Hessian

The Hessian matrix is a square matrix of second-order partial derivatives of a scalar-valued function, providing information about the curvature of the function in multiple dimensions. For a function f: R^n → R, the Hessian H is defined as H = [∂²f/∂x_i∂x_j] for i, j = 1 to n. The Hessian is crucial in optimization, as it helps determine the nature of critical points (i.e., whether they are local minima, maxima, or saddle points) through the second derivative test. In machine learning, the Hessian is utilized in second-order optimization methods, such as Newton's method, which can converge faster than first-order methods by incorporating curvature information into the parameter update process.