Misplaced Pages

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89-500: Since the updates of the BFGS curvature matrix do not require matrix inversion , its computational complexity is only O ( n 2 ) {\displaystyle {\mathcal {O}}(n^{2})} , compared to O ( n 3 ) {\displaystyle {\mathcal {O}}(n^{3})} in Newton's method . Also in common use is L-BFGS , which

178-824: A Riemannian manifold and ∇ {\displaystyle \nabla } its Levi-Civita connection . Let f : M → R {\displaystyle f:M\to \mathbb {R} } be a smooth function. Define the Hessian tensor by Hess ⁡ ( f ) ∈ Γ ( T ∗ M ⊗ T ∗ M )  by  Hess ⁡ ( f ) := ∇ ∇ f = ∇ d f , {\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

267-459: A candidate maximum or minimum . 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 . {\displaystyle (-1)^{m+1}.} A sufficient condition for a local minimum is that all of these minors have the sign of ( − 1 ) m . {\displaystyle (-1)^{m}.} (In

356-510: A field K (e.g., the field ⁠ R {\displaystyle \mathbb {R} } ⁠ of real numbers). The following statements are equivalent, i.e., they are either all true or all false for any given matrix: Furthermore, the following properties hold for an invertible matrix A : The rows of the inverse matrix V of a matrix U are orthonormal to the columns of U (and vice versa interchanging rows for columns). To see this, suppose that UV = VU = I where

445-620: A convex target. However, some real-life applications (like Sequential Quadratic Programming methods) routinely produce negative or nearly-zero curvatures. This can occur when optimizing a nonconvex target or when employing a trust-region approach instead of a line search. It is also possible to produce spurious values due to noise in the target. In such cases, one of the so-called damped BFGS updates can be used (see ) which modify s k {\displaystyle \mathbf {s} _{k}} and/or y k {\displaystyle \mathbf {y} _{k}} in order to obtain

534-445: A field is singular if and only if its determinant is zero. Singular matrices are rare in the sense that if a square matrix's entries are randomly selected from any bounded region on the number line or complex plane , the probability that the matrix is singular is 0, that is, it will "almost never" be singular. Non-square matrices, i.e. m -by- n matrices for which m ≠ n , do not have an inverse. However, in some cases such

623-661: A local Taylor expansion of a function. That is, y = f ( x + Δ x ) ≈ f ( x ) + ∇ f ( x ) T Δ x + 1 2 Δ x T H ( x ) Δ x {\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}

712-409: A matrix may have a left inverse or right inverse . If A is m -by- n and the rank of A is equal to n , ( n ≤ m ), then A has a left inverse, an n -by- m matrix B such that BA = I n . If A has rank m ( m ≤ n ), then it has a right inverse, an n -by- m matrix B such that AB = I m . While the most common case is that of matrices over

801-464: A more robust update. Notable open source implementations are: Notable proprietary implementations include: Matrix inversion In linear algebra , an invertible matrix is a square matrix which has an inverse . In other words, if some other matrix is multiplied by the invertible matrix, the result can be multiplied by an inverse to undo the operation. Invertible matrices are the same size as their inverse. An n -by- n square matrix A

890-400: A positive-definite or negative-definite Hessian cannot apply here since a bordered Hessian can neither be negative-definite nor positive-definite, as z T H z = 0 {\displaystyle \mathbf {z} ^{\mathsf {T}}\mathbf {H} \mathbf {z} =0} if z {\displaystyle \mathbf {z} } is any vector whose sole non-zero entry

979-2385: A scalar f ( x ) ∈ R . {\displaystyle f(\mathbf {x} )\in \mathbb {R} .} If all second-order partial derivatives of f {\displaystyle f} exist, then the Hessian matrix H {\displaystyle \mathbf {H} } of f {\displaystyle f} is a square n × n {\displaystyle n\times n} matrix, usually defined and arranged as H f = [ ∂ 2 f ∂ x 1 2 ∂ 2 f ∂ x 1 ∂ x 2 ⋯ ∂ 2 f ∂ x 1 ∂ x n ∂ 2 f ∂ x 2 ∂ x 1 ∂ 2 f ∂ x 2 2 ⋯ ∂ 2 f ∂ x 2 ∂ x n ⋮ ⋮ ⋱ ⋮ ∂ 2 f ∂ x n ∂ x 1 ∂ 2 f ∂ x n ∂ x 2 ⋯ ∂ 2 f ∂ x n 2 ] . {\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,

