Approximate Nearest Subspace Search with applications to pattern recognition

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Approximate Nearest Subspace Searchwith
applications to pattern recognition

• Ronen Basri Tal Hassner Lihi Zelnik-Manor
• Weizmann Institute
  Caltech

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Subspaces in Computer Vision

• Illumination

• Faces

• Objects

• Viewpoint, Motion

• Dynamic textures

Zelnik-Manor Irani, PAMI06
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Nearest Subspace Search
Query
Which is the Nearest Subspace?
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Sequential Search
Database
n subspaces
d dimensions
k subspace dimension
Sequential search O(ndk)
Too slow!!
Is there a sublinear solution?
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A Related ProblemNearest Neighbor Search
Database
n points
d dimensions
Sequential search O(nd)
There is a sublinear solution!
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Approximate NN

• Tree search (KD-trees)
• Locality Sensitive Hashing

r
(1?)r
Query Logarithmic Preprocessing
O(dn)
Fast!!
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Is it possible to speed-up Nearest Subspace
Search?
Existing point-based methods cannot be applied
Tree search
LSH
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Our Suggested Approach

• Reduction to points
• Works for both
• linear and affine spaces

Run time
Database size
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Problem Definition
Find Mapping
Independent mappings
Monotonic in distance
A linear function of original distance
Apply standard point ANN to u,v
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Finding a Reduction
Feeling lucky?
We are lucky !!
Constants?
Depends on query
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Basic Reduction
Want minimize ?/?
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Geometry of Basic Reduction
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Improving the Reduction
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Final Reduction
constants
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Can We Do Better?
If ?0
Trivial mapping
Additive Constant is Inherent
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Final Mapping Geometry
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ANS Complexities
Linear in n
Preprocessing O(nkd2)
Log in n
Query O(d2)TANN(n,d2)
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Dimensionality May be Large

• Embedding in d2
• Might need to use small e
• Current solution
• Use random projections (use Johnson-Lindenstrauss
  Lemma)
• Repeat several times and select the nearest

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Synthetic Data
Varying dimension
Varying database size
Sequential
Sequential
Our
Our
Run time
Run time
dimension
Database size
n5000, k4
d60, k4
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Face Recognition (YaleB)
Database
64 illuminations k9 subspaces
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Face Recognition Result
Wrong Match
Wrong Person
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Retiling with Patches
Wanted
Patch database
Query
Approx Image
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Retiling with Subspaces
Wanted
Subspace database
Query
Approx Image
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Patches ANN 0.6sec
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Subspaces ANS 1.2 sec
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Patches ANN 0.6sec
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Subspaces ANS 1.2 sec
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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..

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THANK YOU