Holonomic Gradient Descent and its Application to Fisher-Bingham Integral

The gradient descent is a general method to find a local minimum of a smooth function f(z1, . . . , zd). The method utilizes the observation that f(p) decreases if one goes from a point z = p to a “nice” direction, which is usually −(∇f)(p). As textbooks on optimizations present (see, e.g., [5], [16]), we have a lot of achievements on this method and its variations. We suggest a new variation of the gradient descent, which works for real valued holonomic functions f(z1, . . . , zd) and is a d-variable generalization of Euler’s method for solving ordinary differential equations numerically and finding a local minimum of the function. We show an application of our method to directional statistics. In fact, it is our motivating problem to develop the new method. A function f is called a holonomic function, roughly speaking, if f satisfies a system of linear differential equations