SECTION 10

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1068-413: Is a homogeneous polynomial in three variables, the equation f = 0 {\displaystyle f=0} 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 9 inflection points, since the Hessian determinant

1157-402: Is a null set , that is, has Lebesgue measure zero. This is true because singular matrices are the roots of the determinant function. This is a continuous function because it is a polynomial in the entries of the matrix. Thus in the language of measure theory , almost all n -by- n matrices are invertible. Furthermore, the set of n -by- n invertible matrices is open and dense in

1246-439: Is a limited-memory version of BFGS that is particularly suited to problems with very large numbers of variables (e.g., >1000). The BFGS-B variant handles simple box constraints.. The BFGS matrix also admits a compact representation , which makes it better suited for large constrained problems. The algorithm is named after Charles George Broyden , Roger Fletcher , Donald Goldfarb and David Shanno . The optimization problem

1335-421: Is a non-invertible matrix We can see the rank of this 2-by-2 matrix is 1, which is n − 1 ≠ n , so it is non-invertible. Consider the following 2-by-2 matrix: The matrix B {\displaystyle \mathbf {B} } is invertible. To check this, one can compute that det B = − 1 2 {\textstyle \det \mathbf {B} =-{\frac {1}{2}}} , which

1424-505: Is a nonlinear objective function. From an initial guess x 0 ∈ R n {\displaystyle \mathbf {x} _{0}\in \mathbb {R} ^{n}} and an initial guess of the Hessian matrix B 0 ∈ R n × n {\displaystyle B_{0}\in \mathbb {R} ^{n\times n}} the following steps are repeated as x k {\displaystyle \mathbf {x} _{k}} converges to

1513-519: Is a polynomial of degree 3. The Hessian matrix of a convex function is positive semi-definite . Refining this property allows us to test whether a critical point x {\displaystyle x} is a local maximum, local minimum, or a saddle point, as follows: If the Hessian is positive-definite at x , {\displaystyle x,} then f {\displaystyle f} attains an isolated local minimum at x . {\displaystyle x.} If

1602-422: Is called invertible (also nonsingular , nondegenerate or rarely regular ) if there exists an n -by- n square matrix B such that A B = B A = I n , {\displaystyle \mathbf {AB} =\mathbf {BA} =\mathbf {I} _{n},} where I n denotes the n -by- n identity matrix and the multiplication used is ordinary matrix multiplication . If this

1691-519: Is called an involutory matrix . The adjugate of a matrix A can be used to find the inverse of A as follows: If A is an invertible matrix, then It follows from the associativity of matrix multiplication that if for finite square matrices A and B , then also Over the field of real numbers, the set of singular n -by- n matrices, considered as a subset of ⁠ R n × n , {\displaystyle \mathbb {R} ^{n\times n},} ⁠

1780-400: Is convenient to find a suitable starting seed: Victor Pan and John Reif have done work that includes ways of generating a starting seed. Newton's method is particularly useful when dealing with families of related matrices that behave enough like the sequence manufactured for the homotopy above: sometimes a good starting point for refining an approximation for the new inverse can be

1869-594: Is first created with the left side being the matrix to invert and the right side being the identity matrix . Then, Gaussian elimination is used to convert the left side into the identity matrix, which causes the right side to become the inverse of the input matrix. For example, take the following matrix: A = ( − 1 3 2 1 − 1 ) . {\displaystyle \mathbf {A} ={\begin{pmatrix}-1&{\tfrac {3}{2}}\\1&-1\end{pmatrix}}.} The first step to compute its inverse

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1958-555: Is given by Hessian matrix 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

2047-609: 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 example Levi pseudoconvexity . When dealing with holomorphic functions, we could consider the Hessian matrix ( ∂ 2 f ∂ z j ∂ z k ) j , k . {\displaystyle \left({\frac {\partial ^{2}f}{\partial z_{j}\partial z_{k}}}\right)_{j,k}.} Let ( M , g ) {\displaystyle (M,g)} be

2136-433: 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 . Such approximations may use

