Waves from a sparse set of source hidden in additive noise are observed by a sensor array. We treat the estimation of the sparse set of sources as a generalized complex-valued LASSO problem. The corresponding dual problem is formulated and it is shown that the dual solution is useful for selecting the regularization parameter of the LASSO when the number of sources is given. The solution path o...

We introduce a novel scheme for choosing the regularization parameter in high-dimensional linear regression with Lasso. This scheme, inspired by Lepski’s method for bandwidth selection in non-parametric regression, is equipped with both optimal finite-sample guarantees and a fast algorithm. In particular, for any design matrix such that the Lasso has low sup-norm error under an “oracle choice” ...

The group Lasso is an extension of the Lasso for feature selection on (predefined) non-overlapping groups of features. The non-overlapping group structure limits its applicability in practice. There have been several recent attempts to study a more general formulation, where groups of features are given, potentially with overlaps between the groups. The resulting optimization is, however, much ...

Many statistical machine learning techniques sacrifice convenient computational structures to gain estimation robustness and modeling flexibility. In this paper, we study this fundamental tradeoff through a SQRT-Lasso problem for sparse linear regression and sparse precision matrix estimation in high dimensions. We explain how novel optimization techniques help address these computational chall...

In this paper, we analyze the performance of Lasso-TD, a modification of LSTD in which the projection operator is defined as a Lasso problem. We first show that Lasso-TD is guaranteed to have a unique fixed point and its algorithmic implementation coincides with the recently presented LARS-TD and LC-TD methods. We then derive two bounds on the prediction error of Lasso-TD in the Markov design s...

The generalized lasso problem penalizes the `1 norm of a matrix D times the coefficient vector to be modeled, and has a wide range of applications, dictated by the choice of D. Special cases include the trend filtering and fused lasso problem classes. We consider in this talk highly efficient implementations of the generalized lasso dual path algorithm of Tibshirani and Taylor [1]. This covers ...

We consider regression problems where the number of predictors greatly exceeds the number of observations. We propose a method for variable selection that first estimates the regression function, yielding a “preconditioned” response variable. The primary method used for this initial regression is supervised principal components. Then we apply a standard procedure such as forward stepwise select...

We consider the problem of model selection and estimation in sparse high dimensional linear regression models with strongly correlated variables. First, we study the theoretical properties of the dual Lasso solution, and we show that joint consideration of the Lasso primal and its dual solutions are useful for selecting correlated active variables. Second, we argue that correlation among active...

A Large Scale Simulation Optimization (LASSO) framework is being developed by the authors. Linux clusters are the target platform for the framework, specifically cluster resources on the NSF TeraGrid. The framework is designed in a modular fashion that simplifies coupling with simulation model executables, allowing application of simulation optimization approaches across problem domains. In thi...

In many linear regression problems, explanatory variables are activated in groups or clusters; group lasso has been proposed for regression in such cases. This paper studies the nonasymptotic regression performance of group lasso using `1/`2 regularization for arbitrary (random or deterministic) design matrices. In particular, the paper establishes under a statistical prior on the set of nonzer...