Distance between finite sets of points

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Denote the collection of all finite subsets in $mathbbR^d$ as $mathcalS = S subseteq mathbbR^d: $. What are ways to define distance metrics on $mathcalS$ that can be efficiently computed? For instance, one could define
$$
d(A, B) = frac1 sum_xin Asum_yin B |x - y|_2^2
$$
but I'm not sure if the triangle inequality is satisfied?







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  • Could you be precise about what you mean by a distance metric?
    – Matt
    Aug 9 at 21:54










  • @Matt: I mean a metric that satisfies the requirements: en.wikipedia.org/wiki/Metric_(mathematics)
    – p-value
    Aug 9 at 21:59










  • Could you use the discrete metric?
    – Matt
    Aug 9 at 22:02










  • @Matt: What to is your definition of the discrete metric?
    – p-value
    Aug 9 at 22:03










  • If $x = y$ then $d(x,y) = 0$. Otherwise, $d(x,y) = 1$.
    – Matt
    Aug 9 at 22:04














up vote
1
down vote

favorite












Denote the collection of all finite subsets in $mathbbR^d$ as $mathcalS = S subseteq mathbbR^d: $. What are ways to define distance metrics on $mathcalS$ that can be efficiently computed? For instance, one could define
$$
d(A, B) = frac1 sum_xin Asum_yin B |x - y|_2^2
$$
but I'm not sure if the triangle inequality is satisfied?







share|cite|improve this question




















  • Could you be precise about what you mean by a distance metric?
    – Matt
    Aug 9 at 21:54










  • @Matt: I mean a metric that satisfies the requirements: en.wikipedia.org/wiki/Metric_(mathematics)
    – p-value
    Aug 9 at 21:59










  • Could you use the discrete metric?
    – Matt
    Aug 9 at 22:02










  • @Matt: What to is your definition of the discrete metric?
    – p-value
    Aug 9 at 22:03










  • If $x = y$ then $d(x,y) = 0$. Otherwise, $d(x,y) = 1$.
    – Matt
    Aug 9 at 22:04












up vote
1
down vote

favorite









up vote
1
down vote

favorite











Denote the collection of all finite subsets in $mathbbR^d$ as $mathcalS = S subseteq mathbbR^d: $. What are ways to define distance metrics on $mathcalS$ that can be efficiently computed? For instance, one could define
$$
d(A, B) = frac1 sum_xin Asum_yin B |x - y|_2^2
$$
but I'm not sure if the triangle inequality is satisfied?







share|cite|improve this question












Denote the collection of all finite subsets in $mathbbR^d$ as $mathcalS = S subseteq mathbbR^d: $. What are ways to define distance metrics on $mathcalS$ that can be efficiently computed? For instance, one could define
$$
d(A, B) = frac1 sum_xin Asum_yin B |x - y|_2^2
$$
but I'm not sure if the triangle inequality is satisfied?









share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked Aug 9 at 21:47









p-value

19910




19910











  • Could you be precise about what you mean by a distance metric?
    – Matt
    Aug 9 at 21:54










  • @Matt: I mean a metric that satisfies the requirements: en.wikipedia.org/wiki/Metric_(mathematics)
    – p-value
    Aug 9 at 21:59










  • Could you use the discrete metric?
    – Matt
    Aug 9 at 22:02










  • @Matt: What to is your definition of the discrete metric?
    – p-value
    Aug 9 at 22:03










  • If $x = y$ then $d(x,y) = 0$. Otherwise, $d(x,y) = 1$.
    – Matt
    Aug 9 at 22:04
















  • Could you be precise about what you mean by a distance metric?
    – Matt
    Aug 9 at 21:54










  • @Matt: I mean a metric that satisfies the requirements: en.wikipedia.org/wiki/Metric_(mathematics)
    – p-value
    Aug 9 at 21:59










  • Could you use the discrete metric?
    – Matt
    Aug 9 at 22:02










  • @Matt: What to is your definition of the discrete metric?
    – p-value
    Aug 9 at 22:03










  • If $x = y$ then $d(x,y) = 0$. Otherwise, $d(x,y) = 1$.
    – Matt
    Aug 9 at 22:04















Could you be precise about what you mean by a distance metric?
– Matt
Aug 9 at 21:54




Could you be precise about what you mean by a distance metric?
– Matt
Aug 9 at 21:54












@Matt: I mean a metric that satisfies the requirements: en.wikipedia.org/wiki/Metric_(mathematics)
– p-value
Aug 9 at 21:59




@Matt: I mean a metric that satisfies the requirements: en.wikipedia.org/wiki/Metric_(mathematics)
– p-value
Aug 9 at 21:59












Could you use the discrete metric?
– Matt
Aug 9 at 22:02




Could you use the discrete metric?
– Matt
Aug 9 at 22:02












@Matt: What to is your definition of the discrete metric?
– p-value
Aug 9 at 22:03




@Matt: What to is your definition of the discrete metric?
– p-value
Aug 9 at 22:03












If $x = y$ then $d(x,y) = 0$. Otherwise, $d(x,y) = 1$.
– Matt
Aug 9 at 22:04




If $x = y$ then $d(x,y) = 0$. Otherwise, $d(x,y) = 1$.
– Matt
Aug 9 at 22:04















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