# What is meant by subspace clustering in MFA?

The basic idea of MFA is to perform subspace clustering by assuming the covariance structure for each component of the form, $$\Sigma_i = \Lambda_i \Lambda_i^T + \Psi_i$$, where $$\Lambda_i \in \mathbb{R}^{D\times d}$$, is the factor loadings matrix with $$d < D$$ for parsimonious representation of the data, and $$Ψ_i$$ is the diagonal noise matrix. Note that the mixture of probabilistic principal component analysis (MPPCA) model is a special case of MFA with the distribution of the errors assumed to be isotropic with $$Ψ_i = Iσ_i^2$$.

What is meant by subspace clustering here, and how does $$\Sigma_i = \Lambda_i \Lambda_i^T + \Psi_i$$ accomplish the same? I understand that this is a dimensionality reduction technique since $$\text{rank}(\Lambda_i) \leq d < D$$. It'd be great if someone could help me understand more, and/or suggest resources I could look into for learning about this as an absolute beginner.

From what I understand, $$x = \Lambda z + u$$ is one factor-analyzer (right?), i.e. the generative model in maximum likelihood factor analysis. This paper goes on to define a mixture of factor-analyzers indexed by $$\omega_j$$, where $$j = 1,...,m$$. The generative model now obeys the distribution $$P(x) = \sum_{i=1}^m \int P(x|z,\omega_j)P(z|\omega_j)P(\omega_j)dz$$ where, $$P(z|\omega_j) = P(z) = \mathcal{N}(0,I)$$. How does this help/achieve the desired objective? Why take the sum from $$1$$ to $$m$$? Where is subspace clustering happening, and what's happening on a high-level when we are using this mixture of factor-analyzers?