ECCV2010: distance function and metric learning part 1

Distance Functions and Metric Learning: Part 1 12/1/2011 version. Updated Version: http://www.cs.huji.ac.il/~ofirpele/DFML_ECCV2010_tutorial/ Michael Werman Ofir Pele The Hebrew University of Jerusalem

Rough Outline Bin-to-Bin Distances. Cross-Bin Distances: Quadratic-Form (aka Mahalanobis) / Quadratic-Chi. The Earth Mover’s Distance. Perceptual Color Differences. Hands-On Code Example.

Metric ² N o n - n e g a t i v i t y : D ( P ; Q ) ¸ 0 ² I d e n t i t y o f i n d i s c e r n i b l e s : D ( P ; Q ) = 0 i ® P = Q ² S y m m e t r y : D ( P ; Q ) = D ( Q ; P ) ² S u b a d d i t i v i t y ( t r i a n g l e i n e q u a l i t y ) : D ( P ; Q ) · D ( P ; K ) + D ( K ; Q )

Pseudo-Metric (aka Semi-Metric) ² N o n - n e g a t i v i t y : D ( P ; Q ) ¸ 0 ² P r o p e r t y c h a n g e d t o : D ( P ; Q ) = 0 i f P = Q ² S y m m e t r y : D ( P ; Q ) = D ( Q ; P ) ² S u b a d d i t i v i t y ( t r i a n g l e i n e q u a l i t y ) : D ( P ; Q ) · D ( P ; K ) + D ( K ; Q )

Metric A n o b j e c t i s m o s t s i m i l a r t o i t s e l f ² N o n - n e g a t i v i t y : D ( P ; Q ) ¸ 0 ² I d e n t i t y o f i n d i s c e r n i b l e s : D ( P ; Q ) = 0 i ® P = Q

Metric ² S y m m e t r y : D ( P ; Q ) = D ( Q ; P ) ² S u b a d d i t i v i t y ( t r i a n g l e i n e q u a l i t y ) : D ( P ; Q ) · D ( P ; K ) + D ( K ; Q ) U s e f u l f o r m a n y a l g o r i t h m s

Minkowski-Form Distances L p ( P ; Q ) = Ã X i j P i ¡ Q i j p ! 1 p

Minkowski-Form Distances L 2 ( P ; Q ) = s X i ( P i ¡ Q i ) 2 L 1 ( P ; Q ) = X i j P i ¡ Q i j L 1 ( P ; Q ) = m a x i j P i ¡ Q i j

Kullback-Leibler Divergence Information theoretic origin. Non symmetric. K L ( P ; Q ) = X i P i l o g P i Q i Q i = 0 ?

Jensen-Shannon Divergence J S ( P ; Q ) = 1 2 K L ( P ; M ) + 1 2 K L ( Q ; M ) M = 1 2 ( P + Q ) J S ( P ; Q ) = 1 2 X i P i l o g 2 P i P i + Q i + 1 2 X i Q i l o g 2 Q i P i + Q i

Jensen-Shannon Divergence Information Theoretic origin. Symmetric. is a metric. J S ( P ; Q ) = 1 2 K L ( P ; M ) + 1 2 K L ( Q ; M ) M = 1 2 ( P + Q ) p J S

Jensen-Shannon Divergence Using Taylor extension and some algebra: J S ( P ; Q ) = 1 X n = 1 1 2 n ( 2 n ¡ 1 ) X i ( P i ¡ Q i ) 2 n ( P i + Q i ) 2 n ¡ 1 = 1 2 X i ( P i ¡ Q i ) 2 ( P i + Q i ) + 1 1 2 X i ( P i ¡ Q i ) 4 ( P i + Q i ) 3 + : : :

Histogram Distance Statistical origin. Experimentally results are very similar to JS. Reduces the effect of large bins. is a metric Â 2 Â 2 ( P ; Q ) = 1 2 X i ( P i ¡ Q i ) 2 ( P i + Q i ) p Â 2

Histogram Distance < Â 2 Â 2 ( ; ) Â 2 ( ; )

Histogram Distance > Â 2 L 1 ( ; ) L 1 ( ; )

Histogram Distance Experimentally better than . Â 2 L 2 Â 2 ( P ; Q ) = 1 2 X i ( P i ¡ Q i ) 2 ( P i + Q i )

Bin-to-Bin Distances Bin-to-Bin distances such as L2 are sensitive to quantization: > L 1 ( ; ) L 1 ; L 2 ; Â 2 L 1 ( ; )

#bins #bins Bin-to-Bin Distances robustness distinctiveness robustness distinctiveness > L 1 ( ; ) L 1 ( ; )

#bins #bins Bin-to-Bin Distances robustness distinctiveness robustness distinctiveness Can we achieve robustness and distinctiveness ?