2225-490: Is initialized with B 0 = I {\displaystyle B_{0}=I} , the first step will be equivalent to a gradient descent , but further steps are more and more refined by B k {\displaystyle B_{k}} , the approximation to the Hessian. The first step of the algorithm is carried out using the inverse of the matrix B k {\displaystyle B_{k}} , which can be obtained efficiently by applying

2314-445: Is invertible and its inverse is given by where Q is the square ( N × N ) matrix whose i th column is the eigenvector q i {\displaystyle q_{i}} of A , and Λ is the diagonal matrix whose diagonal entries are the corresponding eigenvalues, that is, Λ i i = λ i . {\displaystyle \Lambda _{ii}=\lambda _{i}.} If A

2403-446: Is its first. The second derivative test consists here of sign restrictions of the determinants of a certain set of n − m {\displaystyle n-m} submatrices of the bordered Hessian. Intuitively, the m {\displaystyle m} constraints can be thought of as reducing the problem to one with n − m {\displaystyle n-m} free variables. (For example,

2492-573: Is known as symmetric rank-one method, which does not guarantee the positive definiteness . In order to maintain the symmetry and positive definiteness of B k + 1 {\displaystyle B_{k+1}} , the update form can be chosen as B k + 1 = B k + α u u ⊤ + β v v ⊤ {\displaystyle B_{k+1}=B_{k}+\alpha \mathbf {u} \mathbf {u} ^{\top }+\beta \mathbf {v} \mathbf {v} ^{\top }} . Imposing

2581-524: Is non-degenerate, and called a Morse critical point of f . {\displaystyle f.} The Hessian matrix plays an important role in Morse theory and catastrophe theory , because its kernel and eigenvalues 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 of

2670-416: Is non-zero. As an example of a non-invertible, or singular, matrix, consider the matrix The determinant of C {\displaystyle \mathbf {C} } is 0, which is a necessary and sufficient condition for a matrix to be non-invertible. Gaussian elimination is a useful and easy way to compute the inverse of a matrix. To compute a matrix inverse using this method, an augmented matrix

2759-405: Is simpler than the general case. In one variable, the Hessian contains exactly one second derivative; if it is positive, then x {\displaystyle x} is a local minimum, and if it is negative, then x {\displaystyle 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

Broyden–Fletcher–Goldfarb–Shanno algorithm - Misplaced Pages Continue

2848-403: Is symmetric, Q is guaranteed to be an orthogonal matrix , therefore Q − 1 = Q T . {\displaystyle \mathbf {Q} ^{-1}=\mathbf {Q} ^{\mathrm {T} }.} Furthermore, because Λ is a diagonal matrix, its inverse is easy to calculate: If matrix A is positive definite , then its inverse can be obtained as where L

2937-1030: Is that the process of Gaussian elimination can be viewed as a sequence of applying left matrix multiplication using elementary row operations using elementary matrices ( E n {\displaystyle \mathbf {E} _{n}} ), such as E n E n − 1 ⋯ E 2 E 1 A = I . {\displaystyle \mathbf {E} _{n}\mathbf {E} _{n-1}\cdots \mathbf {E} _{2}\mathbf {E} _{1}\mathbf {A} =\mathbf {I} .} Applying right-multiplication using A − 1 , {\displaystyle \mathbf {A} ^{-1},} we get E n E n − 1 ⋯ E 2 E 1 I = I A − 1 . {\displaystyle \mathbf {E} _{n}\mathbf {E} _{n-1}\cdots \mathbf {E} _{2}\mathbf {E} _{1}\mathbf {I} =\mathbf {I} \mathbf {A} ^{-1}.} And

3026-515: Is the determinant of A , C is the matrix of cofactors, and C represents the matrix transpose . The cofactor equation listed above yields the following result for 2 × 2 matrices. Inversion of these matrices can be done as follows: This is possible because 1/( ad − bc ) is the reciprocal of the determinant of the matrix in question, and the same strategy could be used for other matrix sizes. The Cayley–Hamilton method gives A computationally efficient 3 × 3 matrix inversion

3115-506: Is the gradient ( ∂ f ∂ x 1 , … , ∂ f ∂ x n ) . {\displaystyle \left({\frac {\partial f}{\partial x_{1}}},\ldots ,{\frac {\partial f}{\partial x_{n}}}\right).} Computing and storing the full Hessian matrix takes Θ ( n 2 ) {\displaystyle \Theta \left(n^{2}\right)} memory, which

