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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
23rd March 2011 3EDBT 2011, Uppsala, Sweden
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
23rd March 2011 4EDBT 2011, Uppsala, Sweden
Applications
› QBIC project (Querying Images by content)
› 2D & 3D shapes
› Protein structures
› MindReader
Advanced
› SQFD (Signature QFD)
23rd March 2011 5EDBT 2011, Uppsala, Sweden
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
23rd March 2011 6EDBT 2011, Uppsala, Sweden
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)
23rd March 2011 7EDBT 2011, Uppsala, Sweden
QFDEuclidean (L2)
distance
Correlated
dimensions
Independent
dimensions
Expensive – O(n2) Cheap – O(n)
23rd March 2011 8EDBT 2011, Uppsala, Sweden
Transform QFD space
› L2 instead of QFD
› Preserving distances (homeomorphism)
23rd March 2011 10EDBT 2011, Uppsala, Sweden
Transform QFD space
› L2 instead of QFD
› Preserving distances (homeomorphism)
23rd March 2011 11EDBT 2011, Uppsala, Sweden
Rotate (weighted L2 space)
Scale (L2 space)
Transformation matrix B
› obtained by Cholesky decomposition:
A = BBT
23rd March 2011 13EDBT 2011, Uppsala, Sweden
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
23rd March 2011 14EDBT 2011, Uppsala, Sweden
Application of QMap in MAM› Sequential (SEQ) file
› Pivot Table
› M-tree
1,000,000 images (512 dimensional RGB histogram)
Actions› Indexing
› Querying
23rd March 2011 15EDBT 2011, Uppsala, Sweden
Method (model) “Winner”
SEQ file (QFD)
SEQ file (QMap) QFD
Pivot Table (QFD)
Pivot Table(QMap) QMap
M-tree (QFD)
M-tree (QMap) QMap
23rd March 2011 16EDBT 2011, Uppsala, Sweden
23rd March 2011 17EDBT 2011, Uppsala, Sweden
Method (model) “Winner”
SEQ file (QFD)
SEQ file (QMap) QMap
Pivot Table (QFD)
Pivot Table(QMap) QMap
M-tree (QFD)
M-tree (QMap) QMap
23rd March 2011 18EDBT 2011, Uppsala, Sweden
23rd March 2011 19EDBT 2011, Uppsala, Sweden
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
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