Supervised probabilistic principal component analysis

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Linear dimensionality reduction: survey, insights, and generalizations (2014, background): “As a first extension, one might have incomplete class labels; Yu et al. (2006) extends LDA (with a probabilistic PCA framework; see Section 3.2.2) to the semi-supervised setting where not all points are labeled.”
Relational learning via collective matrix factorization (2008, methodology): “Principal components analysis, which factors a doubly centered matrix under squared loss, has also been extended to the three-factor schema [36].”
A Unified View of Matrix Factorization Models (2008, methodology): “Principal components analysis, which factors a doubly centered matrix under squared loss, has also been extended to the three-factor schema [38].”
Bayesian exponential family projections for coupled data sources (2010, methodology): “…followed by a semi-parametric formulation applicable to even more flexible distributions (Sa- jama and Orlitsky, 2004), by more efficient algorithms converging into the global optimum (Guo and Schuurmans, 2008), and supervised PCA (both exponential family and standard; Yu et al. 2006; Guo 2009).”