2.1 The 2-ary case: sketch of mossel’s proof

Let $\mu$ be a distribution on $\Sigma \times \Gamma$ such that $\mathsf{supp}(\mu )$ is not linearly embeddable. It is easily seen that the non-linearity condition, in this special $2$ -ary case, is same as saying that $\mathsf{supp}(\mu )$ , viewed as a bipartite graph $G_\mu$ on the vertex set $\Sigma \cup \Gamma$ , is connected. Indeed, if this graph were disconnected, with components $C_0 \cup D_0, \ldots , C_{r-1} \cup D_{r-1}$ , then an embedding $\sigma :C_j \to j$ , $\gamma : D_j \to -j$ is an embedding of $\Sigma$ and $\Gamma$ , respectively, into $\mathbb{Z}_r$ and for all $(x,y) \in \mathsf{supp}(\mu )$ (i.e. the edges of the graph $G_\mu$ ), we have $\sigma (x)+\gamma (y) = 0$ in $\mathbb{Z}_r$ .

We intend to show that if $f: \Sigma ^n \to \mathbb{C}, g: \Gamma ^n \to \mathbb{C}$ are $n$ -dimensional $\ell _\infty$ -bounded functions where $g$ has high degree, then $\left | {\mathop {\mathbb{E}}_{({\textbf {x}}, \textbf {y}) \sim \mu ^{\otimes n}}{\left [ {f({\textbf {x}}) g(\textbf {y})} \right ]}} \right |$ is small. For simplicity of exposition, we assume that $g$ in fact has full degree $n$ .Footnote ⁶ In this case, we are able to show that $\left | {\mathop {\mathbb{E}}_{({\textbf {x}},\textbf {y}) \sim \mu ^{\otimes n}}{\left [ {f({\textbf {x}}) g(\textbf {y})} \right ]}} \right | \leqslant (1-\tau )^n \|f\|_2 \|g \|_2$ for some constant $\tau = \tau (\mu ) \gt 0$ . We emphasise here that one gets an upper bound in terms of the $\ell _2$ -norm of the functions. This of course implies an upper bound in terms of the $\ell _\infty$ -norms. Thus we really do not need the $n$ -dimensional functions to be $\ell _\infty$ -bounded in the $2$ -ary case. This is one aspect (among many) in which the $3$ -ary case is fundamentally different, where one does need the $n$ -dimensional functions to be $\ell _\infty$ -bounded (as we will soon demonstrate via an example).

Continuing the consideration of the $2$ -ary case, the proof proceeds in two steps: first establishing a base case inequality (for $n=1$ ) and then observing that the inequality tensorizes, leading to an inductive proof and the desired bound for the general case of $n$ -dimensional functions. The base case inequality is necessarily an $\ell _2$ -inequality and this fact is essential for the inductive proof (and the same holds in the $3$ -ary case).

Towards stating the base case inequality, let $f:\Sigma \to \mathbb{C}, g: \Gamma \to \mathbb{C}$ be functions. By Cauchy-Schwarz,

\begin{equation*}\left | {\mathop {\mathbb{E}}_{(x,y) \sim \mu }{\left [ {f(x) g(y)} \right ]}} \right | \leqslant \|f\|_2 \|g\|_2.\end{equation*}

We refer to this essentially trivial inequality as the (base case) sanity check inequality. The inequality that is actually needed is that when ${\mathop {\mathbb{E}}_{}{\left [ {f} \right ]}} = {\mathop {\mathbb{E}}_{}{\left [ {g} \right ]}}=0$ , we in fact have the improvement

(4) \begin{equation} \left | {\mathop {\mathbb{E}}_{(x,y) \sim \mu }{\left [ {f(x) g(y)} \right ]}} \right | \leqslant (1-\tau ) \|f\|_2 \|g\|_2, \quad \quad {\mathop {\mathbb{E}}_{}{\left [ {f} \right ]}} = {\mathop {\mathbb{E}}_{}{\left [ {g} \right ]}}=0, \end{equation}

for some constant $\tau = \tau (\mu ) \gt 0$ . It is not difficult to see that this follows from the connectedness of the distribution $\mu$ (or equivalently the graph $G_\mu$ ), but we skip the proof. An equivalent way to express the inequality is that the operator $T: \tilde {L}_2(\Gamma ; \mu _y) \to \tilde {L}_2(\Sigma ; \mu _x)$ defined as $Tg (x) = {\mathop {\mathbb{E}}_{(x', y) \sim \mu }{\left [ {g(y)|x'=x} \right ]}}$ has operator norm at most $1-\tau$ . Here $\tilde {L}_2(\Gamma ; \mu _y)$ denotes the subspace of $L_2(\Gamma ;\,\mu _y)$ consisting of those functions $g$ for which ${\mathop {\mathbb{E}}_{}{\left [ {g} \right ]}}=0$ (and similarly for $\tilde {L}_2(\Sigma ;\, \mu _x)$ ). The operator norm of $T$ , denoted $\|T\| = \max _{g: {\mathop {\mathbb{E}}_{}{\left [ {g} \right ]}}=0} \|Tg\|_2/\|g\|_2$ , is at most $1-\tau$ according to the equivalent interpretation of the inequality (4), which can then be derived as:

\begin{equation*}\left | {\mathop {\mathbb{E}}_{(x,y) \sim \mu }{\left [ {f(x)g(y)} \right ]}} \right | = \left | \left \langle f, Tg \right \rangle \right | \leqslant \|f\|_2 \| Tg\|_2 \leqslant \|f \|_2 \|T\| \|g\|_2 \leqslant (1-\tau ) \|f\|_2 \|g\|_2. \end{equation*}

Now we consider the $n$ -dimensional case. Let $f: \Sigma ^n \to \mathbb{R}, g: \Gamma ^n \to \mathbb{R}$ be $n$ -dimensional functions. As mentioned before, we assume that $g$ has full degree, which amounts to saying that $g \in \tilde {L}_2(\Gamma ;\,\mu _y)^{\otimes n}$ . In this case, it follows directly that

\begin{equation*} \left | {\mathop {\mathbb{E}}_{({\textbf {x}}, \textbf {y}) \sim \mu ^{\otimes n}}{\left [ {f({\textbf {x}}) g(\textbf {y})} \right ]}} \right | \leqslant (1-\tau )^n \|f\|_2 \|g \|_2, \end{equation*}

using the well-known fact that the operator norm is multiplicative (i.e. it tensorizes), namely that $\|T^{\otimes n}\| = \|T\|^n \leqslant (1-\tau )^n$ . Using this fact, one immediately concludes that

\begin{equation*} \left | {\mathop {\mathbb{E}}_{({\textbf {x}}, \textbf {y}) \sim \mu ^{\otimes n}}{\left [ {f({\textbf {x}}) g(\textbf {y})} \right ]}} \right | = \left | \left \langle f, T^{\otimes n}g\right \rangle \right | \leqslant \|f\|_2 \| T^{\otimes n} g \|_2 \leqslant \|f\|_2 \| T^{\otimes n}\| \|g\|_2 \leqslant (1-\tau )^n \|f \|_2 \|g \|_2, \end{equation*}

as desired. If one wishes, one can prove the multiplicativity of operator norm by induction and view the overall proof as an inductive proof, using the base case inequality (4) and ’gaining’ a factor $1-\tau$ in each step of the induction. While we don’t demonstrate it here, we mention it because the proof for the $3$ -ary case proceeds along similar lines, albeit with many conceptual and technical hurdles. Therein, it is rather challenging even to formulate the ’correct’ base case inequality.

2.2 Towards 3-ary base case: restoring sanity first

Moving onto the $3$ -ary case, let $\mu$ be a distribution on $\Sigma \times \Gamma \times \Phi$ such that $\mathsf{supp}(\mu )$ is not linearly embeddable. One hopes to write down a suitable base case inequality and use it towards an inductive proof. However, it turns out that even the sanity check inequality fails in general! That is, for $f:\Sigma \to \mathbb{C}, g: \Gamma \to \mathbb{C}, h:\Phi \to \mathbb{C}$ , while we desire a base case inequality (say when ${\mathop {\mathbb{E}}_{}{\left [ {f} \right ]}}=0$ ) of the form

(5) \begin{equation} \left | {\mathop {\mathbb{E}}_{(x,y,z) \sim \mu }{\left [ {f(x) g(y) h(z)} \right ]}} \right | \leqslant (1-\tau ) \|f\|_2 \|g\|_2 \|h\|_2, \end{equation}

it may actually happen that

\begin{equation*} \left | {\mathop {\mathbb{E}}_{(x,y,z) \sim \mu }{\left [ {f(x) g(y) h(z)} \right ]}} \right | \gt \|f\|_2 \|g\|_2 \|h\|_2. \end{equation*}

In other words, we may not even have the upper bound of $\|f\|_2 \|g\|_2 \|h\|_2$ in the $3$ -ary case whereas the corresponding upper bound in the $2$ -ary case is the essentially trivial application of Cauchy-Schwarz! Here is an example.

Suppose that $\Sigma = \Gamma = \Phi$ , $|\Sigma | = m \geqslant 54$ , and $\mu$ has a probability mass of $1-\varepsilon$ uniformly spread on the triples $\{(x,x,x)|x \in \Sigma \}$ and the remaining probability mass of $\varepsilon$ uniformly spread on all the remaining triples in $\Sigma ^3$ . Clearly, $\mathsf{supp}(\mu ) = \Sigma ^3$ and hence $\mu$ is not linearly embeddable. The marginals of $\mu$ are uniform on $\Sigma$ . We can certainly construct a function $f: \Sigma \to \mathbb{R}$ such that ${\mathop {\mathbb{E}}_{}{\left [ {f(x)} \right ]}}=0$ and ${\mathop {\mathbb{E}}_{}{\left [ {f(x)^3} \right ]}} \gt \|f\|_2^3$ . For instance, $f$ could take the values $2m, -m, -m$ at three distinct points in $\Sigma$ and zero at the remaining points in $\Sigma$ . In this case, ${\mathop {\mathbb{E}}_{}{\left [ {f(x)} \right ]}}=0, {\mathop {\mathbb{E}}_{}{\left [ {f(x)^2} \right ]}} = 6m$ , and ${\mathop {\mathbb{E}}_{}{\left [ {f(x)^3} \right ]}} = 6m^2$ , and thus ${\mathop {\mathbb{E}}_{}{\left [ {f(x)^3} \right ]}} \geqslant \sqrt {m/6} \cdot \| f \|_2^3 \geqslant 3 \,\|f\|_2^3$ . Letting $f=g=h$ and recalling that the triples $(x,x,x)$ receive $1-\varepsilon$ of the probability mass, it follows that

\begin{equation*} {\mathop {\mathbb{E}}_{(x,y,z) \sim \mu }{\left [ {f(x) g(y) h(z)} \right ]}} \geqslant (1-\varepsilon ) {\mathop {\mathbb{E}}_{}{\left [ {f(x)^3} \right ]}} - \varepsilon \cdot O_m(1) \geqslant 2 \cdot \|f\|_2^3 = 2 \cdot \|f\|_2 \|g\|_2 \|h\|_2, \end{equation*}

by making $\varepsilon$ sufficiently small. This example also shows that in order to claim the desired bound for $n$ -dimensional functions as in Equation (3), we must use the fact that the functions are $\ell _\infty$ -bounded! Indeed, consider the same example here and let $n$ -dimensional functions $\tilde {f} = \tilde {g} = \tilde {h}: \Sigma ^n \to \mathbb{R}$ be all equal to $f^{\otimes n}/\| f^{\otimes n} \|_2$ . Then these have all $\ell _2$ -norm $1$ , whereas

\begin{equation*} {\mathop {\mathbb{E}}_{({\textbf {x}}, \textbf {y}, \textbf {z})\sim \mu ^{\otimes n}}{\left [ {\tilde {f}({\textbf {x}})\tilde {g}(\textbf {y})\tilde {h}(\textbf {z})} \right ]}} = {\mathop {\mathbb{E}}_{(x,y,z) \sim \mu }{\left [ {f(x) g(y) h(z)} \right ]}}^n \cdot \frac {1}{\|f \|_2^{3n} } \geqslant 2^n. \end{equation*}

We thus face a seemingly intractable hurdle and a contradictory set of constraints: (i) we do need the $\ell _\infty$ -boundedness of the $n$ -dimensional functions, (ii) an inductive proof is some form of tensorization argument and hence inherently an $\ell _2$ -proof; consequently, the intermediate functions arising during the induction can only be assumed to have $\ell _2$ norm at most $1$ , (iii) the inductive argument requires a base case $\ell _2$ -inequality such as (5) which actually happens to fail miserably!

