A review on the Adaptive-Ridge Algorithm with several extensions

The Adaptive Ridge Algorithm is an iterative algorithm designed for variable selection. It is also known under the denomination of Iteratively Reweighted Least-Squares Algorithm in the communities of Compressed Sensing and Sparse Signals Recovery. Besides, it can also be interpreted as an optimization algorithm dedicated to the minimization of possibly nonconvex \(\ell ^q\) penalized energies (with \(0<q<2\)). In the literature, this algorithm can be derived using various mathematical approaches, namely Half Quadratic Minimization, Majorization-Minimization, Alternating Minimization or Local Approximations. In this work, we will show how the Adaptive Ridge Algorithm can be simply derived and analyzed from a single equation, corresponding to a variational reformulation of the \(\ell ^q\) penalty. We will describe in detail how the Adaptive Ridge Algorithm can be numerically implemented and we will perform a thorough experimental study of its parameters. We will also show how the variational formulation of the \(\ell ^q\) penalty combined with modern duality principles can be used to design an interesting variant of the Adaptive Ridge Algorithm dedicated to the minimization of quadratic functions over (nonconvex) \(\ell ^q\) balls.