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I'm interested in the interpretation of the solution to the factor analysis problem with a 2-norm equality constraint on the columns of the loadings matrix. I plan to decompose $\mathbf{X}_i \in \mathbb{R}^{d}$ as $\mathbf{E}\mathbf{F}_i + \boldsymbol{\epsilon}_i$, where $\mathbf{E} \in \mathbb{R}^{d \times p}$ is the loadings matrix, $\mathbf{F}_i \in \mathbb{R}^{p}$ are the factors, and $\boldsymbol{\epsilon}_i \in \mathbb{R}^d \sim \mathcal{N}(\mathbf{0}, \mathbf{B})$.

We certainly need some constraint on $\mathbf{E}$ or $\mathbf{F}_i$ to make the parameters identifiable. For example, PCA is simply a factor analysis problem in which the constraint $\mathbf{E}^T\mathbf{E} = \mathbf{I}$ must be satisfied. If I solve for $\mathbf{E}$ and $\mathbf{F}_i$ such that the each column of $\mathbf{E}$ has 2-norm equal to $1$, is there some special interpretation of the factors like there is in PCA (where each factor is independent)? Each column of $\mathbf{E}$ lies on the unit ball, but what implications does that have on the solution?

Vivek Subramanian
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  • Factor analysis does not solve for (unambiguously estimates) factor values F, it estimates only loadings. Factor scores can be computed afterwards with the help of the loadings but are only approximations. Unlike PCs, true Fs are not linear combinations of variables. – ttnphns May 15 '15 at 10:05
  • Also, it is not a good idea to call eigenvectors "loadings". In PCA, loadings after extraction are the eigenvectors scaled up to the respective eigenvalues, so the inner product matrix of the loadings is diagonal, not identity. – ttnphns May 15 '15 at 10:11

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