  Among the flourishing research of weakly supervised learning (WSL), we
recognize the lack of a unified interpretation of the mechanism behind the
weakly supervised scenarios, let alone a systematic treatment of the risk
rewrite problem, a crucial step in the empirical risk minimization approach. In
this paper, we introduce a framework providing a comprehensive understanding
and a unified methodology for WSL. The formulation component of the framework,
leveraging a contamination perspective, provides a unified interpretation of
how weak supervision is formed and subsumes fifteen existing WSL settings. The
induced reduction graphs offer comprehensive connections over WSLs. The
analysis component of the framework, viewed as a decontamination process,
provides a systematic method of conducting risk rewrite. In addition to the
conventional inverse matrix approach, we devise a novel strategy called
marginal chain aiming to decontaminate distributions. We justify the
feasibility of the proposed framework by recovering existing rewrites reported
in the literature.
