Hessian matrix explained
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, 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". The Hessian is sometimes denoted by H or, ambiguously, by ∇2.
Definitions and properties
Suppose
is a function taking as input a vector
and outputting a scalar
If all second-order
partial derivatives of
exist, then the Hessian matrix
of
is a square
matrix, usually defined and arranged as
That is, the entry of the th row and the th column is
If furthermore the second partial derivatives are all continuous, the Hessian matrix is a symmetric matrix by the symmetry of second derivatives.
The determinant of the Hessian matrix is called the .[1]
The Hessian matrix of a function
is the transpose of 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
inflection points, since the Hessian determinant is a polynomial of degree
Second-derivative test
See main article: Second partial derivative test.
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.
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 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 (the vector of the partial derivatives) of a function
is zero at some point
then
has a (or) at
The
determinant of the Hessian at
is called, in some contexts, a
discriminant. 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, 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. 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 \left( | \partialf |
\partialx1 |
,\ldots,
\right).
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.
[5]
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.
[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]
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). It can be used in normal mode analysis to calculate the different molecular frequencies in infrared spectroscopy.[8] It can also be used in local sensitivity and statistical diagnostics.[9]
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
[10] 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.
[11] 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
f\left(x1,x2,1-x1-x2\right)
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.
[12] 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. 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
f\colon\Complexn\to\Complex,
and write
f\left(z1,\ldots,zn\right).
Then the generalized Hessian is
| \partial2f |
\partialzi\partial\overline{zj |
}. 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 and
its
Levi-Civita connection. 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 differential. 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
Further reading
- Book: Lewis, David W.. Matrix Theory. Singapore. World Scientific. 1991. 978-981-02-0689-5. registration.
- Book: Jan R.. Magnus. Heinz. Neudecker. Matrix Differential Calculus : With Applications in Statistics and Econometrics. New York. Wiley. Revised. 1999. 0-471-98633-X. The Second Differential. 99–115.
Notes and References
- Book: Binmore. Ken. Kenneth Binmore. Davies. Joan. 2007. Calculus Concepts and Methods. 717598615. 978-0-521-77541-0. Cambridge University Press. 190.
- Book: Callahan, James J.. Advanced Calculus: A Geometric View. 2010. Springer Science & Business Media. 978-1-4419-7332-0. 248. en.
- Book: Recent Developments in General Relativity. Casciaro. B.. Fortunato. D.. Francaviglia. M.. Masiello. A.. 2011. Springer Science & Business Media. 9788847021136. 178. en.
- Book: Domenico P. L. Castrigiano. Sandra A. Hayes. Catastrophe theory. 2004. Westview Press. 978-0-8133-4126-2. 18.
- Book: Nocedal. Jorge. Jorge Nocedal. Wright. Stephen. 2000. Numerical Optimization. 978-0-387-98793-4. Springer Verlag.
- Pearlmutter. Barak A.. Fast exact multiplication by the Hessian. Neural Computation. 6. 1. 1994. 10.1162/neco.1994.6.1.147. 147–160. 1251969 .
- 10.1016/j.tcs.2019.09.002. O.M.. Shir. A. Yehudayoff. On the covariance-Hessian relation in evolution strategies. Theoretical Computer Science. 801. 157–174. Elsevier. 2020. free. 1806.03674.
- Mott. Adam J.. Rez. Peter. December 24, 2014. Calculation of the infrared spectra of proteins. European Biophysics Journal. en. 44. 3. 103–112. 10.1007/s00249-014-1005-6. 25538002 . 2945423 . 0175-7571.
- Liu. Shuangzhe . Leiva. Victor. Zhuang. Dan. Ma. Tiefeng. Figueroa-Zúñiga. Jorge I.. March 2022. Matrix differential calculus with applications in the multivariate linear model and its diagnostics. Journal of Multivariate Analysis . 188. 104849. 10.1016/j.jmva.2021.104849. free.
- Web site: Econ 500: Quantitative Methods in Economic Analysis I. October 7, 2004. Arne. Hallam. Iowa State.
- Book: Neudecker. Heinz. Magnus. Jan R.. Matrix Differential Calculus with Applications in Statistics and Econometrics. John Wiley & Sons. New York. 978-0-471-91516-4. 1988. 136.
- Book: Chiang, Alpha C.. Fundamental Methods of Mathematical Economics. McGraw-Hill. Third. 1984. 386. 978-0-07-010813-4.