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

[ShNaYkHs]

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

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

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

 

[mmwave]

I would suggest considering the following conditions for a new set C_2 to satisfy in order to minimize the value of \Phi(A_2) and ensure \Phi(A_2) \leq \Phi(A):

1. The subset C_2 should still contain k nearest points to each point in A_2, as removing a point from A may change the k nearest points for other points in A_2.

2. The subset C_2 should also maintain a similar distribution of points as C in terms of their distances from each other and from the points in A_2. This means that the average distance from each point in A_2 to its k nearest points in C_2 should be similar to the average distance from each point in A_2 to its k nearest points in C.

3. The subset C_2 should minimize the overall distance between the points in A_2 and C_2. This can be achieved by selecting points in C_2 that are closer to the points in A_2, without significantly changing the distribution of points in C_2.

4. Additionally, the subset C_2 should be chosen randomly from A_2, as this will ensure that the selection process is unbiased and does not favor certain points over others.

By following these conditions, it is possible to choose a subset C_2 from A_2 that satisfies \Phi(A_2) \leq \Phi(A). However, it is important to note that this may not always be possible, as the value of \Phi(A) may depend on the specific distribution and arrangement of points in A. In such cases, it may be necessary to modify the conditions or consider alternative methods for selecting a subset C_2 that minimizes \Phi(A_2).