Hessian matrix
It describes the local curvature of a function of many variables.The Hessian matrix was developed in the 19th century by the German mathematician Ludwig Otto Hesse and later named after him.It follows by Bézout's theorem that a cubic plane curve has at most 9 inflection points, since the Hessian determinant is a polynomial of degree 3.The Hessian matrix of a convex function is positive semi-definite.For positive-semidefinite and negative-semidefinite Hessians the test is inconclusive (a critical point where the Hessian is semidefinite but not definite may be a local extremum or a saddle point).The second-derivative test for functions of one and two variables is simpler than the general case.is a local maximum; if it is zero, then the test is inconclusive.Equivalently, the second-order conditions that are sufficient for a local minimum or maximum can be expressed in terms of the sequence of principal (upper-leftmost) minors (determinants of sub-matrices) of the Hessian; these conditions are a special case of those given in the next section for bordered Hessians for constrained optimization—the case in which the number of constraints is zero.Specifically, the sufficient condition for a minimum is that all of these principal minors be positive, while the sufficient condition for a maximum is that the minors alternate in sign, with theThe Hessian matrix plays an important role in Morse theory and catastrophe theory, because its kernel and eigenvalues allow classification of the critical points.[2][3][4] The determinant of the Hessian matrix, when evaluated at a critical point of a function, is equal to the Gaussian curvature of the function considered as a manifold.Hessian matrices are used in large-scale optimization problems within Newton-type methods because they are the coefficient of the quadratic term of a local Taylor expansion of a function.Computing and storing the full Hessian matrix takesmemory, which is infeasible for high-dimensional functions such as the loss functions of neural nets, conditional random fields, and other statistical models with large numbers of parameters.[5] Such approximations may use the fact that an optimization algorithm uses the Hessian only as a linear operatorand proceed by first noticing that the Hessian also appears in the local expansion of the gradient:(While simple to program, this approximation scheme is not numerically stable sincehas to be made small to prevent error due to the[6]) Notably regarding Randomized Search Heuristics, the evolution strategy's covariance matrix adapts to the inverse of the Hessian matrix, up to a scalar factor and small random fluctuations.This result has been formally proven for a single-parent strategy and a static model, as the population size increases, relying on the quadratic approximation.[7] The Hessian matrix is commonly used for expressing image processing operators in image processing and computer vision (see the Laplacian of Gaussian (LoG) blob detector, the determinant of Hessian (DoH) blob detector and scale space).It can be used in normal mode analysis to calculate the different molecular frequencies in infrared spectroscopy.[9] A bordered Hessian is used for the second-derivative test in certain constrained optimization problems.The second derivative test consists here of sign restrictions of the determinants of a certain set ofSpecifically, sign conditions are imposed on the sequence of leading principal minors (determinants of upper-left-justified sub-matrices) of the bordered Hessian, for which the firstA sufficient condition for a local minimum is that all of these minors have the sign ofAs the object of study in several complex variables are holomorphic functions, that is, solutions to the n-dimensional Cauchy–Riemann conditions, we usually look on the part of the Hessian that contains information invariant under holomorphic changes of coordinates.is holomorphic, then its complex Hessian matrix is identically zero, so the complex Hessian is used to study smooth but not holomorphic functions, see for example Levi pseudoconvexity.When dealing with holomorphic functions, we could consider the Hessian matrixwhere this takes advantage of the fact that the first covariant derivative of a function is the same as its ordinary differential.