]
Revised: [Table]

Original: [Equation]
Revised: [Equation]

Original: [Table]
Revised: [Table]

Original: [Equation]
Revised: [Equation]

Original: [Table]
Revised: [Table]

Original: [Equation]
Revised: [Equation]

Original: which concludes the proof of the ﬁrst bound.
Revised: which concludes the proof of the ﬁrst bound.

Original: Step 2.
Revised: Step 2.

Original: Bounding
Revised: Bounding

Original: [Equation]
Revised: [Equation]

Original: Notice that we can write,
Revised: Notice that we can write,

Original: [Equation]
Revised: [Equation]

Original: where we denoted R (cid:96) =
Revised: where we denoted R (cid:96) =

Original: (cid:80) S ∈C (cid:96) r ( S ) 2 and Ξ
Revised: (cid:80) S ∈C (cid:96) r ( S ) 2 and Ξ

Original: [Equation]
Revised: [Equation]

Original: R
Revised: R

Original: [Equation]
Revised: [Equation]

Original: S
Revised: S

Original: [Equation]
Revised: [Equation]

Original: x
Revised: x

Original: [Equation]
Revised: [Equation]

Original: Then, by Lemma 2, we get for any u ≥ s , sup i ∈
Revised: Then, by Lemma 2, we get for any u ≥ s , sup i ∈

Original: [Equation]
Revised: [Equation]

Original: n
Revised: n

Original: [Equation]
Revised: [Equation]

Original: We conclude following the same argument as in the proof of Proposition 9 in Mei et al.
Revised: We conclude following the same argument as in the proof of Proposition 9 in Mei et al.

Original: (2021b).
Revised: (2021b).

Original: Lemma 2.
Revised: Lemma 2.

Original: Let (cid:96) ≥ 2 be an integer.
Revised: Let (cid:96) ≥ 2 be an integer.

Original: Deﬁne Υ ( d ) (cid:96) : Q d → R and Ξ ( d ) (cid:96) : Q d → R to be
Revised: Deﬁne Υ ( d ) (cid:96) : Q d → R and Ξ ( d ) (cid:96) : Q d → R to be

Original: [Equation]
Revised: [Equation]

Original: (cid:88)
Revised: (cid:88)

Original: S ∈C (cid:96) r ( S
Revised: S ∈C (cid:96) r ( S

Original: [Equation]
Revised: [Equation]

Original: ( x
Revised: ( x

Original: [Equation] [Equation]
Revised: [Equation] [Equation]

Original: [Equation] [Equation]
Revised: [Equation] [Equation]

Original: (cid:96) (cid:88)
Revised: (cid:96) (cid:88)

Original: [Equation] [Equation]
Revised: [Equation] [Equation]

Original: x
Revised: x

Original: [Equation]
Revised: [Equation]

Original: where R (cid:96) =
Revised: where R (cid:96) =

Original: (cid:80) S ∈C (cid:96) r ( S ) 2 .
Revised: (cid:80) S ∈C (cid:96) r ( S ) 2 .

Original: Let n ≤ q p for some ﬁxed p .
Revised: Let n ≤ q p for some ﬁxed p .

Original: Then, for ( x i ) i ∈ [ n ] i.i.d.
Revised: Then, for ( x i ) i ∈ [ n ] i.i.d.

Original: ∼ Unif( Q d ) , we have
Revised: ∼ Unif( Q d ) , we have

Original: [Equation]
Revised: [Equation]

Original: where E θ [Υ ( d ) (cid:96) ( θ )]
Revised: where E θ [Υ ( d ) (cid:96) ( θ )]

Original: =
Revised: =

Original: E x [Ξ ( d ) (cid:96) ( x )] = 1 .
Revised: E x [Ξ ( d ) (cid:96) ( x )] = 1 .

Original: Proof
Revised: Proof

Original: [Equation]
Revised: [Equation]

Original: . Step 1. Bounding
Revised: . Step 1. Bounding

Original: [Equation]
Revised: [Equation]

Original: Deﬁne F (cid:96) : Q d → R to be
Revised: Deﬁne F (cid:96) : Q d → R to be

Original: [Equation]
Revised: [Equation]

Original: Notice that F (cid:96) ( x ) is a degree 2 (cid:96) polynomial and therefore satisﬁes the hypercontractivity property.
Revised: Notice that F (cid:96) ( x ) is a degree 2 (cid:96) polynomial and therefore satisﬁes the hypercontractivity property.

