1. Ronen Basri Tal Hassner Lihi Zelnik-Manor Weizmann Institute Caltech
Approximate Nearest Subspace Search with applications to pattern recognition
Ronen Basri Tal Hassner Lihi Zelnik-Manor
Weizmann Institute Caltech

2. Subspaces in Computer Vision
Basri & Jacobs, PAMI’03
Illumination
Faces
Nayar et al., IUW’96
Objects
Viewpoint, Motion
Dynamic textures …
Zelnik-Manor & Irani, PAMI’06

3. Nearest Subspace Search
Query
Which is the Nearest Subspace?

4. Is there a sublinear solution?
Sequential Search
Database
n subspaces
d dimensions
k subspace
dimension
Sequential search: O(ndk)
Too slow!!
Is there a sublinear solution?

5. A Related Problem: Nearest Neighbor Search
Database
n points
d dimensions
Sequential search: O(nd)
There is a sublinear solution!

6. Approximate NN r (1+)r Fast!! Tree search (KD-trees)
Locality Sensitive Hashing r
(1+)r
Query: Logarithmic
Preprocessing: O(dn)
Fast!!

7. Is it possible to speed-up Nearest Subspace Search?
Existing point-based methods cannot be applied
LSH
Tree search

8. Our Suggested Approach
Reduction to points
Works for both
linear and affine spaces
Sequential
Our
Run time
Database size

9. Problem Definition Find Mapping Apply standard point ANN to u,v
Independent mappings
Monotonic in distance
A linear function of original distance
Apply standard point ANN to u,v

10. Finding a Reduction We are lucky !! Feeling lucky? Constants?
Depends on query

11. Basic Reduction
Want: minimize /

12. Geometry of Basic Reduction
Query
Lies on a cone
Database
Lies on a sphere
and on a hyper-plane

13. Improving the Reduction

14. Final Reduction
= constants

15. Additive Constant is Inherent
Can We Do Better?
If =0
Trivial mapping
Additive Constant is Inherent

16. Final Mapping Geometry

17. ANS Complexities Linear in n Log in n Preprocessing: O(nkd2)
Query: O(d2)+TANN(n,d2)

18. Dimensionality May be Large
Embedding in d2
Might need to use small ε
Current solution:
Use random projections (use Johnson-Lindenstrauss Lemma)
Repeat several times and select the nearest

19. Synthetic Data n=5000, k=4 d=60, k=4 Varying database size
Varying dimension
Sequential
Sequential
Our
Our
Run time
Run time
Database size
dimension
d=60, k=4
n=5000, k=4

20. Face Recognition (YaleB)
Database
64 illuminations
k=9 subspaces
Query:
New illumination

21. Face Recognition Result
Wrong Match
Wrong Person
True NS
Approx NS

22. Retiling with Patches
Wanted
Query
Patch database
Approx Image

23. Retiling with Subspaces
Wanted
Subspace database
Query
Approx Image

24. Patches +
ANN
~0.6sec

25. Subspaces +
ANS
~1.2 sec

26. Patches +
ANN
~0.6sec

27. Subspaces +
ANS
~1.2 sec

28. Summary Fast, approximate nearest subspace search
Reduction to point ANN
Useful applications in computer vision
Disadvantages:
Embedding in d2
Additive constant 
Other methods?
Additional applications?
A lot more to be done…..

29. THANK YOU