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Hessian matrix

From Wikipedia, the free encyclopedia
Matrix of second derivatives
Part of a series of articles about
Calculus
∫abf′(t)dt=f(b)−f(a){\displaystyle \int _{a}^{b}f'(t)\,dt=f(b)-f(a)}

Inmathematics, theHessian matrix,Hessian or (less commonly)Hesse matrix is asquare matrix of second-orderpartial derivatives of a scalar-valuedfunction, orscalar field. It describes the localcurvature of a function of many variables. The Hessian matrix was developed in the 19th century by the German mathematicianLudwig 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

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Supposef:Rn→R{\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} } is a function taking as input a vectorx∈Rn{\displaystyle \mathbf {x} \in \mathbb {R} ^{n}} and outputting a scalarf(x)∈R.{\displaystyle f(\mathbf {x} )\in \mathbb {R} .} If all second-orderpartial derivatives off{\displaystyle f} exist, then the Hessian matrixH{\displaystyle \mathbf {H} } off{\displaystyle f} is a squaren×n{\displaystyle n\times n} matrix, usually defined and arranged asHf=[∂2f∂x12∂2f∂x1∂x2⋯∂2f∂x1∂xn∂2f∂x2∂x1∂2f∂x22⋯∂2f∂x2∂xn⋮⋮⋱⋮∂2f∂xn∂x1∂2f∂xn∂x2⋯∂2f∂xn2].{\displaystyle \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(Hf)i,j=∂2f∂xi∂xj.{\displaystyle (\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 matrix by thesymmetry of second derivatives.

Thedeterminant of the Hessian matrix is called theHessian determinant.[1]

The Hessian matrix of a functionf{\displaystyle f} is the transpose of theJacobian matrix of thegradient of the functionf{\displaystyle f}; that is:H(f(x))=J(∇f(x))T.{\displaystyle \mathbf {H} (f(\mathbf {x} ))=\mathbf {J} (\nabla f(\mathbf {x} ))^{\mathsf {T}}.}

Applications

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Inflection points

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Iff{\displaystyle f} is ahomogeneous polynomial in three variables, the equationf=0{\displaystyle f=0} is theimplicit equation of aplane projective curve. Theinflection points of the curve are exactly the non-singular points where the Hessian determinant is zero. It follows byBézout's theorem that acubic plane curve has at most 9 inflection points, since the Hessian determinant is a polynomial of degree 3.

Second-derivative test

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Main article:Second partial derivative test

The Hessian matrix of aconvex function ispositive semi-definite. Refining this property allows us to test whether acritical pointx{\displaystyle x} is a local maximum, local minimum, or a saddle point, as follows:

If the Hessian ispositive-definite atx,{\displaystyle x,} thenf{\displaystyle f} attains an isolated local minimum atx.{\displaystyle x.} If the Hessian isnegative-definite atx,{\displaystyle x,} thenf{\displaystyle f} attains an isolated local maximum atx.{\displaystyle x.} If the Hessian has both positive and negativeeigenvalues, thenx{\displaystyle x} is asaddle point forf.{\displaystyle 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 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, thenx{\displaystyle x} is a local minimum, and if it is negative, thenx{\displaystyle x} is a local maximum; if it is zero, then the test is inconclusive. In two variables, thedeterminant 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 the1×1{\displaystyle 1\times 1} minor being negative.

Critical points

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If thegradient (the vector of the partial derivatives) of a functionf{\displaystyle f} is zero at some pointx,{\displaystyle \mathbf {x} ,} thenf{\displaystyle f} has acritical point (orstationary point) atx.{\displaystyle \mathbf {x} .} Thedeterminant of the Hessian atx{\displaystyle \mathbf {x} } is called, in some contexts, adiscriminant. If this determinant is zero thenx{\displaystyle \mathbf {x} } is called adegenerate critical point off,{\displaystyle f,} or anon-Morse critical point off.{\displaystyle f.} Otherwise it is non-degenerate, and called aMorse critical point off.{\displaystyle f.}

The Hessian matrix plays an important role inMorse theory andcatastrophe theory, because itskernel andeigenvalues 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 theGaussian 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. (SeeGaussian curvature § Relation to principal curvatures.)