3204-437: Is the lower triangular Cholesky decomposition of A , and L * denotes the conjugate transpose of L . Writing the transpose of the matrix of cofactors , known as an adjugate matrix , can also be an efficient way to calculate the inverse of small matrices, but this recursive method is inefficient for large matrices. To determine the inverse, we calculate a matrix of cofactors: so that where | A |

3293-403: Is the case, then the matrix B is uniquely determined by A , and is called the (multiplicative) inverse of A , denoted by A . Matrix inversion is the process of finding the matrix which when multiplied by the original matrix gives the identity matrix. Over a field , a square matrix that is not invertible is called singular or degenerate . A square matrix with entries in

3382-451: 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

3471-494: Is the secant equation. The curvature condition s k ⊤ y k > 0 {\displaystyle \mathbf {s} _{k}^{\top }\mathbf {y} _{k}>0} should be satisfied for B k + 1 {\displaystyle B_{k+1}} to be positive definite, which can be verified by pre-multiplying the secant equation with s k T {\displaystyle \mathbf {s} _{k}^{T}} . If

3560-404: Is the so-called complex Hessian, which is the matrix ( ∂ 2 f ∂ z j ∂ z ¯ k ) j , k . {\displaystyle \left({\frac {\partial ^{2}f}{\partial z_{j}\partial {\bar {z}}_{k}}}\right)_{j,k}.} Note that if f {\displaystyle f}

3649-430: Is to create the augmented matrix ( − 1 3 2 1 0 1 − 1 0 1 ) . {\displaystyle \left({\begin{array}{cc|cc}-1&{\tfrac {3}{2}}&1&0\\1&-1&0&1\end{array}}\right).} Call the first row of this matrix R 1 {\displaystyle R_{1}} and

Broyden–Fletcher–Goldfarb–Shanno algorithm - Misplaced Pages Continue

3738-585: Is to minimize f ( x ) {\displaystyle f(\mathbf {x} )} , where x {\displaystyle \mathbf {x} } is a vector in R n {\displaystyle \mathbb {R} ^{n}} , and f {\displaystyle f} is a differentiable scalar function. There are no constraints on the values that x {\displaystyle \mathbf {x} } can take. The algorithm begins at an initial estimate x 0 {\displaystyle \mathbf {x} _{0}} for

3827-487: Is updated iteratively at each stage, and ∇ f ( x k ) {\displaystyle \nabla f(\mathbf {x} _{k})} is the gradient of the function evaluated at x k . A line search in the direction p k is then used to find the next point x k +1 by minimizing f ( x k + γ p k ) {\displaystyle f(\mathbf {x} _{k}+\gamma \mathbf {p} _{k})} over

3916-481: 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 . It can also be used in local sensitivity and statistical diagnostics. A bordered Hessian is used for the second-derivative test in certain constrained optimization problems. Given

4005-755: The Sherman–Morrison formula to the step 5 of the algorithm, giving This can be computed efficiently without temporary matrices, recognizing that B k − 1 {\displaystyle B_{k}^{-1}} is symmetric, and that y k T B k − 1 y k {\displaystyle \mathbf {y} _{k}^{\mathrm {T} }B_{k}^{-1}\mathbf {y} _{k}} and s k T y k {\displaystyle \mathbf {s} _{k}^{\mathrm {T} }\mathbf {y} _{k}} are scalars, using an expansion such as Therefore, in order to avoid any matrix inversion,

4094-400: The gradient (the vector of the partial derivatives) of a function f {\displaystyle f} is zero at some point x , {\displaystyle \mathbf {x} ,} then f {\displaystyle f} has a critical point (or stationary point ) at x . {\displaystyle \mathbf {x} .} The determinant of

4183-456: The inverse of the Hessian can be approximated instead of the Hessian itself: H k = def B k − 1 . {\displaystyle H_{k}{\overset {\operatorname {def} }{=}}B_{k}^{-1}.} From an initial guess x 0 {\displaystyle \mathbf {x} _{0}} and an approximate inverted Hessian matrix H 0 {\displaystyle H_{0}}

4272-422: The real or complex numbers, all these definitions can be given for matrices over any algebraic structure equipped with addition and multiplication (i.e. rings ). However, in the case of a ring being commutative , the condition for a square matrix to be invertible is that its determinant is invertible in the ring, which in general is a stricter requirement than it being nonzero. For a noncommutative ring ,

