23,893 research outputs found
Robust high-dimensional precision matrix estimation
The dependency structure of multivariate data can be analyzed using the
covariance matrix . In many fields the precision matrix
is even more informative. As the sample covariance estimator is singular in
high-dimensions, it cannot be used to obtain a precision matrix estimator. A
popular high-dimensional estimator is the graphical lasso, but it lacks
robustness. We consider the high-dimensional independent contamination model.
Here, even a small percentage of contaminated cells in the data matrix may lead
to a high percentage of contaminated rows. Downweighting entire observations,
which is done by traditional robust procedures, would then results in a loss of
information. In this paper, we formally prove that replacing the sample
covariance matrix in the graphical lasso with an elementwise robust covariance
matrix leads to an elementwise robust, sparse precision matrix estimator
computable in high-dimensions. Examples of such elementwise robust covariance
estimators are given. The final precision matrix estimator is positive
definite, has a high breakdown point under elementwise contamination and can be
computed fast
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
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