5 research outputs found
Distinct Distances in Graph Drawings
The \emph{distance-number} of a graph is the minimum number of distinct
edge-lengths over all straight-line drawings of in the plane. This
definition generalises many well-known concepts in combinatorial geometry. We
consider the distance-number of trees, graphs with no -minor, complete
bipartite graphs, complete graphs, and cartesian products. Our main results
concern the distance-number of graphs with bounded degree. We prove that
-vertex graphs with bounded maximum degree and bounded treewidth have
distance-number in . To conclude such a logarithmic upper
bound, both the degree and the treewidth need to be bounded. In particular, we
construct graphs with treewidth 2 and polynomial distance-number. Similarly, we
prove that there exist graphs with maximum degree 5 and arbitrarily large
distance-number. Moreover, as increases the existential lower bound on
the distance-number of -regular graphs tends to
Small Transformers Compute Universal Metric Embeddings
We study representations of data from an arbitrary metric space
in the space of univariate Gaussian mixtures with a transport metric (Delon and
Desolneux 2020). We derive embedding guarantees for feature maps implemented by
small neural networks called \emph{probabilistic transformers}. Our guarantees
are of memorization type: we prove that a probabilistic transformer of depth
about and width about can bi-H\"{o}lder embed any -point
dataset from with low metric distortion, thus avoiding the curse
of dimensionality. We further derive probabilistic bi-Lipschitz guarantees,
which trade off the amount of distortion and the probability that a randomly
chosen pair of points embeds with that distortion. If 's geometry
is sufficiently regular, we obtain stronger, bi-Lipschitz guarantees for all
points in the dataset. As applications, we derive neural embedding guarantees
for datasets from Riemannian manifolds, metric trees, and certain types of
combinatorial graphs. When instead embedding into multivariate Gaussian
mixtures, we show that probabilistic transformers can compute bi-H\"{o}lder
embeddings with arbitrarily small distortion.Comment: 42 pages, 10 Figures, 3 Table