A New Approach to Compositional Data Analysis using \(L^{\infty}\)-normalization with Applications to Vaginal Microbiome

We introduce a novel approach to compositional data analysis based on $L^{\infty}$-normalization, addressing challenges posed by zero-rich high-throughput data. Traditional methods like Aitchison's transformations require excluding zeros, conflicting with the reality that omics datasets contain structural zeros that cannot be removed without violating inherent biological structures. Such datasets exist exclusively on the boundary of compositional space, making interior-focused approaches fundamentally misaligned. We present a family of $L^p$-normalizations, focusing on $L^{\infty}$-normalization due to its advantageous properties. This approach identifies compositional space with the $L^{\infty}$-simplex, represented as a union of top-dimensional faces called $L^{\infty}$-cells. Each cell consists of samples where one component's absolute abundance equals or exceeds all others, with a coordinate system identifying it with a d-dimensional unit cube. When applied to vaginal microbiome data, $L^{\infty}$-decomposition aligns with established Community State Types while offering advantages: each $L^{\infty}$-CST is named after its dominating component, has clear biological meaning, remains stable under sample changes, resolves cluster-based issues, and provides a coordinate system for exploring internal structure. We extend homogeneous coordinates through cube embedding, mapping data into a d-dimensional unit cube. These embeddings can be integrated via Cartesian product, providing unified representations from multiple perspectives. While demonstrated through microbiome studies, these methods apply to any compositional data.