Wasserstein Distributionally Robust Graph Learning via Algorithm Unrolling

Abstract

In this paper, we consider inferring the underlying graph topology from smooth graph signals. Most existing approaches learn graphs by minimizing a well-designed empirical risk using the observed data, which may be prone to data uncertainty that arises from noisy measurements and limited observability. Therefore, the learned graphs may be unreliable and exhibit poor out-of-sample performance. To enhance the robustness to data uncertainty, we propose a smoothness-based graph learning framework from a distributionally robust perspective, which is equivalent to solving an $\mathrm{inf-sup}$ problem. However, learning graphs directly in this way is challenging since (i) the $\mathrm{inf-sup}$ problem is intractable, and (ii) many parameters need to be manually determined. To address these issues, we first reformulate the $\mathrm{inf-sup}$ problem into a tractable one, where robustness is achieved via a regularizer. Theoretically, we show that the regularizer can improve generalization of the proposed graph estimator by bounding the out-of-sample risks. We then propose an algorithm based on the ADMM framework to solve the induced problem and further unroll it into a neural network. All parameters are determined automatically and simultaneously by training the unrolled network. Extensive experiments on both synthetic and real-world data demonstrate that our approach can achieve superior and more robust performance than existing models on different observed signals.