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Having a set of points $X\in\mathbb{R}^{n\times d}$ and a similarity matrix $S\in \mathbb{R}^{n\times m}$, I am interested in an approximation of the second set of points $U\in \mathbb{R}^{m\times d}$.

$S$ is constructed using some known measurement between the sets $X$ and $U$, e.g. Gaussian RBF or an Euclidean Distance:

$$Z_{ik} = \exp\left(-\frac{||X_i - U_k||^2}{2\sigma^2}\right)$$

Is the task of finding the set $U$ achievable using e.g. Gradient Descent method?

kjetil b halvorsen
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zunder
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  • Did you try SVD and then do out-of-sample extension to get your $U$ points mapped in that latent space? – Vladislavs Dovgalecs Dec 06 '16 at 19:30
  • @xeon I didn't try this. Could you elaborate a bit more please? I guess I need to apply SVD on $S$, but how exactly should I do the out-of-sample extension then? Is there some useful paper on that topic? – zunder Dec 07 '16 at 08:50
  • related, not a dupe: https://stats.stackexchange.com/questions/345152/does-mercers-theorem-work-in-reverse – Sycorax Aug 13 '18 at 18:26

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