Descent vector

Starting at an SP along the descent vector (direction of negative... 

As a consequence of proposition 3 we obtain In the vicinity of a minimizer of E the Newton vectors and the steepest descent vectors always point to the minimizer. In the vicinity of a saddle point the Newton vectors always point to the saddle point whereas a steepest descent vector point to the saddle point only if E is convex along that vector. This observation forms the basis for a modified Newton-like method which looks for stationary points of prescribed type (see Sect.2.4.3). 

If the minimizer p of each function is equal to the steepest descent vector we obtain the steepest descent method,... 

Thus, the angle (p between the steepest descent vector and the... 

A vector p satisfying the condition (48) is called a descent vector at X. By the above lemma, each vector pew forming an angle of less than 90 with the steepest descent vector (negative gradient) is a descent vector. So the steepest descent vector (p = -g(x)) can be regarded as the limit case. 

A vector p lR is a descent vector at x if and only if there is a positive definite matrix M such that... 

Steepest Descent is the simplest method of optimization. The direction of steepest descen t. g, is just th e n egative of the gradien t vector ... 

Vector quantities, such as a magnetic field or the gradient of electron density, can be plotted as a series of arrows. Another technique is to create an animation showing how the path is followed by a hypothetical test particle. A third technique is to show flow lines, which are the path of steepest descent starting from one point. The flow lines from the bond critical points are used to partition regions of the molecule in the AIM population analysis scheme. 

The gradient can be used to optimize the weight vector according to the method of steepest-descent ... 

We mentioned above that a typical problem for a Boltzman Machine is to obtain a set of weights such that the states of the visible neurons take on some desired probability distribution. For example, the task may he to teach the net to learn that the first component of an Ai-component input vector has value +1 40% of the time. To accompli.sh this, a Boltzman Machine uses the familiar gradient-descent technique, but not on the energy of the net instead, it maximizes the relative entropy of the system. 

The choice ya = ra is the method of steepest descent. If the ya are taken to be the vectors et in rotation the method turns out to be the Gauss-Seidel iteration. If each ya is taken to be that e, for which e ra is greatest, the method is the method of relaxation (often attributed to Southwell but actually known to Gauss). An alternative choice is the et for which the reduction Eq. (2-10) in norm is greatest. 

It is in fact not advisable always to take the steepest descent, rather the search should be performed along directions which are conjugate to each other. Two unit vectors gj and 2 are conjugate with respect to a matrix A if they satisfy... 

The steepest-descent method does converge towards the expected solution but convergence is slow in the vicinity of the minimum. In order to scale variations, we can use a second-order method. The most straightforward method consists in applying the Newton-Raphson scheme to the gradient vector of the function/to be minimized. Since the gradient is zero at the minimum we can use the updating scheme... 

The pseudo-inverse for the calculation of the shift vector in equation (4.67) has been computed traditionally as J+= (J Jp1 J. Adding a certain number, the Marquardt parameter mp, to the diagonal elements of the square matrix J J prior to its inversion, has two consequences (a) it shortens the shift vector 8p and (b) it turns its direction towards steepest descent. The larger the Marquardt parameter, the larger is the effect. In matrix formulation, we can write ... 

Xi2( ) The 100(1 -ac)% point of the chi-squared distribution with 1 degree of freedom iji A parameter setting the relative contributions of the linearization and steepest descent methods in determining the correction vector b of Eq. (45)... 

All line searches start by defining a descent direction. Consider sill vectors z which fulfill the condition... 

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