Numerous penalization based methods have been proposed for fitting a traditional linear regression model in which the number of predictors, p, is large relative to the number of observations, n. Most of these approaches assume sparsity in the underlying coefficients and perform some form of variable selection. Recently, some of this work has been extended to non-linear additive regression models. However, in many contexts one wishes to allow for the possibility of interactions among the predictors. This poses serious statistical and computational difficulties when p is large, as the number of candidate interaction terms is of order p 2 . We introduce a new approach, "Variable selection using Adaptive Non-linear Interaction Structures in High dimensions" ( VAN-ISH), that is based on a penalized least squares criterion and is designed for high dimensional non-linear problems. Our criterion is convex and enforces the heredity constraint, in other words if an interaction term is added to the model, then the corresponding main effects are automatically included. We provide theoretical conditions under which VANISH will select the correct main effects and interactions. These conditions suggest that VANISH should outperform certain natural competitors when the true interaction structure is sufficiently sparse. Detailed simulation results are also provided, demonstrating that VAN-ISH is computationally efficient and can be applied to non-linear models involving thousands of terms while producing superior predictive performance over other approaches.

Summary. We propose a new algorithm, DASSO, for fitting the entire coefficient path of the Dantzig selector with a similar computational cost to the least angle regression algorithm that is used to compute the lasso. DASSO efficiently constructs a piecewise linear path through a sequential simplex-like algorithm, which is remarkably similar to the least angle regression algorithm. Comparison of the two algorithms sheds new light on the question of how the lasso and Dantzig selector are related. In addition, we provide theoretical conditions on the design matrix X under which the lasso and Dantzig selector coefficient estimates will be identical for certain tuning parameters. As a consequence, in many instances, we can extend the powerful non-asymptotic bounds that have been developed for the Dantzig selector to the lasso. Finally, through empirical studies of simulated and real world data sets we show that in practice, when the bounds hold for the Dantzig selector, they almost always also hold for the lasso.

We suggest a new method, called Functional Additive Regression, or FAR, for efficiently performing high-dimensional functional regression. FAR extends the usual linear regression model involving a functional predictor, $X(t)$, and a scalar response, $Y$, in two key respects. First, FAR uses a penalized least squares optimization approach to efficiently deal with high-dimensional problems involving a large number of functional predictors. Second, FAR extends beyond the standard linear regression setting to fit general nonlinear additive models. We demonstrate that FAR can be implemented with a wide range of penalty functions using a highly efficient coordinate descent algorithm. Theoretical results are developed which provide motivation for the FAR optimization criterion. Finally, we show through simulations and two real data sets that FAR can significantly outperform competing methods.Comment: Published at http://dx.doi.org/10.1214/15-AOS1346 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

This paper introduces new methods to analyze the changing progression of COVID-19 cases to deaths in different waves of the pandemic. First, an algorithmic approach partitions each country or state's COVID-19 time series into a first wave and subsequent period. Next, offsets between case and death time series are learned for each country via a normalized inner product. Combining these with additional calculations, we can determine which countries have most substantially reduced the mortality rate of COVID-19. Finally, our paper identifies similarities in the trajectories of cases and deaths for European countries and U.S. states. Our analysis refines the popular conception that the mortality rate has greatly decreased throughout Europe during its second wave of COVID-19; instead, we demonstrate a bifurcation in which wealthier Western European countries have managed their mortality rate more successfully. A similar distinction exists in the United States, where Northeastern states have been the most successful in the country.

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