高斯牛頓法和Levengerg-Marquardt迭代都用來解決非線性最小二乘問題(nonlinear least square)。

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The?Gauss–Newton algorithm?is a method used to solve?non-linear least squares?problems. It is a modification of?Newton's method?for finding a?minimum?of a?function. Unlike Newton's method, the Gauss–Newton algorithm can only be used to minimize a sum of squared function values, but it has the advantage that second derivatives, which can be challenging to compute, are not required.

Description

Given?m?functions?r?= (r₁, …,?r_m) of?n?variables?β?=?(β₁, …,?β_n), with?m?≥?n, the Gauss–Newton algorithm iteratively?finds the minimum of the sum of squares

Starting with an initial guess??for the minimum, the method proceeds by the iterations

where, if?r?and?β?are?column vectors, the entries of the?Jacobian matrix?are

and the symbol??denotes the?matrix transpose.

If?m?=?n, the iteration simplifies to

which is a direct generalization of?Newton's method?in one dimension.

In data fitting, where the goal is to find the parameters?β?such that a given model function?y?=?f(x,?β) best fits some data points (x_i,?y_i), the functions?r_iare the?residuals

Then, the Gauss-Newton method can be expressed in terms of the Jacobian?J_f?of the function?f?as

Notes

The assumption?m?≥?n?in the algorithm statement is necessary, as otherwise the matrix?J_r^TJ_r?is not invertible (rank(J_r^TJ_r)=rank(J_r))and the normal equations (Δ in the "derivation from Newton's method" part)?cannot be solved (at least uniquely).

The Gauss–Newton algorithm can be derived by?linearly approximating?the vector of functions?r_i. Using?Taylor's theorem, we can write at every iteration:

with??The task of finding Δ minimizing the sum of squares of the right-hand side, i.e.,

,

is a?linear least squares?problem, which can be solved explicitly, yielding the normal equations in the algorithm.

The normal equations are?m?linear simultaneous equations in the unknown increments, Δ. They may be solved in one step, using?Cholesky decomposition, or, better, the?QR factorization?of?J_r. For large systems, an?iterative method, such as the?conjugate gradient?method, may be more efficient. If there is a linear dependence between columns of?J_r, the iterations will fail as?J_r^TJ_r?becomes singular.

Derivation from Newton's method

In what follows, the Gauss–Newton algorithm will be derived from?Newton's method?for function optimization via an approximation. As a consequence, the rate of convergence of the Gauss–Newton algorithm can be quadratic under certain regularity conditions. In general (under weaker conditions), the convergence rate is linear.

The recurrence relation for Newton's method for minimizing a function?S?of parameters,?, is

where?g?denotes the?gradient vector?of?S?and?H?denotes the?Hessian matrix?of?S. Since?, the gradient is given by

Elements of the Hessian are calculated by differentiating the gradient elements,?, with respect to?

The Gauss–Newton method is obtained by ignoring the second-order derivative terms (the second term in this expression). That is, the Hessian is approximated by

where??are entries of the Jacobian?J_r. The gradient and the approximate Hessian can be written in matrix notation as

These expressions are substituted into the recurrence relation above to obtain the operational equations

Convergence of the Gauss–Newton method is not guaranteed in all instances. The approximation

that needs to hold to be able to ignore the second-order derivative terms may be valid in two cases, for which convergence is to be expected.

The function values??are small in magnitude, at least around the minimum.

The functions are only "mildly" non linear, so that??is relatively small in magnitude.

Improved version

With the Gauss–Newton method the sum of squares?S?may not decrease at every iteration. However, since Δ is a descent direction, unless??is a stationary point, it holds that??for all sufficiently small?. Thus, if divergence occurs, one solution is to employ a fraction,?, of the increment vector, Δ in the updating formula

.

In other words, the increment vector is too long, but it points in "downhill", so going just a part of the way will decrease the objective function?S. An optimal value for??can be found by using a?line search?algorithm, that is, the magnitude of??is determined by finding the value that minimizes?S, usually using a?direct search method?in the interval?.

In cases where the direction of the shift vector is such that the optimal fraction,?, is close to zero, an alternative method for handling divergence is the use of the?Levenberg–Marquardt algorithm, also known as the "trust region?method".^[1]?The normal equations are modified in such a way that the increment vector is rotated towards the direction of?steepest descent,

,

where?D?is a positive diagonal matrix. Note that when?D?is the identity matrix and?, then?, therefore the?direction?of Δ approaches the direction of the negative gradient?.

The so-called Marquardt parameter,?, may also be optimized by a line search, but this is inefficient as the shift vector must be re-calculated every time??is changed. A more efficient strategy is this. When divergence occurs increase the Marquardt parameter until there is a decrease in S. Then, retain the value from one iteration to the next, but decrease it if possible until a cut-off value is reached when the Marquardt parameter can be set to zero; the minimization of?S?then becomes a standard Gauss–Newton minimization.

The (non-negative) damping factor, , is adjusted at each iteration. If reduction of?S?is rapid, a smaller value can be used, bringing the algorithm closer to the?Gauss–Newton algorithm, whereas if an iteration gives insufficient reduction in the residual, can be increased, giving a step closer to the gradient descent direction. Note that the?gradient?of?S?with respect to?δ?equals?. Therefore, for large values of?, the step will be taken approximately in the direction of the gradient. If either the length of the calculated step,?δ, or the reduction of sum of squares from the latest parameter vector,?β?+?δ, fall below predefined limits, iteration stops and the last parameter vector,?β, is considered to be the solution.

Levenberg's algorithm has the disadvantage that if the value of damping factor, , is large, inverting?J^TJ?+?I?is not used at all. Marquardt provided the insight that we can scale each component of the gradient according to the curvature so that there is larger movement along the directions where the gradient is smaller. This avoids slow convergence in the direction of small gradient. Therefore, Marquardt replaced the identity matrix,?I, with the diagonal matrix consisting of the diagonal elements of?J^TJ, resulting in the Levenberg–Marquardt algorithm:

.

Related algorithms

In a?quasi-Newton method, such as that due to?Davidon, Fletcher and Powell?or Broyden–Fletcher–Goldfarb–Shanno (BFGS method) an estimate of the full Hessian,, is built up numerically using first derivatives??only so that after?n?refinement cycles the method closely approximates to Newton's method in performance. Note that quasi-Newton methods can minimize general real-valued functions, whereas Gauss-Newton, Levenberg-Marquardt, etc. fits only to nonlinear least-squares problems.

Another method for solving minimization problems using only first derivatives is?gradient descent. However, this method does not take into account the second derivatives even approximately. Consequently, it is highly inefficient for many functions, especially if the parameters have strong interactions.

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