4,469 research outputs found
Conic Optimization Theory: Convexification Techniques and Numerical Algorithms
Optimization is at the core of control theory and appears in several areas of
this field, such as optimal control, distributed control, system
identification, robust control, state estimation, model predictive control and
dynamic programming. The recent advances in various topics of modern
optimization have also been revamping the area of machine learning. Motivated
by the crucial role of optimization theory in the design, analysis, control and
operation of real-world systems, this tutorial paper offers a detailed overview
of some major advances in this area, namely conic optimization and its emerging
applications. First, we discuss the importance of conic optimization in
different areas. Then, we explain seminal results on the design of hierarchies
of convex relaxations for a wide range of nonconvex problems. Finally, we study
different numerical algorithms for large-scale conic optimization problems.Comment: 18 page
A dual framework for low-rank tensor completion
One of the popular approaches for low-rank tensor completion is to use the
latent trace norm regularization. However, most existing works in this
direction learn a sparse combination of tensors. In this work, we fill this gap
by proposing a variant of the latent trace norm that helps in learning a
non-sparse combination of tensors. We develop a dual framework for solving the
low-rank tensor completion problem. We first show a novel characterization of
the dual solution space with an interesting factorization of the optimal
solution. Overall, the optimal solution is shown to lie on a Cartesian product
of Riemannian manifolds. Furthermore, we exploit the versatile Riemannian
optimization framework for proposing computationally efficient trust region
algorithm. The experiments illustrate the efficacy of the proposed algorithm on
several real-world datasets across applications.Comment: Aceepted to appear in Advances of Nueral Information Processing
Systems (NIPS), 2018. A shorter version appeared in the NIPS workshop on
Synergies in Geometric Data Analysis 201
The Graphical Lasso: New Insights and Alternatives
The graphical lasso \citep{FHT2007a} is an algorithm for learning the
structure in an undirected Gaussian graphical model, using
regularization to control the number of zeros in the precision matrix
{\B\Theta}={\B\Sigma}^{-1} \citep{BGA2008,yuan_lin_07}. The {\texttt R}
package \GL\ \citep{FHT2007a} is popular, fast, and allows one to efficiently
build a path of models for different values of the tuning parameter.
Convergence of \GL\ can be tricky; the converged precision matrix might not be
the inverse of the estimated covariance, and occasionally it fails to converge
with warm starts. In this paper we explain this behavior, and propose new
algorithms that appear to outperform \GL.
By studying the "normal equations" we see that, \GL\ is solving the {\em
dual} of the graphical lasso penalized likelihood, by block coordinate ascent;
a result which can also be found in \cite{BGA2008}.
In this dual, the target of estimation is \B\Sigma, the covariance matrix,
rather than the precision matrix \B\Theta. We propose similar primal
algorithms \PGL\ and \DPGL, that also operate by block-coordinate descent,
where \B\Theta is the optimization target. We study all of these algorithms,
and in particular different approaches to solving their coordinate
sub-problems. We conclude that \DPGL\ is superior from several points of view.Comment: This is a revised version of our previous manuscript with the same
name ArXiv id: http://arxiv.org/abs/1111.547
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