The Quadratic-Form Histogram Distance is the similarity between bin i and j. If is the inverse of the covariance matrix, QF is called Mahalanobis distance. A i j = s X i j ( P i ¡ Q i ) ( P j ¡ Q j ) A i j ( 1 ) Q F A ( P ; Q ) = q ( P ¡ Q ) T A ( P ¡ Q ) ( 1 ) A

The Quadratic-Form Histogram Distance Q F A ( P ; Q ) = q ( P ¡ Q ) T A ( P ¡ Q ) = s X i j ( P i ¡ Q i ) ( P j ¡ Q j ) A i j A = I = s X i j ( P i ¡ Q i ) 2 = L 2 ( P ; Q )

The Quadratic-Form Histogram Distance Does not reduce the effect of large bins. Alleviates the quantization problem. Linear time computation in # non zero . A i j Q F A ( P ; Q ) = q ( P ¡ Q ) T A ( P ¡ Q )

The Quadratic-Form Histogram Distance If A is positive-semidefinite then QF is a pseudo-metric. Q F A ( P ; Q ) = q ( P ¡ Q ) T A ( P ¡ Q )

The Quadratic-Form Histogram Distance If A is positive-definite then QF is a metric. Q F A ( P ; Q ) = q ( P ¡ Q ) T A ( P ¡ Q )

The Quadratic-Form Histogram Distance Q F A ( P ; Q ) = q ( P ¡ Q ) T A ( P ¡ Q ) = q ( P ¡ Q ) T W T W ( P ¡ Q ) = L 2 ( W P ; W Q )

The Quadratic-Form Histogram Distance We assume there is a linear transformation that makes bins independent There are cases where this is not true e.g. COLOR Q F A ( P ; Q ) = q ( P ¡ Q ) T A ( P ¡ Q ) = L 2 ( W P ; W Q )

The Quadratic-Form Histogram Distance Converting distance to similarity (Hafner et. al 95): Q F A ( P ; Q ) = q ( P ¡ Q ) T A ( P ¡ Q ) A i j = 1 ¡ D i j m a x i j ( D i j )

The Quadratic-Form Histogram Distance Converting distance to similarity (Hafner et. al 95): Q F A ( P ; Q ) = q ( P ¡ Q ) T A ( P ¡ Q ) A i j = e ¡ ® D ( i ; j ) m a x i j ( D ( i ; j ) ) I f ® i s l a r g e e n o u g h , A w i l l b e p o s i t i v e - d e ¯ n i t i v e

The Quadratic-Form Histogram Distance Learning the similarity matrix: part 2 of this tutorial. Q F A ( P ; Q ) = q ( P ¡ Q ) T A ( P ¡ Q )

The Quadratic-Chi Histogram Distance is the similarity between bin i and j. Generalizes and . Reduces the effect of large bins. Alleviates the quantization problem. Linear time computation in # non zero . Â 2 Q C A m ( P ; Q ) = v u u t X i j ( P i ¡ Q i ) ( P j ¡ Q j ) A i j ( P c ( P c + Q c ) A c i ) m ( P c ( P c + Q c ) A c j ) m A i j A i j Q F

The Quadratic-Chi Histogram Distance is non-negative if A is positive-semidefinite. Symmetric. Triangle inequality unknown. If we define and , QC is continuous. Q C A m ( P ; Q ) = v u u t X i j ( P i ¡ Q i ) ( P j ¡ Q j ) A i j ( P c ( P c + Q c ) A c i ) m ( P c ( P c + Q c ) A c j ) m 0 · m < 1 0 0 = 0

Similarity-Matrix-Quantization-Invariant Property ( , ) D ) ( , = D

Sparseness-Invariant = D ) ( , Empty bins D ( , )

The Quadratic-Chi Histogram Distance Code f u n c t i o n d i s t = Q u a d r a t i c C h i ( P , Q , A , m ) Z = ( P + Q ) * A ; % 1 c a n b e a n y n u m b e r a s Z _ i = = 0 i f f D _ i = 0 Z ( Z = = 0 ) = 1 ; Z = Z . ^ m ; D = ( P - Q ) . / Z ; % m a x i s r e d u n d a n t i f A i s % p o s i t i v e - s e m i d e f i n i t e d i s t = s q r t ( m a x ( D * A * D ' , 0 ) ) ;

The Quadratic-Chi Histogram Distance Code f u n c t i o n d i s t = Q u a d r a t i c C h i ( P , Q , A , m ) Z = ( P + Q ) * A ; % 1 c a n b e a n y n u m b e r a s Z _ i = = 0 i f f D _ i = 0 Z ( Z = = 0 ) = 1 ; Z = Z . ^ m ; D = ( P - Q ) . / Z ; % m a x i s r e d u n d a n t i f A i s % p o s i t i v e - s e m i d e f i n i t e d i s t = s q r t ( m a x ( D * A * D ' , 0 ) ) ; T i m e C o m p l e x i t y : O ( # A i j 6 = 0 )

