Statistical Optimal Transport posed as Learning Kernel Embedding
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This work takes the novel approach of posing the statistical Optimal Transport (OT) problem as that of learning the transport plan's kernel mean embedding. The key advantage is that the estimates for the embeddings of the marginals can now be employed directly, leading to a dimension-free sample complexity for the proposed transport plan and transport map estimators. Also, because of the implicit smoothing in the kernel embeddings, the proposed estimators can perform out-of-sample estimation. Interestingly, the proposed formulation employs an MMD based regularization to avoid overfitting, which is complementary to existing ϕ-divergence (entropy) based regularization techniques. An appropriate representer theorem is presented that leads to a fully kernelized formulation and hence the same formulation can be used to perform continuous/semi-discrete/discrete OT in any non-standard domain (as long as universal kernels in those domains are known). Finally, an ADMM based algorithm is presented for solving the kernelized formulation efficiently. Empirical results show that the proposed estimator outperforms discrete OT based estimator in terms of transport map accuracy.
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