In
mathematic
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
s, the Hessian matrix or Hessian is a
square matrix
In mathematics, a square matrix is a matrix with the same number of rows and columns. An ''n''-by-''n'' matrix is known as a square matrix of order Any two square matrices of the same order can be added and multiplied.
Square matrices are often ...
of second-order
partial derivative
In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). Pa ...
s of a scalar-valued
function, or
scalar field
In mathematics and physics, a scalar field is a function associating a single number to every point in a space – possibly physical space. The scalar may either be a pure mathematical number ( dimensionless) or a scalar physical quantity ...
. 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-order
partial derivative
In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). Pa ...
s of
exist, 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,
If furthermore the second partial derivatives are all continuous, the Hessian matrix is a
symmetric matrix by the
symmetry of second derivatives
In mathematics, the symmetry of second derivatives (also called the equality of mixed partials) refers to the possibility of interchanging the order of taking partial derivatives of a function
:f\left(x_1,\, x_2,\, \ldots,\, x_n\right)
of ''n ...
.
The
determinant
In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if a ...
of the Hessian matrix is called the .
The Hessian matrix of a function
is the
Jacobian matrix
In vector calculus, the Jacobian matrix (, ) of a vector-valued function of several variables is the matrix of all its first-order partial derivatives. When this matrix is square, that is, when the function takes the same number of variable ...
of the
gradient
In vector calculus, the gradient of a scalar-valued differentiable function of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p is the "direction and rate of fastest increase". If the gr ...
of the function
; that is:
Applications
Inflection points
If
is a
homogeneous polynomial
In mathematics, a homogeneous polynomial, sometimes called quantic in older texts, is a polynomial whose nonzero terms all have the same degree. For example, x^5 + 2 x^3 y^2 + 9 x y^4 is a homogeneous polynomial of degree 5, in two variables; ...
in three variables, the equation
is the
implicit equation of a
plane projective curve. The
inflection point
In differential calculus and differential geometry, an inflection point, point of inflection, flex, or inflection (British English: inflexion) is a point on a smooth plane curve at which the curvature changes sign. In particular, in the case ...
s 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
inflection points, since the Hessian determinant is a polynomial of degree
Second-derivative test
The Hessian matrix of a
convex function
In mathematics, a real-valued function is called convex if the line segment between any two points on the graph of the function lies above the graph between the two points. Equivalently, a function is convex if its epigraph (the set of poi ...
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
eigenvalue
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denote ...
s, then
is a
saddle point
In mathematics, a saddle point or minimax point is a point on the surface of the graph of a function where the slopes (derivatives) in orthogonal directions are all zero (a critical point), but which is not a local extremum of the functi ...
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.
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 simpler than the general case. In one variable, the Hessian contains exactly 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
In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if a ...
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
minor being negative.
Critical points
If the
gradient
In vector calculus, the gradient of a scalar-valued differentiable function of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p is the "direction and rate of fastest increase". If the gr ...
(the vector of the partial derivatives) of a function
is zero at some point
then
has a (or ) at
The
determinant
In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if a ...
of the Hessian at
is called, in some contexts, a
discriminant
In mathematics, the discriminant of a polynomial is a quantity that depends on the coefficients and allows deducing some properties of the roots without computing them. More precisely, it is a polynomial function of the coefficients of the orig ...
. If this determinant is zero then
is called a of
or a of
Otherwise it is non-degenerate, and called a of
The Hessian matrix plays an important role in
Morse theory and
catastrophe theory
In mathematics, catastrophe theory is a branch of bifurcation theory in the study of dynamical systems; it is also a particular special case of more general singularity theory in geometry.
Bifurcation theory studies and classifies phenomena c ...
, because its
kernel and
eigenvalue
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denote ...
s 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
In differential geometry, the Gaussian curvature or Gauss curvature of a surface at a point is the product of the principal curvatures, and , at the given point:
K = \kappa_1 \kappa_2.
The Gaussian radius of curvature is the reciprocal of .
...
of the function considered as a manifold. The eigenvalues of the Hessian at that point are the principal curvatures of the function, and the eigenvectors are the principal 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
In vector calculus, the gradient of a scalar-valued differentiable function of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p is the "direction and rate of fastest increase". If the gr ...
Computing and storing the full Hessian matrix takes
memory, which is infeasible for high-dimensional functions such as the
loss function
In mathematical optimization and decision theory, a loss function or cost function (sometimes also called an error function) is a function that maps an event or values of one or more variables onto a real number intuitively representing some "cos ...
s of
neural nets,
conditional random fields, and other
statistical model
A statistical model is a mathematical model that embodies a set of statistical assumptions concerning the generation of sample data (and similar data from a larger population). A statistical model represents, often in considerably idealized form ...
s 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
In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a Map (mathematics), mapping V \to W between two vect ...
and proceed by first noticing that the Hessian also appears in the local expansion of the gradient:
Letting
for some scalar
this gives
that is,
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
term, but decreasing it loses precision in the first term.)
