Presentation Transcript

1. VLSH: Voronoi-based Locality Sensitive Hashing Sung-eui Yoon Authors: Lin Loi, Jae-PilHeo, Junghwan Lee, and Sung-Eui Yoon KAIST http://sglab.kaist.ac.kr/VLSH/
2. Main Goals Provide efficient nearest neighbor search for various motion planners, PRM and RRT Works with high-dimensional data sets Supports a diverse set of distance metrics
3. Nearest Neighbor Search in Motion Planning Generate collision free samples For each sample, find near neighbors
4. K-Nearest Neighbor Search Example with K = 6 data points q: query point
5. Approximate K-Nearest Neighbor Search (Ak-NNS) Allow approximation factor, ε > 1, to the distance, d, of the exact NN Ak-NNS q q d dε
6. Main Contributions Proposes VLSH that embeds points in space and uses localized LSHs depending on data distributions VLSH supports: Arbitrary distance metrics used in motion planning Fast approximate nearest neighbor search with high accuracy
7. Previous Work Spatial subdivision data structures (e.g., kd-trees) Suffer from the curse of dimensionality [Weber et al. 1998] Culling techniques using the triangle inequality [Chavez et al. 01] Geometric Near Access Tree (GNAT) [Brin, 95] Used widely at OOPSMP and OMPL GPU based acceleration [Pan et al., 10]
8. Previous Work on LSH and Embedding Locality Sensitivity Hashing (LSH) Fast algorithms for high-dimensional NNS problems [Datar and Indyk 2004] Supports for a limited set of motion planning distance metrics (e.g., Euler angles) [Pan et al. 2010] Embedding Well studied topic [Indyk et al., 04] Embed motion planning spaces to the Euclidean space [Plaku and Kavraki 2006]
9. Background on LSH Randomly generate a projection vector Project points onto vector Bin the projected points to a segment, whose width is w, i.e. quantization factor All the data in a bin has the same hash code Quantization factor w
10. Background on LSH Query point Multiple projections NN of : Data points g1 g2 g3
11. Issues of LSH LSH works with and some of its variants Does not support arbitrary distance metrics used in motion planning, e.g., scaled Euclidean distance metric and swept volume LSH is data independent LSH is ineffective for handling datasets with irregular distributions
12. An Example a1 The number of data points in each bin can vary a lot! a0 w
13. VLSH: Voronoi-based Locality Sensitive Hashing Consists of two steps: Embedding to the Euclidean space Invoking a localized LSH
14. Phase 1: Embedding Pick pivot points from the data set Compute distance to all pivot points in the MP space for defining an embedded point, v’ Use L2 metric as a distance metric between embedded points Support arbitrary motion planning metrics with a low distortion [Bourgain 85] p1 : pivot point d(p1,v) d(p2,v) p2 p0 d(p0,v) : sample point v v’= (d(p0,v), d(p1,v), d(p2,v))
15. Phase 2: Invoking a Local LSH During embedding process, a point can be associated with its closest pivot Pivot points implicitly construct Voronoi regions Voronoi diagram Pivot points Data points
16. Implicit Voronoi Region of a Pivot Construct a localized LSH for points contained in each Voronoi region of a pivot point Use a localized quantization factor Explicit construction for Voronoi regions is not necessary Assigns a point to its closest pivot
17. Expanding Voronoi Regions Considering only points within the Voronoi region of each pivot results in disconnected graphs Points near Voronoi regions are not connected
18. Expanding Voronoi Regions Use an extended Voronoi region of a pivot Contains nearby points, whose second closest pivot is the current one Expansion amount, Contain percentile of those nearby points Large expanding results higher accuracy, but bigger memory overhead 60% to 90% works well
19. Results w/ and w/o Expansion Well-connected graphs (Ak-NNS w/ expansion) Disconnected graphs (Ak-NNS w/o expansion)
20. Query-Time Algorithm Compute the embedded point from a query point Find the closest pivot point and use its localized LSH Return candidate nearest neighbors located in hash-buckets of the localized LSH
21. Test Configurations Intel i7 3.3 GHz CPU with C++ implementation Compare our method against LSH and GNAT (OOPSMP) Test them with samples from a PRM planner 15 nearest neighbor search, i.e. k = 15 10 pivots for our method
22. Benchmarks Wiper: 6 dimensions (1 robot for the wiper) Bug trap: 24 dimensions: 4 rod robots 36 dimensions: 6 rod robots
23. Wiper: Performance Evaluation VLSH vs. GNAT (Em): 3.7x faster VLSH vs. LSH (Em): 2.6x faster
24. Wiper: Quality Evaluation Measure fractional distance error (fde) [Plaku and Kavraki 06] Measure difference between computed approximate and ground truth results Lower values indicate more accurate results
25. Results: Bug trap, 24 dim. VLSH vsGNAT (Em): 3.2x faster VLSH vsLSH (Em): Up to 1.6x faster
26. Results: Bug trap, 36 dim. VLSH vsGNAT (Em): 3.6x faster VLSH vsLSH (Em): Up to 1.4x faster
27. Conclusions Fast approximate nearest neighbor search algorithm for high-dimensional motion planning problems Achieve up to 3.7x faster running time over prior approaches Supports all distance metrics and consider data distributions for higher accuracy
28. Limitation and Future Work Memory overhead Duplicate points in our method Overall it is not significant, since it takes tens of MB in the testes cases Support RRTs that dynamically generate data points Supports GPUs for higher performance
29. Acknowledgements Anonymous reviewers Our funding agency Project webpage: http://sglab.kaist.ac.kr/VLSH/