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Measuring node proximity on large scale networks is a fundamental building block in many application domains, ranging from computer vision, e-commerce, social networks, software engineering, disaster management to biology and epidemiology. The state of the art (e.g., random walk based methods) typically assumes the input network is given a priori, with the known network topology and the associated edge weights. A few recent works aim to further infer the optimal edge weights based on the side information. This paper generalizes the challenge in multiple dimensions, aiming to learn optimal networks for node proximity measures. First (optimization scope), our proposed formulation explores a much larger parameter space, so that it is able to simultaneously infer the optimal network topology and the associated edge weights. This is important as a noisy or missing edge could greatly mislead the network …