Original: For any m ≥ 1 , there exists C > 0 such that
Revised: For any m ≥ 1 , there exists C > 0 such that

Original: [Equation]
Revised: [Equation]

Original: Let us bound the right hand side.
Revised: Let us bound the right hand side.

Original: We have
Revised: We have

Original: [Equation]
Revised: [Equation]

Original: where B 1 = i + S , B 2 = j + S , B 3 = i (cid:48)
Revised: where B 1 = i + S , B 2 = j + S , B 3 = i (cid:48)

Original: + S (cid:48) and B 4 = j (cid:48)
Revised: + S (cid:48) and B 4 = j (cid:48)

Original: + S (cid:48) , and we denoted ω ( B 1 , B 2 , B 3 , B 4 ) =
Revised: + S (cid:48) , and we denoted ω ( B 1 , B 2 , B 3 , B 4 ) =

Original: E
Revised: E

Original: [Equation]
Revised: [Equation]

Original: Notice that ω ( B 1 , B 2 , B 3 , B 4 ) =
Revised: Notice that ω ( B 1 , B 2 , B 3 , B 4 ) =

Original: if B 1 ∆ B 2 = B 3 ∆ B 4 (the symmetric difference) and 0 otherwise.
Revised: if B 1 ∆ B 2 = B 3 ∆ B 4 (the symmetric difference) and 0 otherwise.

Original: In other words, every elements in B 1 ∪ B 2 ∪ B 3 ∪ B 4 appears exactly in 2 or 4 of these sets.
Revised: In other words, every elements in B 1 ∪ B 2 ∪ B 3 ∪ B 4 appears exactly in 2 or 4 of these sets.

Original: Let us ﬁx i ∈
Revised: Let us ﬁx i ∈

Original: [ d ] and S ∈ C (cid:96) , and bound (cid:88)
Revised: [ d ] and S ∈ C (cid:96) , and bound (cid:88)

Original: [Equation]
Revised: [Equation]

Original: Denote | B 1 ∆ B 2 | = 2 k with 1 ≤ k ≤ (cid:96) .
Revised: Denote | B 1 ∆ B 2 | = 2 k with 1 ≤ k ≤ (cid:96) .

Original: In order for ω ( B 1 , B 2 , B 3 , B 4 ) = 1 , B 3 must contain exactly k points in B 1 ∆ B 2 while B 4 must contain the remaining k points.
Revised: In order for ω ( B 1 , B 2 , B 3 , B 4 ) = 1 , B 3 must contain exactly k points in B 1 ∆ B 2 while B 4 must contain the remaining k points.

Original: • Case k < (cid:96) .
Revised: • Case k < (cid:96) .

Original: There are at most (cid:96) 2 ways of choosing j such that B 1 ∩ B 2 (cid:54) = ∅ .
Revised: There are at most (cid:96) 2 ways of choosing j such that B 1 ∩ B 2 (cid:54) = ∅ .

Original: Fixing j (i
Revised: Fixing j (i

Original: e., B 1 and B 2 ) and S (cid:48) , then there are 2 k(cid:96) ways of choosing i (cid:48) and 2 k(cid:96) ways of choosing j (cid:48)
Revised: e., B 1 and B 2 ) and S (cid:48) , then there are 2 k(cid:96) ways of choosing i (cid:48) and 2 k(cid:96) ways of choosing j (cid:48)

Original: such that B 3 ∩ ( B 1 ∆ B 2 ) (cid:54) = ∅ and B 4 ∩ ( B 1 ∆ B 2 ) (cid:54) = ∅ .
Revised: such that B 3 ∩ ( B 1 ∆ B 2 ) (cid:54) = ∅ and B 4 ∩ ( B 1 ∆ B 2 ) (cid:54) = ∅ .

Original: Hence the contribution of these terms in Eq.
Revised: Hence the contribution of these terms in Eq.

Original: (82) is upper bounded by
Revised: (82) is upper bounded by

Original: [Equation]
Revised: [Equation]

Original: • Case k = (cid:96) .
Revised: • Case k = (cid:96) .

Original: There are at most d ways of choosing j .
Revised: There are at most d ways of choosing j .