Notably regarding Randomized Search Heuristics, the
evolution strategy's covariance matrix adapts to the inverse of the Hessian matrix,
up to Two mathematical objects ''a'' and ''b'' are called equal up to an equivalence relation ''R''
* if ''a'' and ''b'' are related by ''R'', that is,
* if ''aRb'' holds, that is,
* if the equivalence classes of ''a'' and ''b'' with respect to ''R'' ...
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.
Other applications
The Hessian matrix is commonly used for expressing image processing operators in
image processing
An image is a visual representation of something. It can be two-dimensional, three-dimensional, or somehow otherwise feed into the visual system to convey information. An image can be an artifact, such as a photograph or other two-dimensio ...
and
computer vision
Computer vision is an interdisciplinary scientific field that deals with how computers can gain high-level understanding from digital images or videos. From the perspective of engineering, it seeks to understand and automate tasks that the human ...
(see the
Laplacian of Gaussian
In computer vision, blob detection methods are aimed at detecting regions in a digital image that differ in properties, such as brightness or color, compared to surrounding regions. Informally, a blob is a region of an image in which some proper ...
(LoG) blob detector,
the determinant of Hessian (DoH) blob detector and
scale space). The Hessian matrix can also be used in
normal mode
A normal mode of a dynamical system is a pattern of motion in which all parts of the system move sinusoidally with the same frequency and with a fixed phase relation. The free motion described by the normal modes takes place at fixed frequencies. ...
analysis to calculate the different molecular frequencies in
infrared spectroscopy
Infrared spectroscopy (IR spectroscopy or vibrational spectroscopy) is the measurement of the interaction of infrared radiation with matter by absorption, emission, or reflection. It is used to study and identify chemical substances or functi ...
.
Generalizations
Bordered Hessian
A 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
If there are, say,
constraints then the zero in the upper-left corner is an
block of zeros, and there are
border rows at the top and
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
if
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
submatrices of the bordered Hessian. Intuitively, the
constraints can be thought of as reducing the problem to one with
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
leading principal minors are neglected, the smallest minor consisting of the truncated first
rows and columns, the next consisting of the truncated first
rows and columns, and so on, with the last being the entire bordered Hessian; if
is larger than
then the smallest leading principal minor is the Hessian itself.
There are thus
minors to consider, each evaluated at the specific point being considered as a
candidate maximum or minimum. A sufficient condition for a local is that these minors alternate in sign with the smallest one having the sign of
A sufficient condition for a local is that all of these minors have the sign of
(In the unconstrained case of
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 that is,
then the collection of second partial derivatives is not a
matrix, but rather a third-order
tensor
In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other 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 theory of functions of several complex variables is the branch of mathematics dealing with complex-valued functions. The name of the field dealing with the properties of function of several complex variables is called several complex variable ...
, the Hessian may be generalized. Suppose
and write
Then the generalized Hessian is
If
satisfies the n-dimensional
Cauchy–Riemann conditions, then the complex Hessian matrix is identically zero.
Generalizations to Riemannian manifolds
Let
be a
Riemannian manifold
In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real, smooth manifold ''M'' equipped with a positive-definite inner product ''g'p'' on the tangent space ...
and
its
Levi-Civita connection
In Riemannian or pseudo Riemannian geometry (in particular the Lorentzian geometry of general relativity), the Levi-Civita connection is the unique affine connection on the tangent bundle of a manifold (i.e. affine connection) that preserves ...
. Let
be a smooth function. Define the Hessian tensor by
where this takes advantage of the fact that the first covariant derivative of a function is the same as its ordinary derivative. Choosing local coordinates
gives a local expression for the Hessian as
where
are the
Christoffel symbols of the connection. Other equivalent forms for the Hessian are given by
See also
* The determinant of the Hessian matrix is a covariant; see
Invariant of a binary form
*
Polarization identity
In linear algebra, a branch of mathematics, the polarization identity is any one of a family of formulas that express the inner product of two vectors in terms of the norm of a normed vector space.
If a norm arises from an inner product the ...
, useful for rapid calculations involving Hessians.
*
*
Notes
Further reading
*
*
External links
*
*
{{Matrix classes
Differential operators
Matrices
Morse theory
Multivariable calculus
Singularity theory