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Selecting a subset from a set such that a given quantity is minimized

  1. Nov 29, 2013 #1
    Let [itex]A[/itex] be is a set of some [itex]p[/itex]-dimensional points [itex]x \in \mathbb{R}^p[/itex]. Let [itex]d_x^A[/itex] denote the mean Euclidean distance from the point [itex]x[/itex] to its [itex]k[/itex] nearest points in [itex]A[/itex] (others than [itex]x[/itex]). Let [itex]C \subset A[/itex] be a subset of points chosen randomly from [itex]A[/itex]. We have [itex]\Phi(A) = \sum_{x \in A} d_x^C[/itex].

    Now suppose that I remove a point [itex]c'[/itex] from [itex]A[/itex], I get a new set [itex]A_2 = A \setminus \{x'\}[/itex].

    **Question:**
    Which condition should a new set [itex]C_2 \subset A_2[/itex] satisfies, in order to have [itex]\Phi(A_2) = \sum_{x \in A_2} d_x^{C_2} \leq \Phi(A)[/itex] ? In other words, how can I choose a subset [itex]C_2[/itex] from [itex]A_2[/itex] such that [itex]\Phi(A_2) \leq \Phi(A)[/itex] ?
     
  2. jcsd
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