4,095 research outputs found
Local Tomography of Large Networks under the Low-Observability Regime
This article studies the problem of reconstructing the topology of a network
of interacting agents via observations of the state-evolution of the agents. We
focus on the large-scale network setting with the additional constraint of
observations, where only a small fraction of the agents can be
feasibly observed. The goal is to infer the underlying subnetwork of
interactions and we refer to this problem as . In order to
study the large-scale setting, we adopt a proper stochastic formulation where
the unobserved part of the network is modeled as an Erd\"{o}s-R\'enyi random
graph, while the observable subnetwork is left arbitrary. The main result of
this work is establishing that, under this setting, local tomography is
actually possible with high probability, provided that certain conditions on
the network model are met (such as stability and symmetry of the network
combination matrix). Remarkably, such conclusion is established under the
- , where the cardinality of the observable
subnetwork is fixed, while the size of the overall network scales to infinity.Comment: To appear in IEEE Transactions on Information Theor
Markov models for fMRI correlation structure: is brain functional connectivity small world, or decomposable into networks?
Correlations in the signal observed via functional Magnetic Resonance Imaging
(fMRI), are expected to reveal the interactions in the underlying neural
populations through hemodynamic response. In particular, they highlight
distributed set of mutually correlated regions that correspond to brain
networks related to different cognitive functions. Yet graph-theoretical
studies of neural connections give a different picture: that of a highly
integrated system with small-world properties: local clustering but with short
pathways across the complete structure. We examine the conditional independence
properties of the fMRI signal, i.e. its Markov structure, to find realistic
assumptions on the connectivity structure that are required to explain the
observed functional connectivity. In particular we seek a decomposition of the
Markov structure into segregated functional networks using decomposable graphs:
a set of strongly-connected and partially overlapping cliques. We introduce a
new method to efficiently extract such cliques on a large, strongly-connected
graph. We compare methods learning different graph structures from functional
connectivity by testing the goodness of fit of the model they learn on new
data. We find that summarizing the structure as strongly-connected networks can
give a good description only for very large and overlapping networks. These
results highlight that Markov models are good tools to identify the structure
of brain connectivity from fMRI signals, but for this purpose they must reflect
the small-world properties of the underlying neural systems
Covariance Estimation: The GLM and Regularization Perspectives
Finding an unconstrained and statistically interpretable reparameterization
of a covariance matrix is still an open problem in statistics. Its solution is
of central importance in covariance estimation, particularly in the recent
high-dimensional data environment where enforcing the positive-definiteness
constraint could be computationally expensive. We provide a survey of the
progress made in modeling covariance matrices from two relatively complementary
perspectives: (1) generalized linear models (GLM) or parsimony and use of
covariates in low dimensions, and (2) regularization or sparsity for
high-dimensional data. An emerging, unifying and powerful trend in both
perspectives is that of reducing a covariance estimation problem to that of
estimating a sequence of regression problems. We point out several instances of
the regression-based formulation. A notable case is in sparse estimation of a
precision matrix or a Gaussian graphical model leading to the fast graphical
LASSO algorithm. Some advantages and limitations of the regression-based
Cholesky decomposition relative to the classical spectral (eigenvalue) and
variance-correlation decompositions are highlighted. The former provides an
unconstrained and statistically interpretable reparameterization, and
guarantees the positive-definiteness of the estimated covariance matrix. It
reduces the unintuitive task of covariance estimation to that of modeling a
sequence of regressions at the cost of imposing an a priori order among the
variables. Elementwise regularization of the sample covariance matrix such as
banding, tapering and thresholding has desirable asymptotic properties and the
sparse estimated covariance matrix is positive definite with probability
tending to one for large samples and dimensions.Comment: Published in at http://dx.doi.org/10.1214/11-STS358 the Statistical
Science (http://www.imstat.org/sts/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Longitudinal LASSO: Jointly Learning Features and Temporal Contingency for Outcome Prediction
Longitudinal analysis is important in many disciplines, such as the study of
behavioral transitions in social science. Only very recently, feature selection
has drawn adequate attention in the context of longitudinal modeling. Standard
techniques, such as generalized estimating equations, have been modified to
select features by imposing sparsity-inducing regularizers. However, they do
not explicitly model how a dependent variable relies on features measured at
proximal time points. Recent graphical Granger modeling can select features in
lagged time points but ignores the temporal correlations within an individual's
repeated measurements. We propose an approach to automatically and
simultaneously determine both the relevant features and the relevant temporal
points that impact the current outcome of the dependent variable. Meanwhile,
the proposed model takes into account the non-{\em i.i.d} nature of the data by
estimating the within-individual correlations. This approach decomposes model
parameters into a summation of two components and imposes separate block-wise
LASSO penalties to each component when building a linear model in terms of the
past measurements of features. One component is used to select features
whereas the other is used to select temporal contingent points. An accelerated
gradient descent algorithm is developed to efficiently solve the related
optimization problem with detailed convergence analysis and asymptotic
analysis. Computational results on both synthetic and real world problems
demonstrate the superior performance of the proposed approach over existing
techniques.Comment: Proceedings of the 21th ACM SIGKDD International Conference on
Knowledge Discovery and Data Mining. ACM, 201
Sparse Multivariate Factor Regression
We consider the problem of multivariate regression in a setting where the
relevant predictors could be shared among different responses. We propose an
algorithm which decomposes the coefficient matrix into the product of a long
matrix and a wide matrix, with an elastic net penalty on the former and an
penalty on the latter. The first matrix linearly transforms the
predictors to a set of latent factors, and the second one regresses the
responses on these factors. Our algorithm simultaneously performs dimension
reduction and coefficient estimation and automatically estimates the number of
latent factors from the data. Our formulation results in a non-convex
optimization problem, which despite its flexibility to impose effective
low-dimensional structure, is difficult, or even impossible, to solve exactly
in a reasonable time. We specify an optimization algorithm based on alternating
minimization with three different sets of updates to solve this non-convex
problem and provide theoretical results on its convergence and optimality.
Finally, we demonstrate the effectiveness of our algorithm via experiments on
simulated and real data
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