Use in optimization

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Hessian matrices are used in large-scaleoptimization problems withinNewton-type methods because they are the coefficient of the quadratic term of a localTaylor expansion of a function. That is,y=f(x+Δx)≈f(x)+∇f(x)TΔx+12ΔxTH(x)Δx{\displaystyle y=f(\mathbf {x} +\Delta \mathbf {x} )\approx f(\mathbf {x} )+\nabla f(\mathbf {x} )^{\mathsf {T}}\Delta \mathbf {x} +{\frac {1}{2}}\,\Delta \mathbf {x} ^{\mathsf {T}}\mathbf {H} (\mathbf {x} )\,\Delta \mathbf {x} }where∇f{\displaystyle \nabla f} is thegradient(∂f∂x1,…,∂f∂xn).{\displaystyle \left({\frac {\partial f}{\partial x_{1}}},\ldots ,{\frac {\partial f}{\partial x_{n}}}\right).} Computing and storing the full Hessian matrix takesΘ(n2){\displaystyle \Theta \left(n^{2}\right)} memory, which is infeasible for high-dimensional functions such as theloss functions ofneural nets,conditional random fields, and otherstatistical models with large numbers of parameters. For such situations,truncated-Newton andquasi-Newton algorithms have been developed. The latter family of algorithms use approximations to the Hessian; one of the most popular quasi-Newton algorithms isBFGS.[5]

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

LettingΔx=rv{\displaystyle \Delta \mathbf {x} =r\mathbf {v} } for some scalarr,{\displaystyle r,} this givesH(x)Δx=H(x)rv=rH(x)v=∇f(x+rv)−∇f(x)+O(r2),{\displaystyle \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,H(x)v=1r[∇f(x+rv)−∇f(x)]+O(r){\displaystyle \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 sincer{\displaystyle r} has to be made small to prevent error due to theO(r){\displaystyle {\mathcal {O}}(r)} term, but decreasing it loses precision in the first term.[6])

Notably regarding Randomized Search Heuristics, theevolution 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

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The Hessian matrix is commonly used for expressing image processing operators inimage processing andcomputer vision (see theLaplacian of Gaussian (LoG) blob detector,the determinant of Hessian (DoH) blob detector andscale space). It can be used innormal mode analysis to calculate the different molecular frequencies ininfrared spectroscopy.[8] It can also be used in local sensitivity and statistical diagnostics.[9]

