Data-driven formulation of natural laws by recursive-LASSO-based symbolic regression

Recursive $\ell_{1,\infty}$ Group lasso

We introduce a recursive adaptive group lasso algorithm for real-time penalized least squares prediction that produces a time sequence of optimal sparse predictor coefficient vectors. At each time index the proposed algorithm computes an exact update of the optimal $\ell_{1,\infty}$-penalized recursive least squares (RLS) predictor. Each update minimizes a convex but nondifferentiable function optimization problem. We develop an online homotopy method to reduce the computational complexity. Numerical simulations demonstrate that the proposed algorithm outperforms the $\ell_1$ regularized RLS a…

Using the LASSO's Dual for Regularization in Sparse Signal Reconstruction from Array Data

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 of the complex-valued LASSO is analyzed. For a given number of sources, the corresponding regularization parameter is determined by an order-recursive algorithm and two iterative algorithms that are based…

Estimation And Selection Via Absolute Penalized Convex Minimization And Its Multistage Adaptive Applications

The $\ell_1$-penalized method, or the Lasso, has emerged as an important tool for the analysis of large data sets. Many important results have been obtained for the Lasso in linear regression which have led to a deeper understanding of high-dimensional statistical problems. In this article, we consider a class of weighted $\ell_1$-penalized estimators for convex loss functions of a general form, including the generalized linear models. We study the estimation, prediction, selection and sparsity properties of the weighted $\ell_1$-penalized estimator in sparse, high-dimensional settings where t…

三重/双重去偏Lasso

Triple/Double-Debiased Lasso

本文提出一种三重（或双重去偏）Lasso估计器，用于高维线性回归模型中低维参数的推断。该估计器基于满足一阶和二阶Neyman正交条件的矩函数，消除正则化引起的偏倚。通过渐近线性表示证明其余项更小，从而提升小样本推断和置信区间的准确性。

Sparse recovery with unknown variance: a LASSO-type approach

We address the issue of estimating the regression vector $β$ in the generic $s$-sparse linear model $y = Xβ+z$, with $β\in\R^{p}$, $y\in\R^{n}$, $z\sim\mathcal N(0,\sg^2 I)$ and $p> n$ when the variance $\sg^{2}$ is unknown. We study two LASSO-type methods that jointly estimate $β$ and the variance. These estimators are minimizers of the $\ell_1$ penalized least-squares functional, where the relaxation parameter is tuned according to two different strategies. In the first strategy, the relaxation parameter is of the order $\chσ \sqrt{\log p}$, where $\chσ^2$ is the empirical variance. %The res…

Autonomous Tracking and State Estimation with Generalised Group Lasso

We address the problem of autonomous tracking and state estimation for marine vessels, autonomous vehicles, and other dynamic signals under a (structured) sparsity assumption. The aim is to improve the tracking and estimation accuracy with respect to classical Bayesian filters and smoothers. We formulate the estimation problem as a dynamic generalised group Lasso problem and develop a class of smoothing-and-splitting methods to solve it. The Levenberg--Marquardt iterated extended Kalman smoother-based multi-block alternating direction method of multipliers (LM-IEKS-mADMM) algorithms are based …