The paper "DAGs with NO TEARS: Continuous Optimization for Structure Learning" by Xun Zheng, Bryon Aragam, Pradeep Ravikumar, and Eric P. Xing introduces a novel approach to learning the structure of directed acyclic graphs (DAGs) by formulating the problem as a continuous optimization over real matrices. This approach avoids the combinatorial complexity of traditional methods, which often rely on local heuristics to enforce the acyclicity constraint. The key contribution is a smooth and exact characterization of acyclicity, expressed through a function \( h(W) \) that is zero if and only if the matrix \( W \) represents an acyclic graph. This function is used to replace the combinatorial acyclicity constraint in the optimization problem, making it amenable to standard numerical optimization algorithms. The resulting method, called NOTEARS (Non-combinatorial Optimization via Trace Exponential and Augmented LagRangian for Structure learning), is shown to outperform existing methods in both theoretical and empirical evaluations, without imposing structural assumptions on the graph. The implementation of NOTEARS is open-source and available at <https://github.com/xunzheng/notears>.The paper "DAGs with NO TEARS: Continuous Optimization for Structure Learning" by Xun Zheng, Bryon Aragam, Pradeep Ravikumar, and Eric P. Xing introduces a novel approach to learning the structure of directed acyclic graphs (DAGs) by formulating the problem as a continuous optimization over real matrices. This approach avoids the combinatorial complexity of traditional methods, which often rely on local heuristics to enforce the acyclicity constraint. The key contribution is a smooth and exact characterization of acyclicity, expressed through a function \( h(W) \) that is zero if and only if the matrix \( W \) represents an acyclic graph. This function is used to replace the combinatorial acyclicity constraint in the optimization problem, making it amenable to standard numerical optimization algorithms. The resulting method, called NOTEARS (Non-combinatorial Optimization via Trace Exponential and Augmented LagRangian for Structure learning), is shown to outperform existing methods in both theoretical and empirical evaluations, without imposing structural assumptions on the graph. The implementation of NOTEARS is open-source and available at <https://github.com/xunzheng/notears>.