Encoding vectors with one unique number

Question

Is it possible to encode a set of elements (in the form of a vector) with some unique number, which can then be converted back to the original vector?

For example, in terms of this encoding, vectors (for example) $v_1=[1,2,3,4,5]$ and $v_2 = [1,2,3,5,4]$ must be characterized by two different numbers, so the Euclidean, Manhattan metric and other norms cannot be applied.

Also, sets include elements of a different order, for example $v_3=[100,10^{-5},1,0.2,-10^{-30}]$

In addition, if the unique number encoding the set of parameters should not be too large, so that a small change in the set can be easily tracked on a plot on a logarithmic scale.

Are there encodings that meet these requirements?

Answer 1

If you just want to convert something to a number, just take some type of encoding of the vector and store it as a string of numbers.

To provide a better answer, I assume that your problem has some geometric considerations. So the question becomes if there exists a function $f: \mathbb{R}^d \to \mathbb{R}$, $d > 1$, where

• $f$ is injective, so different vectors get mapped to different numbers;
• $f$ is "not too large", but by using $g : x \mapsto 1/x$, $\mathbb{R}$ can be viewed as the interval $(-1,1)$ with an attached "infinity". So this shouldn't be an issue.
• The representation is continuous, so that small change in representation reflect small change in represented vector. In this case $f^{-1}$ is continuous.

If we require both $f$ and $f^{-1}$ are continuous, then it is a homeomorphism between $\mathbb{R}$ and $\mathbb{R}^d$, which is impossible.

If we only require $f^{-1}$ to be continuous, we are talking about space-filling curves. For an example, check out Peano curve.

For bounded region, if we consider the Peano curve of a bounded cell, then we can represent a number by the position of it on the curve. This should be computable in principle since the "relative position" on the curve should stabilize, although the curve itself gets infinitely long as we taking finer approximations.

One related idea is to repeatedly partition of the region to $n$ components, that are isomorphic, up to scaling, to the original space. Then you could have a base-$n$ representation of almost every point in the region.

For an example, the Cantor Set generates a tertiary representation of its elements.

For a bounded region in $\mathbb{R}^d$, you could try partitioning the region to $2^d$ components at each step. Then it will give you a base-$2^d$ representation of the region.