Notes

Chapter 10: Processes of Perception and Analysis

Section 12: Human Thinking


[Memory analogs with] numerical data

In situations where pieces of data can be thought of as points in space similarity can often be defined in terms of spatial distance. And this means that around every point corresponding to a piece of data in memory there is a region of points that can be considered more similar to that point than to any other. Picture (a) shows a so-called Voronoi diagram (see page 1038) obtained in this way in two dimensions. Particularly in higher dimensions, it becomes rather difficult in practice to determine for certain which existing point is closest to some new point. But to do it approximately is considerably easier. One approach, illustrated in picture (b), is to use a d-dimensional tree. Another approach, illustrated in picture (c), is to set up a continuous function with minima at the existing points, and then to search for the closest minimum. In most cases, this search will be done using some iterative scheme such as Newton's method; the result is that the boundaries between regions typically take on an intricate nested form. (The case shown corresponds to iteration of the map z->z-(z^3-1)/(3z^2) corresponding to Newton's method for finding the complex roots of z^3==1.)

The pictures below show how one can build up a kind of memory landscape by successively adding points. In a first approximation, the regions considered similar to a particular minimum are delimited by sharp watersheds corresponding to local maxima in the landscape. But if an iterative scheme for minimization is used, these watersheds are typically no longer sharp, but take on a local nested structure, much as in picture (c) above.

In numbers earlier digits are traditionally considered more important than later ones, and this allows numbers to be arranged in a simple one-dimensional sequence. But in strings where each element is considered equally important, no such layout is possible. A vague approximation, perhaps useful for some applications, is nevertheless to use a space-filling curve (see page 893).


From Stephen Wolfram: A New Kind of Science [citation]