SlowGuess/NeurIPS_2023_OCR huggingface dataset:
$$
\mathrm {M D} ^ {2} (\mathbf {x} \mid \mathbf {y}) \stackrel {\text {d e f}} {=} E \left\{\left(\mathbf {x} _ {1} - E \mathbf {x}\right) ^ {\mathrm {T}} \left(\mathbf {x} _ {2} - E \mathbf {x}\right) L \left(\mathbf {y} _ {1}, \mathbf {y} _ {2}\right) \right\},
$$
where  $(\mathbf{x}_1,\mathbf{y}_1)$  and  $(\mathbf{x}_2,\mathbf{y}_2)$  are independent copies of  $(\mathbf{x},\mathbf{y})$ . We show in Lemma 1 that,  $\mathrm{MD}^2 (\mathbf{x}\mid$ $\mathbf{y}) = 0$  if and only if  $E(\mathbf{x}\mid \mathbf{y}) = 0$ . That is,  $\mathrm{MD}^2 (\mathbf{x}\mid \mathbf{y})$  measures the degree of conditional mean of  $\mathbf{x}$  given  $\mathbf{y}$ , which quantifies the difference between  $E(\mathbf{x}\mid \mathbf{y})$  and  $E\mathbf{x}$ .
Lemma 1. Assume the kernel  $L(\mathbf{y}_1, \mathbf{y}_2) = l(\| \mathbf{y}_1 - \mathbf{y}_2 \| / \gamma_{\mathbf{y}})$  is a characteristic, bounded continuous, and real-valued positive definite function, then  $\mathrm{MD}^2(\mathbf{x} \mid \mathbf{y}) = 0$  if and only if  $E(\mathbf{x} \mid \mathbf{y}) = 0$  almost surely.
We remark here that characteristic condition in Lemma 1 is used to guarantee that the HSIC is nonnegative and equals zero if and only if  $\mathbf{x}$  and  $\mathbf{y}$  are independent. One may refer to (Sriperumbudur et al., 2010) for more details about this condition. Furthermore, we let  $k_0^{(i)}$  be the  $i$ -th derivative of  $k_0(\cdot)$  evaluated at  $E\| \mathbf{x}_1^* -\mathbf{x}_2^*\| ^2$ , and  $l_0^{(j)}$  is the  $j$ -th derivative of  $l_{0}(\cdot)$  evaluated at  $E\| \mathbf{y}_1^* -\mathbf{y}_2^*\| ^2$ . Let  $\mathbf{x}^{\otimes n}$  be the  $n$ -th Kronecker power of  $\mathbf{x}$ , which is defined as  $\mathbf{x}^{\otimes n} = \mathbf{x}\otimes \mathbf{x}^{\otimes (n - 1)},\mathbf{x}^{\otimes 1} = \mathbf{x}$ , and  $\otimes$  denotes the Kronecker product. In addition, we use  $\| \cdot \| _F$  to represent the Frobenius norm of a matrix. Then we expand HSIC(x,y) in the following Theorem.
