Hessian matrix

Inmathematics,theHessian matrix,Hessianor (less commonly)Hesse matrixis asquare matrixof second-orderpartial derivativesof a scalar-valuedfunction,orscalar field.It describes the localcurvatureof a function of many variables. The Hessian matrix was developed in the 19th century by the German mathematicianLudwig Otto Hesseand 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$f:\mathbb {R} ^{n}\to \mathbb {R}$is a function taking as input a vector$\mathbf {x} \in \mathbb {R} ^{n}$and outputting a scalar$f(\mathbf {x} )\in \mathbb {R}.$If all second-orderpartial derivativesof$f$exist, then the Hessian matrix$\mathbf {H}$of$f$is a square$n\times n$matrix, usually defined and arranged as $$\mathbf {H} _{f}={\begin{bmatrix}{\dfrac {\partial ^{2}f}{\partial x_{1}^{2}}}&{\dfrac {\partial ^{2}f}{\partial x_{1}\,\partial x_{2}}}&\cdots &{\dfrac {\partial ^{2}f}{\partial x_{1}\,\partial x_{n}}}\\[2.2ex]{\dfrac {\partial ^{2}f}{\partial x_{2}\,\partial x_{1}}}&{\dfrac {\partial ^{2}f}{\partial x_{2}^{2}}}&\cdots &{\dfrac {\partial ^{2}f}{\partial x_{2}\,\partial x_{n}}}\\[2.2ex]\vdots &\vdots &\ddots &\vdots \\[2.2ex]{\dfrac {\partial ^{2}f}{\partial x_{n}\,\partial x_{1}}}&{\dfrac {\partial ^{2}f}{\partial x_{n}\,\partial x_{2}}}&\cdots &{\dfrac {\partial ^{2}f}{\partial x_{n}^{2}}}\end{bmatrix}}.$$ That is, the entry of theith row and thejth column is $$(\mathbf {H} _{f})_{i,j}={\frac {\partial ^{2}f}{\partial x_{i}\,\partial x_{j}}}.$$

If furthermore the second partial derivatives are all continuous, the Hessian matrix is asymmetric matrixby thesymmetry of second derivatives.

Thedeterminantof the Hessian matrix is called theHessian determinant.

The Hessian matrix of a function$f$is the transpose of theJacobian matrixof thegradientof the function$f$;that is:$\mathbf {H} (f(\mathbf {x} ))=\mathbf {J} (\nabla f(\mathbf {x} ))^{T}.$

Applications

Inflection points

If$f$is ahomogeneous polynomialin three variables, the equation$f=0$is theimplicit equationof aplane projective curve.Theinflection pointsof the curve are exactly the non-singular points where the Hessian determinant is zero. It follows byBézout's theoremthat acubic plane curvehas at most$9$inflection points, since the Hessian determinant is a polynomial of degree$3.$

Second-derivative test

Main article:Second partial derivative test

The Hessian matrix of aconvex functionispositive semi-definite.Refining this property allows us to test whether acritical point$x$is a local maximum, local minimum, or a saddle point, as follows:

If the Hessian ispositive-definiteat$x,$then$f$attains an isolated local minimum at$x.$If the Hessian isnegative-definiteat$x,$then$f$attains an isolated local maximum at$x.$If the Hessian has both positive and negativeeigenvalues,then$x$is asaddle pointfor$f.$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 ofMorse theory.

Thesecond-derivative testfor 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$x$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, thedeterminantcan 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$1\times 1$minor being negative.

Critical points

If thegradient(the vector of the partial derivatives) of a function$f$is zero at some point$\mathbf {x},$then$f$has acritical point(orstationary point) at$\mathbf {x}.$Thedeterminantof the Hessian at$\mathbf {x}$is called, in some contexts, adiscriminant.If this determinant is zero then$\mathbf {x}$is called adegenerate critical pointof$f,$or anon-Morse critical pointof$f.$Otherwise it is non-degenerate, and called aMorse critical pointof$f.$

The Hessian matrix plays an important role inMorse theoryandcatastrophe theory,because itskernelandeigenvaluesallow classification of the critical points.

The determinant of the Hessian matrix, when evaluated at a critical point of a function, is equal to theGaussian curvatureof 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. (SeeGaussian curvature § Relation to principal curvatures.)