We now show how to overcome this hurdle step-by-step. This is achieved in a round-about manner, by carefully transforming the distribution and the alphabet $(\Sigma \times \Gamma \times \Phi ,\mu )$ to another distribution and alphabet $(\tilde {\Sigma } \times \tilde {\Gamma } \times \tilde {\Phi } ,\tilde {\mu })$ . Formally, we show that

• • If $\mu$ was not linearly embeddable to begin with, then $\tilde {\mu }$ isn’t either.

• • If Lemma 2.1 (i.e. our Main Lemma/Result) holds for $\tilde {\mu }$ , then it also holds for $\mu$ .

In this sense, we are able to reduce our task of proving the lemma for the original distribution $\mu$ to proving the same lemma for the new distribution $\tilde {\mu }$ . In fact, there will be a series of such transformations. The (first) transformation will ensure that the marginal of $\tilde {\mu }$ on $\tilde {\Gamma } \times \tilde {\Phi }$ is a uniform, product distribution. Once we have this additional property, we at least have the (base case) sanity check inequality as demonstrated next. For the sake of notational convenience, we rename the new distribution and the alphabet again as $(\Sigma \times \Gamma \times \Phi ,\mu )$ and assume that the marginal of $\mu$ on $\Gamma \times \Phi$ is a uniform, product distribution. If so, it is easily seen that we get the (base case) sanity check inequality, namely that for $f:\Sigma \to \mathbb{C}, g: \Gamma \to \mathbb{C}, h:\Phi \to \mathbb{C}$ , we have

\begin{equation*}\left | {\mathop {\mathbb{E}}_{(x,y,z) \sim \mu }{\left [ {f(x) g(y) h(z)} \right ]}} \right | \leqslant \|f\|_2 \|g\|_2 \|h\|_2. \end{equation*}

Indeed, by Cauchy-Schwarz,

(6) \begin{eqnarray} \left | {{\mathop {\mathbb{E}}_{(x,y,z) \sim \mu }{\left [ {f(x) g(y) h(z)} \right ]}}} \right |^2 & \leqslant & {\mathop {\mathbb{E}}_{x\sim \mu _x}{\left [ {\left | {f(x)} \right |^2} \right ]}}{\mathop {\mathbb{E}}_{(y,z)\sim \mu _{y,z}}{\left [ {\left | {g(y)} \right |^2 \left | {h(z)} \right |^2} \right ]}} \nonumber \\ & = & {\mathop {\mathbb{E}}_{x\sim \mu _x}{\left [ {\left | {f(x)} \right |^2} \right ]}} {\mathop {\mathbb{E}}_{y\sim \mu _y}{\left [ {\left | {g(y)} \right |^2} \right ]}} {\mathop {\mathbb{E}}_{z\sim \mu _z}{\left [ {\left | {h(z)} \right |^2} \right ]}} \nonumber \\ & = & \|f\|_2^2 \|g\|_2^2 \|h\|_2^2, \end{eqnarray}

where in the second step, we used the property that $(y,z)$ are uniform and independent! It is also possible to ensure (after the transformation) another property of $\mu$ that is quite convenient: for all pairs $(y,z) \in \Gamma \times \Phi$ , there is a unique $x \in \Sigma$ such that $(x,y,z) \in \mathsf{supp}(\mu )$ (we then say that $(y,z)$ determine $x$ ). The details of this transformation and related proofs appear in Section 8; some of its ingredients are borrowed from authors’ earlier work [Reference Bhangale, Khot and Minzer5].

2.3 The $3$ -ary relaxed base case: overcoming the horn-SAT obstruction

We will henceforth assume that the distribution $\mu$ on $\Sigma \times \Gamma \times \Phi$ has no linear embedding and has uniform marginal on $\Gamma \times \Phi$ . Now that we at least have the sanity check inequality, we ask ourselves whether we can claim the desired base case inequality as below:

Question 2.2. (Desired, Hypothetical Base Case Inequality:) If $\mu$ has no linear embedding and has uniform marginal on $\Gamma \times \Phi$ , is it necessarily the case that for $f:\Sigma \to \mathbb{C}, g:\Gamma \to \mathbb{C}, h:\Phi \to \mathbb{C}$ ,

(7) \begin{equation} \left | {\mathop {\mathbb{E}}_{(x,y,z) \sim \mu }{\left [ {f(x) g(y) h(z)} \right ]}} \right | \leqslant (1-\tau (\theta )) \|f\|_2 \|g\|_2 \|h\|_2, \quad \quad |{\mathop {\mathbb{E}}_{}{\left [ {f} \right ]}}| \leqslant (1-\theta )\|f\|_2. \end{equation}

To avoid the trivial case when $f, g, h$ are all constant functions, we added here the condition that $f$ is non-constant and has some variance, the condition captured by the requirement $|{\mathop {\mathbb{E}}_{}{\left [ {f} \right ]}}| \leqslant (1-\theta )\|f\|_2$ .

We note that such a base case inequality seems necessary towards an inductive proof since one hopes to ’gain’ a factor of $1-\tau$ in each step of the induction. However it turns out that such an inequality need not necessarily hold and there could be an obstruction that we refer to as the Horn-SAT obstruction (and this is the only possible obstruction).

The condition $f(x)=g(y)h(z)$ for Boolean functions is equivalent to the conjunction of clauses $\overline {f(x)} \vee g(y)$ , $\overline {f(x)} \vee h(z)$ , $f(x) \vee \overline {g(y)} \vee \overline {h(z)}$ . These are all Horn-SAT clauses (i.e. having at most one positive literal), explaining the term Horn-SAT embedding. We now make several remarks towards understanding how a Horn-SAT embedding is an obstruction towards the desired inequality (7) and how it is the only possible obstruction.

• • First, we note that having a Horn-SAT embedding violates inequality (7). Indeed, since $f(x)=g(y)h(z)$ in $\mathsf{supp}(\mu )$ and $(y,z)$ are uniform and independent, we have $\|f\|_2 = \|g\|_2 \| h\|_2$ and then

  \begin{equation*}{\mathop {\mathbb{E}}_{(x,y,z) \sim \mu }{\left [ {f(x) g(y) h(z)} \right ]}} = {\mathop {\mathbb{E}}_{(y,z) \sim \mu _{y,z}}{\left [ {g(y)^2 h(z)^2} \right ]}} = \|g\|_2^2 \|h\|_2^2 = \|f\|_2 \|g\|_2 \|h\|_2.\end{equation*}
  One also notes that since $f$ is Boolean and non-constant, it does have constant variance.

• • Secondly, we note that if the inequality (7) is not possible, then there is necessarily a Horn-SAT embedding. A sketch of the proof is as follows. For a fixed $\theta$ , suppose that there are functions that violate the inequality for all $\tau \to 0$ . Then by standard compactness argument, there are functions $f: \Sigma \to \mathbb{C}$ , $g: \Gamma \to \mathbb{C}$ , $h: \Phi \to \mathbb{C}$ , such that

  \begin{equation*}{\mathop {\mathbb{E}}_{(x,y,z) \sim \mu }{\left [ {f(x) g(y) h(z)} \right ]}} = \|f\|_2 \|g\|_2 \|h\|_2,\end{equation*}
  that is, achieving an exact equality. This means that the application of Cauchy-Schwarz in Equation (6) must be tight and therefore $f(x)= s \overline {g(y)h(z)}$ in $\mathsf{supp}(\mu )$ (as equality of complex numbers), where $s\in \mathbb{C}$ is a complex number of absolute value $1$ . If $f(x)$ is always non-zero, then so are $g(y)$ and $h(z)$ . In this case, one can choose a branch of the logarithm function and get an embedding into addition modulo $2\pi i$ , which is an Abelian group.Footnote ⁷ One concludes therefore that $f(x)$ takes the zero value for some $x \in \Sigma$ and of course also takes a non-zero value for some $x' \in \Sigma$ . We can now define the Horn-SAT embedding by turning $f(x), g(y), h(z)$ into Boolean $1$ if the value is non-zero and Boolean $0$ if the value is zero!

• • In the definition, if $f$ is non-constant, then so must be $g$ and $h$ . Let’s suppose on the contrary that $g$ is constant (the same proof applies for $h$ ). If $g \equiv 0$ , then the condition $f(x)=g(y)h(z)$ implies that $f \equiv 0$ , reaching a contradiction. If $g \equiv 1$ , then one concludes that $f(x)=h(z)$ for all $(x,z) \in \mathsf{supp}(\mu _{x,z})$ . Since $\mu$ is not linearly embeddable, its marginals are not linearly embeddable either.Footnote ⁸ In particular, $\mu _{x,z}$ has no linear embedding and hence is connected, implying that both $f$ and $h$ are constant, again a contradiction.

Considering these remarks, if $\mu$ does not have a Horn-SAT embedding, then we do have the base case inequality (7) and we can hope to carry out the induction. However, if $\mu$ does have a Horn-SAT embedding as in Definition 2.3, then the embedding serves as a violation of the inequality and we are stuck with a similar hurdle as before. The Horn-SAT embedding leads to $n$ -dimensional functions $\tilde {f} = f^{\otimes n}/\| f^{\otimes n}\|$ , $\tilde {g} = g^{\otimes n}/\| g^{\otimes n}\|$ , $\tilde {h} = h^{\otimes n}/\| h^{\otimes n}\|$ , with $\ell _2$ -norm $1$ , and

\begin{equation*} {\mathop {\mathbb{E}}_{({\textbf {x}}, \textbf {y}, \textbf {z})\sim \mu ^{\otimes n}}{\left [ {\tilde {f}({\textbf {x}})\tilde {g}(\textbf {y})\tilde {h}(\textbf {z})} \right ]}} = 1.\end{equation*}

As before, this hinders the possibility of proving the $n$ -dimensional inequality (3) by induction: there is no base case inequality and there is a counter-example if one allows functions to have $\ell _2$ norm $1$ instead of $\ell _\infty$ norm $1$ .

We overcome this hurdle in a similar manner as before, albeit with even more subtleness. We carefully transform the distribution and the alphabet $(\Sigma \times \Gamma \times \Phi ,\mu )$ to another distribution and alphabet $(\tilde {\Sigma } \times \tilde {\Gamma } \times \tilde {\Phi } ,\tilde {\mu })$ . Formally, we show that

• • If Lemma 2.1 (i.e. our Main Lemma/Result) holds for $\tilde {\mu }$ , then it also holds for $\mu$ .

• • All the key properties of $\mu$ are retained by $\tilde {\mu }$ which has further additional properties.

In this sense, we are able to reduce our task of proving the lemma for the original distribution $\mu$ to proving the same lemma for the new distribution $\tilde {\mu }$ . Now we state what additional properties $\tilde {\mu }$ has. For the sake of notational convenience, we rename the new distribution and the alphabet as $(\Sigma \times \Gamma \times \Phi , \mu )$ again. The key additional property is stated below, referred to as the relaxed base case inequality.