4361-441: The topological space of all n -by- n matrices. Equivalently, the set of singular matrices is closed and nowhere dense in the space of n -by- n matrices. In practice however, one may encounter non-invertible matrices. And in numerical calculations , matrices which are invertible, but close to a non-invertible matrix, can still be problematic; such matrices are said to be ill-conditioned . An example with rank of n − 1

4450-554: The Gauss–Jordan algorithm which has been contaminated by small errors due to imperfect computer arithmetic . The Cayley–Hamilton theorem allows the inverse of A to be expressed in terms of det( A ) , traces and powers of A : where n is size of A , and tr( A ) is the trace of matrix A given by the sum of the main diagonal . The sum is taken over s and the sets of all k l ≥ 0 {\displaystyle k_{l}\geq 0} satisfying

4539-415: The Hessian at x {\displaystyle \mathbf {x} } is called, in some contexts, a discriminant . If this determinant is zero then x {\displaystyle \mathbf {x} } is called a degenerate critical point of f , {\displaystyle f,} or a non-Morse critical point of f . {\displaystyle f.} Otherwise it

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4628-427: The Hessian is negative-definite at x , {\displaystyle x,} then f {\displaystyle f} attains an isolated local maximum at x . {\displaystyle x.} If the Hessian has both positive and negative eigenvalues , then x {\displaystyle x} is a saddle point for f . {\displaystyle f.} Otherwise

4717-775: The Hessian matrix is a symmetric matrix by the symmetry of second derivatives . The determinant of the Hessian matrix is called the Hessian determinant . The Hessian matrix of a function f {\displaystyle f} is the transpose of the Jacobian matrix of the gradient of the function f {\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}}.} If f {\displaystyle f}

4806-482: 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 × 1 {\displaystyle 1\times 1} minor being negative. If

4895-409: The already obtained inverse of a previous matrix that nearly matches the current matrix, for example, the pair of sequences of inverse matrices used in obtaining matrix square roots by Denman–Beavers iteration ; this may need more than one pass of the iteration at each new matrix, if they are not close enough together for just one to be enough. Newton's method is also useful for "touch up" corrections to

4984-411: The approximate Hessian at stage k is updated by the addition of two matrices: Both U k {\displaystyle U_{k}} and V k {\displaystyle V_{k}} are symmetric rank-one matrices, but their sum is a rank-two update matrix. BFGS and DFP updating matrix both differ from its predecessor by a rank-two matrix. Another simpler rank-one method

5073-419: The augumented matrix by combining A with I and applying Gaussian elimination . The two portions will be transformed using the same sequence of elementary row operations. When the left portion becomes I , the right portion applied the same elementary row operation sequence will become A . A generalization of Newton's method as used for a multiplicative inverse algorithm may be convenient, if it

5162-727: The collection of second partial derivatives is not a n × n {\displaystyle n\times n} matrix, but rather a third-order tensor . This can be thought of as an array of m {\displaystyle m} Hessian matrices, one for each component of f {\displaystyle \mathbf {f} } : H ( f ) = ( H ( f 1 ) , H ( f 2 ) , … , H ( f m ) ) . {\displaystyle \mathbf {H} (\mathbf {f} )=\left(\mathbf {H} (f_{1}),\mathbf {H} (f_{2}),\ldots ,\mathbf {H} (f_{m})\right).} This tensor degenerates to

5251-415: The entry of the i th row and the j th column is ( H f ) i , j = ∂ 2 f ∂ x i ∂ x j . {\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,

5340-1906: The fact that an optimization algorithm uses the Hessian only as a linear operator H ( 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 + Δ x ) = ∇ f ( x ) + H ( x ) Δ x + O ( ‖ Δ x ‖ 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})} Letting Δ x = r v {\displaystyle \Delta \mathbf {x} =r\mathbf {v} } for some scalar r , {\displaystyle r,} this gives H ( x ) Δ x = H ( x ) r v = r H ( x ) v = ∇ f ( x + r v ) − ∇ f ( x ) + O ( r 2 ) , {\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 = 1 r [ ∇ f ( x + r v ) − ∇ 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