Theorem 3. Assume (A1) and (A2) hold true. Then under the alternative hypothesis,
1. when  $p \to \infty$  and  $q$  remains fixed as  $n \to \infty$ , if  $E(\mathbf{x}^{\otimes t} \mid \mathbf{y}) = E(\mathbf{x}^{\otimes t})$  hold true for all  $t < s$  for some  $s \in \mathbb{N}^{+}$ , then HSIC(x,y) =  $O(p^{-s\kappa_{\mathbf{x}} / 2})$ , and
$$
\mathrm {H S I C} (\mathbf {x}, \mathbf {y}) = k _ {0} ^ {(s)} \sum_ {2 a + c = s} \frac {(- 2) ^ {c}}{a ! a ! c !} \mathrm {M D} ^ {2} (\mathbf {x} ^ {* \otimes c} \| \mathbf {x} ^ {*} \| ^ {2 a} | \mathbf {y}) + o (p ^ {- s \kappa_ {\mathbf {x}} / 2}),
$$
2. when  $p \to \infty$  and  $q \to \infty$  as  $n \to \infty$ , if  $\operatorname{cov}(\mathbf{x}^{\otimes t_1}, \mathbf{y}^{\otimes t_2}) \neq \mathbf{0}$  only when  $t_1 \geq s_1$  and  $t_2 \geq s_2$  for some  $s_1, s_2 \in \mathbb{N}^+$ , then  $\mathrm{HSIC}(\mathbf{x}, \mathbf{y}) = O(p^{-s_1 \kappa_{\mathbf{x}} / 2} q^{-s_2 \kappa_{\mathbf{y}} / 2})$ , and
$$
\begin{array}{l} \operatorname {H S I C} (\mathbf {x}, \mathbf {y}) = \sum_ {2 a _ {1} + c _ {1} = s _ {1}} \sum_ {2 a _ {2} + c _ {2} = s _ {2}} \frac {k _ {0} ^ {(s _ {1})} (- 2) ^ {c _ {1}}}{a _ {1} ! a _ {1} ! c _ {1} !} \frac {l _ {0} ^ {(s _ {2})} (- 2) ^ {c _ {2}}}{a _ {2} ! a _ {2} ! c _ {2} !} \left\| \operatorname {c o v} \{\| \mathbf {x} ^ {*} \| ^ {2 a _ {1}} \mathbf {x} ^ {* \otimes c _ {1}}, \| \mathbf {y} ^ {*} \| ^ {2 a _ {2}} \mathbf {y} ^ {* \otimes c _ {2}} ^ {T} \} \right\| _ {F} ^ {2} \\ + o (p ^ {- s _ {1} \kappa_ {\mathbf {x}} / 2} q ^ {- s _ {2} \kappa_ {\mathbf {y}} / 2}). \\ \end{array}
$$
Theorem 3 enables us to gain some insights into the HSIC in high dimensions. First of all, because when  $2a + c = s$ ,  $\mathbf{x}^{*\otimes c}\| \mathbf{x}^*\|^{2a}$  is the  $s$ -th polynomial for  $\mathbf{x}$ , and  $\mathrm{MD}^2 (\mathbf{x}^{*\otimes c}\| \mathbf{x}^*\|^{2a}\mid \mathbf{y})$  measures the departure from the conditional mean independence of  $\mathbf{x}$  given  $\mathbf{y}$ . Then the first part of Theorem 3 implies that when  $p\to \infty$  and  $q$  remains fixed as  $n\to \infty$ , if  $E(\mathbf{x}^{\otimes t}\mid \mathbf{y}) = E(\mathbf{x}^{\otimes t})$  hold true for all  $t < s$  for some  $s\in \mathbb{N}^+$ , the HSIC quantifies the departure from  $E(\mathbf{x}^{\otimes s}\mid \mathbf{y}) = E(\mathbf{x}^{\otimes s})$ . In other words, when only one dimension of the covariates diverges to infinity, the HSIC first measures the conditional mean of  $\mathbf{x}$  given  $\mathbf{y}$ . If there is no such mean dependence, it turns to measure the conditional mean of  $\mathbf{x}^{\otimes 2}$  given  $\mathbf{y}$ , etc. Similarly, when  $p\to \infty$  and  $q\to \infty$  as  $n\to \infty$ , the second part of Theorem 3 reveals that, the HSIC may first search for the covariance between  $\mathbf{x}$  given  $\mathbf{y}$ . If such covariance equals zero, it then turns to search for the covariance between  $\mathbf{x}^{\otimes 2}$  and  $\mathbf{y}$ , or the covariance between  $\mathbf{x}$  and  $\mathbf{y}^{\otimes 2}$ , etc. Therefore, Theorem 3 characterizes the changing process of the ability to measure nonlinear dependence in high dimensions for HSIC. As a comparison, Zhu et al. (2020) only showed that, when both dimensions diverge much faster than the sample size, the HSIC can only capture the marginal linear dependence. From this point of view, our result is a generalization of theirs.
