The extension of convolutional neural networks to irregular domains has pavedthe way to promising graph data analysis methods. It has however come at theexpense of a reduced representation power, as most of these new network archi-tectures can only learn isotropic filters and therefore often underfit the trainingdata. In this work, we propose a method for building anisotropic filters whenlearning representations of signals on a cartesian product graph. Instead of learn-ing directly on the product graph, we factorize it and learn different filters foreach factor, which is beneficial both in terms of computational cost and expressiv-ity of the filters. We show experimentally that anisotropic Laplacian polynomialsindeed outperform their isotropic counterpart on image classification and matrixcompletion tasks.