83,004 research outputs found
Fast Approximate -Means via Cluster Closures
-means, a simple and effective clustering algorithm, is one of the most
widely used algorithms in multimedia and computer vision community. Traditional
-means is an iterative algorithm---in each iteration new cluster centers are
computed and each data point is re-assigned to its nearest center. The cluster
re-assignment step becomes prohibitively expensive when the number of data
points and cluster centers are large.
In this paper, we propose a novel approximate -means algorithm to greatly
reduce the computational complexity in the assignment step. Our approach is
motivated by the observation that most active points changing their cluster
assignments at each iteration are located on or near cluster boundaries. The
idea is to efficiently identify those active points by pre-assembling the data
into groups of neighboring points using multiple random spatial partition
trees, and to use the neighborhood information to construct a closure for each
cluster, in such a way only a small number of cluster candidates need to be
considered when assigning a data point to its nearest cluster. Using complexity
analysis, image data clustering, and applications to image retrieval, we show
that our approach out-performs state-of-the-art approximate -means
algorithms in terms of clustering quality and efficiency
Geometrical complexity of data approximators
There are many methods developed to approximate a cloud of vectors embedded
in high-dimensional space by simpler objects: starting from principal points
and linear manifolds to self-organizing maps, neural gas, elastic maps, various
types of principal curves and principal trees, and so on. For each type of
approximators the measure of the approximator complexity was developed too.
These measures are necessary to find the balance between accuracy and
complexity and to define the optimal approximations of a given type. We propose
a measure of complexity (geometrical complexity) which is applicable to
approximators of several types and which allows comparing data approximations
of different types.Comment: 10 pages, 3 figures, minor correction and extensio
Principal components analysis in the space of phylogenetic trees
Phylogenetic analysis of DNA or other data commonly gives rise to a
collection or sample of inferred evolutionary trees. Principal Components
Analysis (PCA) cannot be applied directly to collections of trees since the
space of evolutionary trees on a fixed set of taxa is not a vector space. This
paper describes a novel geometrical approach to PCA in tree-space that
constructs the first principal path in an analogous way to standard linear
Euclidean PCA. Given a data set of phylogenetic trees, a geodesic principal
path is sought that maximizes the variance of the data under a form of
projection onto the path. Due to the high dimensionality of tree-space and the
nonlinear nature of this problem, the computational complexity is potentially
very high, so approximate optimization algorithms are used to search for the
optimal path. Principal paths identified in this way reveal and quantify the
main sources of variation in the original collection of trees in terms of both
topology and branch lengths. The approach is illustrated by application to
simulated sets of trees and to a set of gene trees from metazoan (animal)
species.Comment: Published in at http://dx.doi.org/10.1214/11-AOS915 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Hashing for Similarity Search: A Survey
Similarity search (nearest neighbor search) is a problem of pursuing the data
items whose distances to a query item are the smallest from a large database.
Various methods have been developed to address this problem, and recently a lot
of efforts have been devoted to approximate search. In this paper, we present a
survey on one of the main solutions, hashing, which has been widely studied
since the pioneering work locality sensitive hashing. We divide the hashing
algorithms two main categories: locality sensitive hashing, which designs hash
functions without exploring the data distribution and learning to hash, which
learns hash functions according the data distribution, and review them from
various aspects, including hash function design and distance measure and search
scheme in the hash coding space
K-nearest Neighbor Search by Random Projection Forests
K-nearest neighbor (kNN) search has wide applications in many areas,
including data mining, machine learning, statistics and many applied domains.
Inspired by the success of ensemble methods and the flexibility of tree-based
methodology, we propose random projection forests (rpForests), for kNN search.
rpForests finds kNNs by aggregating results from an ensemble of random
projection trees with each constructed recursively through a series of
carefully chosen random projections. rpForests achieves a remarkable accuracy
in terms of fast decay in the missing rate of kNNs and that of discrepancy in
the kNN distances. rpForests has a very low computational complexity. The
ensemble nature of rpForests makes it easily run in parallel on multicore or
clustered computers; the running time is expected to be nearly inversely
proportional to the number of cores or machines. We give theoretical insights
by showing the exponential decay of the probability that neighboring points
would be separated by ensemble random projection trees when the ensemble size
increases. Our theory can be used to refine the choice of random projections in
the growth of trees, and experiments show that the effect is remarkable.Comment: 15 pages, 4 figures, 2018 IEEE Big Data Conferenc
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