When the HSIC is used to test for statistical independence, Theorem 2 and Proposition 1 ensure that, when  $n^{1/2} p^{\kappa_{\mathbf{x}}/2} q^{\kappa_{\mathbf{y}}/2}$  HSIC(x,y) → ∞, the HSIC based test would have asymptotic power 1. To make this condition hold true, the first part of Theorem 3 implies that when  $p \to \infty$  and  $q$  remains fixed as  $n \to \infty$ , if there only exists the conditional mean of  $\mathbf{x}^{\otimes s}$  given  $\mathbf{y}$ , it is required that  $p^{(s-1)\kappa_{\mathbf{x}}} = o(n)$  because HSIC(x,y) is of order  $O(p^{-s\kappa_{\mathbf{x}}/2})$  in this case. In other words, when  $n = O\{p^{(s-1)\kappa_{\mathbf{x}}}\}$ , the HSIC based test can have nontrivial power only when there exists the  $i$ -th polynomial mean dependence of  $\mathbf{x}$  given  $\mathbf{y}$  for some  $i < s$ . Similarly, when both  $p$  and  $q$  diverges as  $n \to \infty$ , the second part of Theorem 3, when there only exists the covariance between  $\mathbf{x}^{\otimes s_1}$  and  $\mathbf{y}^{\otimes s_2}$ , the test based on HSIC can have nontrivial power only when  $p^{(s_1-1)\kappa_{\mathbf{x}}} q^{(s_2-1)\kappa_{\mathbf{y}}} = o(n)$ . Because  $\kappa_{\mathbf{x}}$  represents the degree of dependence within covariates, Theorem 3 successfully connects the association type between  $\mathbf{x}$  and  $\mathbf{y}$ , with the sample size, the high dimensionality, and the dependence structures within covariates. We remark here that Gao et al. (2021) only justified that the test based on distance correlation can have power approaching 1 when  $p = q = o(n^{1/2})$  if there exist a pure nonlinear dependence between  $\mathbf{x}$  and  $\mathbf{y}$ . Therefore, their result is in spirit a special case of ours.
5 Numerical Studies
5.1 Simulations
In this subsection, we conduct some simulation studies to validate our theoretical conclusions on the HSIC based test in high dimensions. In particular, in Example 1, we show that, as long as the dimensions are not very small, the normal approximation can approximate the null distribution quite well, although it requires the dimension to diverge to infinity in theory. In Example 2, we fix the dimension of  $\mathbf{y}$  and increase the dimension of  $\mathbf{x}$ , and investigate the finite sample power performance of the HSIC based tests under various mean dependence types. In Example 3, we increase the dimensions of  $\mathbf{x}$  and  $\mathbf{y}$  simultaneously, and inspect how the empirical power is influenced by the covariate dimensions under different relationships.
Throughout the simulations, we choose two kinds of commonly used kernels to implement the HSIC based tests, i.e., Gaussian and Laplacian. For the Gaussian kernel, we set  $K(\mathbf{x}_1,\mathbf{x}_2) =$
$\exp \{-\| \mathbf{x}_1 - \mathbf{x}_2\| ^2 /(2\gamma_{\mathbf{x}}^2)\}$  and  $L(\mathbf{y}_1,\mathbf{y}_2) = \exp \{-\| \mathbf{y}_1 - \mathbf{y}_2\| ^2 /(2\gamma_{\mathbf{y}}^2)\}$ . While for the Laplacian kernel, we set  $L(\mathbf{x}_1,\mathbf{x}_2) = \exp (-\| \mathbf{x}_1 - \mathbf{x}_2\| /\gamma_{\mathbf{x}})$  and  $L(\mathbf{y}_1,\mathbf{y}_2) = \exp (-\| \mathbf{y}_1 - \mathbf{y}_2\| /\gamma_{\mathbf{y}})$ . For both choices of kernels, we set the bandwidth parameters to be  $\gamma_{\mathbf{x}} = c_0\gamma_{\mathbf{x}}^m$  and  $\gamma_{\mathbf{y}} = c_0\gamma_{\mathbf{y}}^m$  and vary  $c_{0}$  from 0.5 to 2. Here,  $\gamma_{\mathbf{z}}^{m}$  represents the median of  $\{\| \mathbf{z}_i - \mathbf{z}_j\| _1\leq i < j\leq n\}$ , and  $\mathbf{z}$  is either  $\mathbf{x}$  or  $\mathbf{y}$ . Due to space constraints, we present results specifically for the case when  $\gamma_{\mathbf{x}} = \gamma_{\mathbf{x}}^{m}$  and  $\gamma_{\mathbf{y}} = \gamma_{\mathbf{y}}^{m}$ . The remaining results are summarized in the Supplementary Material. Notably, our findings remain consistent across different values of  $\gamma_{\mathbf{x}}$  and  $\gamma_{\mathbf{y}}$ , showing the stability and reliability of our approach.
