As stated several times, all defining aspects of information vectors can be represented mathematically. Therefore all similar defining aspects of information vectors can be evaluated and compared mathematically. Some of these comparison strategies are evident and some are more complex. In this section we discuss images and sounds that may provide more challenging as far as mathematical evaluation and comparison.
Imagine a computer program that has to determine which 2″x4″ lumber pieces are longer than one another. I would imagine that a program like that would not have to be excessively complicated because the objects in question (stud lumber) are pretty simple. Now imagine a computer program that has to process 1000 faces and indicate which most closely resemble each other. It seems that program would be considerably more complicated. But the fundamental mechanics for both algorithms are similar.
This document presents a recurring theme of the mathematical basis in all aspects of intelligence. The outline of any two-dimensional or three-dimensional shape can be represented by a series of mathematical formulas, some more complicated than others. Some simple examples to make the point more tangible:
- A straight line can be represented by the equation: y = m x + b where m represents the slope of the line and b represents the point at which the line crosses the y axis
- A circle centered on the origin of a Cartesian XY grid can be represented by the equation: r(2) = x(2) + y(2) where r is the radius of the circle
- Variations of the circle equation produce ellipses
- Constraining these equations to different x and y ranges will produce line segments and arcs
Referring back to the program that determined longer lumber pieces, if both pieces were lined up in parallel at a certain “x” coordinate, the longer piece would terminate at an “x” coordinate that is greater than that of the shorter piece
Simple polygons are sets of line segments with different defining equations attached to one another. For example, a square could be defined by these four equations:
- y = 2; for x = 0 to 2
- x = 0; for y = 0 to 2
- y = 0; for x = 0 to 2
- x = 2; for y = 0 to 2
If I wanted to define a square twice as high and wide (four times the area) as the one defined above, I could replace every 2 in the equations with a 4. In this case, it’s pretty simple to mathematically represent and compare the two information vectors.
I will demonstrate one more example that involves a slightly more abstract mathematical practice. Here is an image you have probably never seen before:
Here is another image you have probably never seen before:
How would you compare those two images in relationship to one another? You most likely recognized that the second image is a rotated representation of the first. You probably didn’t have to rotate the page to figure that out. Two-dimensional and three-dimensional rotation of images is actually performed digitally using linear algebra matrix multiplications. This is how many computer games handle the changing of visual perspective of an object as the context rotates. Without going into more detail (http://en.wikipedia.org/wiki/Rotation_matrix), the point is that your brain does some version of this matrix multiplication in order to compare images with their rotated variations.
It would be interesting to learn more about similar studies done with animals: training a mouse to know that a button labeled with a certain irregular shape will deliver food. Then place that mouse in a new cage in which that irregular shape is rotated by some significant amount. Will it recognize the shape (do the rotational math in its mind) and push the button?
I am going to revisit this topic of information vector comparison after I discuss the idea of concepts.