Use in optimization

Hessian matrices are used in large-scaleoptimizationproblems withinNewton-type methods because they are the coefficient of the quadratic term of a localTaylor expansionof a function. That is, $$y=f(\mathbf {x} +\Delta \mathbf {x} )\approx f(\mathbf {x} )+\nabla f(\mathbf {x} )^{\mathrm {T} }\Delta \mathbf {x} +{\frac {1}{2}}\,\Delta \mathbf {x} ^{\mathrm {T} }\mathbf {H} (\mathbf {x} )\,\Delta \mathbf {x}$$ where$\nabla f$is thegradient$\left({\frac {\partial f}{\partial x_{1}}},\ldots,{\frac {\partial f}{\partial x_{n}}}\right).$Computing and storing the full Hessian matrix takes$\Theta \left(n^{2}\right)$memory, which is infeasible for high-dimensional functions such as theloss functionsofneural nets,conditional random fields,and otherstatistical modelswith large numbers of parameters. For such situations,truncated-Newtonandquasi-Newtonalgorithms have been developed. The latter family of algorithms use approximations to the Hessian; one of the most popular quasi-Newton algorithms isBFGS.

Such approximations may use the fact that an optimization algorithm uses the Hessian only as alinear operator$\mathbf {H} (\mathbf {v} ),$and proceed by first noticing that the Hessian also appears in the local expansion of the gradient: $$\nabla f(\mathbf {x} +\Delta \mathbf {x} )=\nabla f(\mathbf {x} )+\mathbf {H} (\mathbf {x} )\,\Delta \mathbf {x} +{\mathcal {O}}(\|\Delta \mathbf {x} \|^{2})$$

Letting$\Delta \mathbf {x} =r\mathbf {v}$for some scalar$r,$this gives $$\mathbf {H} (\mathbf {x} )\,\Delta \mathbf {x} =\mathbf {H} (\mathbf {x} )r\mathbf {v} =r\mathbf {H} (\mathbf {x} )\mathbf {v} =\nabla f(\mathbf {x} +r\mathbf {v} )-\nabla f(\mathbf {x} )+{\mathcal {O}}(r^{2}),$$ that is, $$\mathbf {H} (\mathbf {x} )\mathbf {v} ={\frac {1}{r}}\left[\nabla f(\mathbf {x} +r\mathbf {v} )-\nabla f(\mathbf {x} )\right]+{\mathcal {O}}(r)$$ 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$r$has to be made small to prevent error due to the${\mathcal {O}}(r)$term, but decreasing it loses precision in the first term.)

Notably regarding Randomized Search Heuristics, theevolution strategy's covariance matrix adapts to the inverse of the Hessian matrix,up toa 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 inimage processingandcomputer vision(see theLaplacian of Gaussian(LoG) blob detector,the determinant of Hessian (DoH) blob detectorandscale space). It can be used innormal modeanalysis to calculate the different molecular frequencies ininfrared spectroscopy.It can also be used in local sensitivity and statistical diagnostics.

Generalizations

Bordered Hessian

Abordered Hessianis used for the second-derivative test in certain constrained optimization problems. Given the function$f$considered previously, but adding a constraint function$g$such that$g(\mathbf {x} )=c,$the bordered Hessian is the Hessian of theLagrange function$\Lambda (\mathbf {x},\lambda )=f(\mathbf {x} )+\lambda [g(\mathbf {x} )-c]:$ $$\mathbf {H} (\Lambda )={\begin{bmatrix}{\dfrac {\partial ^{2}\Lambda }{\partial \lambda ^{2}}}&{\dfrac {\partial ^{2}\Lambda }{\partial \lambda \partial \mathbf {x} }}\\\left({\dfrac {\partial ^{2}\Lambda }{\partial \lambda \partial \mathbf {x} }}\right)^{\mathsf {T}}&{\dfrac {\partial ^{2}\Lambda }{\partial \mathbf {x} ^{2}}}\end{bmatrix}}={\begin{bmatrix}0&{\dfrac {\partial g}{\partial x_{1}}}&{\dfrac {\partial g}{\partial x_{2}}}&\cdots &{\dfrac {\partial g}{\partial x_{n}}}\\[2.2ex]{\dfrac {\partial g}{\partial x_{1}}}&{\dfrac {\partial ^{2}\Lambda }{\partial x_{1}^{2}}}&{\dfrac {\partial ^{2}\Lambda }{\partial x_{1}\,\partial x_{2}}}&\cdots &{\dfrac {\partial ^{2}\Lambda }{\partial x_{1}\,\partial x_{n}}}\\[2.2ex]{\dfrac {\partial g}{\partial x_{2}}}&{\dfrac {\partial ^{2}\Lambda }{\partial x_{2}\,\partial x_{1}}}&{\dfrac {\partial ^{2}\Lambda }{\partial x_{2}^{2}}}&\cdots &{\dfrac {\partial ^{2}\Lambda }{\partial x_{2}\,\partial x_{n}}}\\[2.2ex]\vdots &\vdots &\vdots &\ddots &\vdots \\[2.2ex]{\dfrac {\partial g}{\partial x_{n}}}&{\dfrac {\partial ^{2}\Lambda }{\partial x_{n}\,\partial x_{1}}}&{\dfrac {\partial ^{2}\Lambda }{\partial x_{n}\,\partial x_{2}}}&\cdots &{\dfrac {\partial ^{2}\Lambda }{\partial x_{n}^{2}}}\end{bmatrix}}={\begin{bmatrix}0&{\dfrac {\partial g}{\partial \mathbf {x} }}\\\left({\dfrac {\partial g}{\partial \mathbf {x} }}\right)^{\mathsf {T}}&{\dfrac {\partial ^{2}\Lambda }{\partial \mathbf {x} ^{2}}}\end{bmatrix}}$$

