Linear unit-tests for invariance discovery

There is an increasing interest in algorithms to learn invariant correlations across training environments. A big share of the current proposals find theoretical support in the causality literature but, how useful are they in practice? The purpose of this note is to propose six linear low-dimensional problems --"unit tests"-- to evaluate different types of out-of-distribution generalization in a precise manner. Following initial experiments, none of the three recently proposed alternatives passes all tests.