\vspace{-5pt}
\section{Experiments}\label{sec:main-exp}

We evaluate PolyILR on standard microbiome and single-cell benchmarks. Our goals are to demonstrate PolyILR provides: \textbf{(G1)} valid ILR representations (\S\ref{sec:exp-representation}), \textbf{(G2)} stable feature selection, unlike PhILR with arbitrary binarization (\S\ref{sec:exp-stability}), \textbf{(G3)} interpretable features grounded in the tree (\S\ref{sec:exp-interpretability}), and \textbf{(G4)} structured inference by tree subparts (\S\ref{sec:exp-tree}).

\vspace{-5pt}
\subsection{Setup}\label{sec:exp-setup}

\textbf{Datasets.} We use three large datasets from two domains. For microbiome: \texttt{HMP} (Human Microbiome Project; 4,743 samples, 402 taxa) \citep{human2012structure} and \texttt{cMD3} (curatedMetagenomicData v3; 20,238 samples from 86 studies, 2,047 taxa) \citep{pasolli2017accessible}, with taxonomies from NCBI. For single-cell biology: \texttt{DISCO} (Database of Immune Single-Cell Omics; 751 sample-level composition profiles derived from ${\sim}$5.3M cells, 62--99 cell types) \citep{li2022disco}, with cell types organized by the Cell Ontology. Tasks include predictions on body site (5--18 sites), westernization, age category, healthy vs.\ disease (microbiome), and healthy vs.\ leukemia/HCC (single-cell). {\color{black}As with any log-ratio method, PolyILR requires zero handling before the coordinate transform; we use a small additive pseudocount per dataset (Appendix~\ref{appdx:zeros}).} 

\textbf{Methods.}
We compare PolyILR against CLR (same geometry, no tree alignment) and {\color{black}unweighted} PhILR with random binarization (tree-aligned, but {\color{black}defined on binary trees, so polytomies must be resolved arbitrarily)}. We use random forest (RF), SVM, and logistic regression (LR) with 5-fold cross-validation. {\color{black}Additional experiments, hyperparameter, and dataset construction details are in Appendix~\ref{appdx:exp}.}

\begin{remark}PolyILR is a representation (coordinate system), {\bf not} a task-specific model. So, {\em any} downstream analysis may be applied to the resulting coordinates. Conclusions and scientific validity depend on appropriate statistical methodology.
\end{remark}

\vspace{-5pt}
\subsection{Representation Validity}\label{sec:exp-representation}
We verify PolyILR is a geometrically valid representation \textbf{(G1)}. CLR projects compositions into the tangent space $\mathcal{H}$, spanned by PolyILR coordinates (Thm.~\ref{thm:polyilr}). Table~\ref{tab:repr-stability} (top) supports this equivalence: {\color{black}CLR, PhILR, and PolyILR yield identical SVM and LR accuracy, as expected from isometry. RF accuracy varies modestly across representations (within $\pm$1.5\% on most tasks), since RF is sensitive to the choice of axes; differences in either direction are consistent with the geometric equivalence.}

\vspace{-5pt}
\subsection{Feature selection is stable}\label{sec:exp-stability}
A key advantage of PolyILR over PhILR is \emph{canonical} decomposition on any tree \textbf{(G2)}. PhILR requires binarizing polytomies, making feature selection unstable across binarizations with \emph{no correct choice}. We measure stability via: (i) \emph{index stability} (Jaccard similarity of top-$K$ feature indices across runs) measuring if the same coordinate positions are selected; and (ii) \emph{semantic stability} (similarity of the corresponding taxonomic/ontological contrasts) measuring whether selected features represent the same biological comparisons regardless of index. The latter is fairer to PhILR as it ignores arbitrary index assignment. For PhILR, we vary the random binarization of the same polytomous tree across runs; for PolyILR, (i) and (ii) coincide since coordinates are canonical given the tree. Table~\ref{tab:repr-stability} (bottom) shows PolyILR achieves high stability (0.43--0.92) while PhILR collapses (near 0) under both metrics across all three datasets. Even when comparing semantically, PhILR's artificial binary splits yield different partitions across binarizations, confirming that the instability is structural (Figure~\ref{fig:binary-vs-poly}).