Original: Furthermore, for j ﬁxed, there are at most (cid:0) 2 (cid:96)(cid:96) (cid:1) ways of choosing B 3 and B 4 such that B 3 ∪ B 

Revised: Furthermore, for j ﬁxed, there are at most (cid:0) 2 (cid:96)(cid:96) (cid:1) ways of choosing B 3 and B 4 such that B 3 ∪ B

Original: =
Revised: =

Original: B 1 ∪ B 2 (note that B 1 ∩ B 2 = ∅ and therefore B 3 ∩ B 4 = ∅ ).
Revised: B 1 ∪ B 2 (note that B 1 ∩ B 2 = ∅ and therefore B 3 ∩ B 4 = ∅ ).

Original: Hence the contribution of these terms in Eq.
Revised: Hence the contribution of these terms in Eq.

Original: (82) is upper bounded by
Revised: (82) is upper bounded by

Original: [Equation]
Revised: [Equation]

Original: where we used that r ( S (cid:48) ) ≤ q .
Revised: where we used that r ( S (cid:48) ) ≤ q .

Original: Combining Eqs.
Revised: Combining Eqs.

Original: (83) and (84) and using there are dB ( Q q ; (cid:96) ) choices for i and S
Revised: (83) and (84) and using there are dB ( Q q ; (cid:96) ) choices for i and S

Original: [Equation]
Revised: [Equation]

Original: we get
Revised: we get

Original: [Equation] [Equation]
Revised: [Equation] [Equation]

Original: where we used that (cid:96) ≥ 2 and B ( Q q ; (cid:96) )
Revised: where we used that (cid:96) ≥ 2 and B ( Q q ; (cid:96) )

Original: =
Revised: =

Original: [Table]
Revised: [Table]

Original: Using Eq.
Revised: Using Eq.

Original: (81), we deduce
Revised: (81), we deduce

Original: [Equation]
Revised: [Equation]

Original: Using Markov’s inequality and taking m sufﬁciently small yield Eq.
Revised: Using Markov’s inequality and taking m sufﬁciently small yield Eq.

Original: [Equation]
Revised: [Equation]

Original: The second bound (79) is obtained very similarly.
Revised: The second bound (79) is obtained very similarly.

Original: Deﬁne G (cid:96) : Q d → R to be
Revised: Deﬁne G (cid:96) : Q d → R to be

Original: [Equation]
Revised: [Equation]

Original: Then, we have
Revised: Then, we have

Original: [Equation]
Revised: [Equation]

Original: Further notice that following the same computation as in Eq.
Revised: Further notice that following the same computation as in Eq.

Original: (70), we get
Revised: (70), we get

Original: [Equation]
Revised: [Equation]

Original: Hence, the same computation as for F (cid:96) in step 1 yields
Revised: Hence, the same computation as for F (cid:96) in step 1 yields

Original: [Equation]
Revised: [Equation]

Original: where we used that (cid:96) ≥ 2 .
Revised: where we used that (cid:96) ≥ 2 .

Original: We deduce Eq.
Revised: We deduce Eq.

Original: (79) similarly to step 1.
Revised: (79) similarly to step 1.

Original: Lemma 3 (Hanson-Wright inequality) .
Revised: Lemma 3 (Hanson-Wright inequality) .

Original: There exists a universal constant c > 0 , such that for any t > 0 and q 1 /δ ≥ d ≥ q ∈ N for some δ > 0 , when x ∈ Unif( Q d ) , we have
Revised: There exists a universal constant c > 0 , such that for any t > 0 and q 1 /δ ≥ d ≥ q ∈ N for some δ > 0 , when x ∈ Unif( Q d ) , we have

Original: [Equation]
Revised: [Equation]

Original: where we recall that x ( k )
Revised: where we recall that x ( k )

Original: =
Revised: =

Original: ( x k , . . .
Revised: ( x k , . . .

Original: , x k + q − 1 ) .
Revised: , x k + q − 1 ) .

Original: Proof of Lemma 3.
Revised: Proof of Lemma 3.