Example 1. In this example, we examine the normal approximation accuracy under the null hypothesis. Towards this goal, we simply generate  $\mathbf{x} = (X_1,\dots ,X_p)^\mathrm{T}\in \mathbb{R}^p$  from a multivariate normal distribution with mean zero and covariance matrix  $\boldsymbol{\Sigma}_{\mathbf{x}} = \mathbf{I}_{p\times p}$ . Then we independently generate  $\mathbf{y} = (Y_{1},\ldots ,Y_{q})^{\mathrm{T}}\in \mathbb{R}^{q}$  from another a multivariate normal distribution whose coordinates follow iid standard normal distributions. We fix the sample size to be 100 and consider two scenarios for the covariate dimensions. In the first scenario, we fix  $q = 1$  and vary  $p$  from  $\{5,25,100\}$ . In the second scenario, we set  $p = q = d$  and vary  $d$  from  $\{2,5,10\}$ . For each setting, we repeat the experiments 5000 times and compare the empirical null distributions of the test statistics,  $\{2^{-1 / 2}n\mathrm{hCorr}_n^2 (\mathbf{x},\mathbf{y})\}$ , with the standard normal distribution. Specifically, we display the kernel density curves of the test statistics for the two scenarios in Figures 1 and 2, respectively.
(A):  $p = 5$
(B):  $p = 25$
Figure 1: The kernel densities of the test statistics under the null hypothesis computed from 5000 simulations. We fix  $q = 1$  and vary  $p$  from \{5, 25, 100\}. The horizontal axes represent the observed values of the test statistics, and the vertical axes represent the kernel densities of those values. We choose two different kernels to implement the tests, i.e., Gaussian (dashed line) and Laplacian (dotted line). The solid line is the reference curve, which is the density of the standard normal distribution.
(C):  $p = 100$
According to Figures 1 and 2, we can see that, when the dimensions are small, the normal distribution cannot approximate the null distribution well. For example, when  $p = q = 2$ , the empirical null distributions for the tests with both Gaussian and Laplacian kernels are quite distinct from the reference curve in Figure 2 (A). However, as the dimensions increase, the accuracy of the normal approximation for the null distribution becomes quite precise, regardless of the choice of the kernels. For example, in both Figures 1 and 2, the empirical null distributions are almost the same as the reference curve as long as  $pq$  are not smaller than 25.
Example 2. In this example, we investigate how the empirical power is influenced by the dimension when only one of the covariate dimensions increases. For simplicity, we fix the sample size to be  $n = 100$ , as that in Example 1. As for the covariate dimensions, we fix  $q = 1$  and vary  $p$  from \{30, 50, 100, 200, 500, 1000\}. We consider three models to inspect the power performances. For Models (I) and (II), we generate  $\mathbf{x}$  from a multivariate normal distribution with mean zero and covariance matrix  $\Sigma_{\mathbf{x}} = (0.5^{|i - j|})_{p\times p}$ . Then we generate an independent error term  $\varepsilon$  that follows standard normal distribution. The univariate response  $Y$  is generated through
Model (I):  $Y = X_{1} + \ldots + X_{p} + \varepsilon$ ;
Model (II):  $Y = X_1^2 + \ldots + X_p^2 + \varepsilon$ .
