In mathematics, the Hessian matrix, Hessian or (less commonly) Hesse matrix is a square matrix of second-order partial derivatives of a scalar-valued function...

gradient of a scalar function of several variables has a special name: the Hessian matrix, which in a sense is the "second derivative" of the function in question...

Laplacian, sometimes ∇ 2 {\displaystyle \nabla ^{2}} also represents the Hessian matrix. The former refers to the inner product of ∇ {\displaystyle \nabla }...

entries. Therefore, specifically tailored matrix algorithms can be used in network theory. The Hessian matrix of a differentiable function f : R n → R...

analysis Hessian automatic differentiation Hessian equations, partial differential equations (PDEs) based on the Hessian matrix Hessian pair or Hessian duad...

several real variables that is twice differentiable, then if its Hessian matrix (matrix of its second partial derivatives) is positive-definite at a point...

requires the Jacobian matrix of all partial derivatives of a multivariate function when used to search for zeros or the Hessian matrix when used for finding...

information. It does so by gradually improving an approximation to the Hessian matrix of the loss function, obtained only from gradient evaluations (or approximate...

derivative of an n {\displaystyle n} -dimensional function, known as the Hessian matrix. When examining a function in a neighborhood of a point, one can discard...

covariance matrix amounts to learning a second order model of the underlying objective function similar to the approximation of the inverse Hessian matrix in...

covariance matrix (also known as auto-covariance matrix, dispersion matrix, variance matrix, or variance–covariance matrix) is a square matrix giving the...

definiteness of the Hessian matrix: If the Hessian is positive definite at a critical point, then the point is a local minimum; if the Hessian matrix is negative...

function's Hessian matrix at that point: if the Hessian is indefinite, then that point is a saddle point. For example, the Hessian matrix of the function...

function. Like the original BFGS, L-BFGS uses an estimate of the inverse Hessian matrix to steer its search through variable space, but where BFGS stores a...

{(L_{xx}-L_{yy})^{2}+4L_{xy}^{2}}}\right)} denote the eigenvalues of the Hessian matrix. The unsigned Hessian feature strength measure D 1 , n o r m L {\displaystyle D_{1...

second partial derivatives exist and are continuous. The Hessian matrix H of f is the 2 × 2 matrix of partial derivatives of f: H ( x , y ) = [ f x x ( x...

considering the eigenvalues of the Hessian matrix of second derivatives. A critical point at which the Hessian matrix is nonsingular is said to be nondegenerate...

invariants, and geometry. The Hessian matrix, the Hesse normal form, the Hesse configuration, the Hessian group, Hessian pairs, Hesse's theorem, Hesse...

function and Hessian matrix, the partial likelihood can be maximized using the Newton-Raphson algorithm. The inverse of the Hessian matrix, evaluated at...

maximum depends on whether the matrix of second-order partial and cross-partial derivatives, the so-called Hessian matrix H ( θ ^ ) = [ ∂ 2 ℓ ∂ θ 1 2 |...

Laplace operator, a differential operator often denoted by the symbol ∇2 Hessian matrix, sometimes denoted by ∇2 Aitken's delta-squared process, a numerical...

Uses the Hessian (transpose to Jacobian) definition of vector and matrix derivatives. Petersen, Kaare Brandt; Pedersen, Michael Syskind. The Matrix Cookbook...

}L=t^{2}\left(L_{xx}L_{yy}-L_{xy}^{2}\right)} where H L {\displaystyle HL} denotes the Hessian matrix of the scale-space representation L {\displaystyle L} and then detecting...

of the 2 x 2 Hessian matrix of f (being the product of the eigenvalues of the Hessian). (Recall that the Hessian is a symmetric matrix of second derivatives...

the measure for thresholding is computed from the Hessian matrix instead of a second-moment matrix. In this step, each keypoint is assigned one or more...

identified among the stationary points from the definiteness of the bordered Hessian matrix. The great advantage of this method is that it allows the optimization...

second derivative with the inverse of the Hessian matrix (different authors use different notation for the Hessian, including f ″ ( x ) = ∇ 2 f ( x ) = H...

mathematics, k-Hessian equations (or Hessian equations for short) are partial differential equations (PDEs) based on the Hessian matrix. More specifically...

functions Partial derivative Disk integration Gabriel's horn Jacobian matrix Hessian matrix Curvature Green's theorem Divergence theorem Stokes' theorem Vector...

x_{i}}}\right).} In other words, the matrix of the second-order partial derivatives, known as the Hessian matrix, is a symmetric matrix. Sufficient conditions for...