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I have to methods of projecting random samples in $\mathbb{R}^n$ onto a manifold defined by $C(q)=0$, which is a lower-dimensional subset. Now, samples in $\mathbb{R}^n$ are uniformly distributed. However, when projecting on to the manifold, the distribution is not guaranteed to be uniform. The two projection methods will provide different distributions.

My question is, what is a good measure of distribution that will enable comparison between the two methods? In simple words, I have two high-dimensional data-sets, how can I compare their distributions?

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Here are some interesting theoretical notions of distance between two such distributions:

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  • $\begingroup$ Total variation will be difficult to use in a discrete setting... $\endgroup$
    – Benoît Kloeckner
    Commented Jan 12, 2018 at 8:34
  • $\begingroup$ @BenoîtKloeckner Do you mean "difficult" in the sense that one can not apply the definition without problem or that the notion is not useful? Isn't the total variation distance here just the dual norm w.r.t the sup-norm? Then the TV distance would be the easy to calculate but maybe not useful… $\endgroup$
    – Dirk
    Commented Jan 12, 2018 at 10:43
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    $\begingroup$ Two discrete measure tend to be at TV distance $1$ (or close to one) regardless of their geometrical feature, so it will be useless. The issue is that TV distance does not take into account proximity at all, it only counts the fraction of mass which coincide exactly between the two measures. $\endgroup$
    – Benoît Kloeckner
    Commented Jan 12, 2018 at 20:16

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