We remark that if $\mu$ did not have a Horn-SAT embedding, no transformation is needed, and one can simply take $\Sigma ' = \Sigma$ in the above definition. However in general there might be a Horn-SAT embedding and the transformation would be needed. The transformation is rather subtle and while we do consider it to be one of the key ideas, we skip the discussion here and refer to Section 8.3 for details. To summarise, we reduce the task of proving our Main Lemma 2.1 to the same task with the additional property that $\mu$ satisfies the relaxed base case inequality, that is, to the task of proving the lemma stated below. In the following lemma, properties numbered 1 and 2 are as before, 3 and 4 can be assumed from the authors’ earlier work as discussed in Section 2.2, and that numbered 5 is the key relaxed base case inequality.

Lemma 2.5. (Main Analytical Lemma under Relaxed Base Case Inequality) Suppose $\left | {\Sigma } \right |,\left | {\Gamma } \right |,\left | {\Phi } \right |\leqslant m$ and $\mu$ is a distribution over $\Sigma \times \Gamma \times \Phi$ such that:

1. 1. $\mu (x,y,z)\geqslant \alpha$ for some $\alpha \gt 0$ and all $(x,y,z)\in \textsf { supp}(\mu )$ .

2. 2. $\mathsf{supp}(\mu )$ cannot be linearly embedded.

3. 3. The marginal $\mu _{y,z}$ is uniform and independent over $\Gamma \times \Phi$ .

4. 4. For all $(y,z)\in \Gamma \times \Phi$ , there is a unique $x\in \Sigma$ such that $(x,y,z)\in \textsf { supp}(\mu )$ (i.e. $y,z$ determine $x$ ).

5. 5. $\mu$ satisfies the relaxed base case inequality as in Definition 2.4.

Then for all $\varepsilon \gt 0$ , there are $\xi , \delta \gt 0$ such that the following holds. If $f\colon \Sigma ^n\to \mathbb{C}$ , $g\colon \Gamma ^n \to \mathbb{C}$ and $h\colon \Phi ^{n}\to \mathbb{C}$ are $1$ -bounded functions satisfying that either $\textsf { Stab}_{1-\xi }(g)\leqslant \delta$ or $\textsf { Stab}_{1-\xi }(h)\leqslant \delta$ , then we have that

\begin{equation*} \left | {{\mathop {\mathbb{E}}_{({\textbf {x}},\textbf {y},\textbf {z})\sim \mu ^{\otimes n}}{\left [ {f({\textbf {x}})g(\textbf {y})h(\textbf {z})} \right ]}}} \right |\leqslant \varepsilon . \end{equation*}

2.4 The inductive argument (without the horn-SAT obstruction)

Armed with the “correct” relaxed base case inequality, we now give an overview of the inductive proof (of Lemma 2.5). It is instructive and less cumbersome to first consider the special case when there is no Horn-SAT embedding and we already have the base case inequality as in (7). We will indicate how to incorporate the relaxed base case inequality later. Formal proofs appear in Sections 4, 5, 6, and 7.

So let us focus on this special case and assume the base case inequality (7) holds. The inductive proof proceeds in several steps. We emphasise again that an inductive proof must necessarily work with $\ell _2$ norms of functions that arise as intermediate functions during the induction and we have no control over their $\ell _\infty$ norms.Footnote ⁹ We are given that either $g$ or $h$ has essentially high degree, so let’s say this holds for $g$ , formalised in terms of its low-stability. The first step towards the inductive proof is to note that it is sufficient (and necessary as far as our proof goes) to focus on the case when $f,g, h$ are homogeneous functions. We will skip details regarding how this is sufficient towards the general case. Therefore let’s assume that $f,g,h$ are homogeneous and define the parameter

\begin{equation*} \beta _{n,d_1,d_2,d_3} = \sup _{f,g,h} \frac {\left | {{\mathop {\mathbb{E}}_{({\textbf {x}},\textbf {y},\textbf {z})\sim \mu ^{\otimes n}}{\left [ {f({\textbf {x}})g(\textbf {y})h(\textbf {z})} \right ]}}} \right |}{\| f \|_2\| g \|_2\| h \|_2}, \end{equation*}

where the maximum is taken over all $f\colon \Sigma ^n\to \mathbb{C}$ , $g\colon \Gamma ^n\to \mathbb{C}$ , $h\colon \Phi ^n\to \mathbb{C}$ homogeneous of degrees $d_1,d_2,d_3$ respectively. Since we assumed that $g$ had high degree, we think of $d_2$ as (roughly) the largest among the degrees. Indeed, it is sufficient to consider the case when $d_1,d_3\leqslant 10d_2$ , and we make this assumption skipping the details. We will be able to show an exponential decay, namely

\begin{equation*} \beta _{n,d_1,d_2,d_3}\leqslant (1-\Omega _{\alpha ,m}(1))^{d_2},\end{equation*}

completing the proof. We now describe how this exponential decay is proved. First, we reduce the dimension $n$ so that $n \leqslant O(d_2)$ . Then comes the core inductive argument, where we ’gain’ a factor $1-\Omega _{\alpha ,m}(1)$ in each step of the induction, reducing the degree $d_2$ by one, until we have reduced it to say $\frac {d_2}{2}$ .

Reducing dimension: We show here that it is sufficient to consider the case when $n\leqslant O(d_2)$ (and we already assume that $d_1, d_3 \leqslant 10d_2$ ). The idea is as follows. As long as $n \gg d_2$ , we can find a coordinate $i\in [n]$ which has very small ’influence’ on $f,g$ and $h$ ; assume without loss of generality that this co-ordinate is $i=n$ . If the influence was zero, then $f$ , $g$ , and $h$ would only be functions of the first $n-1$ co-ordinates, and hence we would conclude that $\beta _{n,d_1,d_2,d_3}\leqslant \beta _{n-1,d_1,d_2,d_3}$ , making ’progress’ in reducing $n$ . However in general, that influence may be very small but still non-zero. In that case one may write the decompositions

\begin{equation*}f = f_1 + f', \quad g = g_1 + g', \quad h = h_1 + h', \end{equation*}

where $f_1,g_1,h_1$ depend only on the first $n-1$ co-ordinates, and $f', g', h'$ do depend on the $n^{th}$ co-ordinate but have very small $\ell _2$ -norm (which is precisely what influence is). Since $f',g',h'$ have very small norm, one doesn’t expect them to contribute much, and one still hopes to deduce that $\beta _{n,d_1,d_2,d_3}\leqslant \beta _{n-1,d_1,d_2,d_3}$ . Alas, this doesn’t quite work. While their contribution is very small, it is still non-zero, and a naive application of this idea would only give $\beta _{n,d_1,d_2,d_3}\leqslant \beta _{n-1,d_1,d_2,d_3} + o(1)$ , and the $o(1)$ error terms will keep accumulating in successive inductive steps. To overcome this difficulty, we perform a more detailed analysis, and need more refined decompositions of $f$ , $g$ , and $h$ . For the sake of simplicity, we consider only a specialised scenario that allows us to write

\begin{equation*} f = f_1 + f_2 f_2', \quad g = g_1 + g_2 g_2', \quad h = h_1 + h_2 h_2', \end{equation*}

where $f_1,g_1,h_1$ depend only on the first $n-1$ coordinates and have the same degrees as $f,g, h$ , the functions $f_2,g_2,h_2$ also depend only on the first $n-1$ coordinates but have degrees one less than $f,g, h$ respectively, and $f_2',g_2',h_2'$ are functions that only depend on the last coordinate and have very small $\ell _2$ -norm. Using this decomposition, we can write

\begin{align*} {\mathop {\mathbb{E}}_{({\textbf {x}},\textbf {y},\textbf {z})\sim \mu ^{\otimes n}}{\left [ {f({\textbf {x}})g(\textbf {y})h(\textbf {z})} \right ]}} &= {\mathop {\mathbb{E}}_{({\textbf {x}},\textbf {y},\textbf {z})\sim \mu ^{\otimes n-1}}{\left [ {f_1({\textbf {x}})g_1(\textbf {y})h_1(\textbf {z})} \right ]}}\\ &\quad+ {\mathop {\mathbb{E}}_{({\textbf {x}},\textbf {y},\textbf {z})\sim \mu ^{\otimes n-1}}{\left [ {f_2({\textbf {x}})g_2(\textbf {y})h_1(\textbf {z})} \right ]}} {\mathop {\mathbb{E}}_{({\textbf {x}},\textbf {y},\textbf {z})\sim \mu }{\left [ {f_2'({\textbf {x}})g_2'(\textbf {y})} \right ]}}\\ &\quad+ {\mathop {\mathbb{E}}_{({\textbf {x}},\textbf {y},\textbf {z})\sim \mu ^{\otimes n-1}}{\left [ {f_2({\textbf {x}})g_1(\textbf {y})h_2(\textbf {z})} \right ]}} {\mathop {\mathbb{E}}_{({\textbf {x}},\textbf {y},\textbf {z})\sim \mu }{\left [ {f_2'({\textbf {x}})h_2'(\textbf {z})} \right ]}}\\ &\quad+ {\mathop {\mathbb{E}}_{({\textbf {x}},\textbf {y},\textbf {z})\sim \mu ^{\otimes n-1}}{\left [ {f_2({\textbf {x}})g_2(\textbf {y})h_2(\textbf {z})} \right ]}} {\mathop {\mathbb{E}}_{({\textbf {x}},\textbf {y},\textbf {z})\sim \mu }{\left [ {f_2'({\textbf {x}})g_2'(\textbf {y})h_2'(\textbf {z})} \right ]}}\\ &\quad+\text{Other terms}. \end{align*}

The other terms are zero thanks to the fact that $\mu _{y,z}$ is uniform and independent. The first term is the dominant term, the second and the third terms constitute as error terms, and the fourth term can be ignored when compared to the second and third terms. Roughly speaking, the reason is that if $\varepsilon$ denotes the small norm of $f_2',g_2',h_2'$ , then the corresponding expectations are of the order $\varepsilon ^2$ in the second and third terms, and of the order $\varepsilon ^3$ in the fourth term.

The second and third terms are error terms, which however cannot be ignored altogether (as said before) and require care. Skipping many details, it turns out that the key is to bound the expectation

\begin{equation*}{\mathop {\mathbb{E}}_{({\textbf {x}},\textbf {y},\textbf {z})\sim \mu }{\left [ {f_2'({\textbf {x}})(g_2'(\textbf {y}) + h_2'(\textbf {z}))} \right ]}}.\end{equation*}

This can be upper bounded by $(1-\Omega (1)) \|f_2'\|_2 \sqrt {\|g_2'\|_2^2 + \|h_2'\|_2^2}$ . We emphasise here that this is an inequality on functions of a single co-ordinate. It is referred to as the additive base case inequality (see Lemma 3.18). Using this bound, one can obtain an effective enough bound on the second and third terms above, somehow recover the loss from these error terms and get that $\beta _{n,d_1,d_2,d_3}\leqslant \beta _{n-1,d_1,d_2,d_3}$ as desired.

