89,854 research outputs found
Optimal Clustering under Uncertainty
Classical clustering algorithms typically either lack an underlying
probability framework to make them predictive or focus on parameter estimation
rather than defining and minimizing a notion of error. Recent work addresses
these issues by developing a probabilistic framework based on the theory of
random labeled point processes and characterizing a Bayes clusterer that
minimizes the number of misclustered points. The Bayes clusterer is analogous
to the Bayes classifier. Whereas determining a Bayes classifier requires full
knowledge of the feature-label distribution, deriving a Bayes clusterer
requires full knowledge of the point process. When uncertain of the point
process, one would like to find a robust clusterer that is optimal over the
uncertainty, just as one may find optimal robust classifiers with uncertain
feature-label distributions. Herein, we derive an optimal robust clusterer by
first finding an effective random point process that incorporates all
randomness within its own probabilistic structure and from which a Bayes
clusterer can be derived that provides an optimal robust clusterer relative to
the uncertainty. This is analogous to the use of effective class-conditional
distributions in robust classification. After evaluating the performance of
robust clusterers in synthetic mixtures of Gaussians models, we apply the
framework to granular imaging, where we make use of the asymptotic
granulometric moment theory for granular images to relate robust clustering
theory to the application.Comment: 19 pages, 5 eps figures, 1 tabl
Semantic distillation: a method for clustering objects by their contextual specificity
Techniques for data-mining, latent semantic analysis, contextual search of
databases, etc. have long ago been developed by computer scientists working on
information retrieval (IR). Experimental scientists, from all disciplines,
having to analyse large collections of raw experimental data (astronomical,
physical, biological, etc.) have developed powerful methods for their
statistical analysis and for clustering, categorising, and classifying objects.
Finally, physicists have developed a theory of quantum measurement, unifying
the logical, algebraic, and probabilistic aspects of queries into a single
formalism. The purpose of this paper is twofold: first to show that when
formulated at an abstract level, problems from IR, from statistical data
analysis, and from physical measurement theories are very similar and hence can
profitably be cross-fertilised, and, secondly, to propose a novel method of
fuzzy hierarchical clustering, termed \textit{semantic distillation} --
strongly inspired from the theory of quantum measurement --, we developed to
analyse raw data coming from various types of experiments on DNA arrays. We
illustrate the method by analysing DNA arrays experiments and clustering the
genes of the array according to their specificity.Comment: Accepted for publication in Studies in Computational Intelligence,
Springer-Verla
Noisy Subspace Clustering via Thresholding
We consider the problem of clustering noisy high-dimensional data points into
a union of low-dimensional subspaces and a set of outliers. The number of
subspaces, their dimensions, and their orientations are unknown. A
probabilistic performance analysis of the thresholding-based subspace
clustering (TSC) algorithm introduced recently in [1] shows that TSC succeeds
in the noisy case, even when the subspaces intersect. Our results reveal an
explicit tradeoff between the allowed noise level and the affinity of the
subspaces. We furthermore find that the simple outlier detection scheme
introduced in [1] provably succeeds in the noisy case.Comment: Presented at the IEEE Int. Symp. Inf. Theory (ISIT) 2013, Istanbul,
Turkey. The version posted here corrects a minor error in the published
version. Specifically, the exponent -c n_l in the success probability of
Theorem 1 and in the corresponding proof outline has been corrected to
-c(n_l-1
Self-organized model of cascade spreading
We study simultaneous price drops of real stocks and show that for high drop
thresholds they follow a power-law distribution. To reproduce these collective
downturns, we propose a minimal self-organized model of cascade spreading based
on a probabilistic response of the system elements to stress conditions. This
model is solvable using the theory of branching processes and the mean-field
approximation. For a wide range of parameters, the system is in a critical
state and displays a power-law cascade-size distribution similar to the
empirically observed one. We further generalize the model to reproduce
volatility clustering and other observed properties of real stocks.Comment: 8 pages, 6 figure
Joint ranking and clustering based on Markov Chain transition probabilities learned from data
The focus of this thesis is to develop a Markov Chain based framework
for joint ranking and clustering of a dataset without
the need for critical user-defined hyper-parameters.
Joint ranking and clustering may be useful in several respects,
and may give additional insight for the data analyst,
as opposed to the traditional separate ranking and
clustering procedures.
By coupling Markov chain theory with recent advances in kernel
methods using the so-called probabilistic cluster kernel,
we are able to learn the transition probabilities from the
inherent structures in the data in a near parameter-free approach.
The theory developed in this thesis is applied to several
real world datasets of different types with
promising results
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