Original: For any k (cid:54) = l , denote A =
Revised: For any k (cid:54) = l , denote A =

Original: ( a ij )
Revised: ( a ij )

Original: i,j ∈ [ d ] the matrix with a ( k + i ) , ( l + i ) =
Revised: i,j ∈ [ d ] the matrix with a ( k + i ) , ( l + i ) =

Original: 1 for i = 0 , . . .
Revised: 1 for i = 0 , . . .

Original: , q − 1 and a ij = 0 otherwise, such that (cid:104) x , Ax (cid:105) =
Revised: , q − 1 and a ij = 0 otherwise, such that (cid:104) x , Ax (cid:105) =

Original: (cid:104) x ( k ) , x ( l ) (cid:105) .
Revised: (cid:104) x ( k ) , x ( l ) (cid:105) .

Original: Note that we have (cid:107) A (cid:107) F = √ q , (cid:107) A (cid:107) op ≤ 1 and E [ (cid:104) x , Ax (cid:105) ]
Revised: Note that we have (cid:107) A (cid:107) F = √ q , (cid:107) A (cid:107) op ≤ 1 and E [ (cid:104) x , Ax (cid:105) ]

Original: = 0 .
Revised: = 0 .

Original: By Hanson-Wright inequality of vectors with independent sub-Gaussian entries (for example, see Theorem 1.1 in Rudelson & Vershynin (2013)), we have
Revised: By Hanson-Wright inequality of vectors with independent sub-Gaussian entries (for example, see Theorem 1.1 in Rudelson & Vershynin (2013)), we have

Original: [Equation]
Revised: [Equation]

Original: Taking the union bound over k (cid:54)
Revised: Taking the union bound over k (cid:54)

Original: =
Revised: =

Original: l concludes the proof.
Revised: l concludes the proof.

Original: D T ECHNICAL BACKGROUND OF FUNCTION SPACES ON THE HYPERCUBE
Revised: D T ECHNICAL BACKGROUND OF FUNCTION SPACES ON THE HYPERCUBE

Original: Fourier analysis on the hypercube is a well studied subject O’Donnell (2014).
Revised: Fourier analysis on the hypercube is a well studied subject O’Donnell (2014).

Original: The purpose of this section is to introduce some notations and objects that are useful in the statement and proofs in the main text.
Revised: The purpose of this section is to introduce some notations and objects that are useful in the statement and proofs in the main text.

Original: D. F OURIER BASIS
Revised: D. F OURIER BASIS

Original: Denote Q d = {− 1 , +1 } d the hypercube in d dimension, and τ d to the uniform probability measure on Q d .
Revised: Denote Q d = {− 1 , +1 } d the hypercube in d dimension, and τ d to the uniform probability measure on Q d .

Original: All the functions will be assumed to be elements of L 2 ( Q d , τ d ) (which contains all the bounded functions f : Q d → R ), with scalar product and norm denoted as (cid:104)· , ·(cid:105) L 2 and (cid:107) · (cid:107) L 2 : (cid:104)
Revised: All the functions will be assumed to be elements of L 2 ( Q d , τ d ) (which contains all the bounded functions f : Q d → R ), with scalar product and norm denoted as (cid:104)· , ·(cid:105) L 2 and (cid:107) · (cid:107) L 2 : (cid:104)

Original: [Equation]
Revised: [Equation]

Original: ≡
Revised: ≡

Original: [Equation]
Revised: [Equation]

Original: Notice that L 2 ( Q d , τ d ) is a 2 n dimensional linear space.
Revised: Notice that L 2 ( Q d , τ d ) is a 2 n dimensional linear space.

Original: By analogy with the spherical case we decompose L 2 ( Q d , τ d ) as a direct sum of d + 1 linear spaces obtained from polynomials of degree (cid:96) = 0 , . . .
Revised: By analogy with the spherical case we decompose L 2 ( Q d , τ d ) as a direct sum of d + 1 linear spaces obtained from polynomials of degree (cid:96) = 0 , . . .

Original: , d
Revised: , d

Original: L 2 ( Q d , τ d ) =
Revised: L 2 ( Q d , τ d ) =

Original: d (cid:77) (cid:96) =
Revised: d (cid:77) (cid:96) =

Original: V d,(cid:96) .
Revised: V d,(cid:96) .

Original: For each (cid:96) ∈ { 0 , . . .
Revised: For each (cid:96) ∈ { 0 , . . .