5429-1109: The fact that the first covariant derivative of a function is the same as its ordinary differential. Choosing local coordinates { x i } {\displaystyle \left\{x^{i}\right\}} gives a local expression for the Hessian as Hess ⁡ ( f ) = ∇ i ∂ j f   d x i ⊗ d x j = ( ∂ 2 f ∂ x i ∂ x j − Γ i j k ∂ f ∂ x k ) d x i ⊗ d x j {\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}} where Γ i j k {\displaystyle \Gamma _{ij}^{k}} are

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5518-502: The first term. ) 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. The Hessian matrix is commonly used for expressing image processing operators in image processing and computer vision (see

5607-414: The following steps are repeated as x k {\displaystyle \mathbf {x} _{k}} converges to the solution: In statistical estimation problems (such as maximum likelihood or Bayesian inference), credible intervals or confidence intervals for the solution can be estimated from the inverse of the final Hessian matrix . However, these quantities are technically defined by

5696-4874: The function f {\displaystyle f} considered previously, but adding a constraint function g {\displaystyle g} such that g ( x ) = c , {\displaystyle g(\mathbf {x} )=c,} the bordered Hessian is the Hessian of the Lagrange function Λ ( x , λ ) = f ( x ) + λ [ g ( x ) − c ] {\displaystyle \Lambda (\mathbf {x} ,\lambda )=f(\mathbf {x} )+\lambda [g(\mathbf {x} )-c]} : H ( Λ ) = [ ∂ 2 Λ ∂ λ 2 ∂ 2 Λ ∂ λ ∂ x ( ∂ 2 Λ ∂ λ ∂ x ) T ∂ 2 Λ ∂ x 2 ] = [ 0 ∂ g ∂ x 1 ∂ g ∂ x 2 ⋯ ∂ g ∂ x n ∂ g ∂ x 1 ∂ 2 Λ ∂ x 1 2 ∂ 2 Λ ∂ x 1 ∂ x 2 ⋯ ∂ 2 Λ ∂ x 1 ∂ x n ∂ g ∂ x 2 ∂ 2 Λ ∂ x 2 ∂ x 1 ∂ 2 Λ ∂ x 2 2 ⋯ ∂ 2 Λ ∂ x 2 ∂ x n ⋮ ⋮ ⋮ ⋱ ⋮ ∂ g ∂ x n ∂ 2 Λ ∂ x n ∂ x 1 ∂ 2 Λ ∂ x n ∂ x 2 ⋯ ∂ 2 Λ ∂ x n 2 ] = [ 0 ∂ g ∂ x ( ∂ g ∂ x ) T ∂ 2 Λ ∂ x 2 ] {\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

5785-415: 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 Gaussian curvature § Relation to principal curvatures .) Hessian matrices are used in large-scale optimization problems within Newton -type methods because they are the coefficient of the quadratic term of

5874-511: The function is not strongly convex , then the condition has to be enforced explicitly e.g. by finding a point x k +1 satisfying the Wolfe conditions , which entail the curvature condition, using line search. Instead of requiring the full Hessian matrix at the point x k + 1 {\displaystyle \mathbf {x} _{k+1}} to be computed as B k + 1 {\displaystyle B_{k+1}} ,

5963-451: 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 {\displaystyle r} has to be made small to prevent error due to the O ( r ) {\displaystyle {\mathcal {O}}(r)} term, but decreasing it loses precision in

6052-553: The identity matrix on the left side and the inverse matrix on the right: ( 1 0 2 3 0 1 2 2 ) . {\displaystyle \left({\begin{array}{cc|cc}1&0&2&3\\0&1&2&2\end{array}}\right).} Thus, A − 1 = ( 2 3 2 2 ) . {\displaystyle \mathbf {A} ^{-1}={\begin{pmatrix}2&3\\2&2\end{pmatrix}}.} The reason it works

6141-406: The inverse of a square matrix in some instances, where a set of orthogonal vectors (but not necessarily orthonormal vectors) to the columns of U are known. In which case, one can apply the iterative Gram–Schmidt process to this initial set to determine the rows of the inverse V . A matrix that is its own inverse (i.e., a matrix A such that A = A , and consequently A = I ),

6230-413: The last being the entire bordered Hessian; if 2 m + 1 {\displaystyle 2m+1} is larger than n + m , {\displaystyle n+m,} then the smallest leading principal minor is the Hessian itself. There are thus n − m {\displaystyle n-m} minors to consider, each evaluated at the specific point being considered as