The core induction:We now show the core inductive step giving the exponential decay, namely that $\beta _{n,d_1,d_2,d_3}\leqslant (1-\Omega _{\alpha ,m}(1))^{d_2}$ . We assume that $n \leqslant O(d_2)$ as discussed and that $d_1, d_3 \leqslant 10d_2$ . Skipping details, it is sufficient to assume further that $d_1 \geqslant \Omega (d_2)$ as well. It follows from these assumptions that average influence of a coordinate on $f$ is $\frac {d_1}{n}\geqslant \Omega (1)$ . Let us assume that the coordinate $n$ has influence $\Omega (1)$ on $f$ . For the sake of simplicity, consider furthermore only a specialised scenario that allows us to write $f$ , $g$ , and $h$ as

\begin{equation*} f = f_1 f_1', \quad g = g_1 g_1' \quad h = h_1 h_1', \end{equation*}

where $f_1, g_1, h_1$ depend only on the first $n-1$ co-ordinates and have degrees one less than $f, g, h$ , and the functions $f_1',g_1',h_1'$ depend only on the single coordinate $n$ , and $f_1'$ has constant norm (which amounts to the said influence). In this case, we would have that

\begin{equation*} {\mathop {\mathbb{E}}_{({\textbf {x}},\textbf {y},\textbf {z})\sim \mu ^{\otimes n}}{\left [ {f({\textbf {x}})g(\textbf {y})h(\textbf {z})} \right ]}} = {\mathop {\mathbb{E}}_{({\textbf {x}},\textbf {y},\textbf {z})\sim \mu ^{\otimes n-1}}{\left [ {f_1({\textbf {x}})g_1(\textbf {y})h_1(\textbf {z})} \right ]}} {\mathop {\mathbb{E}}_{({\textbf {x}},\textbf {y},\textbf {z})\sim \mu }{\left [ {f_1'({\textbf {x}})g_1'(\textbf {y})h_1'(\textbf {z})} \right ]}}. \end{equation*}

By the inductive hypothesis, the first term is at most $\beta _{n-1, d_1-1, d_2-1, d_3-1}$ and by the base case inequality, $|{\mathop {\mathbb{E}}_{({\textbf {x}},\textbf {y},\textbf {z})\sim \mu }{\left [ {f_1'({\textbf {x}})g_1'(\textbf {y})h_1'(\textbf {z})} \right ]}}|\leqslant \lambda = 1 - \Omega (1)$ . Hence we get that

\begin{equation*} \beta _{n,d_1,d_2,d_3}\leqslant \lambda \beta _{n-1,d_1-1,d_2-1,d_3-1}, \end{equation*}

as desired, and iterating this gives an exponential decay.

In general, the main complication is that $f$ , $g$ , and $h$ need not take the specialised form as above, and instead one has to decompose them in a more complicated manner (amounting to decomposing a tensor into a sum of mutually orthogonal rank one tensors). Using a more complicated argument (but vaguely similar in spirit) one can still recover that $\beta _{n,d_1,d_2,d_3}\leqslant \lambda \beta _{n-1,d_1-1,d_2-1,d_3-1}$ .

2.5 The inductive argument (incorporating the relaxed base case inequality)

As discussed before, in general the base case inequality (7) does not hold and we are able to use only the relaxed base case inequality in Definition 2.4. We now indicate the main modification necessary in the inductive proof, skipping most other details from this overview.

Let $\Sigma ' \subseteq \Sigma$ be the subset that exhibits the relaxed base case inequality in Definition 2.4. We consider the effective influence and effective degree of the function $f: \Sigma ^n \to \mathbb{R}$ . We recall that the standard influence of the $i^{th}$ co-ordinate is

\begin{equation*} {\mathop {\mathbb{E}}_{\substack {{\textbf {x}}_{-i} \\ x_i, x_i' \in \Sigma }}{\left [ { \left | {f({\textbf {x}}_{-i}, x_i) - f({\textbf {x}}_{-i}, x_i')} \right |^2 } \right ]}}. \end{equation*}

That is, the influence is the variance of the function on the $i^{th}$ coordinate after randomly restricting the rest of the coordinates. We define the effective influence as

\begin{equation*} {\mathop {\mathbb{E}}_{\substack {{\textbf {x}}_{-i}\\ x_i, x_i' \in \Sigma '}}{\left [ { \left | {f({\textbf {x}}_{-i}, x_i) - f({\textbf {x}}_{-i}, x_i')} \right |^2 } \right ]}}, \end{equation*}

which is similar, except that the variance is considered only over the subset $\Sigma '$ .

We also indicate the related notion of the effective degree of $f$ . We set up a suitable orthonormal basis $\textbf { B}$ of characters for (single co-ordinate) functions in $L_2(\Sigma ; \mu _x)$ . We ensure that $\textbf { B} = \textbf { B}_1 \cup \textbf { B}_2$ so that characters in $\textbf { B}_1$ span all functions that are constant on $\Sigma '$ (including the All- $1$ function), and characters in $\textbf { B}_2$ are zero outside $\Sigma '$ . The effective degree of a monomial is then the degree when only the characters in $\textbf { B}_2$ are counted towards the degree. The inductive proof is now carried out assuming that $f$ not only has high degree, but also has high effective degree.

We do mention a crucial detail here. We do need to argue that starting with the original $1$ -bounded function $f:\Sigma ^n \to \mathbb{C}$ that has essentially high degree, we can ’reduce’ to the case where it has high effective degree as well. This argument does need that the original functions $f,g,h$ are $\ell _\infty$ -bounded.Footnote ¹⁰ As noted before, Lemmas 2.1, 2.5 could simply be false (for certain distributions $\mu$ ) if only $\ell _2$ -norm of the functions is assumed to be $1$ .

2.6 Organisation

The rest of the paper is organised as follows. We start with preliminaries in Section 3. We set up the necessary machinery in Sections 4 and 5 that are needed to formulate the inductive statement towards proving the main analytical lemma under relaxed base case inequality. The proof of this lemma, which is divided into two parts, spans Sections 6 and 7. Finally, in Section 8 we derive our main analytical lemma, Lemma 2.1, from Lemma 2.5. In this section, we show how to get around the issue of the Horn-SAT embedding.

Section 9 is devoted to proving applications of our main analytical lemma.

3.1 Degrees and homogeneous functions

We start with the definitions of a monomial and a degree- $d$ monimial.

Definition 3.2. Let $\Gamma$ be a finite set, $n\geqslant 1$ , and let $\nu$ be some probability measure over $\Gamma$ . A function $\chi \colon (\Gamma ^n,\nu ^{\otimes n})\to \mathbb{C}$ is a degree $d$ monomial if there are distinct indices $i_1,\ldots ,i_d$ and monomials $\chi _{i_1},\ldots ,\chi _{i_d}\colon \Gamma \to \mathbb{C}$ with respect to $\nu$ , such that

\begin{equation*} \chi (\textbf {y}) = \prod \limits _{j=1}^{d}\chi _{i_j}(y_{i_j}). \end{equation*}

Based on these definitions, we now define homogeneous functions of degree $d$ .

3.2 Efron–Stein decomposition

For a product space $(\Gamma ^n, \nu ^{\otimes })$ , we will use the standard Efron-Stein decomposition. Given a function $g\colon (\Gamma ^{n},\nu ^{\otimes n})\to \mathbb{C}$ , one may write (in a unique manner)

\begin{equation*} g(\textbf {y}) = \sum \limits _{i=0}^{n} g^{=i}(\textbf {y}), \end{equation*}

where $g^{=i}$ is a homogeneous function of degree $i$ . We denote by $V^{=i}(\Gamma ^{n},\nu ^{\otimes n})$ the space of homogeneous functions of degree $i$ , and often omit the domain and the measure if these are clear from the context. Hence $g^{=i}\in V^{=i}$ . The Efron-Stein decomposition is a refinement of the above. For each $i=0,\ldots ,n$ , one may write

\begin{equation*} g^{=i}(\textbf {y}) = \sum \limits _{\substack {S\subseteq [n], \left | {S} \right | = i}} g^{=S}(\textbf {y}), \end{equation*}

where $g^{=S}(\textbf {y})$ is a homogeneous function of degree $i$ whose value on $\textbf {y}$ only depends on $\textbf {y}_S$ , namely the co-ordinates in the set $S$ . We denote by $V^{=S}(\Gamma ^{n},\nu ^{\otimes n})$ the space of degree $\left | {S} \right |$ homogeneous functions depending only on coordinates in $S$ . So, $g^{=S}\in V^{=S}$ . Furthermore, this decomposition is unique, and satisfies that $g^{=S}$ and $g^{=T}$ are orthogonal for any $S\neq T$ , that is,

\begin{equation*} \left \langle g^{=S}, g^{=T}\right \rangle = {\mathop {\mathbb{E}}_{\textbf {y} \sim \nu ^{\otimes n}}{\left [ { g^{=S}(\textbf {y}) \overline {g^{=T}(\textbf {y})} } \right ]}} = 0. \end{equation*}

We define

\begin{equation*} g^{\supseteq T}(\textbf {y}) = \sum \limits _{S\supseteq T} g^{=S}(\textbf {y}), \quad g^{\leqslant d}(\textbf {y}) = \sum \limits _{i=0}^{d} g^{=i}(\textbf {y}), \quad g^{\gt d}(\textbf {y}) = \sum \limits _{i=d+1}^n g^{=i}(\textbf {y}). \end{equation*}

Next, we define the standard notion of the influence of a variable and the total influence of a function as follows:

Definition 3.4. The influence of a variable $i$ on $g\colon (\Gamma ^n,\mu ^{\otimes n})\to \mathbb{C}$ is defined as

\begin{equation*} I_i[g] = {\mathop {\mathbb{E}}_{\textbf {y}_{-i}\sim \mu ^{n-1}, \ a,b\sim \mu }{\left [ {\left |g(\textbf {y}_{-i}, y_i=a) - g(\textbf {y}_{-i}, y_i=b)\right |^2} \right ]}}. \end{equation*}

The following fact relates the total influence with the Efron-Stein decomposition of a function.

Definition 3.7. The variance of $g\colon (\Gamma ^{n},\mu ^{\otimes n})\to \mathbb{C}$ is defined as

\begin{equation*} \textsf { var}(g) = \frac {1}{2}{\mathop {\mathbb{E}}_{\textbf {y},\textbf {y}'\sim \mu ^{\otimes n}}{\left [ {|g(\textbf {y}) - g(\textbf {y}')|^2} \right ]}} ={\mathop {\mathbb{E}}_{\textbf {y}}{\left [ {\left | {g(\textbf {y})} \right |^2} \right ]}} - |{\mathop {\mathbb{E}}_{\textbf {y}}{\left [ {g(\textbf {y})} \right ]}}|^2. \end{equation*}

3.3 The noise operator and stability

For a parameter $\rho \in [0, 1]$ , a measure $\mu$ over $\Sigma$ and a point $x\in \Sigma$ we define the distribution over points that are $\rho$ -correlated with $x$ as: take $y=x$ with probability $\rho$ , and otherwise sample $y\sim \mu$ . We denote the distribution over $\rho$ -correlated points with $x$ as $y\sim {\textrm{T}}_{\rho } x$ . Tensorizing, for $n\in \mathbb{N}$ and $x\in \Sigma ^n$ , the distribution over $\rho$ -correlated points with $x$ is denoted by ${\textrm{T}}_{\rho }^{\otimes n} x$ , and is sampled by taking $y_i\sim {\textrm{T}}_{\rho }x_i$ for each $i$ independently.

We may think of the operator ${\textrm{T}}_{\rho }^{\otimes n}$ as acting on $L_2(\Sigma ^n;\, \mu ^{\otimes n})$ by mapping a function $f$ to the function ${\textrm{T}}_{\rho }^{\otimes n} f$ defined as

\begin{equation*} {\textrm{T}}_{\rho }^{\otimes n} f(x) = {\mathop {\mathbb{E}}_{y\sim {\textrm{T}}_{\rho }^{\otimes n} x}{\left [ {f(y)} \right ]}}. \end{equation*}

We remark that a straightforward computation shows that the spaces $V^{=i}$ are eigenspaces of the operator ${\textrm{T}}_{\rho }^{\otimes n}$ with eigenvalue $\rho ^{i}$ , and in particular one gets the Fourier analytic formula $\textsf { Stab}_{\rho }(f) = \sum \limits _{S}\rho ^{\left | {S} \right |}\| f^{=S} \|_2^2$ .

3.4 A markov chain lemma

We mention below a bound on the second largest eigenvalue of a Markov chain. We include the proof since we could not find this specific bound in literature (we need dependence on $\xi$ to be linear and not quadratic).