\vspace{-5pt}
\subsection{Features are interpretable}\label{sec:exp-interpretability}
PolyILR coordinates are directly interpretable as taxonomic/ontological contrasts \textbf{(G3)}. Table~\ref{tab:contrasts} shows the top-4 features by RF importance{\color{black}, with rank range across 5 runs indicating stability of the ranking}. Each feature is a log-ratio contrast between groups at a specific node (\S\ref{sec:inference}). 

\emph{Scientific interpretation (Table~\ref{tab:contrasts}).} The recovered contrasts are consistent with previously reported observations in the literature. For HMP body sites, Streptococcus vs.\ Lactococcus (3.4\%) reflects known niche specialization within Streptococcaceae across oral subsites~\cite{human2012structure,dewhirst2010human}{\color{black}, with Lactococcus lactis reported as a prevalent lactic-acid bacterium in the gut~\cite{pasolli2020large}}. For westernization, Prevotella vs.\ Bacteroides/Alistipes (1.5--1.8\%) captures the lifestyle axis, with Prevotella enriched in non-Western populations consuming plant-rich diets~\cite{de2010impact,yatsunenko2012human}. For healthy vs.\ disease, Lachnoclostridium (0.6\%) aligns with documented links to colorectal cancer and atherosclerosis~\cite{cai2022integrated,liang2020novel}. For leukemia, myeloid vs.\ erythroid/precursor imbalance (14.5\%) reflects lineage disruption in hematological malignancies~\cite{lowenberg1999acute}. For HCC, venous EC vs.\ LSEC (5.5\%) captures the well-documented dedifferentiation of liver sinusoidal endothelial cells in hepatocellular carcinoma~\cite{sorensen2015liver}.

\emph{Geometric structure.} Figure~\ref{fig:top2} projects \texttt{HMP} samples onto the top-2 coordinates. Unlike PCA, \emph{each axis here is a single interpretable contrast} rather than a linear combination of all features. We do not claim maximal variance explained; rather, biologically meaningful features alone suffice to separate body sites. The linear substructures within classes may reflect shared sparsity: samples with identical zero-count taxa map to parallel manifolds in ILR space.

\input{figure/combined_top2}
\input{table/main_tree_inf_hmp}
\input{table/main_tree_inf_cmd3}
\input{table/main_tree_inf_disco}

\vspace{-5pt}
\subsection{Tree-Level Inference}\label{sec:exp-tree}
PolyILR enables structured hypothesis testing at multiple resolutions \textbf{(G4)}. RF importance can be aggregated by depth, subtree, node, or leaf (see \S\ref{sec:inference}). We report all four levels for \texttt{HMP} (Table~\ref{tab:tree-hmp}) and \texttt{DISCO} (Table~\ref{tab:tree-disco}), but only depth and taxon for \texttt{cMD3} (Table~\ref{tab:tree-cmd3}) whose meta-analytic tree lacks consistent intermediate labels. Subtree-level partitions by root's children (root omitted).

\emph{Scientific interpretation (Tables~\ref{tab:tree-hmp}--\ref{tab:tree-disco}).}
Aggregations agree with known structure. For HMP, Firmicutes (47\%) and Proteobacteria (19\%) dominate body site signals~\cite{human2012structure,costello2009bacterial,ma2024systematic}. At node level, Actinomycetales (11\%) and Lactobacillales (9\%) capture skin vs.\ oral distinctions. For cMD3 westernization, coarse contrasts ($\leq$3) achieve 95.6\% acc., consistent with diet-associated shifts at coarse resolution~\cite{de2010impact,arumugam2011enterotypes}. For DISCO leukemia (single-cell), 95\% of importance concentrates in Immune cells, with T cell (16\%) and T/NK cell (14\%) nodes dominating~\cite{lowenberg1999acute}. For HCC, importance distributes across Immune (79\%), Endothelial (8\%), and Epithelial (4\%) subtrees, reflecting multi-compartment remodeling~\cite{sorensen2015liver}.

We should note that the biological interpretations above are \emph{plausibility checks consistent with prior literature}. Any causal or clinical conclusions will require much deeper analyses beyond the scope of this methodological work.

In summary, PolyILR addresses all goals \textbf{(G1--G4)} while recovering features consistent with known biomarkers. 