The metric sketching problem is defined as follows. Given a metric on n points, and > 0, we wish to produce a small size data structure (sketch) that, given any pair of point indices, recovers the distance between the points up to a 1 + distortion. In this paper we consider metrics induced by 2 and 1 norms whose spread (the ratio of the diameter to the closest pair distance) is bounded by Φ > 0. A well-known dimensionality reduction theorem due to Johnson and Lindenstrauss yields a sketch of size O( −2 log(Φn)n log n), i.e., O( −2 log(Φn) log n) bits per point. We show that this bound is not optimal, and can be substantially improved to O( −2 log(1/ ) · log n + log log Φ) bits per point. Furthermore, we show that our bound is tight up to a factor of log(1/ ).We also consider sketching of general metrics and provide a sketch of size O(n log(1/ ) + log log Φ) bits per point, which we show is optimal.

Recent work shows that the expressive power of Graph Neural Networks (GNNs) in distinguishing non-isomorphic graphs is exactly the same as that of the Weisfeiler-Lehman (WL) graph test. In particular, they show that the WL test can be simulated by GNNs. However, those simulations involve neural networks for the "combine" function of size polynomial or even exponential in the number of graph nodes n, as well as feature vectors of length linear in n. We present an improved simulation of the WL test on GNNs with exponentially lower complexity. In particular, the neural network implementing the combine function in each node has only polylog(n) parameters, and the feature vectors exchanged by the nodes of GNN consists of only O(log n) bits. We also give logarithmic lower bounds for the feature vector length and the size of the neural networks, showing the (near)-optimality of our construction.

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