Draw the Graph, Draw the Negatives: A Local Sampling Framework for Contrastive Estimation

Noise Contrastive Estimation (NCE) provides a powerful framework for learning unnormalized models by contrasting data with negative samples. While classical NCE draws negatives globally from a reference distribution, Conditional NCE (CNCE) instead generates local negatives around each data point. Existing local constructions, however, are largely restricted to binary discrimination and simple negative-sampling schemes. In this work, we propose a general framework for local negative sampling in NCE that extends the binary view to multi-way classification through graphical negative-sampling schemes. We characterize when the geometry of a generation graph hides the origin of the sampling, in which case the proposal terms cancel and the exact Bayes posterior reduces to an energy-only softmax. Our framework recovers an existing method as a hidden-anchor construction and connects graphical negative sampling to contrastive divergence. Building on this framework, we further introduce a new method, Generative Origin Recognition on cyclic Indices Contrastive Estimation (GORI-CE), which generates negatives through a cyclic Gaussian bridge. Experiments on both synthetic and real data demonstrate the effectiveness of our method and highlight negative-sampling geometry as a useful design principle for energy-based learning.