Percentage of smaller dataset with respect to bigger dataset

Question

I have two datasets, which are lists of multidimensional real-valued vectors. One dataset (call it $A=\{x_1, x_2, x_3, ..., x_n\}$ is of a big size, the other (call it $B=\{x_1, x_2, x_3, ..., x_m\}$). Furthermore, the other is far smaller and is a subset of the bigger one ($B \subset A$). The smaller one $B$, comes from some sampling process and what I want to do is to calculate, what fraction of the smaller (obtained from sampling) is in the bigger. Additionally, since those are real-valued vectors, I can't compare them directly one by one, so a clustering algorithm may be employed. Also the size of one dataset is bigger than the other $|A| >> |B|$.

Answer 1

Naive approach: define a similarity or distance function, say for instance cosine similarity.

1. Calculate the similarity score between any pair $(x_i\in A, y_j\in B)$
2. Define a precision level, say $\epsilon=0.000001$. The assumption is that it's extremely unlikely that two vectors would be this close by chance in $A$.
3. For every $y_j\in B$, find the set $c(y_j) = \{ x_i\in A\ |\ sim(x_i,y_j)\geq 1-\epsilon \}$
4. Obtain the union: $C(B)=\{x_i\in A\ |\ \exists y_j\in B: x_i\in c(y_j) \}$

The proportion of elements of $A$ which are "equal" to an element in $B$ is:

$$\frac{|C(B)|}{|A|}$$