A Survey of Numerical Algorithms that can Solve the Lasso Problems

In statistics, the least absolute shrinkage and selection operator (Lasso) is a regression method that performs both variable selection and regularization. There is a lot of literature available, discussing the statistical properties of the regression coefficients estimated by the Lasso method. However, there lacks a comprehensive review discussing the algorithms to solve the optimization problem in Lasso. In this review, we summarize five representative algorithms to optimize the objective function in Lasso, including the iterative shrinkage threshold algorithm (ISTA), fast iterative shrinkage-thresholding algorithms (FISTA), coordinate gradient descent algorithm (CGDA), smooth L1 algorithm (SLA), and path following algorithm (PFA). Additionally, we also compare their convergence rate, as well as their potential strengths and weakness.

Three-dimensional instantaneous orbit map for rotor-bearing system based on a novel multivariable complex variational mode decomposition algorithm

Full spectrum and holospectrum are homogenous information fusion technology developed for the fault diagnosis of rotating machinery and are often used in the analysis of the orbit of rotating machinery. However, both of the techniques are based on Fourier transform, so they can only handle stationary signals, which limits their development. By drawing inspiration from the approach of multivariate variational mode decomposition (MVMD) and the complex-valued signal decomposition, we propose a method called multivariate complex variational mode decomposition (MCVMD) for processing non-stationary complex-valued signals of multi-dimensional bearing surfaces in this work. In particular, the proposed method takes the advantages of the joint information between the complex-valued signals of multi-dimensional bearing surfaces, and owing to this property, we provide its three-dimensional instantaneous orbit map (3D-IOM) to present the overall perspective of the rotor-bearing system and also offer a high-resolution time-full spectrum (Time-FS) to display the forward and backward frequency components of all the bearing surfaces within a time-frequency plane. The effectiveness of the proposed method through both the simulated experiment and the real-life complex-valued signals are shown in this paper.

A Homotopic Method to Solve the Lasso Problems with an Improved Upper Bound of Convergence Rate

In optimization, it is known that when the objective functions are strictly convex and well-conditioned, gradient based approaches can be extremely effective, e.g., achieving the exponential rate in convergence. On the other hand, the existing Lasso-type of estimator in general cannot achieve the optimal rate due to the undesirable behavior of the absolute function at the origin. A homotopic method is to use a sequence of surrogate functions to approximate the $\ell_1$ penalty that is used in the Lasso-type of estimators. The surrogate functions will converge to the $\ell_1$ penalty in the Lasso estimator. At the same time, each surrogate function is strictly convex, which enables provable faster numerical rate of convergence. In this paper, we demonstrate that by meticulously defining the surrogate functions, one can prove faster numerical convergence rate than any existing methods in computing for the Lasso-type of estimators. Namely, the state-of-the-art algorithms can only guarantee $O(1/\epsilon)$ or $O(1/\sqrt{\epsilon})$ convergence rates, while we can prove an $O([\log(1/\epsilon)]^2)$ for the newly proposed algorithm. Our numerical simulations show that the new algorithm also performs better empirically.