Learning to match discrete objects has been a central task in machine learning, often facilitated by a continuous relaxation of the matching structure. However, practical problems entail partial matchings due to missing correspondences, which pose difficulties to the one-to-one matching learning techniques that dominate the state-of-the-art. This paper introduces Gumbel-IPF networks for learning latent partial matchings. At the core of our method is the differentiable Iterative Proportional Fitting (IPF) procedure that biproportionally projects onto the transportation polytope of target marginals. Our theoretical framework also allows drawing samples from the temperature-dependent partial matching distribution. We investigate the properties of common-practice relaxations through the lens of biproportional fitting and introduce a new metric, the empirical prediction shift. Our method’s advantages are demonstrated in experimental results on the semantic keypoints partial matching task on the Pascal VOC, IMC-PT-SparseGM, and CUB2001 datasets.

Multi-graph matching is a prominent structured prediction task, in which the predicted label is constrained to the space of cycle-consistent matchings. While direct loss minimization is an effective method for learning predictors over structured label spaces, it cannot be applied efficiently to the problem at hand, since executing a specialized solver across sets of matching predictions is computationally prohibitive. Moreover, there’s no supervision on the ground-truth matchings over cycle-consistent prediction sets. Our key insight is to strictly enforce the matching constraints in pairwise matching predictions and softly enforce the cycle-consistency constraints by casting them as weighted loss terms, such that the severity of inconsistency with global predictions is tuned by a penalty parameter. Inspired by the classic penalty method, we prove that our method theoretically recovers the optimal multi-graph matching constrained solution. Our method’s advantages are brought to light in experimental results on the popular keypoint matching task on the Pascal VOC and the Willow ObjectClass datasets.

Direct loss minimization is a popular approach for learning predictors over structured label spaces. This approach is computationally appealing as it replaces integration with optimization and allows to propagate gradients in a deep net using loss-perturbed prediction. Recently, this technique was extended to generative models, by introducing a randomized predictor that samples a structure from a randomly perturbed score function. In this work, we interpolate between these techniques by learning the variance of randomized structured predictors as well as their mean, in order to balance between the learned score function and the randomized noise. We demonstrate empirically the effectiveness of learning this balance in structured discrete spaces.