Joint Flow Matching: Consistent Generative and Discriminative Inference from a Shared Flow

We study Joint Flow Matching (JFM), a flow-based framework for learning a joint distribution over paired variables such as images and class labels. JFM transports the variables in opposite directions: an image is transported from noise to data while its label is transported from data to noise. This provides a single probabilistic model in which generation and classification are represented as complementary conditional inferences. We formulate the joint distribution induced by this construction and characterise the relationship between its forward and reverse conditionals. We investigate the resulting model for classification and conditional generation, including confidence calibration and consistency between generated samples and classifier predictions. Experiments on conditional datasets demonstrate competitive classification accuracy, well-calibrated confidence without post-hoc calibration, and classifier-consistent generation. We further explore the shared joint distribution as a basis for interpreting the relationship between discriminative and generative behaviour.