Proof. Note that $\mu$ is the stationary distribution of $\textrm{T}$ . Let $f\colon \Sigma \to \mathbb{R}$ be the eigenvector of $\textrm{T}$ corresponding to $\lambda _2({\textrm{T}})$ . We have,

\begin{equation*}{\mathop {\mathbb{E}}_{{\textbf {x}}\sim \mu }{\left [ {f({\textbf {x}})} \right ]}} = 0, \quad {\mathop {\mathbb{E}}_{{\textbf {x}}\sim \mu }{\left [ {f({\textbf {x}})^2} \right ]}} = 1, \quad \lambda _2({\textrm{T}}) = \left \langle f, {\textrm{T}} f\right \rangle . \end{equation*}

It follows that there is $x\in \Sigma$ such that $\left | {f(x)} \right |\geqslant 1$ , and without loss of generality $f(x)\geqslant 1$ . Since ${\mathop {\mathbb{E}}_{}{\left [ {f} \right ]}}=0$ , there is $y$ such that $f(y)\leqslant 0$ . Since $\textrm{T}$ is connected, there is a path $x=x^0\rightarrow x^1\rightarrow \ldots \rightarrow x^{\ell } = y$ , where $\ell \leqslant \left | {\Sigma } \right |\leqslant m$ , so that $\mu (x^i,x^{i+1}) \gt 0$ for all $i\leqslant \ell -1$ . We note that

\begin{equation*} 1\leqslant \left | {f(x)-f(y)} \right |\leqslant \sum \limits _{i=0}^{\ell -1}\left | {f(x^{i+1}) - f(x^i)} \right |, \end{equation*}

so it follows that there is $i$ such that $\left | {f(x^{i+1}) - f(x^{i})} \right |\geqslant \frac {1}{\ell }\geqslant \frac {1}{m}$ . Therefore,

\begin{equation*} {\mathop {\mathbb{E}}_{({\textbf {x}},\textbf {y}) \sim \mu }{\left [ {(f({\textbf {x}}) - f(\textbf {y}))^2} \right ]}}\geqslant \mu (x,y) \,\frac {1}{m^2}\geqslant \frac {\xi }{m^2}. \end{equation*}

On the other hand, the left hand side is $ 2\| f \|_{2}^2 - 2\langle {f},{{\textrm{T}} f}\rangle$ , so we get that $\lambda _2({\textrm{T}}) = \langle {f},{{\textrm{T}} f}\rangle \leqslant 1-\frac {\xi }{2m^2}$ , and the proof is concluded.

Lemma 3.10. In the setting of Lemma 3.9 , if $f\colon (\Sigma ^n,\mu ^{\otimes n})\to \mathbb{C}$ is any function, then

\begin{equation*} \| {\textrm{T}}^{\otimes n} f^{=S} \|_2\leqslant \left (1-\Omega _{\alpha ,m}(\xi )\right )^{\left | {S} \right |}\| f^{=S} \|_2. \end{equation*}

Proof. Assume without loss of generality that $S=\{1,\ldots ,t\}$ .

\begin{equation*} \| {\textrm{T}}^{\otimes n} f^{=S} \|_2 = \| {\textrm{T}}_1 (({\textrm{T}}_2\circ \ldots \circ {\textrm{T}}_t) f^{=S}) \|_2 \leqslant (1-\Omega _{\alpha ,m}(\xi )) \| ({\textrm{T}}_2\circ \ldots \circ {\textrm{T}}_t) f^{=S} \|_2, \end{equation*}

where we used Lemma 3.9 and the fact that $({\textrm{T}}_2\circ \ldots \circ {\textrm{T}}_t) f^{=S}$ has expectation $0$ over $x_1$ for any setting of the remaining co-ordinates. The proof is concluded by iterating the above inequality over $x_2,\ldots ,x_t$ .

3.5 Effective degrees and effective influences

Fix a distribution $\mu$ on $\Sigma \times \Gamma \times \Phi$ as in Lemma 2.5, and fix $\Sigma '\subseteq \Sigma$ to be the subset evidencing the fact that it satisfies the relaxed base case, Definition 2.4. We may choose an orthonormal basis of $L_2(\Sigma ; \mu _x)$ as $B = B_1\cup B_2$ where $B_1$ consists of functions that are constant on $\Sigma '$ , and $B_2$ consists of functions only supported on $\Sigma '$ and orthogonal to $B_1$ . Thus, a basis for $L_2(\Sigma ^n;\, \mu _x^n)$ is given by

\begin{equation*} B^{\otimes n} = \left \{ \left . \chi ({\textbf {x}}) = \prod \limits _{i}\chi _i(x_i) \;\right \vert \chi _i\in B_1\cup B_2\,\forall i\in [n] \right \}, \end{equation*}

and a given function $f\colon \Sigma ^n\to \mathbb{C}$ can be uniquely written as

\begin{equation*} f({\textbf {x}}) = \sum \limits _{\chi \in B^{\otimes n}}{\widehat {f}(\chi )\,\chi ({\textbf {x}})}, \quad \text{where }\widehat {f}(\chi ) = \langle {f},{\chi }\rangle . \end{equation*}

We now define the effective degree of a character $\chi$ , which is the number of components from $\{\chi _i\}$ that are not constant on $\Sigma '$ .

Definition 3.11. Given $\chi \in B^{\otimes n}$ , we define the effective degree of $\chi$ as

\begin{equation*} \textsf { effdeg}(\chi ) = \left | {\left \{ \left . i\in [n] \;\right \vert \chi _i\in B_2 \right \}} \right |. \end{equation*}

To be compatible with the standard notion of degree, we will introduce a special notation for the trivial character in $B_1$ which is the constant $1$ function on $\Sigma$ , and we denote it by $\chi _{\textsf { const}}$ . Thus, we note that degree of a character $\chi$ is $\left | {\{i \mid \chi _i\neq \chi _{\textsf { const}}\}} \right |$ , and as $\chi _{\textsf { const}}$ is in $B_1$ , one has that $\textsf { effdeg}(\chi )\leqslant \textsf { deg}(\chi )$ for all $\chi \in B^{\otimes n}$ .

We also define the effective influence of a variable analogous to Definition 3.4, except we only resample the assignment to the variable if it belongs to $\Sigma '$ .

Definition 3.12. For a function $f\colon \Sigma ^n\to \mathbb{C}$ and $i\in [n]$ , we define

\begin{equation*}I_{i,\textsf { effective}}[f] = {\mathop {\mathbb{E}}_{({\textbf {x}},\textbf {y})}{\left [ {\left | {f({\textbf {x}}) - f(\textbf {y})} \right |^2} \right ]}},\end{equation*}

where we sample ${\textbf {x}}\sim \mu _x^{\otimes n}$ , take $y_j = x_j$ for all $j\neq i$ , and for the $i^{th}$ co-ordinate, if $x_i\in \Sigma \setminus \Sigma '$ we take $y_i = x_i$ , and if $x_i\in \Sigma '$ , we sample $y_i\in \Sigma '$ independently.

Definition 3.13. For a function $f\colon \Sigma ^n\to \mathbb{C}$ , the total effective influence is defined as

\begin{equation*} I_{\textsf { effective}}[f] = \sum \limits _{i=1}^{n} I_{i, \textsf { effective}}[f]. \end{equation*}

Based on how we defined the effective degree and the effective influences, we have the following fact analogous to Fact 3.6.

Fact 3.14. For a function $f\colon \Sigma ^n\to \mathbb{C}$ and $i\in [n]$ , we have

\begin{equation*} I_{i,\textsf { effective}}[f] = 2\sum \limits _{\chi :\chi _i\in B_2}{\left | {\widehat {f}(\chi )} \right |^2}, \quad I_{\textsf { effective}}[f] = 2\sum \limits _{\chi }{\textsf { effdeg}(\chi )\,\left | {\widehat {f}(\chi )} \right |^2}. \end{equation*}

3.6 High degree is preserved under random restrictions

We will often consider random restrictions of functions and would need to argue that if the original function has high degree, then so is the restricted function w.h.p. We recall that having high degree is formalised in terms of having low-stability. In the lemma below, a random restriction of a function $g: (\Sigma ^n, \mu ^n) \to \mathbb{C}$ includes every co-ordinate in the set $I$ with probability $s$ and then samples each co-ordinate in $I$ according to $\nu _1$ . For the restricted function $g_{I \to \textbf {y}}$ on $[n]\setminus I$ , each co-ordinate has marginal $\nu _2$ . It is easily checked that to get the marginals ’correct’, we need $\mu = s \nu _1 + (1-s) \nu _2$ .

Lemma 3.15. Let $\nu _1,\nu _2$ be distributions over $\Sigma$ whose support is full and the probability of each atom is at least $\alpha$ and $\left | {\Sigma } \right |\leqslant m$ . Let $\mu = s\nu _1 + (1-s)\nu _2$ . Then for some $c = c(\alpha )\gt 0$ we have

\begin{equation*} {\mathop {\mathbb{E}}_{\substack {\left | {I} \right |\sim s n\\ \textbf {y}\sim \nu _1^I}}{\left [ {\textsf { Stab}_{1-\xi }(g_{I\rightarrow \textbf {y}};\, \nu _2)} \right ]}}\leqslant \textsf { Stab}_{1-c(1-s)\xi }(g;\,\mu ). \end{equation*}

Proof. The left hand side is

\begin{equation*} {\mathop {\mathbb{E}}_{({\textbf {x}},{\textbf {x}}')}{\left [ {g({\textbf {x}})\overline {g({\textbf {x}}')}} \right ]}}, \end{equation*}

where $({\textbf {x}},{\textbf {x}}')$ are sampled by taking, for each $i\in [n]$ independently, $x_i = x_i'\sim \nu _1$ with probability $s$ ; otherwise with probability $1-\xi$ we take $x_i = x_i'\sim \nu _2$ , and with probability $\xi$ we take $x_i,x_i'\sim \nu _2$ independently. We denote this Markov chain by $\textrm{T}$ , and note that the stationary distribution of it is $\mu$ , and also that as the support of $\nu _2$ is full, $\textrm{T}$ is connected. Additionally, for every $a,b\in \Sigma$ we have that ${\textrm{T}}(a,b)\geqslant (1-s)\xi \alpha ^2$ . We may now write the above expectation as

\begin{equation*} \langle {g},{{\textrm{T}}^{\otimes n} g}\rangle =\sum \limits _{S,T}\langle {g^{=S}},{{\textrm{T}}^{\otimes n} g^{=T}}\rangle =\sum \limits _{S}\langle {g^{=S}},{{\textrm{T}}^{\otimes n} g^{=S}}\rangle \leqslant \sum \limits _{S}\| g^{=S} \|_2\| {\textrm{T}}^{\otimes n} g^{=S} \|_2. \end{equation*}

Using Lemma 3.10, we get that

\begin{equation*} \langle {g},{{\textrm{T}}^{\otimes n} g}\rangle \leqslant \sum \limits _{S}\left (1-\Omega _{\alpha , m}((1-s)\xi )\right )^{\left | {S} \right |}\| g^{=S} \|_2^2 =\textsf { Stab}_{1-c(1-s)\xi }(g), \end{equation*}

where $c=c(m,\alpha )\gt 0$ .

3.7 Embedding into the infinite cyclic group

In our proofs, towards arriving at a contradiction, we will often get an embedding of $\textsf { supp}(\mu )$ into an infinite cyclic group. Here, we argue that if $\textsf { supp}(\mu )$ cannot be embedded into a finite Abelian group, then it also cannot be embedded into an infinite cyclic group (say, $[0,1)$ with addition mod $1$ ). Therefore, the finiteness of the Abelian group is not essential to the definition of linear embeddability.

