692 research outputs found
Discrete Signal Processing on Graphs: Frequency Analysis
Signals and datasets that arise in physical and engineering applications, as
well as social, genetics, biomolecular, and many other domains, are becoming
increasingly larger and more complex. In contrast to traditional time and image
signals, data in these domains are supported by arbitrary graphs. Signal
processing on graphs extends concepts and techniques from traditional signal
processing to data indexed by generic graphs. This paper studies the concepts
of low and high frequencies on graphs, and low-, high-, and band-pass graph
filters. In traditional signal processing, there concepts are easily defined
because of a natural frequency ordering that has a physical interpretation. For
signals residing on graphs, in general, there is no obvious frequency ordering.
We propose a definition of total variation for graph signals that naturally
leads to a frequency ordering on graphs and defines low-, high-, and band-pass
graph signals and filters. We study the design of graph filters with specified
frequency response, and illustrate our approach with applications to sensor
malfunction detection and data classification
Consensus State Gram Matrix Estimation for Stochastic Switching Networks from Spectral Distribution Moments
Reaching distributed average consensus quickly and accurately over a network
through iterative dynamics represents an important task in numerous distributed
applications. Suitably designed filters applied to the state values can
significantly improve the convergence rate. For constant networks, these
filters can be viewed in terms of graph signal processing as polynomials in a
single matrix, the consensus iteration matrix, with filter response evaluated
at its eigenvalues. For random, time-varying networks, filter design becomes
more complicated, involving eigendecompositions of sums and products of random,
time-varying iteration matrices. This paper focuses on deriving an estimate for
the Gram matrix of error in the state vectors over a filtering window for
large-scale, stationary, switching random networks. The result depends on the
moments of the empirical spectral distribution, which can be estimated through
Monte-Carlo simulation. This work then defines a quadratic objective function
to minimize the expected consensus estimate error norm. Simulation results
provide support for the approximation.Comment: 52nd Asilomar Conference on Signals, Systems, and Computers (Asilomar
2017
Gossip and Distributed Kalman Filtering: Weak Consensus under Weak Detectability
The paper presents the gossip interactive Kalman filter (GIKF) for
distributed Kalman filtering for networked systems and sensor networks, where
inter-sensor communication and observations occur at the same time-scale. The
communication among sensors is random; each sensor occasionally exchanges its
filtering state information with a neighbor depending on the availability of
the appropriate network link. We show that under a weak distributed
detectability condition:
1. the GIKF error process remains stochastically bounded, irrespective of the
instability properties of the random process dynamics; and
2. the network achieves \emph{weak consensus}, i.e., the conditional
estimation error covariance at a (uniformly) randomly selected sensor converges
in distribution to a unique invariant measure on the space of positive
semi-definite matrices (independent of the initial state.)
To prove these results, we interpret the filtered states (estimates and error
covariances) at each node in the GIKF as stochastic particles with local
interactions. We analyze the asymptotic properties of the error process by
studying as a random dynamical system the associated switched (random) Riccati
equation, the switching being dictated by a non-stationary Markov chain on the
network graph.Comment: Submitted to the IEEE Transactions, 30 pages
Finding Non-overlapping Clusters for Generalized Inference Over Graphical Models
Graphical models use graphs to compactly capture stochastic dependencies
amongst a collection of random variables. Inference over graphical models
corresponds to finding marginal probability distributions given joint
probability distributions. In general, this is computationally intractable,
which has led to a quest for finding efficient approximate inference
algorithms. We propose a framework for generalized inference over graphical
models that can be used as a wrapper for improving the estimates of approximate
inference algorithms. Instead of applying an inference algorithm to the
original graph, we apply the inference algorithm to a block-graph, defined as a
graph in which the nodes are non-overlapping clusters of nodes from the
original graph. This results in marginal estimates of a cluster of nodes, which
we further marginalize to get the marginal estimates of each node. Our proposed
block-graph construction algorithm is simple, efficient, and motivated by the
observation that approximate inference is more accurate on graphs with longer
cycles. We present extensive numerical simulations that illustrate our
block-graph framework with a variety of inference algorithms (e.g., those in
the libDAI software package). These simulations show the improvements provided
by our framework.Comment: Extended the previous version to include extensive numerical
simulations. See http://www.ima.umn.edu/~dvats/GeneralizedInference.html for
code and dat
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