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Approximate Nearest Subspace Search with applications to pattern recognitionPowerPoint Presentation

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

Nayar et al., IUW’96

Subspaces in Computer Vision- Illumination

- Faces

- Objects

- Viewpoint, Motion

- Dynamic textures
- …

Zelnik-Manor & Irani, PAMI’06

Sequential Search

Database

nsubspaces

ddimensions

ksubspace

dimension

Sequential search:O(ndk)

Too slow!!

Is there a sublinear solution?

A Related Problem:Nearest Neighbor Search

Database

npoints

ddimensions

Sequential search:O(nd)

There is a sublinear solution!

Approximate NN

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

r

(1+)r

Query: Logarithmic

Preprocessing: O(dn)

Fast!!

Is it possible to speed-up Nearest Subspace Search?

Existing point-based methods cannot be applied

LSH

Tree search

Our

Our Suggested Approach- Reduction to points
- Works for both
linear and affine spaces

Run time

Database size

Problem Definition

Find Mapping

Independent mappings

Monotonic in distance

A linear function of original distance

Apply standard point ANN to u,v

Basic Reduction

Want: minimize /

Final Reduction

= constants

Dimensionality May be Large

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

Synthetic Data

Varying dimension

Varying database size

Sequential

Sequential

Our

Our

Run time

Run time

dimension

Database size

n=5000, k=4

d=60, k=4

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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