I'm aware of this question https://stackoverflow.com/questions/4350215/fastest-nearest-neighbor-algorithm But it's not the same question as I'm asking. Because, Octree and its generalization are only applicable to very small $D$, and they increase exponentially with regard to it. But I'm interested in very high dimensional cases, like $D>1000$. Assume we have a cloud of $N$ points in a $D$ dimensional space. But they might have a lower inherent dimension $d$, i.e., they lie on a $d$ dimensional manifold. Now, what is the most efficient way to compute K-nearest neighbors for each point? I've researched this topic a little bit, but most of the literature focuses on KNN clustering which is slightly different from what I'm asking. First, I don't have any labels here and those methods which rely on labels to reduce complexity are not of any interest. Second, many methods focus on reducing query processing time, which means they pre-process the data, build a database, and quickly process new queries. But I'm interested in reducing the overall time complexity rather than queries. What is the state-of-the-art method and its time complexity?
UPDATE: One might assume that these points are uniformly spread in a hyper-cube. I suppose anything which holds for uniform distribution, to a lesser extent will hold for a smoothly changing distribution.