Original: , d } , consider the Fourier basis { Y ( d ) (cid:96),S } S ⊆
Revised: , d } , consider the Fourier basis { Y ( d ) (cid:96),S } S ⊆

Original: [ d ] , | S | = (cid:96) of degree (cid:96) , where for a set
Revised: [ d ] , | S | = (cid:96) of degree (cid:96) , where for a set

Original: S ⊆
Revised: S ⊆

Original: [ d ] , the basis is given by
Revised: [ d ] , the basis is given by

Original: [Equation]
Revised: [Equation]

Original: i
Revised: i

Original: [Equation]
Revised: [Equation]

Original: x i .
Revised: x i .

Original: It is easy to verify that (notice that x ki = x i if k is odd and x ki
Revised: It is easy to verify that (notice that x ki = x i if k is odd and x ki

Original: = 1 if k is even)
Revised: = 1 if k is even)

Original: [Equation]
Revised: [Equation]

Original: Hence { Y ( d ) (cid:96),S } S ⊆
Revised: Hence { Y ( d ) (cid:96),S } S ⊆

Original: [ d ] , | S | = (cid:96) form an orthonormal basis of V d,(cid:96) and dim( V d,(cid:96) )
Revised: [ d ] , | S | = (cid:96) form an orthonormal basis of V d,(cid:96) and dim( V d,(cid:96) )

Original: =
Revised: =

Original: B ( Q d ; (cid:96) ) =
Revised: B ( Q d ; (cid:96) ) =

Original: (cid:18) d (cid:96) (cid:19) .
Revised: (cid:18) d (cid:96) (cid:19) .

Original: We will omit the superscript ( d ) in Y ( d ) (cid:96),S when clear from the context and write Y S :
Revised: We will omit the superscript ( d ) in Y ( d ) (cid:96),S when clear from the context and write Y S :

Original: = Y ( d ) (cid:96),S .
Revised: = Y ( d ) (cid:96),S .

Original: We denote by P (cid:96) the orthogonal projections to V d,(cid:96) in L 2 ( Q d ) .
Revised: We denote by P (cid:96) the orthogonal projections to V d,(cid:96) in L 2 ( Q d ) .

Original: This can be written in terms of the
Revised: This can be written in terms of the

Original: Fourier basis as
Revised: Fourier basis as

Original: [Equation]
Revised: [Equation]

Original: We also deﬁne P ≤ (cid:96) ≡ (cid:80) (cid:96)k =

Revised: We also deﬁne P ≤ (cid:96) ≡ (cid:80) (cid:96)k =

Original: P k , P >(cid:96) ≡ I − P ≤ (cid:96) =
Revised: P k , P >(cid:96) ≡ I − P ≤ (cid:96) =

Original: (cid:80) ∞
Revised: (cid:80) ∞

Original: k = (cid:96) +1 P k , and P <(cid:96) ≡ P ≤ (cid:96) − 1 , P ≥ (cid:96) ≡ P >(cid:96) − 1 .
Revised: k = (cid:96) +1 P k , and P <(cid:96) ≡ P ≤ (cid:96) − 1 , P ≥ (cid:96) ≡ P >(cid:96) − 1 .

Original: D. H YPERCUBIC G EGENBAUER
Revised: D. H YPERCUBIC G EGENBAUER

Original: We consider the following family of polynomials { Q ( d ) (cid:96) } (cid:96) =0 ,...,d that we will call hypercubic Gegenbauer , or G egenbauer on the d -dimensional hypercube , deﬁned as
Revised: We consider the following family of polynomials { Q ( d ) (cid:96) } (cid:96) =0 ,...,d that we will call hypercubic Gegenbauer , or G egenbauer on the d -dimensional hypercube , deﬁned as

Original: [Equation]
Revised: [Equation]

Original: [Equation]
Revised: [Equation]

Original: Notice that the right hand side only depends on (cid:104) x , y (cid:105) and therefore these polynomials are well deﬁned.
Revised: Notice that the right hand side only depends on (cid:104) x , y (cid:105) and therefore these polynomials are well deﬁned.

Original: In particular,
Revised: In particular,

Original: [Equation]
Revised: [Equation]

Original: Hence { Q ( d ) (cid:96) } (cid:96) =0 ,...,d form an orthogonal basis of L 2 ( {− d, − d + 2 , . . .
Revised: Hence { Q ( d ) (cid:96) } (cid:96) =0 ,...,d form an orthogonal basis of L 2 ( {− d, − d + 2 , . . .

