10 research outputs found
Active Learning of Multiple Source Multiple Destination Topologies
We consider the problem of inferring the topology of a network with
sources and receivers (hereafter referred to as an -by- network), by
sending probes between the sources and receivers. Prior work has shown that
this problem can be decomposed into two parts: first, infer smaller subnetwork
components (i.e., -by-'s or -by-'s) and then merge these components
to identify the -by- topology. In this paper, we focus on the second
part, which had previously received less attention in the literature. In
particular, we assume that a -by- topology is given and that all
-by- components can be queried and learned using end-to-end probes. The
problem is which -by-'s to query and how to merge them with the given
-by-, so as to exactly identify the -by- topology, and optimize a
number of performance metrics, including the number of queries (which directly
translates into measurement bandwidth), time complexity, and memory usage. We
provide a lower bound, , on the number of
-by-'s required by any active learning algorithm and propose two greedy
algorithms. The first algorithm follows the framework of multiple hypothesis
testing, in particular Generalized Binary Search (GBS), since our problem is
one of active learning, from -by- queries. The second algorithm is called
the Receiver Elimination Algorithm (REA) and follows a bottom-up approach: at
every step, it selects two receivers, queries the corresponding -by-, and
merges it with the given -by-; it requires exactly steps, which is
much less than all possible -by-'s. Simulation results
over synthetic and realistic topologies demonstrate that both algorithms
correctly identify the -by- topology and are near-optimal, but REA is
more efficient in practice
Maximum Likelihood Estimation for Linear Gaussian Covariance Models
We study parameter estimation in linear Gaussian covariance models, which are
-dimensional Gaussian models with linear constraints on the covariance
matrix. Maximum likelihood estimation for this class of models leads to a
non-convex optimization problem which typically has many local maxima. Using
recent results on the asymptotic distribution of extreme eigenvalues of the
Wishart distribution, we provide sufficient conditions for any hill-climbing
method to converge to the global maximum. Although we are primarily interested
in the case in which , the proofs of our results utilize large-sample
asymptotic theory under the scheme . Remarkably, our
numerical simulations indicate that our results remain valid for as small
as . An important consequence of this analysis is that for sample sizes , maximum likelihood estimation for linear Gaussian covariance
models behaves as if it were a convex optimization problem
Latent tree models
Latent tree models are graphical models defined on trees, in which only a
subset of variables is observed. They were first discussed by Judea Pearl as
tree-decomposable distributions to generalise star-decomposable distributions
such as the latent class model. Latent tree models, or their submodels, are
widely used in: phylogenetic analysis, network tomography, computer vision,
causal modeling, and data clustering. They also contain other well-known
classes of models like hidden Markov models, Brownian motion tree model, the
Ising model on a tree, and many popular models used in phylogenetics. This
article offers a concise introduction to the theory of latent tree models. We
emphasise the role of tree metrics in the structural description of this model
class, in designing learning algorithms, and in understanding fundamental
limits of what and when can be learned
Link Delay Estimation via Expander Graphs
One of the purposes of network tomography is to infer the status of
parameters (e.g., delay) for the links inside a network through end-to-end
probing between (external) boundary nodes along predetermined routes. In this
work, we apply concepts from compressed sensing and expander graphs to the
delay estimation problem. We first show that a relative majority of network
topologies are not expanders for existing expansion criteria. Motivated by this
challenge, we then relax such criteria, enabling us to acquire simulation
evidence that link delays can be estimated for 30% more networks. That is, our
relaxation expands the list of identifiable networks with bounded estimation
error by 30%. We conduct a simulation performance analysis of delay estimation
and congestion detection on the basis of l1 minimization, demonstrating that
accurate estimation is feasible for an increasing proportion of networks
Active Topology Inference using Network Coding
Our goal is to infer the topology of a network when (i) we can send probes
between sources and receivers at the edge of the network and (ii) intermediate
nodes can perform simple network coding operations, i.e., additions. Our key
intuition is that network coding introduces topology-dependent correlation in
the observations at the receivers, which can be exploited to infer the
topology. For undirected tree topologies, we design hierarchical clustering
algorithms, building on our prior work. For directed acyclic graphs (DAGs),
first we decompose the topology into a number of two-source, two-receiver
(2-by-2) subnetwork components and then we merge these components to
reconstruct the topology. Our approach for DAGs builds on prior work on
tomography, and improves upon it by employing network coding to accurately
distinguish among all different 2-by-2 components. We evaluate our algorithms
through simulation of a number of realistic topologies and compare them to
active tomographic techniques without network coding. We also make connections
between our approach and alternatives, including passive inference, traceroute,
and packet marking
EM's Convergence in Gaussian Latent Tree Models
We study the optimization landscape of the log-likelihood function and the
convergence of the Expectation-Maximization (EM) algorithm in latent Gaussian
tree models, i.e. tree-structured Gaussian graphical models whose leaf nodes
are observable and non-leaf nodes are unobservable. We show that the unique
non-trivial stationary point of the population log-likelihood is its global
maximum, and establish that the expectation-maximization algorithm is
guaranteed to converge to it in the single latent variable case. Our results
for the landscape of the log-likelihood function in general latent tree models
provide support for the extensive practical use of maximum likelihood
based-methods in this setting. Our results for the EM algorithm extend an
emerging line of work on obtaining global convergence guarantees for this
celebrated algorithm. We show our results for the non-trivial stationary points
of the log-likelihood by arguing that a certain system of polynomial equations
obtained from the EM updates has a unique non-trivial solution. The global
convergence of the EM algorithm follows by arguing that all trivial fixed
points are higher-order saddle points
Active topology inference using network coding
Our goal, in this paper, is to infer the topology of a network when (i) we can send probes between sources and receivers at the edge of the network and (ii) intermediate nodes can perform simple network coding operations, i.e., additions. Our key intuition is that network coding introduces topology-dependent correlation in the observations at the receivers, which can be exploited to infer the topology. For undirected tree topologies, we design hierarchical clustering algorithms, building on our prior work in [24]. For directed acyclic graphs (DAGs), first we decompose the topology into a number of two source, two receiver (2-by-2) subnetwork components and then we merge these components to reconstruct the topology. Our approach for DAGs builds on prior work on tomography [36], and improves upon it by employing network coding to accurately distinguish among all different 2-by-2 components. We evaluate our algorithms through simulation of a number of realistic topologies and compare them to active tomographic techniques without network coding. We also make connections between our approach and other alternatives, including passive inference, traceroute, and packet marking