Regularization Paths for Sparse Nonnegative Least Squares Problems with Applications to Life Cycle Assessment Tree Discovery

The nonnegative least squares problems are useful in applications where the physical nature of problem domain permits only additive linear combinations. We discuss the l
₁
-regularized nonnegative least squares (L1-NLS) problem, where l1-regularization is used to induce sparsity. Although l
₁
-regularization has been successfully used in least squares regression, when combined with nonnegativity constraints, developments of algorithms and their understandings have been limited. We propose an algorithm that generates the entire regularization paths of the L1-NLS problem. We prove the correctness of the proposed algorithm and illustrate a novel application in environmental sustainability. The application relates to life cycle assessment (LCA), a technique used to estimate environmental impact during the entire lifetime of a product. We address an inverse problem in LCA. Given environmental impact factors of a target product and of a large library of constituents, the goal is to reverse engineer an inventory tree for the product. Using real-world data sets, we demonstrate how our L1-NLS approach controls the size of discovered trees, and how the full regularization paths effectively illustrate the spectrum of discovered trees with varying sparsity and compositions.