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A new approach to variable selection in least squares problems. (English) Zbl 0962.65036

The 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.

##### MSC:
 65F20 Overdetermined systems, pseudoinverses (numerical linear algebra) 65F25 Orthogonalization (numerical linear algebra) 65C60 Computational problems in statistics 62J05 Linear regression 62-07 Data analysis (statistics)