Hessian matrix

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In mathematics, the Hessian matrix or Hessian is a square matrix of second-order partial derivatives of a function. 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".

Given the real-valued function

f(x_1, x_2, \dots, x_n),\,\!

if all second partial derivatives of f exist and are continuous over the domain of the function, then the Hessian matrix of f is

H(f)_{ij}(\mathbf x) = D_i D_j f(\mathbf x)\,\!

where x = (x1, x2, ..., xn) and Di is the differentiation operator with respect to the ith argument. Thus

H(f) = \begin{bmatrix}
\dfrac{\partial^2 f}{\partial x_1^2} & \dfrac{\partial^2 f}{\partial x_1\,\partial x_2} & \cdots & \dfrac{\partial^2 f}{\partial x_1\,\partial x_n} \\[2.2ex]
\dfrac{\partial^2 f}{\partial x_2\,\partial x_1} & \dfrac{\partial^2 f}{\partial x_2^2} & \cdots & \dfrac{\partial^2 f}{\partial x_2\,\partial x_n} \\[2.2ex]
\vdots & \vdots & \ddots & \vdots \\[2.2ex]
\dfrac{\partial^2 f}{\partial x_n\,\partial x_1} & \dfrac{\partial^2 f}{\partial x_n\,\partial x_2} & \cdots & \dfrac{\partial^2 f}{\partial x_n^2}
\end{bmatrix}.

Because f is often clear from context, H(f)(\mathbf x) is frequently abbreviated to H(\mathbf x).

The Hessian matrix is related to the Jacobian matrix by H(f)(\mathbf x) = J(\nabla \! f)(\mathbf x).

The determinant of the above matrix is also sometimes referred to as the Hessian.[1]

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,

y=f(\mathbf{x}+\Delta\mathbf{x})\approx f(\mathbf{x}) + J(\mathbf{x})\Delta \mathbf{x} +\frac{1}{2} \Delta\mathbf{x}^\mathrm{T} H(\mathbf{x}) \Delta\mathbf{x}

where J is the Jacobian matrix, which is a vector (the gradient) for scalar-valued functions. The full Hessian matrix can be difficult to compute in practice; in such situations, quasi-Newton algorithms have been developed that use approximations to the Hessian. The best-known quasi-Newton algorithm is the BFGS algorithm.[citation needed]

Mixed derivatives and symmetry of the Hessian[edit]

The mixed derivatives of f are the entries off the main diagonal in the Hessian. Assuming that they are continuous, the order of differentiation does not matter (Clairaut's theorem). For example,

\frac {\partial}{\partial x} \left( \frac { \partial f }{ \partial y} \right) =
       \frac {\partial}{\partial y} \left( \frac { \partial f }{ \partial x} \right).

This can also be written

f_{yx} = f_{xy}. \,

In a formal statement: if the second derivatives of f are all continuous in a neighborhood D, then the Hessian of f is a symmetric matrix throughout D; see symmetry of second derivatives.

Critical points[edit]

If the gradient (the vector of the partial derivatives) of a function f is zero at some point x, then f has a critical point (or stationary point) at x. The determinant of the Hessian at x is then called the discriminant. If this determinant is zero then x is called a degenerate critical point of f, or a non-Morse critical point of f. Otherwise it is non-degenerate, and called a Morse critical point of f.

The Hessian matrix plays an important role in Morse theory, because its kernel and eigenvalues allow classification of the critical points.

Second derivative test[edit]

The following test can be applied at a non-degenerate critical point x. If the Hessian is positive definite at x, then f attains a local minimum at x. If the Hessian is negative definite at x, then f attains a local maximum at x. If the Hessian has both positive and negative eigenvalues then x is a saddle point for f (this is true even if x is degenerate). Otherwise the test is inconclusive.

Note that for positive semidefinite and negative semidefinite Hessians the test is inconclusive (yet a conclusion can be made that f is locally convex or concave respectively). 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 x is a local minimum, and if it is negative then x 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.

More generally, 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.

