Approximate, incremental nearest-neighbour algorithm for moving bodies

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This question raises several issues. The bounty will go to an answer which addresses them holistically.

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Here\'s a problem I\'ve been playing with.

Answer 1

A very simple and very fast method from statistics is to use random linear projections. These can help you determine clusters and neighbors very quickly. With more projections, you get more accuracy (addressing your question about errors, I believe).

This paper offers an extensive quantitative analysis of several methods, including a new method (DPES) that is related to RLP.

This paper addresses use of RLP including for distance preservation even in the context of moving points.

This paper addresses RLP for motion planning and details several heuristics.

RLP methods are:

1. Very fast
2. Lead to approximations that are tunable for accuracy and speed
3. Distance and angle preserving (provable)
4. Easily scaled to large dimensions and large #s of objects
5. Useful for dimensionality reduction
6. Lead to compact projections (e.g. one could project into a hierarchical binary partitioning)
7. Flexible: you can project into whatever space you feel would be good for you - usually R^d, but projecting into 2^d (i.e. binary space of dimension d) is also allowable, only subject to a reduction in accuracy for a given # of projections.
8. Statistically interesting

After embedding into a lower dimensional space, neighbor calculations are very easy, as projections that are, say, binned in the same regions (if you bin the projections into a grid) are very likely to be close in the original space.

Although the dimensionality of the original data is small (even 10 is small), the ability to rapidly project into a pre-selected grid is very useful for identifying and counting neighbors.

Finally, you only need to update for those objects whose location (or relative location, if you're centering and scaling the data) have changed.

For related works, look into the Johnson-Lindenstrauss lemma.