  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.
