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There has been significant recent interest in \emph{graph-based nearest neighbor search methods}, many of which are centered on the construction of (approximately) {\em navigable}\/ graphs over high-dimensional point sets. A graph is navigable if we can successfully move from any starting node to any target node using a greedy routing strategy where we always move to the neighbor that is closest to the destination according to the given distance function. The complete graph is obviously navigable for any point set, but the important question for applications is if sparser graphs can be constructed. While this question is fairly well understood in low-dimensions, we establish some of the first upper and lower bounds for high-dimensional point sets. First, we give a simple and efficient way to construct a navigable graph with average degree $$O(\sqrt{n \log n })$$ for any set of $$n$$ points, in any dimension, for any distance function. We compliment this result with a nearly matching lower bound: even under the Euclidean metric in $$O(\log n)$$ dimensions, a random point set has no navigable graph with average degree $$O(n^{\alpha})$$ for any $$\alpha < 1/2$$. Our lower bound relies on sharp anti-concentration bounds for binomial random variables, which we use to show that the near-neighborhoods of a set of random points do not overlap significantly, forcing any navigable graph to have many edges. 

Existing deep-learning approaches to semantic column type annotation (CTA) have important shortcomings: they rely on semantic types which are fixed at training time; require a large number of training samples per type; incur high run-time inference costs; and their performance can degrade when evaluated on novel datasets, even when types remain constant. Large language models have exhibited strong zero-shot classification performance on a wide range of tasks and in this paper we explore their use for CTA. We introduce ArcheType, a simple, practical method for context sampling, prompt serialization, model querying, and label remapping, which enables large language models to solve CTA problems in a fully zero-shot manner. We ablate each component of our method separately, and establish that improvements to context sampling and label remapping provide the most consistent gains. ArcheType establishes a new state-of-the-art performance on zero-shot CTA benchmarks (including three new domain-specific benchmarks which we release along with this paper), and when used in conjunction with classical CTA techniques, it outperforms a SOTA DoDuo model on the fine-tuned SOTAB benchmark. 

Recently, Bessa et al. (PODS 2023) showed that sketches based on coordinated weighted sampling theoretically and empirically outperform popular linear sketching methods like Johnson-Lindentrauss projection and CountSketch for the ubiquitous problem of inner product estimation. We further develop this finding by introducing and analyzing two alternative sampling-based methods. In contrast to the computationally expensive algorithm in Bessa et al., our methods run in linear time (to compute the sketch) and perform better in practice, significantly beating linear sketching on a variety of tasks. For example, they provide state-of-the-art results for estimating the correlation between columns in unjoined tables, a problem that we show how to reduce to inner product estimation in a black-box way. While based on known sampling techniques (threshold and priority sampling) we introduce significant new theoretical analysis to prove approximation guarantees for our methods.