\vspace{-15pt}
\section{Structured Analysis with PolyILR}\label{sec:polyilr-analysis}

We describe how PolyILR coordinates enable structured analysis beyond what standard log-ratio transforms provide. Let $x \in \Delta^{d-1}$ be a composition with components as leaves of $\mathcal{T}$. The PolyILR transform yields $z = \varphi(x) \in \mathbb{R}^{d-1}$.

\vspace{-5pt}
\subsection{Tree-Aligned Coordinates}\label{sec:tree-coords}

\textbf{Coordinate indexing.}
By construction, each coordinate index $j$ corresponds bijectively to a pair $(u, m)$: an internal node $u$ and a contrast index $m \in \{1, \ldots, k_u - 1\}$. The coordinate $z_j = z_{(u,m)}$ is a \emph{balance}---a log-ratio comparing the geometric mean of leaves under child $m+1$ against that under children $1, \ldots, m$ at node $u$ (see \eqref{eq:main_helmert}). This association is intrinsic.

\textbf{Multiscale structure.}
By construction, each internal node $u$ contributes $k_u - 1$ coordinates to the basis $V$ (see \S\ref{sec:construction}). Because these coordinates are orthonormal, the coordinates at distinct nodes span orthogonal subspaces, yielding a disjoint partition of $\mathbb{R}^{d-1}$ indexed by tree nodes. We can thus reason about node $u$ as a unit: do the coordinates at $u$ jointly explain an outcome? Does variation concentrate at $u$? 

This node-level partition extends to coarser groupings. Aggregating nodes by tree depth or by subtree membership yields alternative orthogonal partitions of the same space (Table~\ref{table:tree-subparts}). 
% Standard ILR bases offer no such intrinsic groupings; 
PolyILR inherits them directly from the tree. We illustrate these partitions in Figure~\ref{fig:tree-inference} (Appendix~\ref{appdx:exp}).

\begin{table}[h]
\centering
{\footnotesize
\begin{tabular}{ll}
\toprule
\textbf{Aggregation} & \textbf{Question Answered} \\
\midrule
Node & Which \emph{splits} drive signal? \\
Depth & What \emph{resolution} matters? \\
Subtree & Is an entire \emph{clade} informative? \\
\bottomrule
\end{tabular}
}
\vspace{3pt}
\caption{Tree substructure aggregations enabled by PolyILR.}
\vspace{-15pt}
\label{table:tree-subparts}
\end{table}

\subsection{Implications for Inference}\label{sec:inference}
Given data $\{(x_i, y_i)\}_{i=1}^{N}$ with outcome $y_i$, we transform $z_i = \varphi(x_i)$ and fit any model on $\{(z_i, y_i)\}$. Since $\varphi$ is an isometry, the full geometry is preserved.

\input{table/main_repr_acc_and_stab}
% \input{table/main_repr_acc}
% \input{table/main_stab}
\input{table/main_interp}


\textbf{Feature selection.}
Identifying \emph{which features drive outcomes} is a key scientific goal. In genomics, neuroscience, and microbiome alike, the goal is often not just prediction but understanding which variables matter and why \citep{rudin2019stop, marcos2021applications}. With standard ILR, important coordinates are anonymous indices with no semantics. With PolyILR, when coordinate $j = (u, m)$ is identified as important, we know which node and contrast drive the signal: a log-ratio comparing specific groups of leaves. This interpretability is intrinsic to the representation, no post-hoc processing needed.

\textbf{Tree-level aggregation.}
The multiscale structure of PolyILR enables inference at any tree substructure. Let $\omega_j$ denote importance of coordinate $j$ from any method (e.g., random forest). We aggregate per-coordinate importances over any disjoint set $S$ of coordinates (node, depth, or subtree) via $\omega(S) = \sum_{j \in S} \omega_j$. This aggregation is well-defined because coordinates at distinct nodes span orthogonal subspaces, any partition of the tree into disjoint substructures yields a partition of $\mathbb{R}^{d-1}$, and the corresponding importances sum to the total without double-counting.

\textbf{Leaf-level importance.}
To quantify importance of an individual leaf $\ell$ (e.g., a taxon), we cannot directly aggregate coordinates because coordinates are not exclusive to any single leaf. Instead, we can distribute importance weighted by participation. By construction, $V_{\ell j}$ quantifies how much leaf $\ell$ participates in coordinate $j$. We define $\omega(\ell) = \sum_{j=1}^{d-1} V_{\ell j}^2 \cdot \omega_j$. Since columns of $V$ are unit vectors, $\sum_{\ell} V_{\ell j}^2 = 1$, so leaf importances sum to total importance.