5,114 research outputs found
Fast Candidate Points Selection in the LASSO Path
The LASSO sparse regression method has recently received attention in a variety of applications from image compression techniques to parameter estimation problems. This paper addresses the problem of regularization parameter selection in this method in a general case of complex-valued regressors and bases. Generally, this parameter controls the degree of sparsity or equivalently, the estimated model order. However, with the same sparsity/model order, the smallest regularization parameter is desired. We relate such points to the nonsmooth points in the path of LASSO solutions and give an analytical expression for them. Then, we introduce a numerically fast method of approximating the desired points by a recursive algorithm. The procedure decreases the necessary number of solutions of the LASSO problem dramatically, which is an important issue due to the polynomial computational cost of the convex optimization techniques. We illustrate our method in the context of DOA estimation
The group fused Lasso for multiple change-point detection
We present the group fused Lasso for detection of multiple change-points
shared by a set of co-occurring one-dimensional signals. Change-points are
detected by approximating the original signals with a constraint on the
multidimensional total variation, leading to piecewise-constant approximations.
Fast algorithms are proposed to solve the resulting optimization problems,
either exactly or approximately. Conditions are given for consistency of both
algorithms as the number of signals increases, and empirical evidence is
provided to support the results on simulated and array comparative genomic
hybridization data
Forward stagewise regression and the monotone lasso
We consider the least angle regression and forward stagewise algorithms for
solving penalized least squares regression problems. In Efron, Hastie,
Johnstone & Tibshirani (2004) it is proved that the least angle regression
algorithm, with a small modification, solves the lasso regression problem. Here
we give an analogous result for incremental forward stagewise regression,
showing that it solves a version of the lasso problem that enforces
monotonicity. One consequence of this is as follows: while lasso makes optimal
progress in terms of reducing the residual sum-of-squares per unit increase in
-norm of the coefficient , forward stage-wise is optimal per unit
arc-length traveled along the coefficient path. We also study a condition
under which the coefficient paths of the lasso are monotone, and hence the
different algorithms coincide. Finally, we compare the lasso and forward
stagewise procedures in a simulation study involving a large number of
correlated predictors.Comment: Published at http://dx.doi.org/10.1214/07-EJS004 in the Electronic
Journal of Statistics (http://www.i-journals.org/ejs/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Improved variable selection with Forward-Lasso adaptive shrinkage
Recently, considerable interest has focused on variable selection methods in
regression situations where the number of predictors, , is large relative to
the number of observations, . Two commonly applied variable selection
approaches are the Lasso, which computes highly shrunk regression coefficients,
and Forward Selection, which uses no shrinkage. We propose a new approach,
"Forward-Lasso Adaptive SHrinkage" (FLASH), which includes the Lasso and
Forward Selection as special cases, and can be used in both the linear
regression and the Generalized Linear Model domains. As with the Lasso and
Forward Selection, FLASH iteratively adds one variable to the model in a
hierarchical fashion but, unlike these methods, at each step adjusts the level
of shrinkage so as to optimize the selection of the next variable. We first
present FLASH in the linear regression setting and show that it can be fitted
using a variant of the computationally efficient LARS algorithm. Then, we
extend FLASH to the GLM domain and demonstrate, through numerous simulations
and real world data sets, as well as some theoretical analysis, that FLASH
generally outperforms many competing approaches.Comment: Published in at http://dx.doi.org/10.1214/10-AOAS375 the Annals of
Applied Statistics (http://www.imstat.org/aoas/) by the Institute of
Mathematical Statistics (http://www.imstat.org
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