Towards Fully Covariant Machine Learning

Abstract

Any representation of data involves arbitrary investigator choices. Because those choices are external to the data-generating process, each choice leads to an exact symmetry, corresponding to the group of transformations that takes one possible representation to another. These are the passive symmetries; they include coordinate freedom, gauge symmetry, and units covariance, all of which have led to important results in physics. In machine learning, the most visible passive symmetry is the relabeling or permutation symmetry of graphs. The active symmetries are those that must be established by observation and experiment. They include, for instance, translations invariances or rotation invariances of physical law. These symmetries are the subject of most of the equivariant machine learning literature. Our goal, in this conceptual contribution, is to understand the implications for machine learning of the many passive and active symmetries in play. We discuss dos and don'ts for machine learning practice if passive symmetries are to be respected. We discuss links to causal modeling and argue that the implementation of passive symmetries is particularly valuable when the goal of the learning problem is to generalize out of sample. We conjecture that the implementation of passive symmetries might help machine learning in the same ways that it transformed physics in the twentieth century.