In the present paper, we studied a Dynamic Stochastic Block Model (DSBM) under the assumptions that the connection probabilities, as functions of time, are smooth and that at most s nodes can switch their class memberships between two consecutive time points. We estimate the edge probability tensor by a kernel-type procedure and extract the group memberships of the nodes by spectral clustering. The procedure is computationally viable, adaptive to the unknown smoothness of the functional connection probabilities, to the rate s of membership switching and to the unknown number of clusters. In addition, it is accompanied by non-asymptotic guarantees for the precision of estimation and clustering.
We extend deconvolution in a periodic setting to deal with functional data.
The resulting functional deconvolution model can be viewed as a generalization
of a multitude of inverse problems in mathematical physics where one needs to
recover initial or boundary conditions on the basis of observations from a
noisy solution of a partial differential equation. In the case when it is
observed at a finite number of distinct points, the proposed functional
deconvolution model can also be viewed as a multichannel deconvolution model.
We derive minimax lower bounds for the $L^2$-risk in the proposed functional
deconvolution model when $f(\cdot)$ is assumed to belong to a Besov ball and
the blurring function is assumed to possess some smoothness properties,
including both regular-smooth and super-smooth convolutions. Furthermore, we
propose an adaptive wavelet estimator of $f(\cdot)$ that is asymptotically
optimal (in the minimax sense), or near-optimal within a logarithmic factor, in
a wide range of Besov balls. In addition, we consider a discretization of the
proposed functional deconvolution model and investigate when the availability
of continuous data gives advantages over observations at the asymptotically
large number of points. As an illustration, we discuss particular examples for
both continuous and discrete settings.Comment: Published in at http://dx.doi.org/10.1214/07-AOS552 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
We consider a problem of recovering a high-dimensional vector $\mu$ observed
in white noise, where the unknown vector $\mu$ is assumed to be sparse. The
objective of the paper is to develop a Bayesian formalism which gives rise to a
family of $l_0$-type penalties. The penalties are associated with various
choices of the prior distributions $\pi_n(\cdot)$ on the number of nonzero
entries of $\mu$ and, hence, are easy to interpret. The resulting Bayesian
estimators lead to a general thresholding rule which accommodates many of the
known thresholding and model selection procedures as particular cases
corresponding to specific choices of $\pi_n(\cdot)$. Furthermore, they achieve
optimality in a rather general setting under very mild conditions on the prior.
We also specify the class of priors $\pi_n(\cdot)$ for which the resulting
estimator is adaptively optimal (in the minimax sense) for a wide range of
sparse sequences and consider several examples of such priors.Comment: Published in at http://dx.doi.org/10.1214/009053607000000226 the
Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Mathematical Statistics (http://www.imstat.org
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