# A classifier based on the Mandelbrot set

I am trying to link these somewhat disparate concepts- a binary classifier and a fractal structure as I think that a binary classifier based on such a structure, in particular Mandelbrot set would be considerably more accurate than others- of course random forest classifier is bound to give tough competition for that.

We all are familiar with Mandelbrot set. The fractal boundary separates the complex numbers which remain bounded after a sizable number of iterations of the map

$z_{n+1}= (z_n)^2+c$, where $c$ is a complex constant and $z_{n+1}$, $z_n$ are the values of the complex number at $(n+1)$th and $n$th iterations respectively

and the ones that rush to infinity or "die".

Problem before us is the standard classification problem in machine learning:

Consider a data set where $X_1,X_2,\cdots X_n$ are independent variables-quantitative numerical variables, all of them. And $Y$ be the dependent variable-binary, takes values $(0,1)$.

We want to classify the observations, i.e. we wish to know under what conditions on the values of $X_1,X_2,\cdots X_n$ is the value of $Y$ either $0$ or $1$?

This problem is solved in many ways. One of them is to construct kernel functions as done in SVM. The idea is to link somehow the kernel function with mandelbrot set.

Why mandelbrot set?? because its boundary is fractal, and it has

1. Has there been such a classifier constructed before?

2. Any comments about its accuracy/feasibility etc.? I have not visualized this function in detail, but I do think that such a classifier would have some decisive advantages over learning machines that use commonlu used kernel functions like linear, polynomial, etc.

ADDED: Henning Makholm provided valuable inputs as to the vagueness of my question.

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Your description of the Mandelbrot set is not correct. It sounds like the description of the Julia set for $c=0$, i.e., the circle. The Mandelbrot set is the set of all $c$ such that the iteration $z_{n+1}= z_n^2+c$ does not go to infinity when started with $z_0=0$. –  lhf Dec 13 '11 at 11:59
The core of your question about classifiers does not make any sense to me. What are you trying to classify? Why do you think the Mandelbrot set has anything to do with this task? –  lhf Dec 13 '11 at 12:02
@ lhf: ok, point taken. but My question is about classifiers like support vectors that learn from data and classify future data points accordingly. Very often, the accuracy of such classifiers depends upon the kernel functions chosen. With this background, if we take a look at the mandelbrot set, we see (as told by you) that The Mandelbrot set is the set of all c such that the iteration (z^2)_n+1=(z^2)_n+ c does not go to infinity when started with z0=0. My feeble hunch is that this fractal boundary property can somehow be utilized towards constructing a more "agile" kernel function. –  NikBels Dec 13 '11 at 13:02
Are you aware of the phenomenon of overfitting? –  Rahul Jan 7 '12 at 20:28
hmm okk. kind of understood what yo'r hinting at. thanks. –  NikBels Jan 7 '12 at 20:36