Hessian matrix

In mathematics, the Hessian matrix or Hessian is a square matrix of second-order partial derivatives of a scalar-valued function, or scalar field. 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. Hesse originally used the term "functional determinants".
Definitions and properties.
Suppose is a function taking as input a vector and outputting a scalar . If all second partial derivatives of exist and are continuous over the domain of the function, then the Hessian matrix of is a square matrix, usually defined and arranged as follows:
or, by stating an equation for the coefficients using indices i and j,
The Hessian matrix is a symmetric matrix, since the hypothesis of continuity of the second derivatives implies that the order of differentiation does not matter (Schwarz's theorem).
The determinant of the Hessian matrix is called the "Hessian determinant".
The Hessian matrix of a function is the Jacobian matrix of the gradient of the function ; that is: .
Applications.
Inflection points.
If is a homogeneous polynomial in three variables, the equation is the implicit equation of a plane projective curve. The inflection points of the curve are exactly the non-singular points where the Hessian determinant is zero. 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.
Second-derivative test.
The Hessian matrix of a convex function is positive semi-definite. Refining this property allows us to test whether a critical point is a local maximum, local minimum, or a saddle point, as follows:
If the Hessian is positive-definite at , then attains an isolated local minimum at . If the Hessian is negative-definite at , then attains an isolated local maximum at . If the Hessian has both positive and negative eigenvalues, then is a saddle point for . Otherwise the test is inconclusive. This implies that at a local minimum the Hessian is positive-semidefinite, and at a local maximum the Hessian is negative-semidefinite.
Note that 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). However, more can be said from the point of view of Morse theory.
The second-derivative test for functions of one and two variables is simple. In one variable, the Hessian contains just one second derivative; if it is positive, then is a local minimum, and if it is negative, then is a local maximum; if it is zero, then the test is inconclusive. In two variables, the determinant can be used, because the determinant is the product of the eigenvalues. If it is positive, then the eigenvalues are both positive, or both negative. If it is negative, then the two eigenvalues have different signs. If it is zero, then the second-derivative 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 the 1×1 minor being negative.
Critical points.
If the gradient (the vector of the partial derivatives) of a function is zero at some point , then has a "critical point" (or "stationary point") at . The determinant of the Hessian at is called, in some contexts, a discriminant. If this determinant is zero then is called a "degenerate critical point" of , or a "non-Morse critical point" of . Otherwise it is non-degenerate, and called a "Morse critical point" of .
The Hessian matrix plays an important role in Morse theory and catastrophe theory, because its kernel and eigenvalues allow classification of the critical points.
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. The eigenvalues of the Hessian at that point are the principle curvatures of the function, and the eigenvectors are the principle directions of curvature. (See .)
Use in optimization.
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. That is,
where is the gradient . Computing and storing the full Hessian matrix takes memory, 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. For such situations, truncated-Newton and quasi-Newton algorithms have been developed. The latter family of algorithms use approximations to the Hessian; one of the most popular quasi-Newton algorithms is BFGS.
Such approximations may use the fact that an optimization algorithm uses the Hessian only as a linear operator , and proceed by first noticing that the Hessian also appears in the local expansion of the gradient:
Letting for some scalar , this gives
i.e.,
so if the gradient is already computed, the approximate Hessian can be computed by a linear (in the size of the gradient) number of scalar operations. (While simple to program, this approximation scheme is not numerically stable since has to be made small to prevent error due to the formula_7 term, but decreasing it loses precision in the first term.)
Other applications.
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). The Hessian matrix can also be used in normal mode analysis to calculate the different molecular frequencies in infrared spectroscopy.
Generalizations.
Bordered Hessian.
A bordered Hessian is used for the second-derivative test in certain constrained optimization problems. Given the function considered previously, but adding a constraint function such that , the bordered Hessian is the Hessian of the Lagrange function formula_8:
If there are, say, "m" constraints then the zero in the upper-left corner is an "m" × "m" block of zeros, and there are "m" border rows at the top and "m" border columns at the left.
The above rules stating that extrema are characterized (among critical points with a non-singular Hessian) by a positive-definite or negative-definite Hessian cannot apply here since a bordered Hessian can neither be negative-definite nor positive-definite, as formula_10 if formula_11 is any vector whose sole non-zero entry is its first.
The second derivative test consists here of sign restrictions of the determinants of a certain set of "n – m" submatrices of the bordered Hessian. Intuitively, one can think of the "m" constraints as reducing the problem to one with "n" – "m" free variables. (For example, the maximization of subject to the constraint can be reduced to the maximization of without constraint.)
Specifically, 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 first 2"m" leading principal minors are neglected, the smallest minor consisting of the truncated first 2"m"+1 rows and columns, the next consisting of the truncated first 2"m"+2 rows and columns, and so on, with the last being the entire bordered Hessian; if 2"m"+1 is larger than "n"+"m", then the smallest leading principal minor is the Hessian itself. There are thus "n"–"m" minors to consider, each evaluated at the specific point being considered as a candidate maximum or minimum. A sufficient condition for a local "maximum" is that these minors alternate in sign with the smallest one having the sign of (–1)"m"+1. A sufficient condition for a local "minimum" is that all of these minors have the sign of (–1)"m". (In the unconstrained case of "m"=0 these conditions coincide with the conditions for the unbordered Hessian to be negative definite or positive definite respectively).
Vector-valued functions.
If is instead a vector field , i.e.
then the collection of second partial derivatives is not a matrix, but rather a third-order tensor. This can be thought of as an array of Hessian matrices, one for each component of :
This tensor degenerates to the usual Hessian matrix when .
Generalization to the complex case.
In the context of several complex variables, the Hessian may be generalized. Suppose formula_14, and we write formula_15. Then one may generalize the Hessian to formula_16. Note that if formula_17 satisfies the n-dimensional Cauchy–Riemann conditions, then the complex Hessian matrix is identically zero.
Generalizations to Riemannian manifolds.
Let formula_18 be a Riemannian manifold and formula_19 its Levi-Civita connection. Let formula_20 be a smooth function. We may define the Hessian tensor
where we have taken advantage of the first covariant derivative of a function being the same as its ordinary derivative. Choosing local coordinates formula_23 we obtain the local expression for the Hessian as
where formula_25 are the Christoffel symbols of the connection. Other equivalent forms for the Hessian are given by