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Learning distance functions between complex objects, such as the Wassersteindistance to compare point sets, is a common goal in machine learningapplications. However, functions on such complex objects (e.g., point sets andgraphs) are often required to be invariant to a wide variety of group actionse.g. permutation or rigid transformation. Therefore, continuous and symmetricproduct functions (such as distance functions) on such complex objects mustalso be invariant to the product of such group actions. We call these functionssymmetric and factor-wise group invariant (or SFGI functions in short). In thispaper, we first present a general neural network architecture for approximatingSFGI functions. The main contribution of this paper combines this generalneural network with a sketching idea to develop a specific and efficient neuralnetwork which can approximate the $p$-th Wasserstein distance between pointsets. Very importantly, the required model complexity is independent of thesizes of input point sets. On the theoretical front, to the best of ourknowledge, this is the first result showing that there exists a neural networkwith the capacity to approximate Wasserstein distance with bounded modelcomplexity. Our work provides an interesting integration of sketching ideas forgeometric problems with universal approximation of symmetric functions. On theempirical front, we present a range of results showing that our newly proposedneural network architecture performs comparatively or better than other models(including a SOTA Siamese Autoencoder based approach). In particular, ourneural network generalizes significantly better and trains much faster than theSOTA Siamese AE. Finally, this line of investigation could be useful inexploring effective neural network design for solving a broad range ofgeometric optimization problems (e.g., $k$-means in a metric space).

Neural approximation of Wasserstein distance via a universal architecture for symmetric and factorwise group invariant functions