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Numerical solutions of a boundary value problem on the sphere using radial basis functions
Boundary value problems on the unit sphere arise naturally in geophysics and
oceanography when scientists model a physical quantity on large scales. Robust
numerical methods play an important role in solving these problems. In this
article, we construct numerical solutions to a boundary value problem defined
on a spherical sub-domain (with a sufficiently smooth boundary) using radial
basis functions (RBF). The error analysis between the exact solution and the
approximation is provided. Numerical experiments are presented to confirm
theoretical estimates
Distributed Representations of Sentences and Documents
Many machine learning algorithms require the input to be represented as a
fixed-length feature vector. When it comes to texts, one of the most common
fixed-length features is bag-of-words. Despite their popularity, bag-of-words
features have two major weaknesses: they lose the ordering of the words and
they also ignore semantics of the words. For example, "powerful," "strong" and
"Paris" are equally distant. In this paper, we propose Paragraph Vector, an
unsupervised algorithm that learns fixed-length feature representations from
variable-length pieces of texts, such as sentences, paragraphs, and documents.
Our algorithm represents each document by a dense vector which is trained to
predict words in the document. Its construction gives our algorithm the
potential to overcome the weaknesses of bag-of-words models. Empirical results
show that Paragraph Vectors outperform bag-of-words models as well as other
techniques for text representations. Finally, we achieve new state-of-the-art
results on several text classification and sentiment analysis tasks
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