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This paper is about supervised and semi-supervised dimensionality reduction (DR) by generating spectral embeddings from multi-output data based on the pairwise proximity information. Two flexible and generic frameworks are proposed to achieve supervised DR (SDR) for multilabel classification. One is able to extend any existing single-label SDR to multilabel via sample duplication, referred to as MESD. The other is a multilabel design framework that tackles the SDR problem by computing weight (proximity) matrices based on simultaneous feature and label information, referred to as MOPE, as a generalization of many current techniques. A diverse set of different schemes for label-based proximity calculation, as well as a mechanism for combining label-based and feature-based weight information by considering information importance and prioritization, are proposed for MOPE. Additionally, we summarize many current spectral methods for unsupervised DR (UDR), single/multilabel SDR, and semi-supervised DR (SSDR) and express them under a common template representation as a general guide to researchers in the field. We also propose a general framework for achieving SSDR by combining existing SDR and UDR models, and also a procedure of reducing the computational cost via learning with a target set of relation features. The effectiveness of our proposed methodologies is demonstrated with experiments with document collections for multilabel text categorization from the natural language processing domain.