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Assume I have a data set $ { \left\{ {x}_{i} \right\} }_{i = 1}^{N} $ which represents the value of each data point.
For each data we have its features $ {f}_{i} \in {\mathbb{R}}^{d} $.

The model I came for the data is $ \arg \min_x {\left( x - \sum_{j \neq i} {w}_{ij} {x}_{j} \right)}^{2} $, where $ {w}_{i, j} = \exp \left( -\frac{1}{2} {\left( {f}_{i} - {f}_{j} \right)}^{T} C \left( {f}_{i} - {f}_{j} \right) \right) $.
This model is idealization of the data, namely ideally the data would have obeyed this law yet it doesn't due to perturbations in data.

Yet the model is correct only if the the correct Mahalanobis Distance Matrix $ C $ is chosen (We don't need its inverse, so lets stick with $ C $).

So if I have a set of data and a set of features of the data, how can I find the matrix $ C $ which minimizes the model?

Formally, what I'm looking for given the data set $ { \left\{ \left( {x}_{i}, {f}_{i} \right) \right\}}_{i = 1}^{N} $:

$$ \arg \min_{C} {\sum_{i = 1}^{N} \left( {x}_{i} - \sum_{j \neq i} exp \left( -\frac{1}{2} {\left( {f}_{i} - {f}_{j} \right)}^{T} C \left( {f}_{i} - {f}_{j} \right) \right) {x}_{j} \right)}^{2} $$

Question: Is it any other defined problem type I can find and read about?
How would you solve it?

I'm using MATLAB.

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The problem is not convex, as one can verify using the simplest possible choice: $N=2, d=1$ and with data $x_1=x_2=1, f_1=1, f_2=0$. In that case, your objective function is $$ F(C) = 2\left(1-\exp(-C/2)\right)^2. $$ Plotting this function shows that it is not convex.

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  • $\begingroup$ This answer is simply incorrect. It is indeed convex but you need to define your range for $C$. Clearly, this function has a global minimum point as $C = 0$ and $F(0) = 0$ and in the range of $[-1,1]$, it is convex. See it here: wolframalpha.com/input/?i=plot+2*%281-exp%28-x%2F2%29%29%5E2+from+-1+to+1. Sorry for awful link, you need to copy it and paste it in your browser. $\endgroup$ – Alone Programmer Jan 12 at 18:23
  • $\begingroup$ @AloneProgrammer: Maybe so, but that restriction is not stated in the original problem. Furthermore, $C$ is supposed to represent a Mahalanobis distance for which I also can't see why it should be restricted to an interval with upper bound 1. Finally, what would be the generalization of your restriction to an interval for the original problem, in which $C$ is a matrix? $\endgroup$ – Wolfgang Bangerth Jan 13 at 4:51
  • $\begingroup$ I don't have any insist on $[-1,1]$ interval, I'm just saying this objective function might have a global minimum and probably in a really small range around that global minimum, this objective function might be convex, but I didn't implement it yet with for example CVXPY. $\endgroup$ – Alone Programmer Jan 13 at 13:25
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    $\begingroup$ @AloneProgrammer: Well, yes, every smooth optimization problem is convex around non-degenerate minima :-) That's the second order sufficient condition. $\endgroup$ – Wolfgang Bangerth Jan 13 at 17:03

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