Sparse Compositional Local Metric Learning

Mahalanobis distance metric learning becomes an especially challenging problem as the dimension of the feature space p is scaled upwards. The number of parameters to optimize grows with space complexity of order O (p 2), making storage infeasible, interpretability poor, and causing the model to have a high tendency to overfit. Additionally, optimization while maintaining feasibility of the solution becomes prohibitively expensive, requiring a projection onto the positive semi-definite cone after every iteration. In addition to the obvious space and computational challenges, vanilla distance metric learning is unable to model complex and multi-modal trends in the data. Inspired by the recent resurgence of Frank-Wolfe style optimization, we propose a new method for sparse compositional local Mahalanobis distance metric learning. Our proposed technique learns a set of distance metrics which are composed of local and global components. We capture local interactions in the feature space, while ensuring that all metrics share a global component, which may act as a regularizer. We optimize our model using an alternating pairwise Frank-Wolfe style algorithm. This serves a dual purpose, we can control the sparsity of our solution, and altogether avoid any expensive projection operations. Finally, we conduct an empirical evaluation of our method with the current state of the art and present the results on five datasets from varying domains.