6,336 research outputs found
Group Lasso estimation of high-dimensional covariance matrices
In this paper, we consider the Group Lasso estimator of the covariance matrix
of a stochastic process corrupted by an additive noise. We propose to estimate
the covariance matrix in a high-dimensional setting under the assumption that
the process has a sparse representation in a large dictionary of basis
functions. Using a matrix regression model, we propose a new methodology for
high-dimensional covariance matrix estimation based on empirical contrast
regularization by a group Lasso penalty. Using such a penalty, the method
selects a sparse set of basis functions in the dictionary used to approximate
the process, leading to an approximation of the covariance matrix into a low
dimensional space. Consistency of the estimator is studied in Frobenius and
operator norms and an application to sparse PCA is proposed
Regularized Estimation of High-dimensional Covariance Matrices.
Many signal processing methods are fundamentally related to the
estimation of covariance matrices. In cases where there are a large
number of covariates the dimension of covariance matrices is much
larger than the number of available data samples. This is especially
true in applications where data acquisition is constrained by limited
resources such as time, energy, storage and bandwidth. This
dissertation attempts to develop necessary components for covariance
estimation in the high-dimensional setting. The dissertation makes
contributions in two main areas of covariance estimation: (1) high
dimensional shrinkage regularized covariance estimation and (2)
recursive online complexity regularized estimation with applications of
anomaly detection, graph tracking, and compressive sensing.
New shrinkage covariance estimation methods are proposed that
significantly outperform previous approaches in terms of mean squared
error. Two multivariate data scenarios are considered: (1)
independently Gaussian distributed data; and (2) heavy tailed
elliptically contoured data. For the former scenario we improve on
the Ledoit-Wolf (LW) shrinkage estimator using the principle of
Rao-Blackwell conditioning and iterative approximation of the
clairvoyant estimator. In the latter scenario, we apply a variance
normalizing transformation and propose an iterative robust LW
shrinkage estimator that is distribution-free within the elliptical
family. The proposed robustified estimator is implemented via fixed
point iterations with provable convergence and unique limit.
A recursive online covariance estimator is proposed for tracking
changes in an underlying time-varying graphical model. Covariance
estimation is decomposed into multiple decoupled adaptive regression
problems. A recursive recursive group lasso is derived using a
homotopy approach that generalizes online lasso methods to group
sparse system identification. By reducing the memory of the objective
function this leads to a group lasso regularized LMS that provably
dominates standard LMS. Finally, we introduce a state-of-the-art
sampling system, the Modulated Wideband Converter (MWC) which is based
on recently developed analog compressive sensing theory. By inferring
the block-sparse structures of the high-dimensional covariance matrix
from a set of random projections, the MWC is capable of achieving
sub-Nyquist sampling for multiband signals with arbitrary carrier
frequency over a wide bandwidth.Ph.D.Electrical Engineering: SystemsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/86396/1/yilun_1.pd
Structure estimation for discrete graphical models: Generalized covariance matrices and their inverses
We investigate the relationship between the structure of a discrete graphical
model and the support of the inverse of a generalized covariance matrix. We
show that for certain graph structures, the support of the inverse covariance
matrix of indicator variables on the vertices of a graph reflects the
conditional independence structure of the graph. Our work extends results that
have previously been established only in the context of multivariate Gaussian
graphical models, thereby addressing an open question about the significance of
the inverse covariance matrix of a non-Gaussian distribution. The proof
exploits a combination of ideas from the geometry of exponential families,
junction tree theory and convex analysis. These population-level results have
various consequences for graph selection methods, both known and novel,
including a novel method for structure estimation for missing or corrupted
observations. We provide nonasymptotic guarantees for such methods and
illustrate the sharpness of these predictions via simulations.Comment: Published in at http://dx.doi.org/10.1214/13-AOS1162 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Block-diagonal covariance selection for high-dimensional Gaussian graphical models
Gaussian graphical models are widely utilized to infer and visualize networks
of dependencies between continuous variables. However, inferring the graph is
difficult when the sample size is small compared to the number of variables. To
reduce the number of parameters to estimate in the model, we propose a
non-asymptotic model selection procedure supported by strong theoretical
guarantees based on an oracle inequality and a minimax lower bound. The
covariance matrix of the model is approximated by a block-diagonal matrix. The
structure of this matrix is detected by thresholding the sample covariance
matrix, where the threshold is selected using the slope heuristic. Based on the
block-diagonal structure of the covariance matrix, the estimation problem is
divided into several independent problems: subsequently, the network of
dependencies between variables is inferred using the graphical lasso algorithm
in each block. The performance of the procedure is illustrated on simulated
data. An application to a real gene expression dataset with a limited sample
size is also presented: the dimension reduction allows attention to be
objectively focused on interactions among smaller subsets of genes, leading to
a more parsimonious and interpretable modular network.Comment: Accepted in JAS
Learning to Discover Sparse Graphical Models
We consider structure discovery of undirected graphical models from
observational data. Inferring likely structures from few examples is a complex
task often requiring the formulation of priors and sophisticated inference
procedures. Popular methods rely on estimating a penalized maximum likelihood
of the precision matrix. However, in these approaches structure recovery is an
indirect consequence of the data-fit term, the penalty can be difficult to
adapt for domain-specific knowledge, and the inference is computationally
demanding. By contrast, it may be easier to generate training samples of data
that arise from graphs with the desired structure properties. We propose here
to leverage this latter source of information as training data to learn a
function, parametrized by a neural network that maps empirical covariance
matrices to estimated graph structures. Learning this function brings two
benefits: it implicitly models the desired structure or sparsity properties to
form suitable priors, and it can be tailored to the specific problem of edge
structure discovery, rather than maximizing data likelihood. Applying this
framework, we find our learnable graph-discovery method trained on synthetic
data generalizes well: identifying relevant edges in both synthetic and real
data, completely unknown at training time. We find that on genetics, brain
imaging, and simulation data we obtain performance generally superior to
analytical methods
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
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