22,592 research outputs found
Time varying undirected graphs
Undirected graphs are often used to describe high dimensional distributions. Under sparsity conditions, the graph can be estimated using â„“ 1 penalization methods. However, current methods assume that the data are independent and identically distributed. If the distribution, and hence the graph, evolves over time then the data are not longer identically distributed. In this paper we develop a nonparametric method for estimating time varying graphical structure for multivariate Gaussian distributions using an â„“ 1 regularization method, and show that, as long as the covariances change smoothly over time, we can estimate the covariance matrix well (in predictive risk) even when p is larg
Time Varying Undirected Graphs
Undirected graphs are often used to describe high dimensional distributions.
Under sparsity conditions, the graph can be estimated using
penalization methods. However, current methods assume that the data are
independent and identically distributed. If the distribution, and hence the
graph, evolves over time then the data are not longer identically distributed.
In this paper, we show how to estimate the sequence of graphs for
non-identically distributed data, where the distribution evolves over time.Comment: 12 pages, 3 figures, to appear in COLT 200
Learning flexible representations of stochastic processes on graphs
Graph convolutional networks adapt the architecture of convolutional neural
networks to learn rich representations of data supported on arbitrary graphs by
replacing the convolution operations of convolutional neural networks with
graph-dependent linear operations. However, these graph-dependent linear
operations are developed for scalar functions supported on undirected graphs.
We propose a class of linear operations for stochastic (time-varying) processes
on directed (or undirected) graphs to be used in graph convolutional networks.
We propose a parameterization of such linear operations using functional
calculus to achieve arbitrarily low learning complexity. The proposed approach
is shown to model richer behaviors and display greater flexibility in learning
representations than product graph methods
Time scale modeling for consensus in sparse directed networks with time-varying topologies
The paper considers the consensus problem in large networks represented by
time-varying directed graphs. A practical way of dealing with large-scale
networks is to reduce their dimension by collapsing the states of nodes
belonging to densely and intensively connected clusters into aggregate
variables. It will be shown that under suitable conditions, the states of the
agents in each cluster converge fast toward a local agreement. Local agreements
correspond to aggregate variables which slowly converge to consensus. Existing
results concerning the time-scale separation in large networks focus on fixed
and undirected graphs. The aim of this work is to extend these results to the
more general case of time-varying directed topologies. It is noteworthy that in
the fixed and undirected graph case the average of the states in each cluster
is time-invariant when neglecting the interactions between clusters. Therefore,
they are good candidates for the aggregate variables. This is no longer
possible here. Instead, we find suitable time-varying weights to compute the
aggregate variables as time-invariant weighted averages of the states in each
cluster. This allows to deal with the more challenging time-varying directed
graph case. We end up with a singularly perturbed system which is analyzed by
using the tools of two time-scales averaging which seem appropriate to this
system
Consensus with Ternary Messages
We provide a protocol for real-valued average consensus by networks of agents
which exchange only a single message from the ternary alphabet {-1,0,1} between
neighbors at each step. Our protocol works on time-varying undirected graphs
subject to a connectivity condition, has a worst-case convergence time which is
polynomial in the number of agents and the initial values, and requires no
global knowledge about the graph topologies on the part of each node to
implement except for knowing an upper bound on the degrees of its neighbors
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