I have a theory that in order to best compress 1D data between 0 and 1 in a lossy way, you should map each data point $i$ to:
$\int_0^i \! P(x) \, \mathrm{d}x$
Where $P(x)$ is the probability density function of the data. The probability density function over the mapped data should be level and being level when you store the values to a finite precision, the amount of redundant information should be minimized.
My question is: how can I generalize this to more dimensions? Say I have a set of points which fit a given distribution and I want to compress points that fit the same distribution.
My initial assessment is that I should incrementally move the points so that they follow a level probability density function, then (with lots of arm waving) it should be possible to traverse the mapping using the proximity to the reference points.
A further constraint is that given data that fits something that looks like a spiral, rather than squashing it together, it should unwind it.
Can anyone shed any light on this?
- I assume that I'm not the first person to talk about such a concept. What is its name?
- Is this a good approach?
- How do you incrementally move the points so that they fit a level probability density function?
- How do you traverse the mapping?
- How do I modify the above approach to fit the added constraint?
edit: To clarify, essentially what I want to do is take a set of points, then warp the space the points are in so that the points are equally spaced.