On the Consistency of Maximum Likelihood Estimation of Probabilistic Principal Component Analysis

TL;DR

This paper proves the consistency of maximum likelihood estimation in probabilistic PCA by addressing the model's rotational symmetry, providing theoretical guarantees for the soundness of ML solutions in high-dimensional data reduction.

Contribution

It introduces a novel topological approach to establish the consistency of ML estimation in PPCA, extending results beyond traditional MLE and including strong covariance estimation.

Findings

ML solution is consistent in quotient space

Strong consistency of ML and covariance estimation established

Results apply to a broad class of estimators

Abstract

Probabilistic principal component analysis (PPCA) is currently one of the most used statistical tools to reduce the ambient dimension of the data. From multidimensional scaling to the imputation of missing data, PPCA has a broad spectrum of applications ranging from science and engineering to quantitative finance. Despite this wide applicability in various fields, hardly any theoretical guarantees exist to justify the soundness of the maximal likelihood (ML) solution for this model. In fact, it is well known that the maximum likelihood estimation (MLE) can only recover the true model parameters up to a rotation. The main obstruction is posed by the inherent identifiability nature of the PPCA model resulting from the rotational symmetry of the parameterization. To resolve this ambiguity, we propose a novel approach using quotient topological spaces and in particular, we show that the maximum likelihood solution is consistent in an appropriate quotient Euclidean space. Furthermore, our consistency results encompass a more general class of estimators beyond the MLE. Strong consistency of the ML estimate and consequently strong covariance estimation of the PPCA model have also been established under a compactness assumption.