| \vspace{-15pt} |
| \section{Structured Analysis with PolyILR}\label{sec:polyilr-analysis} |
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| 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 $z = \varphi(x) \in \mathbb{R}^{d-1}$. |
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| \vspace{-5pt} |
| \subsection{Tree-Aligned Coordinates}\label{sec:tree-coords} |
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| \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. |
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| \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 $u$? |
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| 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}). |
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| PolyILR inherits them directly from the tree. We illustrate these partitions in Figure~\ref{fig:tree-inference} (Appendix~\ref{appdx:exp}). |
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|
| \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} |
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| \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. |
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| \input{table/main_repr_acc_and_stab} |
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| \input{table/main_interp} |
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| \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. |
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| \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. |
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| \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. |
