Algorithm 1 Recursive entropy estimator

1:functioncopulaH($\left\{x_k^i\right\}$, D, N, $level=0$) 2: $H\leftarrow 0$ 3: for k = 1 to D do 4:  $u_k\leftarrow$ rank($x_k$)/N                     ▹ Calculate rank (by sorting) 5:  $H\leftarrow H+$ H1D($\left\{u_k^i\right\}$, N, $level$)                ▹ entropy of marginal k 6: end for 7:   8: if $D=1$ or $N<=$ min #samples then 9:  return H 10: end if 11:   12:                        ▹A is the matrix of pairwise independence 13: $A_{ij}=$ true if $x^i$ and $x^j$ are statistically independent 14: $n_{\mathrm{blocks}}\leftarrow$ # of blocks in A. 15: if $n_{\mathrm{blocks}}>1$ then                         ▹ Split dimensions 16:  for $j=1$ to $n_{\mathrm{blocks}}$ do 17:   $v\leftarrow$ elements in block j 18:   $H\leftarrow H+$ copulaH(${\left\{u_k^i\right\}}_{k\in v}^{i=1\cdots N}$,dim(v),N,$level$) 19:  end for 20:  return H 21: else                             ▹ No independent blocks 22:  $k\leftarrow$ choose a dim for splitting 23:  $L=\left\{i\right|u_k^i\le 1/2\}$ 24:  $\left\{v_j^i\right\}=\left\{2u_j^i\right|i\in L,j=1\cdots D\}$ 25:  $H\leftarrow H+$ copulaH($\left\{v_j^i\right\}$,D,$N/2$,$level+1$) /2 26:   27:  $R=\left\{i\right|u_k^i>1/2\}$ 28:  $\left\{w_j^i\right\}=\{2u_j^i-1|i\in R,j=1\cdots D\}$ 29:  $H\leftarrow H+$ copulaH($\left\{w_j^i\right\}$,D,$N/2$,$level+1$) /2 30: end if 31:end function