23,665 research outputs found
Rates of convergence for robust geometric inference
Distances to compact sets are widely used in the field of Topological Data
Analysis for inferring geometric and topological features from point clouds. In
this context, the distance to a probability measure (DTM) has been introduced
by Chazal et al. (2011) as a robust alternative to the distance a compact set.
In practice, the DTM can be estimated by its empirical counterpart, that is the
distance to the empirical measure (DTEM). In this paper we give a tight control
of the deviation of the DTEM. Our analysis relies on a local analysis of
empirical processes. In particular, we show that the rates of convergence of
the DTEM directly depends on the regularity at zero of a particular quantile
fonction which contains some local information about the geometry of the
support. This quantile function is the relevant quantity to describe precisely
how difficult is a geometric inference problem. Several numerical experiments
illustrate the convergence of the DTEM and also confirm that our bounds are
tight
Gossip Algorithms for Distributed Signal Processing
Gossip algorithms are attractive for in-network processing in sensor networks
because they do not require any specialized routing, there is no bottleneck or
single point of failure, and they are robust to unreliable wireless network
conditions. Recently, there has been a surge of activity in the computer
science, control, signal processing, and information theory communities,
developing faster and more robust gossip algorithms and deriving theoretical
performance guarantees. This article presents an overview of recent work in the
area. We describe convergence rate results, which are related to the number of
transmitted messages and thus the amount of energy consumed in the network for
gossiping. We discuss issues related to gossiping over wireless links,
including the effects of quantization and noise, and we illustrate the use of
gossip algorithms for canonical signal processing tasks including distributed
estimation, source localization, and compression.Comment: Submitted to Proceedings of the IEEE, 29 page
Estimating the geometric median in Hilbert spaces with stochastic gradient algorithms: and almost sure rates of convergence
The geometric median, also called -median, is often used in robust
statistics. Moreover, it is more and more usual to deal with large samples
taking values in high dimensional spaces. In this context, a fast recursive
estimator has been introduced by Cardot, Cenac and Zitt. This work aims at
studying more precisely the asymptotic behavior of the estimators of the
geometric median based on such non linear stochastic gradient algorithms. The
rates of convergence as well as almost sure rates of convergence of
these estimators are derived in general separable Hilbert spaces. Moreover, the
optimal rate of convergence in quadratic mean of the averaged algorithm is also
given
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