\section{Background}

\begin{figure}[t]
    \centering
    \resizebox{0.47\textwidth}{!}{\input{figure/binary-vs-poly}}
    \vspace{-10pt}
    \caption{\emph{Polytomous vs.\ Binarized Tree.} (Left) $\mathcal{T}$ with a polytomous root ($k=3$). PolyILR assigns {\color{black}$k-1=2$ basis vectors (column of $V$) to the root (red)}. (Right) Binarized $\mathcal{T}_b$ required by PhILR (\emph{one coordinate per node}) introduces an artificial node (yellow) encoding an arbitrary grouping of leaves 1 and 2---a choice not justified by original $\mathcal{T}$.}
    \vspace{-15pt}
    \label{fig:binary-vs-poly}
\end{figure}

We introduce the geometry of compositional data and formalize the tree alignment problem (see \citet{aitchison1982statistical}).  

\subsection{Aitchison Geometry}\label{sec:background}

\textbf{Compositional data.}
Compositional data are nonnegative vectors whose totals are uninformative and only relative proportions matter, e.g., microbial abundances (counts with varying sequencing depth) or chemical concentrations (parts of a mixture) \citep{gloor2017microbiome, jackson1997compositional}. We normalize to unit sum, placing data in the open simplex:
\begin{equation}
    \Delta^{d-1} = \left\{ x \in \mathbb{R}^d_{>0} : \sum_{i=1}^d x_i = 1 \right\}.
    \label{eq:main_composition-simplex}
\end{equation}
% \begin{equation}
%     \scalebox{0.9}{$\displaystyle \Delta^{d-1} = \left\{ x \in \mathbb{R}^d_{>0} : \sum_{i=1}^d x_i = 1 \right\}.$}
%     \label{eq:main_composition-simplex}
% \end{equation}
The key constraint is that only ratios $x_i / x_j$ carry information, not absolute values.

\textbf{Geometry is fixed.}
Aitchison geometry \citep{aitchison1982statistical} formalizes this by equipping the set $\Delta^{d-1}$ with perturbation $x \oplus y = \mathcal{C}(x_1 y_1, \ldots, x_d y_d)$ and powering $\alpha \odot x = \mathcal{C}(x_1^\alpha, \ldots, x_d^\alpha)$, where $\mathcal{C}(\cdot)$ is the closure. Under these operations, $(\Delta^{d-1}, \oplus, \odot)$ forms a $(d-1)$-dimensional Hilbert space with inner product:
\[
\langle x, y \rangle_A = \frac{1}{d} \sum_{i=1}^d \sum_{j=1}^d \log \frac{x_i}{x_j} \log \frac{y_i}{y_j}.
\]
The induced Aitchison distance $d_A(x,y) = \|x \ominus y\|_A$ is perturbation-invariant: $d_A(x \oplus z, y \oplus z) = d_A(x,y)$. Note that this geometry is \emph{not a modeling choice}---it is the unique structure respecting compositional invariance.

\subsection{ILR Basis}
\textbf{Basis is a choice.}
The centered log-ratio (CLR) transform maps $x \in \Delta^{d-1}$ (under Aitchison geometry) isometrically to the CLR hyperplane (i.e., \emph{Aitchison tangent space})
\begin{equation}
    \mathcal{H} = \{z \in \mathbb{R}^d : \mathbf{1}^\top z = 0\}
    \label{eq:main_tangent_clr}
\end{equation}
via $\text{clr}(x) = (\log x_1/g(x), \ldots, \log x_d/g(x))$, where $g(x) = (\prod_i x_i)^{1/d}$ is the geometric mean. Any matrix $V \in \mathbb{R}^{d \times d-1}$ whose \emph{columns form an orthonormal basis} of $\mathcal{H}$ yields isometric log-ratio (ILR) coordinates and admits an isometric bijection 
\begin{equation}
    \varphi(x) = V^\top \log x,
    \label{eq:main_ilr_transform}
\end{equation}
from $(\Delta^{d-1}, \langle \cdot, \cdot \rangle_A)$ to $(\mathbb{R}^{d-1}, \langle \cdot, \cdot \rangle_2)$ \citep{egozcue2003isometric}. All such bases $V$ are related by orthogonal transformations. That is, they induce the same geometry but different decompositions of the simplex. The question is thus not whether to use an ILR basis, but \emph{which basis to choose}, and this choice largely determines interpretability.

\textbf{Basis choice controls interpretability.}
This parallels classical signal processing: Fourier bases yield frequency components from translation symmetry \citep{brigham1988fast}; wavelet bases yield scale-localized components from dyadic partitions \citep{mallat2002theory}. Aligning the basis with domain structure produces interpretable coordinates. 