Proof. Suppose towards contradiction it can be. Then there are $\sigma \colon \Sigma \to [0,1)$ , $\phi \colon \Phi \to [0,1)$ and $\gamma \colon \Gamma \to [0,1)$ not all constant such that $\sigma (x) + \phi (y) + \gamma (z) = 0 \pmod {1}$ . Without loss of generality, $\sigma$ is non-constant. Consider the set of numbers $S = \textsf { Image}(\sigma )\cup \textsf { Image}(\phi ) \cup \textsf { Image}(\gamma )$ , let $r = \left | {\Sigma } \right | +\left | {\Phi } \right | + \left | {\Gamma } \right |$ and let $N = N(r)\in \mathbb{N}$ to be determined. Then $\left | {S} \right |\leqslant r$ , so by Dirichlet’s approximation theorem we may find integers $p_i, q$ such that for each $s_i\in S$ we have that $\left | {s_i - \frac {p_i}{q}} \right |\leqslant \frac {1}{q N^{1/r}}$ .

Let

\begin{equation*} \alpha = \min _{x,x' \sigma (x)\neq \sigma (x')}\min _{z\in \mathbb{Z}}\left | {z+\sigma (x)-\sigma (x')} \right |. \end{equation*}

We choose $N = \left (\frac {3}{\alpha }\right )^r$ , define $\sigma '$ by $\sigma '(x) = \frac {p_i}{q}\pmod {1}$ if $\sigma (x) = s_i$ , and similarly define $\phi ',\gamma '$ .

1. 1. First, we show that $\sigma ', \phi ', \gamma '$ is an embedding. Fix $(x,y,z)\in \textsf { supp}(\mu )$ ; then we have

   \begin{equation*} \sigma '(x) + \phi '(y) + \gamma '(z) = \sigma (x)+\phi (y) + \gamma (z) + \Delta , \end{equation*}
   where $\left | {\Delta } \right |\leqslant \frac {3}{q N^{1/r}}$ . Noting that $\sigma (x) + \phi (y) + \gamma (z)$ is an integer (as it is $0$ mod $1$ ), it follows that $\sigma '(x) + \phi '(y) + \gamma '(z)$ is very close to an integer, up to $\frac {3}{q N^{1/r}} \lt 1/q$ . On the other hand, by definition of $\sigma ',\phi ',\gamma '$ it is a number of the form $P/q$ for some integer $P$ , hence it can either be an integer or at least $1/q$ far from one. It follows that it is an integer, so $\sigma '(x) + \phi '(y) + \gamma '(z) = 0\pmod {1}$ .

2. 2. Second, we show that at least one of them is not constant. Indeed, take $x,x'\in \Sigma$ on which $\sigma$ differs, and suppose towards contradiction that $\sigma '(x) - \sigma '(x') = 0$ . Let $i,j$ be such that $\sigma '(x) = p_i/q$ , $\sigma '(x') = p_j/q$ , and $\sigma (x) = s_i$ , $\sigma (x') = s_j$ . Then we get that $(p_i - p_j)/q$ is an integer, and as $s_i - s_j$ is $\Delta$ -close to it for $\left | {\Delta } \right |\leqslant \frac {2}{q N^{1/r}}$ , we get that $s_i - s_j$ is $\frac {\alpha }{3}$ close to an integer. This contradicts the definition of $\alpha$ .

3.8 The additive base case

In this section, we deduce a certain auxiliary (base case) inequality that follows solely under the assumption that the distribution $\mu$ has no linear embedding. We emphasise that it holds irrespective of whether or not $\mu$ has a Horn-SAT embedding. The inequality is used while reducing the dimension and making it comparable to the degree during our inductive proof (as in the overview Section 2.4).

Claim 3.17. Let $\mu$ be a distribution on $\Sigma \times \Gamma \times \Phi$ that has no linear embedding. Then there exists $c_1 = c_1(\mu ) \gt 0$ , such that for $f\colon \Sigma \to \mathbb{C}$ , $g\colon \Phi \to \mathbb{C}$ and $h\colon \Gamma \to \mathbb{C}$ that each have average equal to $0$ we have that

\begin{equation*} \left | {{\mathop {\mathbb{E}}_{(x, y,z)\sim \mu }{\left [ {f(x)(g(y)+h(z))} \right ]}}} \right | \leqslant (1-c_1)\| f \|_2\| g+h \|_2. \end{equation*}

Proof. Assume this is not the case, so that we may find a sequence of functions $(f_m,g_m,h_m)$ such that $\| f_m \|_2 = \| g_m+h_m \|_2 = 1$ , $\mathop {\mathbb{E}}[f_m] = \mathop {\mathbb{E}}[g_m] = \mathop {\mathbb{E}}[h_m] = 0$ , and $\left | {{\mathop {\mathbb{E}}_{(x, y,z)\sim \mu }{\left [ {f_m(x)(g_m(y)+h_m(z))} \right ]}}} \right |\geqslant 1-\frac {1}{m}$ . Passing to subsequences, we may assume that $f_m$ converges to a function $f$ , $g_m$ converges to a function $g$ and $h_m$ converges to a function $h$ , so that we get $\mathop {\mathbb{E}}[f]= \mathop {\mathbb{E}}[g] = \mathop {\mathbb{E}}[h]$ , $\| f \|_2 = \| g+h \|_2=1$ and $\left | {{\mathop {\mathbb{E}}_{(x, y,z)\sim \mu }{\left [ {f(x)(g(y)+h(z))} \right ]}}} \right |\geqslant 1$ . By Cauchy-Schwarz we have that

\begin{equation*} \left | {{\mathop {\mathbb{E}}_{(x,y,z)\sim \mu }{\left [ {f(x)(g(y)+h(z))} \right ]}}} \right | \leqslant \sqrt {{\mathop {\mathbb{E}}_{(x,y,z)\sim \mu }{\left [ {f(x)^2} \right ]}}} \sqrt {{\mathop {\mathbb{E}}_{(x,y,z)\sim \mu }{\left [ {\left | {g(y)+h(z)} \right |^2} \right ]}}} =\| f \|_2\| g+h \|_2 =1, \end{equation*}

hence we get that Cauchy-Schwarz is tight and so $\overline {f(x)} = \theta (g(y)+h(z))$ for some $\theta \in \mathbb{C}$ of absolute value 1, for all $(x,y,z)\in \textsf { supp}(\mu )$ . As the $2$ -norm of $f$ is $1$ and its average is $0$ , the function $f$ is not constant, and so after diving them by a sufficiently large constant and adding a constant (so that their image is contained in [0, 1]), we get that either the real part and the imaginary part of $\overline {f}, -\theta g, -\theta h$ form a non-trivial embedding of $\mu$ . Together with Claim 3.16, this contradicts the assumption that $\mu$ has no linear embedding.

Lemma 3.18. Let $\mu$ be a distribution on $\Sigma \times \Gamma \times \Phi$ that has no linear embedding and $\mu _{y,z}$ is uniform. Then there exists $c_1 = c_1(\mu ) \gt 0$ , such that for $f\colon \Sigma \to \mathbb{C}$ , $g\colon \Phi \to \mathbb{C}$ and $h\colon \Gamma \to \mathbb{C}$ we have that

\begin{equation*} \left | {{\mathop {\mathbb{E}}_{(x, y,z)\sim \mu }{\left [ {f(x)(g(y)+h(z))} \right ]}}} \right | \leqslant \| f \|_2\sqrt {\left | {\mathop {\mathbb{E}}[g] + \mathop {\mathbb{E}}[h]} \right |^2 + (1-c_1)(W_{=1}[g] + W_{=1}[h])}. \end{equation*}

Here $W_{=1}[g]$ denotes the variance ${\mathop {\mathbb{E}}_{}{\left [ {\left | {g} \right |^2} \right ]}}-\left | {{\mathop {\mathbb{E}}_{}{\left [ {g} \right ]}}} \right |^2$ (and similarly for $W_{=1}[h]$ ).

Proof. Write $f = \mathop {\mathbb{E}}[f] + f^{=1}$ , $g=\mathop {\mathbb{E}}[g] + g^{=1}$ and $h = \mathop {\mathbb{E}}[h] + h^{=1}$ so that

\begin{equation*} {\mathop {\mathbb{E}}_{(x, y,z)\sim \mu }{\left [ {f(x)(g(y)+h(z))} \right ]}} =\mathop {\mathbb{E}}[f](\mathop {\mathbb{E}}[g] + \mathop {\mathbb{E}}[h]) + {\mathop {\mathbb{E}}_{(x, y,z)\sim \mu }{\left [ {f^{=1}(x)(g^{=1}(y)+h^{=1}(z))} \right ]}}. \end{equation*}

By Claim 3.17 we get that

\begin{equation*} \left | {{\mathop {\mathbb{E}}_{(x, y,z)\sim \mu }{\left [ {f^{=1}(x)(g^{=1}(y)+h^{=1}(z))} \right ]}}} \right |\leqslant (1-c_1)\| f^{=1} \|_2\| g^{=1} + h^{=1} \|_2, \end{equation*}

and so

\begin{align*} \left | {{\mathop {\mathbb{E}}_{(x, y,z)\sim \mu }{\left [ {f(x)(g(y)+h(z))} \right ]}}} \right | &\leqslant \left | {\mathop {\mathbb{E}}[f]} \right |\left | {\mathop {\mathbb{E}}[g] + \mathop {\mathbb{E}}[h]} \right | + (1-c_1)\| f^{=1} \|_2\| g^{=1} + h^{=1} \|_2\\ &\leqslant \sqrt {\left | {\mathop {\mathbb{E}}[f]} \right |^2 + \| f^{=1} \|_2^2}\sqrt {\left | {\mathop {\mathbb{E}}[g]+\mathop {\mathbb{E}}[h]} \right |^2 + (1-c_1)^2\| g^{=1} + h^{=1} \|_2^2}\\ &=\| f \|_2 \sqrt {\left | {\mathop {\mathbb{E}}[g]+\mathop {\mathbb{E}}[h]} \right |^2 + (1-c_1)^2\| g^{=1} + h^{=1} \|_2^2} \end{align*}

where we used Cauchy-Schwarz. As $\mu _{y,z}$ is uniform we get that $\| g^{=1} + h^{=1} \|_2^2 = \| g^{=1} \|_2^2 + \| h^{=1} \|_2^2 = W_{=1}[g] + W_{=1}[h]$ and the proof is concluded

4.1 Soft truncations

In this section, we define a couple of noise operators that will be used in stating the intermediate lemmas from (8).

The noise operators. Let $\xi \in (0,1]$ be some parameter.

1. 1. We define the operator ${\textrm{T}}_{1-\xi }$ acting on $L_2(\Sigma )$ as follows. Consider the Markov chain on $\Sigma$ that on $x\in \Sigma$ generates $x'\sim {\textrm{T}}_{1-\xi } x$ by: with probability $1-\xi$ we take $x' = x$ , and otherwise we re-samples $x'\sim \mu _x$ . We let ${\textrm{T}}_{1-\xi }$ be the corresponding averaging operator on $L_2(\Sigma )$ , that is, ${\textrm{T}}_{1-\xi } f(x) = {\mathop {\mathbb{E}}_{{\textbf {x}}'\sim {\textrm{T}}_{1-\xi } x}{\left [ {f({\textbf {x}}')} \right ]}}$ .

2. 2. We define analogues of the operator ${\textrm{T}}_{1-\xi }$ on $L_2(\Gamma )$ , $L_2(\Phi )$ in the same way. For notational convenience, we will use the same notation for them, that is, ${\textrm{T}}_{1-\xi }$ , and it will be clear from the context which operator is applied.

