The Balanced-Pairwise-Affinities (BPA) feature transform is designed to upgrade the features of a set of input items to facilitate downstream matching or grouping related tasks. The transformed set encodes a rich representation of high order relations between the instance features. A particular min-cost-max-flow fractional matching problem, whose entropy regularized version can be approximated by an optimal transport (OT) optimization, results in transform which is efficient, differentiable, equivariant, parameterless and probabilistically interpretable. While the Sinkhorn OT solver has been adapted extensively in many contexts, we use it differently by minimizing the cost between a set of features to \textit{itself} and using the transport plan's \textit{rows} as the new representation. Empirically, the transform is highly effective and flexible in its use, consistently improving networks it is inserted into, in a variety of tasks and training schemes. We demonstrate state-of-the-art results in few-shot-classification, unsupervised-image-clustering and person-re-identification.