JohnsonLindenstrausslemma
The Johnson-Lindenstrauss lemma is a result in high-dimensional geometry that states that a set of N points in a high-dimensional Euclidean space can be embedded into a lower-dimensional space such that the distances between the points are approximately preserved. Specifically, for any set of N points in R^d and any epsilon > 0, there exists a linear map f: R^d -> R^k into a k-dimensional space, where k = O(epsilon^-2 log N), such that for all pairs of points x, y, (1-epsilon)||x-y||^2 <= ||f(x) - f(y)||^2 <= (1+epsilon)||x-y||^2. The constant factor in the O notation is also related to epsilon.
This lemma is significant because it shows that many of the geometric properties of a high-dimensional dataset
The proof of the Johnson-Lindenstrauss lemma typically involves constructing a random projection matrix. The entries of
The lemma's utility lies in its ability to reduce the computational complexity of algorithms that depend on