For Model (III), we first generate  $Y \sim N(0,1)$ . Conditional on  $Y$ , we generate  $\mathbf{x}$  through
Model (III):  $\{(X_{2k - 1},X_{2k})^{\mathrm{T}}\mid Y\} \sim N\{\mathbf{0},(\rho_{k,Y}^{|i - j|})_{2\times 2}\}$ $k = 1,\ldots ,p / 2,$
Figure 2: The kernel densities of the test statistics under the null hypothesis computed from 5000 simulations. We set  $p = q = d$  and vary  $d$  from \{2, 5, 10\}. The horizontal axes represent the observed values of the test statistics, and the vertical axes represent the kernel densities of those values. We choose two different kernels to implement the tests, i.e., Gaussian (dashed line) and Laplacian (dotted line). The solid line is the reference curve, which is the density of the standard normal distribution.
Table 1: The empirical powers of different tests for Models (I)-(III) in Example 2. The significance level is 0.05. We fix  $q = 1$  and vary  $p$  from  $\{30,50,100,200,500,1000\}$ .
Model	Test	p
30	50	100	200	500	1000
(I)	Gaussian	1.000	1.000	1.000	1.000	0.998	0.954
Laplacian	1.000	1.000	1.000	1.000	0.994	0.916
DC	1.000	1.000	1.000	1.000	1.000	0.998
(II)	Gaussian	0.934	0.774	0.484	0.318	0.184	0.132
Laplacian	0.998	0.978	0.834	0.596	0.314	0.234
DC	0.974	0.894	0.650	0.412	0.226	0.176
(III)	Gaussian	0.044	0.050	0.060	0.046	0.060	0.068
Laplacian	0.050	0.046	0.054	0.048	0.054	0.060
DC	0.050	0.050	0.052	0.034	0.064	0.044
where  $\rho_{k,Y} = (2\delta_k - 1)\Phi (Y),\delta_k\mathrm{s}$  are iid Bernoulli(0.5) random variables, and  $\Phi (\cdot)$  is the cumulative distribution function of a standard normal distribution. In addition,  $(X_{2k - 1},X_{2k})^{\mathrm{T}}\mathrm{s}$  are mutually independent conditional on  $Y$ . We remark here that in Model (III),  $X_{k}$  is marginally independent of  $Y$  for all  $k = 1,\ldots ,p$ . It can be verified that there exist linear, quadratic and quartic conditional means of  $\mathbf{x}$  given  $Y$  in Models (I), (II) and (III), respectively.
As we have demonstrated in Example 1, when  $q = 1$ , the standard normal distribution can approximate the null distribution pretty well as long as  $p$  is no less than 25, we use the standard normal distribution to obtain  $p$ -values. We repeat each experiment 500 times and report the empirical powers in Table 1. As a comparison, we also include the results for the distance correlation in Table 1. We can observe that all the empirical powers for Model (I) can still approach one, implying that the linear mean dependence can be easily detected, even when the dimension is much higher than the sample size. For example, when  $p = 1000$ , the power for Model (I) when the Gaussian kernel is used is still as high as 0.954. However, when there only exists some higher order of mean dependence, the empirical powers drop significantly. In addition, the performances deteriorate as the order of mean dependence increase. For example, when  $p = 100$  and we choose the Gaussian kernel to implement the test, the empirical power for Model (III) is only 0.060, which is smaller than 0.484, the empirical power for Model (II).
Example 3. In this example, we study how the dimensionality can effect the empirical power when both of the covariate dimensions increase simultaneously. We also fix the sample size to be  $n = 100$ . We set  $p = q = d$  and vary  $d$  from \{6, 10, 20, 50, 100, 200\}. Similar to Example 2, we consider three models in this example. For Models (IV) and (V), we generate  $\mathbf{x}$  the same as that in Models (I) and (II), i.e.,  $\mathbf{x} \sim N\{\mathbf{0}, (0.5^{|i - j|})_{p \times p}\}$ . The independent error terms  $\varepsilon_1, \ldots, \varepsilon_d$  are generated from  $d$  independent standard normal distributions. Then we generate  $\mathbf{y} = (Y_1, \ldots, Y_d)^{\mathrm{T}}$  through
$$
\text {M o d e l} (\mathrm {I V}): \quad Y _ {j} = X _ {j} + \varepsilon_ {j}, \quad j = 1, \dots , d;
$$
Table 2: The empirical powers of different tests for Models (IV)-(VI) in Example 3. The significance level is 0.05. We set  $p = q = d$  and vary  $d$  from  $\{6,10,20,50,100,200\}$ .
