14,576 research outputs found
Sparse permutation invariant covariance estimation
The paper proposes a method for constructing a sparse estimator for the
inverse covariance (concentration) matrix in high-dimensional settings. The
estimator uses a penalized normal likelihood approach and forces sparsity by
using a lasso-type penalty. We establish a rate of convergence in the Frobenius
norm as both data dimension and sample size are allowed to grow, and
show that the rate depends explicitly on how sparse the true concentration
matrix is. We also show that a correlation-based version of the method exhibits
better rates in the operator norm. We also derive a fast iterative algorithm
for computing the estimator, which relies on the popular Cholesky decomposition
of the inverse but produces a permutation-invariant estimator. The method is
compared to other estimators on simulated data and on a real data example of
tumor tissue classification using gene expression data.Comment: Published in at http://dx.doi.org/10.1214/08-EJS176 the Electronic
Journal of Statistics (http://www.i-journals.org/ejs/) by the Institute of
Mathematical Statistics (http://www.imstat.org
A Tensor-Based Dictionary Learning Approach to Tomographic Image Reconstruction
We consider tomographic reconstruction using priors in the form of a
dictionary learned from training images. The reconstruction has two stages:
first we construct a tensor dictionary prior from our training data, and then
we pose the reconstruction problem in terms of recovering the expansion
coefficients in that dictionary. Our approach differs from past approaches in
that a) we use a third-order tensor representation for our images and b) we
recast the reconstruction problem using the tensor formulation. The dictionary
learning problem is presented as a non-negative tensor factorization problem
with sparsity constraints. The reconstruction problem is formulated in a convex
optimization framework by looking for a solution with a sparse representation
in the tensor dictionary. Numerical results show that our tensor formulation
leads to very sparse representations of both the training images and the
reconstructions due to the ability of representing repeated features compactly
in the dictionary.Comment: 29 page
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