3. 3. We define the operators $\textrm{E}_{1-\xi }$ . Let $\Sigma '\subseteq \Sigma$ be evidencing the fact that $\mu$ satisfies the relaxed base case. Consider the Markov chain on $\Sigma$ that on $x\in \Sigma$ generates $x'\sim \textrm{E}_{1-\xi } x$ by: if $x\in \Sigma \setminus \Sigma '$ , we take $x' = x$ . Otherwise, with probability $1-\xi$ we take $x' = x$ , and with probability $\xi$ we resample $x'\sim \mu _x\,|\,\Sigma '$ . We let $\textrm{E}_{1-\xi }$ be the corresponding averaging operator on $L_2(\Sigma )$ , that is, $\textrm{E}_{1-\xi } f(x) = {\mathop {\mathbb{E}}_{x'\sim \textrm{E}_{1-\xi } x}{\left [ {f(x')} \right ]}}$ .

The noise operator $T_{1-\xi }$ when applied on a function $f$ dampens (in the $\ell _2$ measure) the high-degree terms from $f$ . Analogously, as we will see, the operator $E_{1-\xi }$ when applied on a function dampens the high effective degree terms from the function.

4.2 Intermediate lemmas

With these operators, we can now state the relaxed analogues of Lemma 4.1 wherein the degree conditions are replaced with analogous soft truncations (but in return, we still retain the boundedness of the functions).

4.2.1 Softly truncating $g$ from both sides and $f$ from above

In the first relaxation, we softly truncate the degree of the function $g$ from both sides and the degree of the function $f$ from above using the noise operators $T_{1-\xi }$ .

Lemma 4.2. For all $\alpha \gt 0$ , $m\in \mathbb{N}$ , and $M\gt 0$ , there is $\xi _0\gt 0$ such that the following holds for all $0\lt \xi \leqslant \xi _0$ . Suppose $\mu$ is a distribution over $\Sigma \times \Gamma \times \Phi$ satisfying the conditions of Lemma 2.5 , and $f\colon \Sigma ^n\to \mathbb{C}$ , $g\colon \Gamma ^n\to \mathbb{C}$ , $h\colon \Phi ^n\to \mathbb{C}$ are $1$ -bounded functions. Then

\begin{align*} \left | {{\mathop {\mathbb{E}}_{({\textbf {x}},\textbf {y},\textbf {z})\sim \mu ^{\otimes n}}{\left [ {{\textrm{T}}_{1-M\xi /\log (1/\xi )^3}f({\textbf {x}}) ({\textrm{T}}_{1-\xi /2}-{\textrm{T}}_{1-\xi })g(\textbf {y})h(\textbf {z})} \right ]}}} \right | {\lesssim } \frac {1}{\log ^6(1/\xi )}. \end{align*}

4.2.2 Softly truncating the effective degree of $f$ from below

In the next relaxation, we further softly truncate the effective degree of $f$ from below. We do this using the operator $\textrm{E}_{1-\xi }$ .

Lemma 4.3. For all $\alpha \gt 0$ , $m\in \mathbb{N}$ , and $M\gt 0$ , there is $\xi _0\gt 0$ such that the following holds for all $0\lt \xi \leqslant \xi _0$ . Suppose $\mu$ is a distribution over $\Sigma \times \Gamma \times \Phi$ satisfying the conditions of Lemma 2.5 , and $f\colon \Sigma ^n\to \mathbb{C}$ , $g\colon \Gamma ^n\to \mathbb{C}$ , $h\colon \Phi ^n\to \mathbb{C}$ are $1$ -bounded functions. Then

\begin{equation*} \left | {{\mathop {\mathbb{E}}_{({\textbf {x}},\textbf {y},\textbf {z})\sim \mu ^{\otimes n}}{\left [ {({\textrm{I}}-\textrm{E}_{1-M\xi \log (1/\xi )^{100}}){\textrm{T}}_{1-M\xi /\log (1/\xi )^3}f({\textbf {x}}) ({\textrm{T}}_{1-\xi /2}-{\textrm{T}}_{1-\xi })g(\textbf {y})h(\textbf {z})} \right ]}}} \right | {\lesssim } \frac {1}{\log ^{6}(1/\xi )}. \end{equation*}

We defer the proofs of the implications Lemma 4.1 $\implies$ Lemma 4.3 $\implies$ Lemma 4.2 $\implies$ Lemma 2.5 to the Appendix. All these proofs follow from the standard arguments that use the fact that the noise operator essentially gets rid of the high degree part of the functions.

5.1 The parameter $\beta _{n,d_1,d_2,d_3}$

We begin by defining the parameter $\beta _{n,d_1,d_2,d_3}'$ for all $d_1,d_2,d_3,n\in \mathbb{N}$ (which is a close variant of $\beta _{n,d_1,d_2,d_3}$ which will be shortly defined):

\begin{equation*} \beta '_{n,d_1,d_2,d_3} = \max _{\substack { f\colon \Sigma ^n\to \mathbb{C}\text{ degree $d_1$ homogenous}, \\ \textsf { effdeg}(f)\geqslant d_1/\log ^{20}(d_1)\\ g\colon \Gamma ^n\to \mathbb{C}\text{ degree $d_2$ homogenous}, \\ h\colon \Phi ^n\to \mathbb{C}\text{ degree at most $d_3$ homogenous} }} \frac {\left | {{\mathop {\mathbb{E}}_{({\textbf {x}},\textbf {y},\textbf {z})\sim \mu ^{\otimes n}}{\left [ {f({\textbf {x}})g(\textbf {y})h(\textbf {z})} \right ]}}} \right |}{\| f \|_2\| g \|_2\| h \|_2}. \end{equation*}

Using Cauchy-Schwarz, it is clear that $\beta _{n,d_1,d_2,d_3}\leqslant 1$ always. The following lemma asserts that if $d_1,d_2,d_3$ are roughly the same up to poly-logs, then $\beta _{n,d_1,d_2,d_3}$ is actually almost exponentially decaying in $d$ .

Lemma 5.1. For all $K, m\in \mathbb{N}$ and $\alpha \gt 0$ , there are $C_, d_0\in \mathbb{N}$ such that the following holds. Let $\mu$ be a distribution over $\Sigma \times \Gamma \times \Phi$ as in Lemma 4.1, and let $d_1,d_2,d_3\leqslant n$ be such that $d_i\leqslant d_j \log ^{K}(d_j)$ for all $i,j=1,2,3$ , and $d_i\geqslant d_0$ for all $i$ . Then

\begin{equation*} \beta '_{n,\,d_1,\,d_2,\,d_3} \leqslant 2^{-\frac {d_1}{\log ^{C}(d_1)}}. \end{equation*}

We note that Lemma 5.1 immediately implies Lemma 4.1, hence we will focus henceforth on proving that 5.1 is true. We will actually need to adjust the parameter $\beta '_{n,\,d_1,\,d_2,\,d_3}$ a bit and state a statement similar to Lemma 5.1 but stronger. The main difference will be that instead of working with the function $f$ over $\textbf {x}$ , we will work with the function $F(\textbf {y},\textbf {z})$ . Given a function $f$ , as in $\mu$ we have that $\textbf {y},\textbf {z}$ implies $\textbf {x}$ , we may define $F\colon (\Gamma ^{n}\times \Phi ^n,\mu _{y,z}^{\otimes n})\to \mathbb{C}$ by

\begin{equation*} F(\textbf {y},\textbf {z}) = f({\textbf {x}}), \end{equation*}

where $\textbf {x}$ is the unique point in $\Sigma ^{n}$ such that $(x_i,y_i,z_i)\in \textsf { supp}(\mu )$ for all $i\in [n]$ . We denote this as an operator

\begin{equation*} W\colon L_2(\Sigma ^n,\mu _x^{\otimes n})\to L_2(\Gamma ^n\times \Phi ^n,\mu _{y,z}^{\otimes n}),\text{ so that } F = W f. \end{equation*}

It will be more convenient for us to work with $F$ since the distribution over $y,z$ is uniform (whereas the distribution over $x$ may not be), but to facilitate that we need to translate the information that $f$ is homogeneous and has high effective degree.

5.1.1 Setting up the basis to define the effective degree of $F$

Consider any two basis elements $\chi ,\chi '\in B_1\cup B_2$ , and note that

\begin{equation*} \langle {W \chi },{W\chi '}\rangle = \langle {\chi },{\chi '}\rangle = 1_{\chi = \chi '}, \end{equation*}

so ${\left \{ W\chi \right \}}_{B_1\cup B_2}$ form a partial orthonormal basis for $L_2(\Gamma ^n\times \Phi ^n,\mu _{y,z}^{\otimes n})$ . We may complete it to an orthonormal basis using some set ${\left \{ \tilde {\chi } \right \}}_{\tilde {\chi }\in C}$ . Thus, any function $F\colon \Gamma ^n\times \Phi ^n\to \mathbb{C}$ may be written as

\begin{equation*} F(\textbf {y},\textbf {z}) = \sum \limits _{\tilde {\chi }\in (W B_1\cup W B_2\cup C)^{\otimes n}} \widehat {F}(\tilde {\chi })\tilde {\chi }(\textbf {y},\textbf {z}), \end{equation*}

where

\begin{equation*} \tilde {\chi }(\textbf {y},\textbf {z}) =\prod \limits _{i=1}^{n}\tilde {\chi }_i(y_i,z_i), \quad \text{ and } \widehat {F}(\tilde {\chi }) =\langle {F},{\tilde {\chi }}\rangle . \end{equation*}

Noting that the function $W \chi _{\textsf { const}}$ is the all $1$ function, we define the following notion of degree for monomials over $\textbf {y},\textbf {z}$ .

Definition 5.2. The degree of $\tilde {\chi }\in (W B_1\cup W B_2\cup C)^{\otimes n}$ is defined to be the number of coordinates $i$ such that $\tilde {\chi }_i\neq W \chi _{\textsf { const}}$ .

We also define the effective degree.

Definition 5.3. The effective degree of $\tilde {\chi }\in (W B_1\cup W B_2\cup C)^{\otimes n}$ is defined to be the number of coordinates $i$ such that $\tilde {\chi }_i\in W B_2$ .

Finally, we define a property of functions $F(\textbf {y},\textbf {z})$ which is equivalent to it being in the image of the operator $W$ .

Definition 5.4. Consider the graph $G = (V,E)$ whose vertices are $\Gamma \times \Phi$ and $(y,z)$ and $(y',z')$ are adjacent if there is some $x\in \Sigma$ such that $(x,y,z),(x,y',z')\in \textsf { supp}(\mu )$ . Consider the graph $G^{\otimes n} = (V^{\otimes n}, E')$ where $E' = \left \{ \left . ((\textbf {y},\textbf {z}),(\textbf {y}',\textbf {z}')) \;\right \vert ((y_i,z_i), (y_i', z_i'))\in E\,\forall i \right \}$ .

We say a function $F\colon \Gamma ^{n}\times \Phi ^{n}\to \mathbb{C}$ is constant on connected components if $F$ is constant on all of the connected components of $G^{\otimes n}$ .

We have the following claim.

Claim 5.5. For a function $F\colon \Gamma ^{n}\times \Phi ^{n}\to \mathbb{C}$ , the following are equivalent:

1. 1. $F$ is constant on connected components;

2. 2. There is $f\colon \Sigma ^{n}\to \mathbb{C}$ such that $F = W f$ .

3. 3. $\widehat {F}(\chi ) = 0$ for $\chi \not \in (W B_1\cup W B_2)^{\otimes n}$ .