6319-495: The linear Diophantine equation The formula can be rewritten in terms of complete Bell polynomials of arguments t l = − ( l − 1 ) ! tr ⁡ ( A l ) {\displaystyle t_{l}=-(l-1)!\operatorname {tr} \left(A^{l}\right)} as This is described in more detail under Cayley–Hamilton method . If matrix A can be eigendecomposed, and if none of its eigenvalues are zero, then A

6408-657: The maximization of f ( x 1 , x 2 , x 3 ) {\displaystyle f\left(x_{1},x_{2},x_{3}\right)} subject to the constraint x 1 + x 2 + x 3 = 1 {\displaystyle x_{1}+x_{2}+x_{3}=1} can be reduced to the maximization of f ( x 1 , x 2 , 1 − x 1 − x 2 ) {\displaystyle f\left(x_{1},x_{2},1-x_{1}-x_{2}\right)} without constraint.) Specifically, sign conditions are imposed on

6497-406: The normal "real" Hessian is a 2 n × 2 n {\displaystyle 2n\times 2n} matrix. As the object of study in several complex variables are holomorphic functions , that is, solutions to the n-dimensional Cauchy–Riemann conditions , we usually look on the part of the Hessian that contains information invariant under holomorphic changes of coordinates. This "part"

6586-452: The optimal value and proceeds iteratively to get a better estimate at each stage. The search direction p k at stage k is given by the solution of the analogue of the Newton equation: where B k {\displaystyle B_{k}} is an approximation to the Hessian matrix at x k {\displaystyle \mathbf {x} _{k}} , which

6675-504: The right side I A − 1 = A − 1 , {\displaystyle \mathbf {I} \mathbf {A} ^{-1}=\mathbf {A} ^{-1},} which is the inverse we want. To obtain E n E n − 1 ⋯ E 2 E 1 I , {\displaystyle \mathbf {E} _{n}\mathbf {E} _{n-1}\cdots \mathbf {E} _{2}\mathbf {E} _{1}\mathbf {I} ,} we create

6764-665: The rows of V are denoted as v i T {\displaystyle v_{i}^{\mathrm {T} }} and the columns of U as u j {\displaystyle u_{j}} for 1 ≤ i , j ≤ n . {\displaystyle 1\leq i,j\leq n.} Then clearly, the Euclidean inner product of any two v i T u j = δ i , j . {\displaystyle v_{i}^{\mathrm {T} }u_{j}=\delta _{i,j}.} This property can also be useful in constructing

6853-768: The scalar γ > 0. {\displaystyle \gamma >0.} The quasi-Newton condition imposed on the update of B k {\displaystyle B_{k}} is Let y k = ∇ f ( x k + 1 ) − ∇ f ( x k ) {\displaystyle \mathbf {y} _{k}=\nabla f(\mathbf {x} _{k+1})-\nabla f(\mathbf {x} _{k})} and s k = x k + 1 − x k {\displaystyle \mathbf {s} _{k}=\mathbf {x} _{k+1}-\mathbf {x} _{k}} , then B k + 1 {\displaystyle B_{k+1}} satisfies which

6942-899: The secant condition, B k + 1 s k = y k {\displaystyle B_{k+1}\mathbf {s} _{k}=\mathbf {y} _{k}} . Choosing u = y k {\displaystyle \mathbf {u} =\mathbf {y} _{k}} and v = B k s k {\displaystyle \mathbf {v} =B_{k}\mathbf {s} _{k}} , we can obtain: Finally, we substitute α {\displaystyle \alpha } and β {\displaystyle \beta } into B k + 1 = B k + α u u ⊤ + β v v ⊤ {\displaystyle B_{k+1}=B_{k}+\alpha \mathbf {u} \mathbf {u} ^{\top }+\beta \mathbf {v} \mathbf {v} ^{\top }} and get