Generalizations

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Bordered Hessian

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Abordered Hessian is used for the second-derivative test in certain constrained optimization problems. Given the functionf{\displaystyle f} considered previously, but adding a constraint functiong{\displaystyle g} such thatg(x)=c,{\displaystyle g(\mathbf {x} )=c,} the bordered Hessian is the Hessian of theLagrange functionΛ(x,λ)=f(x)+λ[g(x)−c]{\displaystyle \Lambda (\mathbf {x} ,\lambda )=f(\mathbf {x} )+\lambda [g(\mathbf {x} )-c]}:[10]H(Λ)=[∂2Λ∂λ2∂2Λ∂λ∂x(∂2Λ∂λ∂x)T∂2Λ∂x2]=[0∂g∂x1∂g∂x2⋯∂g∂xn∂g∂x1∂2Λ∂x12∂2Λ∂x1∂x2⋯∂2Λ∂x1∂xn∂g∂x2∂2Λ∂x2∂x1∂2Λ∂x22⋯∂2Λ∂x2∂xn⋮⋮⋮⋱⋮∂g∂xn∂2Λ∂xn∂x1∂2Λ∂xn∂x2⋯∂2Λ∂xn2]=[0∂g∂x(∂g∂x)T∂2Λ∂x2]{\displaystyle \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{\displaystyle m} constraints then the zero in the upper-left corner is anm×m{\displaystyle m\times m} block of zeros, and there arem{\displaystyle m} border rows at the top andm{\displaystyle 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, aszTHz=0{\displaystyle \mathbf {z} ^{\mathsf {T}}\mathbf {H} \mathbf {z} =0} ifz{\displaystyle \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 ofn−m{\displaystyle n-m} submatrices of the bordered Hessian.[11] Intuitively, them{\displaystyle m} constraints can be thought of as reducing the problem to one withn−m{\displaystyle n-m} free variables. (For example, the maximization off(x1,x2,x3){\displaystyle f\left(x_{1},x_{2},x_{3}\right)} subject to the constraintx1+x2+x3=1{\displaystyle x_{1}+x_{2}+x_{3}=1} can be reduced to the maximization off(x1,x2,1−x1−x2){\displaystyle 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 first2m{\displaystyle 2m} leading principal minors are neglected, the smallest minor consisting of the truncated first2m+1{\displaystyle 2m+1} rows and columns, the next consisting of the truncated first2m+2{\displaystyle 2m+2} rows and columns, and so on, with the last being the entire bordered Hessian; if2m+1{\displaystyle 2m+1} is larger thann+m,{\displaystyle n+m,} then the smallest leading principal minor is the Hessian itself.[12] There are thusn−m{\displaystyle n-m} minors to consider, each evaluated at the specific point being considered as acandidate maximum or minimum. A sufficient condition for a localmaximum is that these minors alternate in sign with the smallest one having the sign of(−1)m+1.{\displaystyle (-1)^{m+1}.} A sufficient condition for a localminimum is that all of these minors have the sign of(−1)m.{\displaystyle (-1)^{m}.} (In the unconstrained case ofm=0{\displaystyle m=0} these conditions coincide with the conditions for the unbordered Hessian to be negative definite or positive definite respectively).

Vector-valued functions

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Iff{\displaystyle f} is instead avector fieldf:Rn→Rm,{\displaystyle \mathbf {f} :\mathbb {R} ^{n}\to \mathbb {R} ^{m},} that is,f(x)=(f1(x),f2(x),…,fm(x)),{\displaystyle \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 an×n{\displaystyle n\times n} matrix, but rather a third-ordertensor. This can be thought of as an array ofm{\displaystyle m} Hessian matrices, one for each component off{\displaystyle \mathbf {f} }:H(f)=(H(f1),H(f2),…,H(fm)).{\displaystyle \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 whenm=1.{\displaystyle m=1.}

Generalization to the complex case

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In the context ofseveral complex variables, the Hessian may be generalized. Supposef:Cn→C,{\displaystyle f\colon \mathbb {C} ^{n}\to \mathbb {C} ,} and writef(z1,…,zn).{\displaystyle f\left(z_{1},\ldots ,z_{n}\right).} IdentifyingCn{\displaystyle {\mathbb {C} }^{n}} withR2n{\displaystyle {\mathbb {R} }^{2n}}, the normal "real" Hessian is a2n×2n{\displaystyle 2n\times 2n} matrix. As the object of study in several complex variables areholomorphic functions, that is, solutions to the n-dimensionalCauchy–Riemann conditions, we usually look on the part of the Hessian that contains information invariant under holomorphic changes of coordinates. This "part" is the so-called complex Hessian, which is the matrix(∂2f∂zj∂z¯k)j,k.{\displaystyle \left({\frac {\partial ^{2}f}{\partial z_{j}\partial {\bar {z}}_{k}}}\right)_{j,k}.} Note that iff{\displaystyle f} is holomorphic, then its complex Hessian matrix is identically zero, so the complex Hessian is used to study smooth but not holomorphic functions, see for exampleLevi pseudoconvexity. When dealing with holomorphic functions, we could consider the Hessian matrix(∂2f∂zj∂zk)j,k.{\displaystyle \left({\frac {\partial ^{2}f}{\partial z_{j}\partial z_{k}}}\right)_{j,k}.}

