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]: Consider a weighted or unweighted k-nearest neighbor graph that has been built on n data points drawn randomly according to some density p on R. We study the convergence of the shortest path distance in such graphs as the sample size tends to infinity. We prove that for unweighted kNN graphs, this distance converges to an unpleasant distance function on the underlying space whose properties are detrimental to machine learning. We also study the behavior of the shortest path distance in weighted kNN graphs.
[A]: Shortest path distance in random k-nearest neighbor graphs


[Q]: Latent Dirichlet Allocation (LDA) mining thematic structure of documents plays an important role in nature language processing and machine learning areas. However, the probability distribution from LDA only describes the statistical relationship of occurrences in the corpus and usually in practice, probability is not the best choice for feature representations. Recently, embedding methods have been proposed to represent words and documents by learning essential concepts and representations, such as Word2Vec and Doc2Vec. The embedded representations have shown more effectiveness than LDA-style representations in many tasks. In this paper, we propose the Topic2Vec approach which can learn topic representations in the same semantic vector space with words, as an alternative to probability. The experimental results show that Topic2Vec achieves interesting and meaningful results.
[A]: Topic2Vec: Learning Distributed Representations of Topics


[Q]: We formulate a phonetic-prosodic space based on attributes as perceptual observables, rather than articulatory specifications. We propose an alphabet as markers in the phonetic subspace, aiming for a resolution sufficient to support recognition of all spoken languages. The prosodic subspace is made up of directly measurable physical variables. With the proposed alphabet, traditional diphthongs naturally generalize to a broader class of language-neutral phonotactic constraints, indicating a correlation structure similar to that of the traditional sonority-based syllable. We define a stochastic structure on the phone strings based on this diphthongal constraint, and show how a specific spoken language can be defined as a specific set of probability distributions of this stochastic structure. Furthermore, phonological variations within a spoken language can be modeled as varying probability distributions restricted to the phonetic subspace, conditioned on different values in the prosodic subspace.
[A]:
A 10-dimensional Phonetic-prosodic Space and its Stochastic Structure A framework for probabilistic modeling of spoken languages and their phonology