7031-1435: The second row R 2 {\displaystyle R_{2}} . Then, add row 1 to row 2 ( R 1 + R 2 → R 2 ) . {\displaystyle (R_{1}+R_{2}\to R_{2}).} This yields ( − 1 3 2 1 0 0 1 2 1 1 ) . {\displaystyle \left({\begin{array}{cc|cc}-1&{\tfrac {3}{2}}&1&0\\0&{\tfrac {1}{2}}&1&1\end{array}}\right).} Next, subtract row 2, multiplied by 3, from row 1 ( R 1 − 3 R 2 → R 1 ) , {\displaystyle (R_{1}-3\,R_{2}\to R_{1}),} which yields ( − 1 0 − 2 − 3 0 1 2 1 1 ) . {\displaystyle \left({\begin{array}{cc|cc}-1&0&-2&-3\\0&{\tfrac {1}{2}}&1&1\end{array}}\right).} Finally, multiply row 1 by −1 ( − R 1 → R 1 ) {\displaystyle (-R_{1}\to R_{1})} and row 2 by 2 ( 2 R 2 → R 2 ) . {\displaystyle (2\,R_{2}\to R_{2}).} This yields

7120-505: The sequence of leading principal minors (determinants of upper-left-justified sub-matrices) of the bordered Hessian, for which the first 2 m {\displaystyle 2m} leading principal minors are neglected, the smallest minor consisting of the truncated first 2 m + 1 {\displaystyle 2m+1} rows and columns, the next consisting of the truncated first 2 m + 2 {\displaystyle 2m+2} rows and columns, and so on, with

7209-468: The solution: Convergence can be determined by observing the norm of the gradient; given some ϵ > 0 {\displaystyle \epsilon >0} , one may stop the algorithm when | | ∇ f ( x k ) | | ≤ ϵ . {\displaystyle ||\nabla f(\mathbf {x} _{k})||\leq \epsilon .} If B 0 {\displaystyle B_{0}}

7298-407: The term "functional determinants". The Hessian is sometimes denoted by H or, ambiguously, by ∇ . Suppose f : R n → R {\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} } is a function taking as input a vector x ∈ R n {\displaystyle \mathbf {x} \in \mathbb {R} ^{n}} and outputting

7387-500: 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

7476-430: The true Hessian matrix, and the BFGS approximation may not converge to the true Hessian matrix. The BFGS update formula heavily relies on the curvature s k ⊤ y k {\displaystyle \mathbf {s} _{k}^{\top }\mathbf {y} _{k}} being strictly positive and bounded away from zero. This condition is satisfied when we perform a line search with Wolfe conditions on

7565-817: The unconstrained case of m = 0 {\displaystyle m=0} these conditions coincide with the conditions for the unbordered Hessian to be negative definite or positive definite respectively). If f {\displaystyle f} is instead a vector field f : R n → R m , {\displaystyle \mathbf {f} :\mathbb {R} ^{n}\to \mathbb {R} ^{m},} that is, f ( x ) = ( f 1 ( x ) , f 2 ( x ) , … , f m ( x ) ) , {\displaystyle \mathbf {f} (\mathbf {x} )=\left(f_{1}(\mathbf {x} ),f_{2}(\mathbf {x} ),\ldots ,f_{m}(\mathbf {x} )\right),} then

7654-562: The update equation of B k + 1 {\displaystyle B_{k+1}} : Consider the following unconstrained optimization problem minimize x ∈ R n f ( x ) , {\displaystyle {\begin{aligned}{\underset {\mathbf {x} \in \mathbb {R} ^{n}}{\text{minimize}}}\quad &f(\mathbf {x} ),\end{aligned}}} where f : R n → R {\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} }

7743-691: The usual Hessian matrix when m = 1. {\displaystyle m=1.} In the context of several complex variables , the Hessian may be generalized. Suppose f : C n → C , {\displaystyle f\colon \mathbb {C} ^{n}\to \mathbb {C} ,} and write f ( z 1 , … , z n ) . {\displaystyle f\left(z_{1},\ldots ,z_{n}\right).} Identifying C n {\displaystyle {\mathbb {C} }^{n}} with R 2 n {\displaystyle {\mathbb {R} }^{2n}} ,

7832-428: The usual determinant is not defined. The conditions for existence of left-inverse or right-inverse are more complicated, since a notion of rank does not exist over rings. The set of n × n invertible matrices together with the operation of matrix multiplication and entries from ring R form a group , the general linear group of degree n , denoted GL n ( R ) . Let A be a square n -by- n matrix over

7921-400: The zero in the upper-left corner is an m × m {\displaystyle m\times m} block of zeros, and there are m {\displaystyle m} border rows at the top and m {\displaystyle m} border columns at the left. The above rules stating that extrema are characterized (among critical points with a non-singular Hessian) by

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