SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Definition D.5 (Statistical Dimension). For the hypothesis testing problem of Definition D.2, we define the statistical dimension  $\mathrm{SDA}(\mathcal{S},\mu ,n)$  as follows:
$$
\operatorname {S D A} (\mathcal {S}, \mu , n) = \max  \left\{q \in \mathbb {N}: \underset {u, v \sim \mu} {\mathbf {E}} [ | \langle \bar {D} _ {u}, \bar {D} _ {v} \rangle_ {D _ {0}} - 1 |   | E ] \leq \frac {1}{n} f o r a l l e v e n t s E s. t. \underset {u, v \sim \mu} {\mathbf {P r}} [ E ] \geq \frac {1}{q ^ {2}} \right\}.
$$
We will omit writing  $\mu$  when it is clear from the context.
D.2 Proof of Theorem D.4
To prove Theorem D.4, we first need to bound the SDA of our setting. The following lemma translates the  $(\gamma ,\beta)$ -correlation of  $S$  to a lower bound for the statistical dimension of the hypothesis testing problem. The proof is very similar to that of Corollary 8.28 of  $\mathrm{[BBH^{+}20]}$  but it is given below for completeness.
Lemma D.6. Let  $0 < c < 1/2$  and  $d, m \in \mathbb{Z}_+$ . Consider the hypothesis testing problem of Definition D.3. Then, for any  $q \geq 1$ ,
$$
\operatorname {S D A} \left(\mathcal {D}, \left(\frac {p ^ {- 1} \Omega (d) ^ {1 / 2 - c}}{p \left(q ^ {2} / 2 ^ {\Omega \left(d ^ {c / 2}\right)} + 1\right)}\right)\right) \geq q.
$$
Proof. The first part is to calculate the correlation of the set  $S$ . By Theorem 3.2, we know that the set  $S$  is  $(\gamma, \beta)$ -correlated with  $\gamma = p^2\Omega(d)^{c - 1/2}$  and  $\beta = 4p$ .
We next calculate the SDA according to Definition D.5. We denote by  $\bar{E}_{\mathbf{v}}$  the ratios of the density of  $E_{\mathbf{v}}$  to the density of  $R$ . Note that the quantity  $\langle \bar{E}_{\mathbf{u}}, \bar{E}_{\mathbf{v}} \rangle - 1$  used there is equal to  $\langle \bar{E}_{\mathbf{u}} - 1, \bar{E}_{\mathbf{v}} - 1 \rangle$ . Let  $E$  be an event that has  $\mathrm{Pr}_{\mathbf{u}, \mathbf{v} \sim \mu}[E] \geq 1 / q^2$ . For  $d$  sufficiently large we have that
$$
\begin{array}{l} \underset {u, v \sim \mu} {\mathbf {E}} [ | \langle \bar {E} _ {\mathbf {u}}, \bar {E} _ {\mathbf {v}} \rangle - 1 | E ] \leq \min  \left(1, \frac {1}{| S | \Pr [ E ]}\right) \beta + \max  \left(0, 1 - \frac {1}{| S | \Pr [ E ]}\right) \gamma \\ \leq p \left(\frac {q ^ {2}}{2 ^ {\Omega (d ^ {c})}} + \frac {p}{\Omega (d) ^ {1 / 2 - c}}\right) = p \left(\frac {p ^ {- 1} \Omega (d) ^ {1 / 2 - c}}{q ^ {2} / 2 ^ {\Omega (d ^ {c / 2})} + 1}\right) ^ {- 1}, \\ \end{array}
$$
where the first inequality uses that  $\mathbf{Pr}[\mathbf{u} = \mathbf{v}|E] = \mathbf{Pr}[\mathbf{u} = \mathbf{v}, E] / \mathbf{Pr}[E]$  and bounds the numerator in two different ways:  $\mathbf{Pr}[\mathbf{u} = \mathbf{v}, E] / \mathbf{Pr}[E] \leq \mathbf{Pr}[\mathbf{u} = \mathbf{v}] / \mathbf{Pr}[E] = 1 / (|\mathcal{S}| \mathbf{Pr}[E])$  and  $\mathbf{Pr}[\mathbf{u} = \mathbf{v}, E] / \mathbf{Pr}[E] \leq \mathbf{Pr}[E] / \mathbf{Pr}[E] = 1$ .
In  $\mathrm{[BBH^{+}20]}$ , the following relation between SDA and low-degree likelihood ratio is established.
Fact D.7 (Theorem 4.1 of  $[\mathrm{BBH}^+20])$ . Let  $\mathcal{D}$  be a hypothesis testing problem on  $\mathbb{R}^d$  with respect to null hypothesis  $D_0$ . Let  $n, k \in \mathbb{N}$  with  $k$  even. Suppose that for all  $0 \leq n' \leq n$ ,  $\mathrm{SDA}(S, n') \geq 100^k (n / n')^k$ . Then, for all  $r$ ,  $\left\| \mathbf{E}_{u \sim \mu}\left[(\bar{D}_u^{\otimes n})^{\leq r, \Omega(k)}\right] - 1\right\|_{D_0^{\otimes n}}^2 \leq 1$ .
In Lemma D.6 we set  $n = \Omega(d)^{1/2 - c} / p^2$  and  $q = \sqrt{2^{\Omega(d^{c/2})}(n / n')}$ . Then,  $\mathrm{SDA}(\mathcal{S}, n') \geq \sqrt{2^{\Omega(d^{c/2})}(n / n')} \geq (100n / n')^k$  for  $k < d^{c/4}$  and then we apply the theorem above.