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In this MSR Technical Report, I came across a definition of the manifold learning problem:

Given a set of $k$ unlabelled observations $\{v_1,v_2,\dots,v_i,\dots,v_k\}$ with $v_i \in \mathbb{R}^d$ we wish to find a smooth mapping $f:\mathbb{R}^d \rightarrow \in \mathbb{R}^{d'}$, $f(v_i) = v_i'$ such that $d' << d$ and that preserves the observations' relative geodesic distances.

I understand the notation $f:\mathbb{R}^d \rightarrow \mathbb{R}^{d'}$ ; what's the significance of $\in$?

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do you have any reason to believe this isn't a typo? – Jonathan Sep 21 '12 at 17:21
@Jonathan: I suspect it was caused when one of the authors wanted to copy-paste $\mathbb{R}^{d'}$ from something like $x \in \mathbb{R}^{d'}$ and accidentally copy-pasted $\in \mathbb{R}^{d'}$. – Rod Carvalho Sep 21 '12 at 17:32
From my familiarity with the subject I can say it is almost certainly a typo and you can ignore it. f simply maps to vectors of a much lower dimension. – Bitwise Sep 21 '12 at 18:35
up vote 3 down vote accepted

Community verdict: there are typos in the box on page 97.

  • $\mathbf v_i\in\mathbf{R}^d$ should read $\mathbf v_i\in \mathbb{R}^d$
  • $\mathbf f:\mathbb{R}^d\to \in\mathbb{R}^{d'}$ should read $\mathbf f:\mathbb{R}^d\to \mathbb{R}^{d'}$.
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