A new formula for the determinant of $4 \times 4$ matrices

Abstract: In this paper, we present a new formula for the determinant of a $4 \times 4$ matrix. We approach via the sparse optimization problem and derive the formula through the Least Absolute Shrinkage and Selection Operator (LASSO). Our formula has the potential to advance understanding of the algebraic structure of determinants, such as an upper bound of the tensor rank, various notions to measure complexity, and effective computational tools in exterior algebras. We also address several numerical experiments which compare our formula with built-in functions in a computer-algebra system.