Bordered Hessian[edit]

A bordered Hessian is used for the second-derivative test in certain constrained optimization problems. Given the function as before:

f(x_1, x_2, \dots, x_n),

but adding a constraint function such that:

g(x_1, x_2, \dots, x_n) = c,

the bordered Hessian appears as

H(f,g) = \begin{bmatrix}
0 & \dfrac{\partial g}{\partial x_1} & \dfrac{\partial g}{\partial x_2} & \cdots & \dfrac{\partial g}{\partial x_n} \\[2.2ex]
\dfrac{\partial g}{\partial x_1} & \dfrac{\partial^2 f}{\partial x_1^2} & \dfrac{\partial^2 f}{\partial x_1\,\partial x_2} & \cdots & \dfrac{\partial^2 f}{\partial x_1\,\partial x_n} \\[2.2ex]
\dfrac{\partial g}{\partial x_2} & \dfrac{\partial^2 f}{\partial x_2\,\partial x_1} & \dfrac{\partial^2 f}{\partial x_2^2} & \cdots & \dfrac{\partial^2 f}{\partial x_2\,\partial x_n} \\[2.2ex]
\vdots & \vdots & \vdots & \ddots & \vdots \\[2.2ex]
\dfrac{\partial g}{\partial x_n} & \dfrac{\partial^2 f}{\partial x_n\,\partial x_1} & \dfrac{\partial^2 f}{\partial x_n\,\partial x_2} & \cdots & \dfrac{\partial^2 f}{\partial x_n^2}
\end{bmatrix}

If there are, say, m constraints then the zero in the north-west corner is an m × m block of zeroes, 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 z'Hz = 0 if z 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.[2] Intuitively, one can think of the m constraints as reducing the problem to one with n - m free variables. (For example, the maximization of f(x_1,x_2,x_3) subject to the constraint x_1+x_2+x_3=1 can be reduced to the maximization of f(x_1,x_2,1-x_1-x_2) without constraint.)

Specifically,[3] sign conditions are imposed on the sequence of principal minors (determinants of upper-left-justified sub-matrices) of the bordered Hessian, the smallest minor consisting of the truncated first 2m+1 rows and columns, the next consisting of the truncated first 2m+2 rows and columns, and so on, with the last being the entire bordered Hessian. There are thus nm minors to consider. 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[edit]

If f is instead a function from \mathbb{R}^n \to \mathbb{R}^m, i.e.

f(x_1, x_2, \dots, x_n) = (f_1, f_2, \dots, f_m),

then the array of second partial derivatives is not a two-dimensional matrix of size n \times n, but rather a tensor of order 3. This can be thought of as a multi-dimensional array with dimensions m \times n \times n, which degenerates to the usual Hessian matrix for m = 1.

Generalizations to Riemannian manifolds[edit]

Let (M,g) be a Riemannian manifold and \nabla its Levi-Civita connection. Let f:M \to \mathbb{R} be a smooth function. We may define the Hessian tensor

\displaystyle \mbox{Hess}(f) \in \Gamma(T^*M \otimes T^*M) by \mbox{Hess}(f):=\nabla \nabla f = \nabla df,

where we have taken advantage of the first covariant derivative of a function being the same as its ordinary derivative. Choosing local coordinates \{x^i\} we obtain the local expression for the Hessian as

 \mbox{Hess}(f)=\nabla_i\, \partial_j f \ dx^i \!\otimes\! dx^j = \left( \frac{\partial^2 f}{\partial x^i \partial x^j}-\Gamma_{ij}^k \frac{\partial f}{\partial x^k} \right) dx^i \otimes dx^j

where \Gamma^k_{ij} are the Christoffel symbols of the connection. Other equivalent forms for the Hessian are given by

\mbox{Hess}(f)(X,Y)= \langle \nabla_X \mbox{grad}f,Y \rangle and \mbox{Hess}(f)(X,Y)=X(Yf)-df(\nabla_XY).

See also[edit]

Notes[edit]

  1. ^ Binmore, Ken; Davies, Joan (2007). Calculus Concepts and Methods. Cambridge University Press. p. 190. ISBN 9780521775410. OCLC 717598615. 
  2. ^ Neudecker, Heinz; Magnus, Jan R. (1988), Matrix differential calculus with applications in statistics and econometrics, New York: John Wiley & Sons, ISBN 978-0-471-91516-4 , page 136
  3. ^ Chiang, Alpha C., Fundamental Methods of Mathematical Economics, McGraw-Hill, third edition, 1984: p. 386.

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