Algorithm 1.

Algorithm for projected distance correlation with OGA

Require: Observed random sample vectors xn ∈ Rn×p, yn ∈ Rn×q, zn ∈ Rn×r.
1:	for i = 1 to p do
2:	Initialize $r_0=x_n^i$, T0 = ϕ.
3:	for k = 1 to $\left(1+\log q\right)\sqrt{\frac{n}{\log q}}$ do
4:	Update the active set by $T_k=T_{k-1}\cup z_n^{t_k}$, where $t_k=\arg {max}_{j\in \left\{1,\dots ,q\right\}}\left|<r_{k-1},z_n^j>\right|$.
5:	Update the estimate by $f_k=P_{T_k}{x}_n^i$, where $P_{T_k}$ is the orthogonal projection onto Tk.
6:	Update the residue $r_k=x_n^i-f_k$.
7:	Record ${\text{HDIC}}_k=n\log (\frac{{\Vert r_k\Vert }_2^2}{n})+2k\log r$.
8:	end for
9:	Find k* = argminkHDICk.
10:	Select ${\epsilon }_x^i=r_{k^{\ast }}$.
11:	end for
12	Calculate squared sample projected distance correlation $R_n^2\left({\epsilon }_x,{\epsilon }_y\right)$ and the corresponding p-value.