30,755 research outputs found
Large-Scale Kernel Methods for Independence Testing
Representations of probability measures in reproducing kernel Hilbert spaces
provide a flexible framework for fully nonparametric hypothesis tests of
independence, which can capture any type of departure from independence,
including nonlinear associations and multivariate interactions. However, these
approaches come with an at least quadratic computational cost in the number of
observations, which can be prohibitive in many applications. Arguably, it is
exactly in such large-scale datasets that capturing any type of dependence is
of interest, so striking a favourable tradeoff between computational efficiency
and test performance for kernel independence tests would have a direct impact
on their applicability in practice. In this contribution, we provide an
extensive study of the use of large-scale kernel approximations in the context
of independence testing, contrasting block-based, Nystrom and random Fourier
feature approaches. Through a variety of synthetic data experiments, it is
demonstrated that our novel large scale methods give comparable performance
with existing methods whilst using significantly less computation time and
memory.Comment: 29 pages, 6 figure
An introduction to spectral distances in networks (extended version)
Many functions have been recently defined to assess the similarity among
networks as tools for quantitative comparison. They stem from very different
frameworks - and they are tuned for dealing with different situations. Here we
show an overview of the spectral distances, highlighting their behavior in some
basic cases of static and dynamic synthetic and real networks
Efficient Semidefinite Spectral Clustering via Lagrange Duality
We propose an efficient approach to semidefinite spectral clustering (SSC),
which addresses the Frobenius normalization with the positive semidefinite
(p.s.d.) constraint for spectral clustering. Compared with the original
Frobenius norm approximation based algorithm, the proposed algorithm can more
accurately find the closest doubly stochastic approximation to the affinity
matrix by considering the p.s.d. constraint. In this paper, SSC is formulated
as a semidefinite programming (SDP) problem. In order to solve the high
computational complexity of SDP, we present a dual algorithm based on the
Lagrange dual formalization. Two versions of the proposed algorithm are
proffered: one with less memory usage and the other with faster convergence
rate. The proposed algorithm has much lower time complexity than that of the
standard interior-point based SDP solvers. Experimental results on both UCI
data sets and real-world image data sets demonstrate that 1) compared with the
state-of-the-art spectral clustering methods, the proposed algorithm achieves
better clustering performance; and 2) our algorithm is much more efficient and
can solve larger-scale SSC problems than those standard interior-point SDP
solvers.Comment: 13 page
Spectrum of large random reversible Markov chains: two examples
We take on a Random Matrix theory viewpoint to study the spectrum of certain
reversible Markov chains in random environment. As the number of states tends
to infinity, we consider the global behavior of the spectrum, and the local
behavior at the edge, including the so called spectral gap. Results are
obtained for two simple models with distinct limiting features. The first model
is built on the complete graph while the second is a birth-and-death dynamics.
Both models give rise to random matrices with non independent entries.Comment: accepted in ALEA, March 201
On the ubiquity of the Cauchy distribution in spectral problems
We consider the distribution of the values at real points of random functions
which belong to the Herglotz-Pick (HP) class of analytic mappings of the upper
half plane into itself. It is shown that under mild stationarity assumptions
the individual values of HP functions with singular spectra have a Cauchy type
distribution. The statement applies to the diagonal matrix elements of random
operators, and holds regardless of the presence or not of level repulsion, i.e.
applies to both random matrix and Poisson-type spectra.Comment: Slightly revised version: presentation was made more explicit in
places, and additional references were provide
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