Advanced information technology and data analytics promise to enhance the efficiency, sustainability, and economics of the power grid infrastructure. 
Aligned with this goal, we will present two learning paradigms leveraging the link of electricity markets with the underlying physical grids. First, we demonstrate how grid topologies can be revealed using only publicly available market data. This becomes feasible because multiplying the matrix of real-time locational marginal prices (LMPs) 
with the weighted grid Laplacian yields surprisingly a low-rank and sparse matrix product. Secondly, we show how day-ahead price inference can be cast as a low-rank kernel learning task. Congestion patterns 
are uniquely modeled as rank-one components in the matrix of spatio-temporally varying LMPs.