permutationequivariant

Permutation equivariance is a property of functions or models in machine learning and mathematics. A function is said to be permutation equivariant if applying a permutation to its input results in the same permutation being applied to its output. This means that the order of the input elements does not affect the relationship between the input and the output, only the arrangement of the output elements.

For example, if we have a function f that takes a set of items {a, b, c} as input, and produces a set of results {f(a), f(b), f(c)}, then f is permutation equivariant if, for any permutation P of the input set, applying P to the input results in P applied to the output. Specifically, if P swaps 'a' and 'b', such that the input becomes {b, a, c}, the output should become {f(b), f(a), f(c)}. The structure of the transformation is preserved regardless of the input order.

This property is particularly useful when dealing with data where the order of elements is not inherently meaningful, such as sets of points in space, molecules, or collections of objects. Neural network architectures designed to be permutation equivariant, like certain types of graph neural networks or deep sets, can effectively learn from such data by not being distracted by arbitrary input ordering. This allows them to generalize better and require less training data compared to models that are not designed with this property.