Over the past ten years, metric learning allowed the improvement of numerous machine learning approaches that manipulate distances or similarities. In this field, local metric learning has been shown to be very efficient, especially to take into account non linearities in the data and better capture the peculiarities of the application of interest. However, it is well known that local metric learning (i) can entail overfitting and (ii) face difficulties to compare two instances that are assigned to two different local models. In this paper, we address these two issues by introducing a novel metric learning algorithm that linearly combines local models (C2LM). Starting from a partition of the space in regions and a model (a score function) for each region, C2LM defines a metric between points as a weighted combination of the models. A weight vector is learned for each pair of regions, and a spatial regularization ensures that the weight vectors evolve smoothly and that nearby models are favored in the combination. The proposed approach has the particularity of working in a regression setting, of working implicitly at different scales, and of being generic enough so that it is applicable to similarities and distances. We prove theoretical guarantees of the approach using the framework of algorithmic robustness. We carry out experiments with datasets using both distances (perceptual color distances, using Mahalanobis-like distances) and similarities (semantic word similarities, using bilinear forms), showing that C2LM consistently improves regression accuracy even in the case where the amount of training data is small.

Local Metric Approach	C2LM Approach