1,631 research outputs found
Generalized density clustering
We study generalized density-based clustering in which sharply defined
clusters such as clusters on lower-dimensional manifolds are allowed. We show
that accurate clustering is possible even in high dimensions. We propose two
data-based methods for choosing the bandwidth and we study the stability
properties of density clusters. We show that a simple graph-based algorithm
successfully approximates the high density clusters.Comment: Published in at http://dx.doi.org/10.1214/10-AOS797 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
The log-linear group-lasso estimator and its asymptotic properties
We define the group-lasso estimator for the natural parameters of the
exponential families of distributions representing hierarchical log-linear
models under multinomial sampling scheme. Such estimator arises as the solution
of a convex penalized likelihood optimization problem based on the group-lasso
penalty. We illustrate how it is possible to construct an estimator of the
underlying log-linear model using the blocks of nonzero coefficients recovered
by the group-lasso procedure. We investigate the asymptotic properties of the
group-lasso estimator as a model selection method in a double-asymptotic
framework, in which both the sample size and the model complexity grow
simultaneously. We provide conditions guaranteeing that the group-lasso
estimator is model selection consistent, in the sense that, with overwhelming
probability as the sample size increases, it correctly identifies all the sets
of nonzero interactions among the variables. Provided the sequences of true
underlying models is sparse enough, recovery is possible even if the number of
cells grows larger than the sample size. Finally, we derive some central limit
type of results for the log-linear group-lasso estimator.Comment: Published in at http://dx.doi.org/10.3150/11-BEJ364 the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
Markov Properties of Discrete Determinantal Point Processes
Determinantal point processes (DPPs) are probabilistic models for repulsion.
When used to represent the occurrence of random subsets of a finite base set,
DPPs allow to model global negative associations in a mathematically elegant
and direct way. Discrete DPPs have become popular and computationally tractable
models for solving several machine learning tasks that require the selection of
diverse objects, and have been successfully applied in numerous real-life
problems. Despite their popularity, the statistical properties of such models
have not been adequately explored. In this note, we derive the Markov
properties of discrete DPPs and show how they can be expressed using graphical
models.Comment: 9 pages, 1 figur
Consistency of spectral clustering in stochastic block models
We analyze the performance of spectral clustering for community extraction in
stochastic block models. We show that, under mild conditions, spectral
clustering applied to the adjacency matrix of the network can consistently
recover hidden communities even when the order of the maximum expected degree
is as small as , with the number of nodes. This result applies to
some popular polynomial time spectral clustering algorithms and is further
extended to degree corrected stochastic block models using a spherical
-median spectral clustering method. A key component of our analysis is a
combinatorial bound on the spectrum of binary random matrices, which is sharper
than the conventional matrix Bernstein inequality and may be of independent
interest.Comment: Published in at http://dx.doi.org/10.1214/14-AOS1274 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Univariate Mean Change Point Detection: Penalization, CUSUM and Optimality
The problem of univariate mean change point detection and localization based
on a sequence of independent observations with piecewise constant means has
been intensively studied for more than half century, and serves as a blueprint
for change point problems in more complex settings. We provide a complete
characterization of this classical problem in a general framework in which the
upper bound on the noise variance, the minimal spacing
between two consecutive change points and the minimal magnitude of the
changes, are allowed to vary with . We first show that consistent
localization of the change points, when the signal-to-noise ratio , is impossible. In contrast, when
diverges with at the rate of at least
, we demonstrate that two computationally-efficient change
point estimators, one based on the solution to an -penalized least
squares problem and the other on the popular wild binary segmentation
algorithm, are both consistent and achieve a localization rate of the order
. We further show that such rate is minimax
optimal, up to a term
Optimal change point detection and localization in sparse dynamic networks
We study the problem of change point localization in dynamic networks models. We assume that we observe a sequence of independent adjacency matrices of the same size, each corresponding to a realization of an unknown inhomogeneous Bernoulli model. The underlying distribution of the adjacency matrices are piecewise constant, and may change over a subset of the time points, called change points. We are concerned with recovering the unknown number and positions of the change points. In our model setting, we allow for all the model parameters to change with the total number of time points, including the network size, the minimal spacing between consecutive change points, the magnitude of the smallest change and the degree of sparsity of the networks. We first identify a region of impossibility in the space of the model parameters such that no change point estimator is provably consistent if the data are generated according to parameters falling in that region. We propose a computationally-simple algorithm for network change point localization, called network binary segmentation, that relies on weighted averages of the adjacency matrices. We show that network binary segmentation is consistent over a range of the model parameters that nearly cover the complement of the impossibility region, thus demonstrating the existence of a phase transition for the problem at hand. Next, we devise a more sophisticated algorithm based on singular value thresholding, called local refinement, that delivers more accurate estimates of the change point locations. Under appropriate conditions, local refinement guarantees a minimax optimal rate for network change point localization while remaining computationally feasible
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