Local linear approximation with Laplacian smoothing penalty and application in biology

Highly correlated structures appear in various fields, such as biology, biochemistry, and finance, with challenges of dimensionality and sparse estimation. To solve this problem, we propose an algorithm called local linear approximation with the Laplacian smoothing penalty (LLA-LSP). This method produces an accurate and smooth estimate that incorporates the correlation structure among predictors. We compare and discuss the difference between the Laplacian smoothing penalty and the total variance penalty. We prove that this algorithm converges to the oracle solution in a few iterations with a large probability. Numerical results show that the LLA-LSP has good performance in both variable selection and estimation. We apply the proposed algorithm to two biological datasets, a gene expression dataset and a chemical protein dataset, and provide meaningful insights.