# Log-Ratio Propagation on the Simplex: A Theory of Cellwise Contamination for Compositional Data

arXiv: https://arxiv.org/abs/2605.31345 · HTML: https://mlsanity.com/paper/2605.31345/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 11 of its 48 references that are in the map.
- First posted on arXiv 29 May 2026.

## Builds on (most cited first, 11 of 11)

- [Detecting deviating data cells](https://mlsanity.com/paper/1601.07251/) (2016)
- [Aitchison's Compositional Data Analysis 40 Years On: A Reappraisal](https://mlsanity.com/paper/2201.05197/) (2022)
- [Robust estimation of precision matrices under cellwise contamination](https://mlsanity.com/paper/1501.01950/) (2015)
- [The Cellwise Minimum Covariance Determinant Estimator](https://mlsanity.com/paper/2207.13493/) (2022)
- [Challenges of cellwise outliers](https://mlsanity.com/paper/2302.02156/) (2023)
- [MacroPCA: An all-in-one PCA method allowing for missing values as well as cellwise and rowwise outliers](https://mlsanity.com/paper/1806.00954/) (2018)
- [Robust and sparse Gaussian graphical modeling under cell-wise contamination](https://mlsanity.com/paper/1802.05475/) (2018)
- [Robust Principal Components by Casewise and Cellwise Weighting](https://mlsanity.com/paper/2408.13596/) (2024)
- [Cellwise robust and sparse principal component analysis](https://mlsanity.com/paper/2408.15612/) (2024)
- [Cellwise outlier detection in heterogeneous populations](https://mlsanity.com/paper/2409.07881/) (2024)
- [Cellwise and Casewise Robust Covariance in High Dimensions](https://mlsanity.com/paper/2505.19925/) (2025)

## Abstract (arXiv)

Compositional data must be analysed through log-ratios: scale invariance, the defining axiom of the field, leaves no alternative. The centred log-ratio divides by the geometric mean of every part, so a single contaminated component shifts every centred-log-ratio coordinate at once, displacing the log-ratio vector by a fixed amount that no choice of coordinates can reduce. We develop a theory of cellwise contamination on the simplex around this observation. A scale-invariant contamination model built from multiplicative perturbation combines with a propagation theorem showing that corruption of a single raw part induces a rank-one shift of the log-ratio vector, with direction determined by the contrast matrix. The resulting perturbation pattern is not equivalent to any independent cellwise contamination model in log-ratio coordinates -- so standard Euclidean cellwise methods applied to log-ratios are ill-posed under the simplex contamination mechanism. For estimators whose Euclidean cellwise breakdown is witnessed by a column-concentrated configuration -- a class including MCD, S-, τ-, and coordinate-wise M-estimators of location and scatter -- the cellwise breakdown value on the simplex is reduced by the factor (D-1)/D relative to its Euclidean counterpart, a reduction that is tight and arises purely from the normalisation mismatch between nD raw cells and n(D-1) ilr cells. The cellwise influence function for the variation matrix carries a diagnostic fingerprint: contamination of a single part inflates exactly one row and column, identifying the responsible component. These results form the theoretical foundation for cellwise-robust methods on the simplex; a companion paper develops a cellwise-robust PCA estimator that exploits the propagation geometry and demonstrates it on simulated and geochemical data.