In this task, you are given a part of an article. Your task is to generate headline (title) for this text. Preferred headlines are under fifteen words.

[Q]: We introduce a new spatial data structure for high dimensional data called the approximate principal direction tree (APD tree) that adapts to the intrinsic dimension of the data. Our algorithm ensures vector-quantization accuracy similar to that of computationally-expensive PCA trees with similar time-complexity to that of loweraccuracy RP trees. APD trees use a small number of powermethod iterations to find splitting planes for recursively partitioning the data. As such they provide a natural trade-off between the running-time and accuracy achieved by RP and PCA trees. Our theoretical results establish a) strong performance guarantees regardless of the convergence rate of the powermethod and b) that O(log d) iterations suffice to establish the guarantee of PCA trees when the intrinsic dimension is d. We demonstrate this trade-off and the efficacy of our data structure on both the CPU and GPU.
[A]: Approximate Principal Direction Trees


[Q]: Language learning is thought to be a highly complex process. One of the hurdles in learning a language is to learn the rules of syntax of the language. Rules of syntax are often ordered in that before one rule can applied one must apply another. It has been thought that to learn the order of n rules one must go through all n! permutations. Thus to learn the order of 27 rules would require 27! steps or 1.08889x10 steps. This number is much greater than the number of seconds since the beginning of the universe! In an insightful analysis the linguist Block ([Block 86], pp. 62-63, p.238) showed that with the assumption of transitivity this vast number of learning steps reduces to a mere 377 steps. We present a mathematical analysis of the complexity of Block’s algorithm. The algorithm has a complexity of order n given n rules. In addition, we improve Block’s results exponentially, by introducing an algorithm that has complexity of order less than n log n.
[A]: On the complexity of learning a language: An improvement of Block’s algorithm


[Q]: We develop novel firstand second-order features for dependency parsing based on the Google Syntactic Ngrams corpus, a collection of subtree counts of parsed sentences from scanned books. We also extend previous work on surface n-gram features from Web1T to the Google Books corpus and from first-order to second-order, comparing and analysing performance over newswire and web treebanks. Surface and syntactic n-grams both produce substantial and complementary gains in parsing accuracy across domains. Our best system combines the two feature sets, achieving up to 0.8% absolute UAS improvements on newswire and 1.4% on web text.
[A]:
Web-scale Surface and Syntactic n-gram Features for Dependency Parsing