Archive for April, 2007
Dimension is not easy to understand. At the turn of the century it was one of the major problems in mathematics to determine what dimension means and which properties it has. And since then the situation has become somewhat worse because mathematicians have come up with some ten different notions of dimension: topological dimension, Hausdorff dimension, fractal dimension, self-similarity dimension, box-counting dimension, capacity dimension, information dimension, Euclidean dimension, and more. They are all related. Some of them, however, make sense in certain situations, but not at all in others, where alternative definitions are more helpful. Sometimes they all make sense and are the same. Sometimes several make sense but do not agree. The details can be confusing even for a research mathematician.
Everybody firmly believes in it because the mathematicians imagine it is a fact of observation, and observers that it is a theory of mathematics.
So, which is it? To apportion the blame, I suggest that the fact of observation be that “errors” are independent and identically distributed with finite variances. Observers may as well vote for Lyapunov’s set of conditions. I am sure that mathematicians are more than happy to shoulder the responsibility for the resulting Central Limit Theorem.
Hello, world. This is the start of a blog for me.
Why start a blog? Many reasons actually. Improving my writing skills. As a communication platform with other friends around the globe. Getting ideas which may or may not be useful out there. A tool for learning. Trying to get more used to basic Web 2.0 (or is it Web 1.1?) concepts before everyone gets to use Web .
I am impressed by the current group of machine learning blogs like Machine Learning (Theory) and Machine Learning Thoughts among others, and I am actually learning a lot from them. However, this will not be a single topic blog, and I will be touching on different dimensions of life. Categories will hopefully reveal some method in the resulting madness.
By the way, dimensions may crop up on any topic raised here, in a mathematical sense or otherwise. You’ve been warned.