Factor analysis with finite data

TL;DR

This paper develops a method to perform factor analysis reliably when only finite data samples are available, addressing the uncertainty in estimating the covariance matrix.

Contribution

It introduces a strategy to incorporate estimation uncertainty of the covariance matrix into the factor analysis process, improving robustness with finite data.

Findings

Provides a new approach to account for covariance estimation errors

Enhances the reliability of factor analysis with limited data

Offers theoretical guarantees for the proposed method

Abstract

Factor analysis aims to describe high dimensional random vectors by means of a small number of unknown common factors. In mathematical terms, it is required to decompose the covariance matrix $\Sigma$ of the random vector as the sum of a diagonal matrix $D$ | accounting for the idiosyncratic noise in the data | and a low rank matrix $R$ | accounting for the variance of the common factors | in such a way that the rank of $R$ is as small as possible so that the number of common factors is minimal. In practice, however, the matrix $\Sigma$ is unknown and must be replaced by its estimate, i.e. the sample covariance, which comes from a finite amount of data. This paper provides a strategy to account for the uncertainty in the estimation of $\Sigma$ in the factor analysis problem.