Generalizations to Riemannian manifolds

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Let(M,g){\displaystyle (M,g)} be aRiemannian manifold and∇{\displaystyle \nabla } itsLevi-Civita connection. Letf:M→R{\displaystyle f:M\to \mathbb {R} } be a smooth function. Define the Hessian tensor byHess⁡(f)∈Γ(T∗M⊗T∗M) by Hess⁡(f):=∇∇f=∇df,{\displaystyle \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{xi}{\displaystyle \left\{x^{i}\right\}} gives a local expression for the Hessian asHess⁡(f)=∇i∂jf dxi⊗dxj=(∂2f∂xi∂xj−Γijk∂f∂xk)dxi⊗dxj{\displaystyle \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}}whereΓijk{\displaystyle \Gamma _{ij}^{k}} are theChristoffel symbols of the connection. Other equivalent forms for the Hessian are given byHess⁡(f)(X,Y)=⟨∇Xgrad⁡f,Y⟩ and Hess⁡(f)(X,Y)=X(Yf)−df(∇XY).{\displaystyle \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

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References

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  1. ^Binmore, Ken; Davies, Joan (2007).Calculus Concepts and Methods. Cambridge University Press. p. 190.ISBN 978-0-521-77541-0.OCLC 717598615.
  2. ^Callahan, James J. (2010).Advanced Calculus: A Geometric View. Springer Science & Business Media. p. 248.ISBN 978-1-4419-7332-0.
  3. ^Casciaro, B.; Fortunato, D.; Francaviglia, M.; Masiello, A., eds. (2011).Recent Developments in General Relativity. Springer Science & Business Media. p. 178.ISBN 9788847021136.
  4. ^Domenico P. L. Castrigiano; Sandra A. Hayes (2004).Catastrophe theory. Westview Press. p. 18.ISBN 978-0-8133-4126-2.
  5. ^Nocedal, Jorge; Wright, Stephen (2000).Numerical Optimization. Springer Verlag.ISBN 978-0-387-98793-4.
  6. ^Pearlmutter, Barak A. (1994)."Fast exact multiplication by the Hessian"(PDF).Neural Computation.6 (1):147–160.doi:10.1162/neco.1994.6.1.147.S2CID 1251969.
  7. ^Shir, O.M.; A. Yehudayoff (2020)."On the covariance-Hessian relation in evolution strategies".Theoretical Computer Science.801. Elsevier:157–174.arXiv:1806.03674.doi:10.1016/j.tcs.2019.09.002.
  8. ^Mott, Adam J.; Rez, Peter (December 24, 2014)."Calculation of the infrared spectra of proteins".European Biophysics Journal.44 (3):103–112.doi:10.1007/s00249-014-1005-6.ISSN 0175-7571.PMID 25538002.S2CID 2945423.
  9. ^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.doi:10.1016/j.jmva.2021.104849.
  10. ^Hallam, Arne (October 7, 2004)."Econ 500: Quantitative Methods in Economic Analysis I"(PDF).Iowa State.
  11. ^Neudecker, Heinz; Magnus, Jan R. (1988).Matrix Differential Calculus with Applications in Statistics and Econometrics. New York:John Wiley & Sons. p. 136.ISBN 978-0-471-91516-4.
  12. ^Chiang, Alpha C. (1984).Fundamental Methods of Mathematical Economics (Third ed.). McGraw-Hill. p. 386.ISBN 978-0-07-010813-4.

Further reading

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  • Lewis, David W. (1991).Matrix Theory. Singapore: World Scientific.ISBN 978-981-02-0689-5.
  • 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.ISBN 0-471-98633-X.

External links

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