Let ε ∈ (0, 1) and X ⊂ R d be arbitrary with |X| having size n > 1. The Johnson-Lindenstrauss lemma states there exists f :We show that a strictly stronger version of this statement holds, answering one of the main open questions of [MMMR18]: "∀y ∈ X" in the above statement may be replaced with "∀y ∈ R d ", so that f not only preserves distances within X, but also distances to X from the rest of space. Previously this stronger version was only known with the worse bound m = O(ε −4 log n). Our proof is via a tighter analysis of (a specific instantiation of) the embedding recipe of [MMMR18]. * Harvard University. shyamnarayanan@college.harvard.edu. Supported by a PRISE fellowship and a Herchel-Smith Fellowship.† Harvard University. minilek@seas.harvard.edu.
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.
Phenotype robustness, defined as the average mutational robustness of all the genotypes that map to a given phenotype, plays a key role in facilitating neutral exploration of novel phenotypic variation by an evolving population. By applying results from coding theory, we prove that the maximum phenotype robustness occurs when genotypes are organised as bricklayer's graphs, so called because they resemble the way in which a bricklayer would fill in a Hamming graph. The value of the maximal robustness is given by a fractal continuous everywhere but differentiable nowhere sums-of-digits function from number theory. Interestingly, genotype-phenotype (GP) maps for RNA secondary structure and the HP model for protein folding can exhibit phenotype robustness that exactly attains this upper bound. By exploiting properties of the sums-of-digits function, we prove a lower bound on the deviation of the maximum robustness of phenotypes with multiple neutral components from the bricklayer's graph bound, and show that RNA secondary structure phenotypes obey this bound. Finally, we show how robustness changes when phenotypes are coarse-grained and derive a formula and associated bounds for the transition probabilities between such phenotypes.
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