Approximate Nearest Neighbors. CS 510 Lecture #24 April 18 th, 2014

Transcription

1 Approximate Nearest Neighbors CS 510 Lecture #24 April 18 th, 2014

2 How is the assignment going? 2

3 hcp://breckon.eu/toby/demos/videovolumes/ Review: Video as Data Cube Mathematically, the cube can be viewed as a 3 rd -order tensor An example of multi-linear algebra 3

4 Review: Principal Angles Goal: directly compare two videos A & B Method: Model each video as the image subspace spanned by its frames Compute the smallest angle between any vector in A and any vector in B 4

5 Principal Angles (cont.) Mathematically this is computed as: A T A = R A λ A R A T (via PCA) B T B = R B λ B R B T (via PCA) R AT R B = C L λ pa C R (via SVD) Where λ pa are the principal angles C L and C R are the principal vectors of A & B The vectors that form the principal angles True distance measures use all the principal angles E.g. the sum of the principal angles But in practice, just the first works better 5

6 Review: Product Manifold Distances What s so special about time? Every frame is a sample in row/col space Every row is a sample in time/col space Every column is a sample in time/row space Compute principal angles in all 3 spaces Distance is the combination of 3 distances Incorporate motion & appearance 6

7 Problem: Efficiency Comparing two simple videos is quick But every track produces a stream of sliding windows Order of the # of frames in the track Every window needs to be compared to the training samples High accuracy requires many training samples How to reduce the total # of comparisons? Related question: how to quickly cluster videos? 7

8 Nearest Neighbors Goal: Find the nearest sample in a gallery to a novel probe sample Obvious solution: Measure distance from probe to every gallery instance Select instance with smallest distance Obvious problem: O(n) 8

9 Approximate Nearest Neighbors Goal: find nearest sample in gallery As often as possible When wrong, pick sample that is still close O(log (n)) Approach: binary trees Recursively divide feature space Each split divides gallery ~50/50 9

10 ANN Illustrated hcp://www1.cs.columbia.edu/cave/projects/nnsearch/ 10

11 ANN Trees Previous example thresholded feature values to divide feature space Boundaries can be Arbitrary hyperplanes (i.e. diagonal) Non-linear boundaries (i.e. spheres) Example: Hierarchical K-Means Problem: Samples near boundaries cause errors 11

12 Randomized Forests Build multiple ANN trees With different boundaries Requires randomized boundary selection Look up nearest neighbor in each tree Select best Two versions in OpenCV Randomized Hierarchical K-Means FLANN 12

13 Proximity Trees Problem: distance measures are not feature spaces Principal angles Product manifold distances Solution: proximity tree Select pivot sample at random Sort gallery by distance to pivot Split 50/50 nearest/farthest samples Repeat 13

14 Proximity Trees Illustrated Two randomized proximity tree par44ons 14

15 ANNs to BoW Every ANN tree creates a clustering The samples in the same partition are a group Every ANN tree is therefore a codebook Proximity tree construction O(n log(n)) Most ANNs are O(n log(n)) Why not create multiple, randomized codebooks? 15