Original: , d − 2 , d } , ˜ τ 1 d ) where ˜ τ 1 d is the distribution of (cid:104) 1 , x (cid:105) when x ∼ τ d , i.e., ˜ τ 1 d ∼ 2 Bin ( d, 1 / 2) − d/ 2 .
Revised: , d − 2 , d } , ˜ τ 1 d ) where ˜ τ 1 d is the distribution of (cid:104) 1 , x (cid:105) when x ∼ τ d , i.e., ˜ τ 1 d ∼ 2 Bin ( d, 1 / 2) − d/ 2 .

Original: It is easy to check more generally that
Revised: It is easy to check more generally that

Original: [Equation]
Revised: [Equation]

Original: Furthermore, Eq.
Revised: Furthermore, Eq.

Original: (87) imply that —up to
Revised: (87) imply that —up to

Original: a constant— Q ( d ) k
Revised: a constant— Q ( d ) k

Original: ( (cid:104) x , y (cid:105) ) is a representation of the projector onto the subspace of degree- k polynomials
Revised: ( (cid:104) x , y (cid:105) ) is a representation of the projector onto the subspace of degree- k polynomials

Original: [Equation]
Revised: [Equation]

Original: For a function σ ( · / √ d ) ∈
Revised: For a function σ ( · / √ d ) ∈

Original: L 2 ( {− d, − d + 2 , . . .
Revised: L 2 ( {− d, − d + 2 , . . .

Original: , d − 2 , d } , ˜ τ 1 d ) , denote its hypercubic Gegenbauer coefﬁcients ξ d,k ( σ ) to be
Revised: , d − 2 , d } , ˜ τ 1 d ) , denote its hypercubic Gegenbauer coefﬁcients ξ d,k ( σ ) to be

Original: [Equation]
Revised: [Equation]

Original: To any inner-product kernel H d ( x 1 , x 2 ) =
Revised: To any inner-product kernel H d ( x 1 , x 2 ) =

Original: h d ( (cid:104) x 1 , x 2 (cid:105) /d ) , with h d ( · / √ d ) ∈
Revised: h d ( (cid:104) x 1 , x 2 (cid:105) /d ) , with h d ( · / √ d ) ∈

Original: L 2 ( {− d, − d + 2 , . . .
Revised: L 2 ( {− d, − d + 2 , . . .

Original: , d − 2 , d } , ˜ τ 1 d ) , we can associate a self adjoint operator H d : L 2 ( Q d ) →
Revised: , d − 2 , d } , ˜ τ 1 d ) , we can associate a self adjoint operator H d : L 2 ( Q d ) →

Original: L 2 ( Q d ) via
Revised: L 2 ( Q d ) via

Original: [Equation]
Revised: [Equation]

Original: By permutation invariance, the space V k of homogeneous polynomials of degree k is an eigenspace of H d , and we will denote the corresponding eigenvalue by ξ d,k ( h d ) .
Revised: By permutation invariance, the space V k of homogeneous polynomials of degree k is an eigenspace of H d , and we will denote the corresponding eigenvalue by ξ d,k ( h d ) .

Original: In other words H d f ( x ) ≡ (cid:80) qk =
Revised: In other words H d f ( x ) ≡ (cid:80) qk =

Original: [Equation]
Revised: [Equation]

Original: ξ d,k ( h d ) P k f .
Revised: ξ d,k ( h d ) P k f .

Original: The eigenvalues can be computed via
Revised: The eigenvalues can be computed via

Original: [Equation]
Revised: [Equation]

Original: D. H ERMITE POLYNOMIALS
Revised: D. H ERMITE POLYNOMIALS

Original: The Hermite polynomials { He k } k ≥ 0 form an orthogonal basis of L 2 ( R , γ ) , where γ (d x ) =
Revised: The Hermite polynomials { He k } k ≥ 0 form an orthogonal basis of L 2 ( R , γ ) , where γ (d x ) =

Original: e − x 2 / 2 d x/ √ 2 π is the standard Gaussian measure, and He k has degree k .
Revised: e − x 2 / 2 d x/ √ 2 π is the standard Gaussian measure, and He k has degree k .

