25,816 research outputs found
Fast Hierarchical Clustering and Other Applications of Dynamic Closest Pairs
We develop data structures for dynamic closest pair problems with arbitrary
distance functions, that do not necessarily come from any geometric structure
on the objects. Based on a technique previously used by the author for
Euclidean closest pairs, we show how to insert and delete objects from an
n-object set, maintaining the closest pair, in O(n log^2 n) time per update and
O(n) space. With quadratic space, we can instead use a quadtree-like structure
to achieve an optimal time bound, O(n) per update. We apply these data
structures to hierarchical clustering, greedy matching, and TSP heuristics, and
discuss other potential applications in machine learning, Groebner bases, and
local improvement algorithms for partition and placement problems. Experiments
show our new methods to be faster in practice than previously used heuristics.Comment: 20 pages, 9 figures. A preliminary version of this paper appeared at
the 9th ACM-SIAM Symp. on Discrete Algorithms, San Francisco, 1998, pp.
619-628. For source code and experimental results, see
http://www.ics.uci.edu/~eppstein/projects/pairs
Information Recovery from Pairwise Measurements
A variety of information processing tasks in practice involve recovering
objects from single-shot graph-based measurements, particularly those taken
over the edges of some measurement graph . This paper concerns the
situation where each object takes value over a group of different values,
and where one is interested to recover all these values based on observations
of certain pairwise relations over . The imperfection of
measurements presents two major challenges for information recovery: 1)
: a (dominant) portion of measurements are
corrupted; 2) : a significant fraction of pairs are
unobservable, i.e. can be highly sparse.
Under a natural random outlier model, we characterize the , that is, the critical threshold of non-corruption rate
below which exact information recovery is infeasible. This accommodates a very
general class of pairwise relations. For various homogeneous random graph
models (e.g. Erdos Renyi random graphs, random geometric graphs, small world
graphs), the minimax recovery rate depends almost exclusively on the edge
sparsity of the measurement graph irrespective of other graphical
metrics. This fundamental limit decays with the group size at a square root
rate before entering a connectivity-limited regime. Under the Erdos Renyi
random graph, a tractable combinatorial algorithm is proposed to approach the
limit for large (), while order-optimal recovery is
enabled by semidefinite programs in the small regime.
The extended (and most updated) version of this work can be found at
(http://arxiv.org/abs/1504.01369).Comment: This version is no longer updated -- please find the latest version
at (arXiv:1504.01369
Introduction to Grassmann Manifolds and Quantum Computation
Geometrical aspects of quantum computing are reviewed elementarily for
non-experts and/or graduate students who are interested in both Geometry and
Quantum Computation.
In the first half we show how to treat Grassmann manifolds which are very
important examples of manifolds in Mathematics and Physics. Some of their
applications to Quantum Computation and its efficiency problems are shown in
the second half. An interesting current topic of Holonomic Quantum Computation
is also covered.
In the Appendix some related advanced topics are discussed.Comment: Latex File, 28 pages, corrected considerably in the process of
refereeing. to appear in Journal of Applied Mathematic
On the optimality of shape and data representation in the spectral domain
A proof of the optimality of the eigenfunctions of the Laplace-Beltrami
operator (LBO) in representing smooth functions on surfaces is provided and
adapted to the field of applied shape and data analysis. It is based on the
Courant-Fischer min-max principle adapted to our case. % The theorem we present
supports the new trend in geometry processing of treating geometric structures
by using their projection onto the leading eigenfunctions of the decomposition
of the LBO. Utilisation of this result can be used for constructing numerically
efficient algorithms to process shapes in their spectrum. We review a couple of
applications as possible practical usage cases of the proposed optimality
criteria. % We refer to a scale invariant metric, which is also invariant to
bending of the manifold. This novel pseudo-metric allows constructing an LBO by
which a scale invariant eigenspace on the surface is defined. We demonstrate
the efficiency of an intermediate metric, defined as an interpolation between
the scale invariant and the regular one, in representing geometric structures
while capturing both coarse and fine details. Next, we review a numerical
acceleration technique for classical scaling, a member of a family of
flattening methods known as multidimensional scaling (MDS). There, the
optimality is exploited to efficiently approximate all geodesic distances
between pairs of points on a given surface, and thereby match and compare
between almost isometric surfaces. Finally, we revisit the classical principal
component analysis (PCA) definition by coupling its variational form with a
Dirichlet energy on the data manifold. By pairing the PCA with the LBO we can
handle cases that go beyond the scope defined by the observation set that is
handled by regular PCA
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