Model	Test	d
6	10	20	50	100	200
(IV)	Gaussian	1.000	1.000	1.000	1.000	1.000	1.000
Laplacian	1.000	1.000	1.000	1.000	1.000	1.000
DC	1.000	1.000	1.000	1.000	1.000	1.000
(V)	Gaussian	1.000	1.000	0.944	0.440	0.242	0.140
Laplacian	1.000	1.000	1.000	0.904	0.578	0.282
DC	1.000	0.986	0.844	0.400	0.232	0.136
(VI)	Gaussian	0.056	0.064	0.046	0.078	0.038	0.072
Laplacian	0.058	0.064	0.044	0.074	0.040	0.072
DC	0.060	0.066	0.046	0.078	0.040	0.072
Model (V):  $Y_{j} = X_{j}^{2} + \varepsilon_{j}, \quad j = 1, \ldots, d.$
For Model (VI), we first generate all coordinates of  $\mathbf{y}$  independently from standard normal distributions. Then given  $\mathbf{y}$ , we generate  $\mathbf{x}$  through
Model (VI):  $\{(X_{2k - 1},X_{2k})^{\mathrm{T}}\mid \mathbf{y}\} \sim N\{\mathbf{0},(\rho_{k,\mathbf{y}}^{|i - j|})_{2\times 2}\}$ $k = 1,\ldots ,d / 2$
where  $\rho_{k,\mathbf{y}} = (2\delta_k - 1)\Phi (Y_{2k})$  and  $\delta_{k}\mathrm{s}$  are iid Bernoulli(0.5) random variables. Meanwhile, we generate  $(X_{2k - 1},X_{2k})^{\mathrm{T}}\mathrm{s}$  independently conditional on  $\mathbf{y}$ . Similar to Models (I), (II) and (III), there exist covariances between  $\mathbf{y}$  and linear, quadratic and quartic polynomials of  $\mathbf{x}$  in Models (IV), (V) and (VI), respectively.
Following that in Example 2, we use the standard normal distribution to obtain  $p$ -values and repeat each experiment 500 times to report the empirical powers. The corresponding results are summarized in Table 2, from which we observe a similar phenomenon. That is, the empirical powers decay as the dimensions increase, and the decay rates increase when there only exist higher orders of covariances between  $x$  and  $y$ . In addition, according to both Tables 1 and 2, the conclusions for the HSIC with Gaussian and Laplacian kernels also apply to the distance correlation. This is in line with our anticipation because the distance correlation is equivalent to the HSIC with some particular kernel.
5.2 Real Data Application
The prices of gasoline and many other fuels may have important impacts on raw materials and related companies. For example, the petroleum is widely used as upstream raw materials by many chemical industries. We are interested in whether there is any dependency between stock prices of the energy sector and the raw material sector in the U.S. stock market. To see this, we extract the monthly mean stock prices of energy companies as well as raw material companies starting from January 2021 to December 2022 from https://finance.yahoo.com/. According to the Global Industry Classification Standard, we get 345 and 294 companies from raw material and energy sectors, respectively. We remove those stocks whose data are not complete in this period, yielding 224 companies from the raw material sector and 214 companies from the energy sector, respectively. At each month  $t$ ,  $1 \leq t \leq 24$ , we denote the monthly mean stock prices of these companies from the two sectors by  $\mathbf{x}_t = (X_{t1},\dots,X_{tp})^{\mathrm{T}}$  and  $\mathbf{y}_t = (Y_{t1},\dots,Y_{tq})^{\mathrm{T}}$ , where  $p = 224$  and  $q = 214$ . Let  $S_{ti}^{\mathbf{x}} = \log (X_{ti} / X_{t - 1,i})$  and  $S_{tj}^{\mathbf{y}} = \log (Y_{tj} / Y_{t - 1,j})$  be the stock returns at month  $t$  for the  $i$ -th company in the raw material sector, and  $j$ -th company in the energy sector, respectively. We test whether the stock returns in these two sectors are independent using  $\{(s_t^\mathbf{x},s_t^\mathbf{y})\}_{t = 2}^{24}$ , where  $s_t^\mathbf{x} = (S_{t1}^\mathbf{x},\dots,S_{tp}^\mathbf{x})^\mathrm{T}$  and  $s_t^\mathbf{y} = (S_{t1}^\mathbf{y},\dots,S_{tp}^\mathbf{y})^\mathrm{T}$ . We implement the test using HSIC with Gaussian and Laplacian kernels. The resulting  $p$ -values are  $2.031 \times 10^{-10}$  and  $2.749 \times 10^{-9}$ , respectively. Then we can conclude that there exists dependences between the monthly mean stock prices of the energy sector and the raw material sector.
