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Imagine two spaces:

  • An ‘input’ space with dimension $m$.
  • An ‘output’ space with dimension $n$.
  • $m \geq n$

There are points in each of these spaces defined such that some characteristic is defined. The characteristic is defined and valid in both spaces.

An example* might use RGB (Red, Green, Blue) values for the input and HSV (Hue, Saturation, Value) for the output.

(*the actual solution needs to generalize to accomodate arbitrary dimensions)

enter image description here

The characteristic of ‘greeny-ness’ is defined in the input space as the vector: $[0,255,0]$, and in the output space as: $[120,100,100]$.

‘red’, ‘black’, ‘yellow’ and ‘random colour that looks the same in both spaces’ could be similarly defined.

Imagine now that a limited subset of colors has been defined in this way – i.e. there are $p$-pairs of $m$-dimensioned vectors coupled with their corresponding $n$-dimensioned vectors ($m=n=3$ in this case).

The problem:

Given an arbitrary input vector, find (interpolate) the corresponding point in the output space that most exemplifies the 'characteristic' of that point (in the input space).

Using the color example, I might have all 8 corners of an RGB color cube defined as points on the input side – and their corresponding HSV values coupled with them as follows:

$$[0,0,0] \longleftrightarrow[0,0,0]$$$$ [255,0,0] \longleftrightarrow [0,100,100]$$$$ [0,255,0] \longleftrightarrow [120,100,100]$$$$ [255,255,0] \longleftrightarrow [60,100,100]$$$$ [0,0,255] \longleftrightarrow [240,100,100]$$$$ [255,0,255] \longleftrightarrow [300,100,100]$$$$ [0,255,255] \longleftrightarrow [180,100,100]$$$$ [255,255,255] \longleftrightarrow [0,0,100]$$

Given $[128,128,128]$ (‘grey’) as the input point in the input space I’d expect to be able to find [0,0,50] (‘grey’ in HSV) in the output space.

I know that I know that $[128,128,128]$ is right in the middle of the RGB cube with the Euclidean distances to all the 8 points being $ \frac{\sqrt{3}}{2} \times 256$. It's also worth noting that while each RGB values range over 8-bits (256 each ), the HSV values range over 360°, 100 & 100 respectively...

Yes, there are known RGB$\rightarrow $HSV routines - I just use this example as it is easy to visualize - but in the real application the dimensionality would more like 70 input parameters ($m=70$), mapping to 20 output parameters ($n=20$) and possibly up to 50 coupled points defined ($p=50$).

So far I’ve tried:

  • Using inverted Euclidean (or manhattan) norms found on the input side to inform weighted interpolations on the output.

  • Euclidean norms building simultaneous equations (‘hyper-spheres'!) that are solved using non-linear least squares (trilateration in higher dimensions and with over-fitting)

  • Using PCA dimensionality reduction on m to ensure $m=n$

Each of these has had practical success of sorts (especially if the p-coupled pairs are consistent within their spaces, and the more the better).

But there are always examples where the solution falls apart: eg. with $m=n=2$ and $p=4$ and the coupled vectors:

$$[0,0]_{\mathbf{i}_1} \longleftrightarrow[-10,10]_{\mathbf{o}_1}$$$$ [100,0]_{\mathbf{i}_2} \longleftrightarrow [10,-10]_{\mathbf{o}_2}$$$$ [0,100]_{\mathbf{i}_3} \longleftrightarrow [-10,140]_{\mathbf{o}_3}$$$$ [100,100]_{\mathbf{i}_4} \longleftrightarrow [110,110]_{\mathbf{o}_4}$$

(Note this is not an RGB>HSV example)

With least squares the solution (the black dot on the right-hand plot) to the input point: $[10,5]$ should be closer to the pre-defined point 'o1':

enter image description here

(Note how close the input (black point on the left) is to 'i1')

My ad-hoc patches simply lead me to chase my tail - so...

My questions:

While I’m aware that the nature of interpolation excludes precision, I ask:

  1. What approach would be the closest I can get to a solution that generalizes for all dimensions, and inputs ? (within and outside the convex hulls defined in the $p$ coupled points)

  2. Is there some other function of the input point with respect to the points pre-defined in the input space that I can glean information that would allow a more direct solution?

  3. Is a direct and analytical approach even possible, or will I have to rely on measures of success via machine learning methods?

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  • $\begingroup$ As far as I know, color spaces are no vector spaces. $\endgroup$ Commented Feb 5, 2016 at 8:17
  • $\begingroup$ @BjörnFriedrich. Keep in mind the colour spaces are just an example application. As for the tag I'm not 100% sure exactly what a vector-space is myself, but after reading the tag description I figured that people interested in them may have an interest in this question (?). I am keen to remove bad tags though, especially if there is a better tag to put in it's place - I'm open to suggestions ! :) $\endgroup$ Commented Feb 5, 2016 at 9:41
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    $\begingroup$ Have you looked at multidimensional scaling (MDS)? $\endgroup$ Commented Feb 5, 2016 at 22:16
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    $\begingroup$ for general non-linear function fitting $f : [0;1]^n \to [0;1]^m$ from $P$ pairs $(x_i,f(x_i))$ I'd say that there are numerous methods, all related, all converging to the function when $\max ||x_i - x_j|| \to 0$ : en.wikipedia.org/wiki/K-nearest_neighbors_algorithm which is the simplest, en.wikipedia.org/wiki/Kernel_density_estimation and its generalizations, en.wikipedia.org/wiki/Artificial_neural_network ... $\endgroup$
    – reuns
    Commented Feb 7, 2016 at 7:23
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    $\begingroup$ @Lamar Latrell : so you didn't understand kernel density estimation well, it interpolates from the neighbors by giving more confidence to the nearest. try with $K(x-x_i) = e^{-||x-x_i||^2}$. and nested kernel density estimation leads to neural networks : the most general framework for non-linear function approximation $\endgroup$
    – reuns
    Commented Feb 8, 2016 at 0:51

1 Answer 1

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If, no one else answers the bounty (?) I'll go with my own lead:

https://en.wikipedia.org/wiki/Radial_basis_function_network

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