2,322 research outputs found
Testing for Differences in Gaussian Graphical Models: Applications to Brain Connectivity
Functional brain networks are well described and estimated from data with
Gaussian Graphical Models (GGMs), e.g. using sparse inverse covariance
estimators. Comparing functional connectivity of subjects in two populations
calls for comparing these estimated GGMs. Our goal is to identify differences
in GGMs known to have similar structure. We characterize the uncertainty of
differences with confidence intervals obtained using a parametric distribution
on parameters of a sparse estimator. Sparse penalties enable statistical
guarantees and interpretable models even in high-dimensional and low-sample
settings. Characterizing the distributions of sparse models is inherently
challenging as the penalties produce a biased estimator. Recent work invokes
the sparsity assumptions to effectively remove the bias from a sparse estimator
such as the lasso. These distributions can be used to give confidence intervals
on edges in GGMs, and by extension their differences. However, in the case of
comparing GGMs, these estimators do not make use of any assumed joint structure
among the GGMs. Inspired by priors from brain functional connectivity we derive
the distribution of parameter differences under a joint penalty when parameters
are known to be sparse in the difference. This leads us to introduce the
debiased multi-task fused lasso, whose distribution can be characterized in an
efficient manner. We then show how the debiased lasso and multi-task fused
lasso can be used to obtain confidence intervals on edge differences in GGMs.
We validate the techniques proposed on a set of synthetic examples as well as
neuro-imaging dataset created for the study of autism
Regularization and Bayesian Learning in Dynamical Systems: Past, Present and Future
Regularization and Bayesian methods for system identification have been
repopularized in the recent years, and proved to be competitive w.r.t.
classical parametric approaches. In this paper we shall make an attempt to
illustrate how the use of regularization in system identification has evolved
over the years, starting from the early contributions both in the Automatic
Control as well as Econometrics and Statistics literature. In particular we
shall discuss some fundamental issues such as compound estimation problems and
exchangeability which play and important role in regularization and Bayesian
approaches, as also illustrated in early publications in Statistics. The
historical and foundational issues will be given more emphasis (and space), at
the expense of the more recent developments which are only briefly discussed.
The main reason for such a choice is that, while the recent literature is
readily available, and surveys have already been published on the subject, in
the author's opinion a clear link with past work had not been completely
clarified.Comment: Plenary Presentation at the IFAC SYSID 2015. Submitted to Annual
Reviews in Contro
Sparsity-Cognizant Total Least-Squares for Perturbed Compressive Sampling
Solving linear regression problems based on the total least-squares (TLS)
criterion has well-documented merits in various applications, where
perturbations appear both in the data vector as well as in the regression
matrix. However, existing TLS approaches do not account for sparsity possibly
present in the unknown vector of regression coefficients. On the other hand,
sparsity is the key attribute exploited by modern compressive sampling and
variable selection approaches to linear regression, which include noise in the
data, but do not account for perturbations in the regression matrix. The
present paper fills this gap by formulating and solving TLS optimization
problems under sparsity constraints. Near-optimum and reduced-complexity
suboptimum sparse (S-) TLS algorithms are developed to address the perturbed
compressive sampling (and the related dictionary learning) challenge, when
there is a mismatch between the true and adopted bases over which the unknown
vector is sparse. The novel S-TLS schemes also allow for perturbations in the
regression matrix of the least-absolute selection and shrinkage selection
operator (Lasso), and endow TLS approaches with ability to cope with sparse,
under-determined "errors-in-variables" models. Interesting generalizations can
further exploit prior knowledge on the perturbations to obtain novel weighted
and structured S-TLS solvers. Analysis and simulations demonstrate the
practical impact of S-TLS in calibrating the mismatch effects of contemporary
grid-based approaches to cognitive radio sensing, and robust
direction-of-arrival estimation using antenna arrays.Comment: 30 pages, 10 figures, submitted to IEEE Transactions on Signal
Processin
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