Proof. It is clear that the third item implies the second item, and that the second item implies the first item. We next show that the first item implies the third item. Towards this end, assume that $F$ is constant on connected component, and let $\chi \not \in (W B_1\cup W B_2)^{\otimes n}$ . Then $\chi _i \in C$ for some $i=1,\ldots ,n$ , without loss of generality assume that $i=1$ . Then

\begin{equation*} \widehat {F}(\chi ) ={\mathop {\mathbb{E}}_{(\textbf {y}_{-1},\textbf {z}_{-1})\sim \mu _{y,z}^{\otimes n-1}}}\left [ {\prod \limits _{i=2}^{n}\overline {\chi _i(y_i,z_i)} {\mathop {\mathbb{E}}_{(y_1,z_1)\sim \mu _{y,z}}{\left [ {F(\textbf {y},\textbf {z})\overline {\chi _1(y_1,z_1)}} \right ]}}}\right ] . \end{equation*}

Fix $\textbf {y}_{-1} =y_{-1}$ and $\textbf {z}_{-1} = z_{-1}$ ; we show that the inner expectation is $0$ . To see this, first note that since $F_{{\left \{ 2,\ldots ,n \right \}}\rightarrow (y_{-1},z_{-z})}(y_1,z_1)$ , as a function of $y_1,z_1$ , is constant on connected components, it suffices to show that each basis element in $C$ is perpendicular to functions that are constant on connected components. To see that, it suffices to show that for each connected component $C$ of $G$ , the indicator of it $1_C(y_1,z_1)$ is in $\textsf { span}(W B_1\cup W B_2)$ , and we next show this is true.

Indeed, let $C$ be some connected component. Define

\begin{equation*} f(x) = 1_C(y,z) \end{equation*}

where $(y,z)$ are chosen so that $(x,y,z)\in \textsf { supp}(\mu )$ . We note that this is well defined, since if we have two $(y,z)$ and $(y',z')$ such that $(x,y,z)$ and $(x,y',z')$ are both in $\textsf { supp}(\mu )$ , then they lie in the same connected component of $H$ and hence $1_C(y,z) = 1_C(y',z')$ . It follows that $1_C = W f$ , hence $1_C\in \textsf { span}(W B_1\cup W B_2)$ .

The following claim encapsulates the properties we need about $F = W f$ .

Claim 5.6. Suppose that $f\colon \Sigma ^n\to \mathbb{C}$ is a degree $d$ homogeneous function, and for all $\chi$ such that $\widehat {f}(\chi ) \neq 0$ we have $\textsf { effdeg}(\chi )\geqslant d'$ . Then $F = Wf$ is a degree $d$ homogeneous function, it is constant on connected components and for all $\tilde {\chi }$ such that $\widehat {F}(\tilde {\chi })\neq 0$ we have that $\textsf { effdeg}(\chi )\geqslant d'$ .

Proof. Note that writing $f({\textbf {x}}) = \sum \limits _{\chi \in (B_1\cup B_2)^{\otimes n}}\widehat {f}(\chi ) \chi ({\textbf {x}})$ , we get that

\begin{equation*} F(\textbf {y},\textbf {z}) = Wf(\textbf {y},\textbf {z}) = \sum \limits _{\chi \in (B_1\cup B_2)^{\otimes n}}\widehat {f}(\chi ) (W\chi )(\textbf {y},\textbf {z}), \end{equation*}

so we get that $\widehat {F}(\tilde {\chi })$ is non-zero only if $\tilde {\chi }\in (WB_1\cup WB_2)^{\otimes n}$ , and it is equal to $\widehat {f}(\chi )$ where $\tilde {\chi } = W\chi$ . From this and Claim 5.5, the assertions of the claim immediately follow.

5.1.2 The main inductive statement

We are now ready to define $\beta _{n,d_1,d_2,d_3}$ and state a stronger form of Lemma 5.1. We define

\begin{equation*} \beta _{n,d_1,d_2,d_3} = \max _{\substack { F\colon \Gamma ^n\times \Phi ^n\to \mathbb{C}\text{ degree $d_1$ homogenous}, \\ \text{$F$ is constant on connected components},\\ \textsf { effdeg}(F)\geqslant d_1/\log ^{20}(d_1),\\ g\colon \Gamma ^n\to \mathbb{C}\text{ degree $d_2$ homogenous}, \\ h\colon \Phi ^n\to \mathbb{C}\text{ degree at most $d_3$ homogenous}. }} \frac {\left | {{\mathop {\mathbb{E}}_{({\textbf {x}},\textbf {y},\textbf {z})\sim \mu ^{\otimes n}}{\left [ {F(\textbf {y},\textbf {z})g(\textbf {y})h(\textbf {z})} \right ]}}} \right |}{\| F \|_2\| g \|_2\| h \|_2}. \end{equation*}

Lemma 5.7. For all $K, m\in \mathbb{N}$ , and $\alpha \gt 0$ , there are $C_, d_0\in \mathbb{N}$ such that the following holds. Let $\mu$ be a distribution over $\Sigma \times \Gamma \times \Phi$ as in Lemma 4.1 , and let $d_1,d_2,d_3\leqslant n$ be such that $d_i\leqslant d_j \log ^{K}(d_j)$ for all $i,j=1,2,3$ , and $d_i\geqslant d_0$ for all $i$ . Then

\begin{equation*} \beta _{n,d_1,d_2,d_3} \leqslant 2^{-\frac {d_1}{\log ^{C}(d_1)}}. \end{equation*}

Lemma 5.1 follows from Lemma 5.7 by the virtue of Claim 5.6.

The proof of Lemma 5.7 spans Sections 6, 7, and in the rest of the current section we set up some machinery and explain the high level overview of the argument. Our overall approach to the proof of Lemma 5.7 is inductive, and so we will need to be able to write down a function over $n$ variables as linear combination of products of a function of $n-1$ variables and a function over a single variable. Towards this end, we employ the singular-value decomposition as presented in the next section.

Throughout, we will have a partition the set of coordinates $[n]$ into $I\cup J$ where $\left | {I} \right | = n-1$ and $\left | {J} \right | = 1$ , which for now will be arbitrary (we will later explain how to choose it so that it satisfies several additional properties that we need).

5.2 SVD decompositions

In this section, we state claims asserting an SVD decomposition for homogeneous and non-homogeneous functions. The proofs of these claims are deferred to the Appendix. Throughout, $I, J$ is a partition of $[n]$ wherein $\left | {J} \right | = 1$ and $\left | {I} \right | = n-1$ .

5.2.1 The SVD decompositions for homogeneous functions

The following decomposition claim is phrased in terms of the function $g$ . However, it applies to functions over $z$ , as well as to functions over $y,z$ . We will use it for both $g$ and $h$ , and may use it also for $F$ . However, for $F$ we need a few additional properties, which we establish in Claim 5.10.

Claim 5.8. If $g\colon \Gamma ^n\to \mathbb{C}$ is a homogeneous function of degree $d$ and $\| g \|_2=1$ , then we may write

\begin{equation*} g(\textbf {y}) = \sum \limits _{r=1}^{m} \lambda _r g_r(\textbf {y}_I) g_r'(\textbf {y}_J), \end{equation*}

and $R = \left \{ \left . i \;\right \vert \lambda _i\neq 0 \right \}$ where

1. 1. For $r\in R$ , $g_r\colon \Gamma ^{I}\to \mathbb{C}$ is an orthonormal set of functions.

2. 2. If $1\in R$ , then $g_1$ is homogeneous of degree $d$ and for $r\geqslant 2$ in $R$ the function $g_r$ is homogeneous of degree $d-1$ .

3. 3. For $r\in R$ , $g_r'\colon \Gamma ^{J}\to \mathbb{C}$ is an orthonormal set of functions.

4. 4. $g_1'$ is constant.

5. 5. Each $\lambda _i$ is a non-negative real number and $\sum \limits _{r=1}^{m} \lambda _i^2 = 1$ .

Proof. We defer this proof to the Appendix.

A natural question is what can be said about the coefficients $\lambda _i$ in the above SVD decomposition, and indeed this will help us in choosing an appropriate partition. We have

Claim 5.9. Let $g\colon \Gamma ^n\to \mathbb{C}$ is a homogeneous function of degree $d_1$ and $\| g \|_2=1$ , and write

\begin{equation*} g(\textbf {y}) = \sum \limits _{r=1}^{m} \lambda _r g_r(\textbf {y}_I) g_r'(\textbf {y}_J), \end{equation*}

as in Claim 5.8 . If $j$ is the unique variable in the set $J$ in the partition $[n] = I\cup J$ , then

\begin{equation*} \lambda _1^2 = 1 - \frac {1}{2}I_{j}[g]. \end{equation*}

Proof. Consider $I_j[g]$ , and note that

\begin{align*} I_j[g] = {\mathop {\mathbb{E}}_{\substack {\textbf {y}\sim \mu ^{n-1}\\ a, b}}{\left [ {\left | {g(\textbf {y}_I,a) - g(\textbf {y}_I,b)} \right |^2} \right ]}} &= {\mathop {\mathbb{E}}_{\substack {\textbf {y}\sim \mu ^{n-1}\\ a, b}}{\left [ {\left |\sum \limits _{r}\lambda _r g_r(\textbf {y}_I)(g_r'(a) - g_r'(b))\right |^2} \right ]}}\\ &= \sum \limits _{r_1,r_2} \lambda _{r_1}\overline {\lambda _{r_2}} \langle {g_{r_1}},{g_{r_2}}\rangle {\mathop {\mathbb{E}}_{a, b}{\left [ {(g_{r_1}'(a) - g_{r_1}'(b))\overline {(g_{r_2}'(a) - g_{r_2}'(b))}} \right ]}}. \end{align*}

For $r_1\neq r_2$ we have $\langle {g_{r_1}},{g_{r_2}}\rangle = 0$ , so the last sum is equal to

\begin{equation*} \sum \limits _{r}\lambda _r^2 {\mathop {\mathbb{E}}_{a, b}{\left [ {\left | {g_{r}'(a) - g_{r}'(b)} \right |^2} \right ]}} = 2\sum \limits _{r}\lambda _r^2 \textsf { var}(g_r') =2\sum \limits _{r\neq 1}\lambda _r^2, \end{equation*}

as the variance of $g_1'$ is $0$ , and the variance of any other $g_r'$ is $1$ . Hence

\begin{equation*} 1-\frac {I_j[g_r]}{2} = 1-\sum \limits _{r\neq 1}\lambda _r^2 = \lambda _1^2. \end{equation*}

We next state an SVD decomposition statement that addresses the function $F$ .

Claim 5.10. Suppose $F\colon \Gamma ^n\times \Phi ^n\to \mathbb{C}$ is a homogeneous function of degree $d$ which is constant on connected components, $\| F \|_2=1$ , and the effective degree of each monomial in $F$ is at least $d'$ . Then we may write

\begin{equation*} F(\textbf {y},\textbf {z}) = \sum \limits _{t=1}^{m} \gamma _t F_t(\textbf {y}_I,\textbf {z}_I) F_t'(\textbf {y}_J,\textbf {z}_J), \end{equation*}

and $T = \left \{ \left . t \;\right \vert \gamma _t\neq 0 \right \}$ where

1. 1. For $t\in T$ , $F_t\colon \Gamma ^{I}\times \Phi ^{I}\to \mathbb{C}$ is an orthonormal set of functions.

2. 2. If $1\in T$ , then $F_1$ is homogeneous of degree $d$ and for $t\geqslant 2$ in $T$ the function $F_t$ is homogeneous of degree $d-1$ .

3. 3. If $1\in T$ , then in $F_1$ the effective degree of monomial is at least $d'$ and for $T\geqslant 2$ in $T$ the effective degree of each monomial in $F_t$ is at least $d'-1$ .

4. 4. The functions $F_t$ and $F_t'$ are constant on connected components for all $t$ .

5. 5. For $t\in T$ , $F_t'\colon \Gamma ^{J}\times \Phi ^J\to \mathbb{C}$ is an orthonormal set of functions.

6. 6. $F_1'$ is constant.

7. 7. Each $\gamma _t$ is a non-negative real number and $\sum \limits _{t=1}^{m} \gamma _t^2 = 1$ .

Proof. The proof of this claim is also deferred to the Appendix.