If there are, say,$m$constraints then the zero in the upper-left corner is an$m\times m$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$\mathbf {z} ^{\mathsf {T}}\mathbf {H} \mathbf {z} =0$if$\mathbf {z}$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.Intuitively, the$m$constraints can be thought of as reducing the problem to one with$n-m$free variables. (For example, the maximization of$f\left(x_{1},x_{2},x_{3}\right)$subject to the constraint$x_{1}+x_{2}+x_{3}=1$can be reduced to the maximization of$f\left(x_{1},x_{2},1-x_{1}-x_{2}\right)$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$2m$leading principal minors are neglected, 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; if$2m+1$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 acandidate maximum or minimum.A sufficient condition for a localmaximumis that these minors alternate in sign with the smallest one having the sign of$(-1)^{m+1}.$A sufficient condition for a localminimumis 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$f$is instead avector field$\mathbf {f}:\mathbb {R} ^{n}\to \mathbb {R} ^{m},$that is, $$\mathbf {f} (\mathbf {x} )=\left(f_{1}(\mathbf {x} ),f_{2}(\mathbf {x} ),\ldots,f_{m}(\mathbf {x} )\right),$$ then the collection of second partial derivatives is not a$n\times n$matrix, but rather a third-ordertensor.This can be thought of as an array of$m$Hessian matrices, one for each component of$\mathbf {f}$: $$\mathbf {H} (\mathbf {f} )=\left(\mathbf {H} (f_{1}),\mathbf {H} (f_{2}),\ldots,\mathbf {H} (f_{m})\right).$$ This tensor degenerates to the usual Hessian matrix when$m=1.$

Generalization to the complex case

In the context ofseveral complex variables,the Hessian may be generalized. Suppose$f\colon \mathbb {C} ^{n}\to \mathbb {C},$and write$f\left(z_{1},\ldots,z_{n}\right).$Then the generalized Hessian is${\frac {\partial ^{2}f}{\partial z_{i}\partial {\overline {z_{j}}}}}.$If$f$satisfies the n-dimensionalCauchy–Riemann conditions,then the complex Hessian matrix is identically zero.

Generalizations to Riemannian manifolds

Let$(M,g)$be aRiemannian manifoldand$\nabla$itsLevi-Civita connection.Let$f:M\to \mathbb {R}$be a smooth function. Define the Hessian tensor by $$\operatorname {Hess} (f)\in \Gamma \left(T^{*}M\otimes T^{*}M\right)\quad {\text{ by }}\quad \operatorname {Hess} (f):=\nabla \nabla f=\nabla df,$$ 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$\left\{x^{i}\right\}$gives a local expression for the Hessian as $$\operatorname {Hess} (f)=\nabla _{i}\,\partial _{j}f\ dx^{i}\!\otimes \!dx^{j}=\left({\frac {\partial ^{2}f}{\partial x^{i}\partial x^{j}}}-\Gamma _{ij}^{k}{\frac {\partial f}{\partial x^{k}}}\right)dx^{i}\otimes dx^{j}$$ where$\Gamma _{ij}^{k}$are theChristoffel symbolsof the connection. Other equivalent forms for the Hessian are given by $$\operatorname {Hess} (f)(X,Y)=\langle \nabla _{X}\operatorname {grad} f,Y\rangle \quad {\text{ and }}\quad \operatorname {Hess} (f)(X,Y)=X(Yf)-df(\nabla _{X}Y).$$

See also

• The determinant of the Hessian matrix is a covariant; seeInvariant of a binary form
• Polarization identity,useful for rapid calculations involving Hessians.
• Jacobian matrix– Matrix of all first-order partial derivatives of a vector-valued functionPages displaying short descriptions of redirect targets
• Hessian equation

Further reading

• Lewis, David W. (1991).Matrix Theory.Singapore: World Scientific.ISBN978-981-02-0689-5.
• Magnus, Jan R.; Neudecker, Heinz (1999). "The Second Differential".Matrix Differential Calculus: With Applications in Statistics and Econometrics(Revised ed.). New York: Wiley. pp. 99–115.ISBN0-471-98633-X.

External links

• "Hessian of a function",Encyclopedia of Mathematics,EMS Press,2001 [1994]
• Weisstein, Eric W."Hessian".MathWorld.