quadratic form distance (qfd) - univerzita karlovasiret.ms.mff.cuni.cz/skopal/pres/qmap.pdf ·...
TRANSCRIPT
Quadratic Form Distance (QFD)
QMap model
› QFD to L2 Space Transformation
QMap and MAMs
› Experimental evaluation
Conclusion
23rd March 2011 2EDBT 2011, Uppsala, Sweden
Content-based similarity searching
› Pair-wise similarity
› High-dimensional (feature) vectors
Fast query processing
› Efficiency
› Effectiveness
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Similarity measuring:
QFDA (u,v) = √ (u – v)A(u – v)T
u, v feature vectors (1×n)
A similarity matrix/QFD matrix (n×n)
› positive definite (zAzT > 0)
› static / dynamic correlations
› data independent
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Applications
› QBIC project (Querying Images by content)
› 2D & 3D shapes
› Protein structures
› MindReader
Advanced
› SQFD (Signature QFD)
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Transformation approaches
› QBIC system
Lower-bounding (e.g. Faloutsos et. al 1994)
› Contractive reduction techniques
› SVD / KLT decompositions
Combination (e.g. Hafner et. al 1995)
› Transformation to k-dimensional Lp space
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Metric Access Methods (MAMs)
› Effective/efficient similarity searching
› Reduce distance computations
› Complexity depends on distance function
QFD is considered as expensive – O(n2)
› Indexing needed
We show the transformation of QFD
› Obtain cheaper distance function – O(n)
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QFDEuclidean (L2)
distance
Correlated
dimensions
Independent
dimensions
Expensive – O(n2) Cheap – O(n)
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Transform QFD space
› L2 instead of QFD
› Preserving distances (homeomorphism)
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Transform QFD space
› L2 instead of QFD
› Preserving distances (homeomorphism)
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Rotate (weighted L2 space)
Scale (L2 space)
Transformation matrix B
› obtained by Cholesky decomposition:
A = BBT
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Linear
transformation
1. QFDA (u,v) = √ (u – v)A(u – v)T
2. Cholesky decomposition: BBT = A
3. QFD (u,v) = √ (u – v)BBT(u – v)T
4. QFD (u,v) = √ [(u – v)B][(u – v)B]T
5. QFD (u,v) = √ (uB – vB)(uB – vB)T
6. L2 (u’,v’) = √ (u’ – v’)(u’ – v’)T
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Application of QMap in MAM› Sequential (SEQ) file
› Pivot Table
› M-tree
1,000,000 images (512 dimensional RGB histogram)
Actions› Indexing
› Querying
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Method (model) “Winner”
SEQ file (QFD)
SEQ file (QMap) QFD
Pivot Table (QFD)
Pivot Table(QMap) QMap
M-tree (QFD)
M-tree (QMap) QMap
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Method (model) “Winner”
SEQ file (QFD)
SEQ file (QMap) QMap
Pivot Table (QFD)
Pivot Table(QMap) QMap
M-tree (QFD)
M-tree (QMap) QMap
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QMap model
› Space transformation: QFD → L2
› Distance-preserving (homeomorphic)
› Data-independent
› Output is explicitly formulated
QMap model is separated from the
usage of any access methods
› Superior performance
23rd March 2011 20EDBT 2011, Uppsala, Sweden