Original: We will follow the classical normalization (here and below, expectation is with respect to G ∼ N (0 , 1) ):
Revised: We will follow the classical normalization (here and below, expectation is with respect to G ∼ N (0 , 1) ):

Original: [Equation]
Revised: [Equation]

Original: As a consequence, for any function g ∈ L 2 ( R , γ ) , we have the decomposition
Revised: As a consequence, for any function g ∈ L 2 ( R , γ ) , we have the decomposition

Original: [Equation]
Revised: [Equation]

Original: The Hermite polynomials can be obtained as high-dimensional limits of the Gegenbauer polynomials introduced in the previous section.
Revised: The Hermite polynomials can be obtained as high-dimensional limits of the Gegenbauer polynomials introduced in the previous section.

Original: Indeed, the Gegenbauer polynomials (up to a √ d scaling in domain) are constructed by Gram-Schmidt orthogonalization of the monomials { x k } k ≥ 0 with respect to the measure ˜ τ
Revised: Indeed, the Gegenbauer polynomials (up to a √ d scaling in domain) are constructed by Gram-Schmidt orthogonalization of the monomials { x k } k ≥ 0 with respect to the measure ˜ τ

Original: 1 d , while Hermite polynomial are obtained by Gram-Schmidt orthogonalization with respect to γ .
Revised: 1 d , while Hermite polynomial are obtained by Gram-Schmidt orthogonalization with respect to γ .

Original: Since ˜ τ 1 d ⇒ γ (here ⇒ denotes weak convergence), it is immediate to show that, for any ﬁxed integer k ,
Revised: Since ˜ τ 1 d ⇒ γ (here ⇒ denotes weak convergence), it is immediate to show that, for any ﬁxed integer k ,

Original: [Equation]
Revised: [Equation]

Original: Here and below, for P a polynomial, Coeﬀ { P ( x ) } is the vector of the coefﬁcients of P .
Revised: Here and below, for P a polynomial, Coeﬀ { P ( x ) } is the vector of the coefﬁcients of P .

Original: As a consequence, for any ﬁxed integer k , we have
Revised: As a consequence, for any ﬁxed integer k , we have

Original: [Equation]
Revised: [Equation]

Original: [Equation]
Revised: [Equation]

Original: where µ k ( σ ) and ξ d,k ( σ ) are given in Eq.
Revised: where µ k ( σ ) and ξ d,k ( σ ) are given in Eq.

Original: (93) and (89).
Revised: (93) and (89).

Original: D. H YPERCONTRACTIVITY OF UNIFORM DISTRIBUTIONS ON THE HYPERCUBE
Revised: D. H YPERCONTRACTIVITY OF UNIFORM DISTRIBUTIONS ON THE HYPERCUBE

Original: By Holder’s inequality, we have (cid:107) f (cid:107) L p ≤ (cid:107) f (cid:107)
Revised: By Holder’s inequality, we have (cid:107) f (cid:107) L p ≤ (cid:107) f (cid:107)

Original: L q for any f and any p ≤ q .
Revised: L q for any f and any p ≤ q .

Original: The reverse inequality does not hold in general, even up to a constant.
Revised: The reverse inequality does not hold in general, even up to a constant.

Original: However, for some measures, the reverse inequality will hold for some sufﬁciently nice functions.
Revised: However, for some measures, the reverse inequality will hold for some sufﬁciently nice functions.

Original: These measures satisfy the celebrated hypercontractivity properties Gross (1975); Bonami (1970); Beckner (1975; 1992).
Revised: These measures satisfy the celebrated hypercontractivity properties Gross (1975); Bonami (1970); Beckner (1975; 1992).

Original: Lemma 4 (Hypercube hypercontractivity Beckner (1975)) .
Revised: Lemma 4 (Hypercube hypercontractivity Beckner (1975)) .

Original: For any (cid:96) = { 0 , . . .
Revised: For any (cid:96) = { 0 , . . .

Original: , d } and f d ∈
Revised: , d } and f d ∈

Original: L 2 ( Q d ) to be a degree (cid:96) polynomial, then for any integer q ≥ 2 , we have
Revised: L 2 ( Q d ) to be a degree (cid:96) polynomial, then for any integer q ≥ 2 , we have

Original: [Equation]
Revised: [Equation]