1,597 research outputs found
Sparse geometric graphs with small dilation
Given a set S of n points in R^D, and an integer k such that 0 <= k < n, we
show that a geometric graph with vertex set S, at most n - 1 + k edges, maximum
degree five, and dilation O(n / (k+1)) can be computed in time O(n log n). For
any k, we also construct planar n-point sets for which any geometric graph with
n-1+k edges has dilation Omega(n/(k+1)); a slightly weaker statement holds if
the points of S are required to be in convex position
Characterizing the impact of geometric properties of word embeddings on task performance
Analysis of word embedding properties to inform their use in downstream NLP
tasks has largely been studied by assessing nearest neighbors. However,
geometric properties of the continuous feature space contribute directly to the
use of embedding features in downstream models, and are largely unexplored. We
consider four properties of word embedding geometry, namely: position relative
to the origin, distribution of features in the vector space, global pairwise
distances, and local pairwise distances. We define a sequence of
transformations to generate new embeddings that expose subsets of these
properties to downstream models and evaluate change in task performance to
understand the contribution of each property to NLP models. We transform
publicly available pretrained embeddings from three popular toolkits (word2vec,
GloVe, and FastText) and evaluate on a variety of intrinsic tasks, which model
linguistic information in the vector space, and extrinsic tasks, which use
vectors as input to machine learning models. We find that intrinsic evaluations
are highly sensitive to absolute position, while extrinsic tasks rely primarily
on local similarity. Our findings suggest that future embedding models and
post-processing techniques should focus primarily on similarity to nearby
points in vector space.Comment: Appearing in the Third Workshop on Evaluating Vector Space
Representations for NLP (RepEval 2019). 7 pages + reference
Communication tree problems
In this paper, we consider random communication
requirements and several cost
measures for a particular model of tree routing on a
complete network. First
we show that a random tree does not give any approximation.
Then give
approximation algorithms for the case for two random models
of requirements.Postprint (published version
On metric Ramsey-type phenomena
The main question studied in this article may be viewed as a nonlinear
analogue of Dvoretzky's theorem in Banach space theory or as part of Ramsey
theory in combinatorics. Given a finite metric space on n points, we seek its
subspace of largest cardinality which can be embedded with a given distortion
in Hilbert space. We provide nearly tight upper and lower bounds on the
cardinality of this subspace in terms of n and the desired distortion. Our main
theorem states that for any epsilon>0, every n point metric space contains a
subset of size at least n^{1-\epsilon} which is embeddable in Hilbert space
with O(\frac{\log(1/\epsilon)}{\epsilon}) distortion. The bound on the
distortion is tight up to the log(1/\epsilon) factor. We further include a
comprehensive study of various other aspects of this problem.Comment: 67 pages, published versio
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