Algorithm 1 Simulated Angular Dataset Generation Routine. ${\mu }_j$, ${\mathsf{\Sigma }}_j$ are the parameters of the mixture component distribution; $\pi$ is the probability vector assigning weight mixture components; ${\delta }_i$ is the mixture component identifier for each simulated observation.

for $n_{\mathrm{iter}}$ in $[1,\dots ,10]$ do     for $n_{\mathrm{mix}}$ in $[1,2,4,8]$ do         for j in $1,\dots ,n_{\mathrm{mix}}$ do            Generate ${\mu }_j\sim {\mathcal{N}}_{32}\left(\mathbf{0},\mathit{I}\right)$            Generate ${\mathsf{\Sigma }}_j\sim {\mathcal{IW}}_{32}\left(70,70\mathit{I}\right)$         end for         Generate $\pi \sim \mathrm{Dirichlet}\left({\mathbf{10}}_{n_{\mathrm{mix}}}\right)$         for i in $1,\dots ,1000$ do            Generate ${\delta }_i\sim \mathrm{Categorical}\left(\pi \right)$            Generate ${\mathit{X}}_i\sim \mathcal{LN}\left({\mu }_{\left[{\delta }_i\right]},{\mathsf{\Sigma }}_{\left[{\delta }_i\right]}\right)$         end for         for $n_{\mathrm{col}}$ in $[2,4,8,16,24,32]$ do            Project columns 1 to $n_{\mathrm{col}}$ of $\mathbf{X}$ onto ${\mathcal{S}}_{\infty }^{n_{\mathrm{col}}-1}$ and save.         end for     end for end for