235 research outputs found
An offline/online procedure for dual norm calculations of parameterized functionals: empirical quadrature and empirical test spaces
We present an offline/online computational procedure for computing the dual
norm of parameterized linear functionals.
The key elements of the approach are
(i) an empirical test space for the manifold of Riesz elements associated
with the parameterized functional, and
(ii) an empirical quadrature procedure to efficiently deal with
parametrically non-affine terms. We present a number of theoretical results to
identify the different sources of error and to motivate the technique. Finally,
we show the effectiveness of our approach to reduce both offline and online
costs associated with the computation of the time-averaged residual indicator
proposed in [Fick, Maday, Patera, Taddei, Journal of Computational Physics,
2018 (accepted)]
Optimization with Sparsity-Inducing Penalties
Sparse estimation methods are aimed at using or obtaining parsimonious
representations of data or models. They were first dedicated to linear variable
selection but numerous extensions have now emerged such as structured sparsity
or kernel selection. It turns out that many of the related estimation problems
can be cast as convex optimization problems by regularizing the empirical risk
with appropriate non-smooth norms. The goal of this paper is to present from a
general perspective optimization tools and techniques dedicated to such
sparsity-inducing penalties. We cover proximal methods, block-coordinate
descent, reweighted -penalized techniques, working-set and homotopy
methods, as well as non-convex formulations and extensions, and provide an
extensive set of experiments to compare various algorithms from a computational
point of view
The solution path of the generalized lasso
We present a path algorithm for the generalized lasso problem. This problem
penalizes the norm of a matrix D times the coefficient vector, and has
a wide range of applications, dictated by the choice of D. Our algorithm is
based on solving the dual of the generalized lasso, which greatly facilitates
computation of the path. For (the usual lasso), we draw a connection
between our approach and the well-known LARS algorithm. For an arbitrary D, we
derive an unbiased estimate of the degrees of freedom of the generalized lasso
fit. This estimate turns out to be quite intuitive in many applications.Comment: Published in at http://dx.doi.org/10.1214/11-AOS878 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
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