Principal component analysis for high-dimensional compositional data

TL;DR

This paper develops a novel method for principal component analysis tailored to high-dimensional compositional data, addressing the challenges posed by latent basis variables and the simplex constraint, with theoretical guarantees and practical algorithms.

Contribution

It establishes a link between the principal subspace of compositional data and the basis covariance, providing identifiable estimation under high-dimensional settings and proposing efficient algorithms.

Findings

Proposed methods achieve near-oracle performance in simulations.

Derived nonasymptotic error bounds demonstrating the method's accuracy.

Applied to analyze word usage patterns among statisticians.

Abstract

Dimension reduction for high-dimensional compositional data plays an important role in many fields, where the principal component analysis of the basis covariance matrix is of scientific interest. In practice, however, the basis variables are latent and rarely observed, and standard techniques of principal component analysis are inadequate for compositional data because of the simplex constraint. To address the challenging problem, we relate the principal subspace of the centered log-ratio compositional covariance to that of the basis covariance, and prove that the latter is approximately identifiable with the diverging dimensionality under some subspace sparsity assumption. The interesting blessing-of-dimensionality phenomenon enables us to propose the principal subspace estimation methods by using the sample centered log-ratio covariance. We also derive nonasymptotic error bounds for the subspace estimators, which exhibits a tradeoff between identification and estimation. Moreover, we develop efficient proximal alternating direction method of multipliers algorithms to solve the nonconvex and nonsmooth optimization problems. Simulation results demonstrate that the proposed methods perform as well as the oracle methods with known basis. Their usefulness is illustrated through an analysis of word usage pattern for statisticians.