We also compare the RV coefficient with our proposed method in this dataset. The RV coefficient yields a  $p$ -value of  $2.02 \times 10^{-4}$ , which indicates the presence of linear dependences in this dataset. We remark that, in this data set, both covariates dimensions  $p$  and  $q$  are much larger than sample size  $n$ . According to the second part of Theorem 3 and the discussions at the end of Section 4, the HSIC can only have nontrivial power if  $p^{(s_1 - 1)\kappa_{\mathbf{x}}}q^{(s_2 - 1)\kappa_{\mathbf{y}}} = o(n)$ . In the context of our dataset, this
condition is satisfied exclusively when  $s_1 = s_2 = 1$ , signifying a covariance relationship between  $\mathbf{x}$  and  $\mathbf{y}$ . This together with the fact that the test based on HSIC rejected the null hypothesis, we conclude that there exists a linear dependence relationship between  $\mathbf{x}$  and  $\mathbf{y}$ . This is consistent with the RV coefficient result. Moreover, we identify that there is a strong linear relationship between stock returns for two pairs of companies: Denison Mines Corp. and Energy Fuels Inc., and Uranium Energy Corp. and Energy Fuels Inc. The  $R^2$ s for the corresponding linear models are 0.8779 and 0.8718, respectively. This means that the models fit the data very well and that the stock returns move together in a predictable way.
This paper aims to investigate the performance of HSIC in high dimensions from a theoretical perspective. We first prove that the asymptotic null distribution of a rescaled HSIC is a standard normal in the high dimensional setting. Then we derive a general condition for the HSIC based tests to have power asymptotically approaching one. To gain more insights, we decompose this condition and show that it depends on the sample size, the covariate dimensions, the dependence structures within covariates, and the association types between  $\mathbf{x}$  and  $\mathbf{y}$ . We also show that the ability of HSIC to detect nonlinear dependence deteriorates as the dimensions increase. Most existing works focused on either fixed dimensions or dimensions that diverge much faster than the sample size. We fill this gap in the literature and provide a comprehensive picture of the performance of HSIC based tests.
Our theory also reveals how the ability of HSIC to capture nonlinear dependence diminishes in high dimensions. To improve the power performance, Zhu et al. (2020) proposed to aggregate marginal sample HSIC as the test statistic instead of using HSIC over the whole features. However, this approach only measures marginal dependences, and it is well known that marginal independence does not imply joint independence. Moreover, because it sums over all pairs of marginal HSIC, it may suffer from a significant power loss when the alternative is sparse. Developing an independence test that has nontrivial power against all kinds of dependences remains an open question. Therefore, it would be interesting to explore new methods that can enhance power performances in high dimensions.
The authors thank five anonymous reviewers for their comments and suggestions that significantly enhanced the quality of the paper. All authors contribute equally to this paper and their names are listed in an alphabetical order. This work is supported by National Natural Science Foundation of China (12271331 and 72192832), and Program for Innovative Research Team of Shanghai University of Finance and Economics.