A new approach to variable selection in least squares problems.

*(English)*Zbl 0962.65036The least squares technique of solving a large non-homogeneous system of linear algebraic equations is considered within the so called Lasso approach reducing the set of variables (columns of the matrix). The “non-smooth” constraint (demanding that the \(\ell_1\) norm of the solution vector \(x\) is smaller than a constant \(\kappa\)) has a useful property of forcing components of \(x\) to zero when \(\kappa\) is small, but precisely this requirement makes the problem complicated.

The authors introduce and study two complementary methods. Both of them have a finite termination property and both of them may find an efficient implementation via a modified Gram-Schmidt orthogonalization. One of them (a compact descent method) can operate as a probe at a particular \(\kappa\), the other one (viz., homotopy method) is capable to describe the possible selection regimes globally.

The authors introduce and study two complementary methods. Both of them have a finite termination property and both of them may find an efficient implementation via a modified Gram-Schmidt orthogonalization. One of them (a compact descent method) can operate as a probe at a particular \(\kappa\), the other one (viz., homotopy method) is capable to describe the possible selection regimes globally.

Reviewer: Miloslav Znojil (Řež)

##### MSC:

65F20 | Numerical solutions to overdetermined systems, pseudoinverses |

65F25 | Orthogonalization in numerical linear algebra |

65C60 | Computational problems in statistics (MSC2010) |

62J05 | Linear regression; mixed models |

62-07 | Data analysis (statistics) (MSC2010) |