Instability of statistical factor analysis

Factor analysis, a popular method for interpreting multivariate data, models the covariance among $p$ variables as being due to a small number ($k$, $1 \leq k < p$) of hidden variables. A factor analysis of $Y$ can be thought of as an ordered or unordered collection, $F(Y)$, of $k$ linearly independent lines in $\mathbb{R} ^{p}$. Let $\mathcal{Y}'$ be the collection of data sets for which $F(Y)$ is defined. The ``singularities'' of $F$ are those data sets, $Y$, in the closure, $\overline{\mathcal{Y}}'$, at which the limit, $\lim_{Y' \to Y, Y' \in \mathcal{Y}'} F(Y')$, does not exist. $F$ is unstable near its singularities.

Let $\Phi(Y)$ be the direct sum of the lines in $F(Y)$. $\Phi$ determines a $k$-plane bundle, $\eta$, over a subset, $\mathcal{X}$, of $\mathcal{Y}$. If $k > 1$ and $\eta$ is rich enough, $F$ ordered or, at least if $k = 2$ or 3, unordered, must have a singularity at some data set in $\mathcal{X}$. The proofs are applications of algebraic topology. Examples are provided.