The Quadratic-Chi Histogram Distance Code f u n c t i o n d i s t = Q u a d r a t i c C h i ( P , Q , A , m ) Z = ( P + Q ) * A ; % 1 c a n b e a n y n u m b e r a s Z _ i = = 0 i f f D _ i = 0 Z ( Z = = 0 ) = 1 ; Z = Z . ^ m ; D = ( P - Q ) . / Z ; % m a x i s r e d u n d a n t i f A i s % p o s i t i v e - s e m i d e f i n i t e d i s t = s q r t ( m a x ( D * A * D ' , 0 ) ) ; W h a t a b o u t s p a r s e ( e . g . B o W ) h i s t o g r a m s ?

The Quadratic-Chi Histogram Distance Code 0 0 0 0 0 0 0 0 0 4 5 0 -3 P ¡ Q = W h a t a b o u t s p a r s e ( e . g . B o W ) h i s t o g r a m s ? T i m e C o m p l e x i t y : O ( S K )

The Quadratic-Chi Histogram Distance Code 0 0 0 0 0 0 0 0 0 4 5 0 -3 P ¡ Q = # ( P 6 = 0 ) + # ( Q 6 = 0 ) W h a t a b o u t s p a r s e ( e . g . B o W ) h i s t o g r a m s ? T i m e C o m p l e x i t y : O ( S K )

The Quadratic-Chi Histogram Distance Code A v e r a g e o f n o n - z e r o e n t r i e s 0 0 0 0 0 0 0 0 0 4 5 0 -3 P ¡ Q = i n e a c h r o w o f A W h a t a b o u t s p a r s e ( e . g . B o W ) h i s t o g r a m s ? T i m e C o m p l e x i t y : O ( S K )

The Earth Mover’s Distance The Earth Mover’s Distance is defined as the minimal cost that must be paid to transform one histogram into the other, where there is a “ ground distance ” between the basic features that are aggregated into the histogram.

The Earth Mover’s Distance ≠

The Earth Mover’s Distance =

The Earth Mover’s Distance E M D D ( P ; Q ) = m i n F = f F i j g X i ; j F i j D i j s : t : X j F i j = P i X i F i j = Q j X i ; j F i j = X i P i = X j Q j = 1 F i j ¸ 0

The Earth Mover’s Distance E M D D ( P ; Q ) = m i n F = f F i j g P i ; j F i j D i j P i F i j s : t : X j F i j · P i X i F i j · Q j X i ; j F i j = m i n ( X i P i ; X j Q j ) f a l s e F i j ¸ 0

Pele and Werman 08 – , a new EMD definition. The Earth Mover’s Distance \ E M D

Definition: \ E M D D C ( P ; Q ) = m i n F = f F i j g X i ; j F i j D i j + ¯ ¯ ¯ ¯ ¯ ¯ X i P i ¡ X j Q j ¯ ¯ ¯ ¯ ¯ ¯ £ C s : t : X j F i j · P i X i F i j · Q j X i ; j F i j = m i n ( X i P i ; X j Q j ) F i j ¸ 0 f a l s e

S u p p l i e r E M D D e m a n d e r D i j = 0 P D e m a n d e r · P S u p p l i e r

D e m a n d e r S u p p l i e r P D e m a n d e r · P S u p p l i e r D i j = C \ E M D

When to Use When the total mass of two histograms is important. E M D ¡ ; ¢ = E M D ¡ ; ¢ \ E M D

When to Use When the total mass of two histograms is important. ¡ ; ¢ < ¡ ; ¢ E M D ¡ ; ¢ = E M D ¡ ; ¢ \ E M D \ E M D \ E M D

When to Use When the difference in total mass between histograms is a distinctive cue. E M D ¡ ; ¢ = E M D ¡ ; ¢ = 0 \ E M D

When to Use When the difference in total mass between histograms is a distinctive cue. ¡ ; ¢ < ¡ ; ¢ E M D ¡ ; ¢ = E M D ¡ ; ¢ = 0 \ E M D \ E M D \ E M D

When to Use If ground distance is a metric: ² E M D i s a m e t r i c o n l y f o r n o r m a l i z e d h i s t o g r a m s . ² \ E M D i s a m e t r i c f o r a l l h i s t o g r a m s ( C ¸ 1 2 ¢ ) . \ E M D

C = 1 D i j = ( 0 i f i = j 2 o t h e r w i s e \ E M D D C ( P ; Q ) = m i n F = f F i j g X i ; j F i j D i j + ¯ ¯ ¯ ¯ ¯ ¯ X i P i ¡ X j Q j ¯ ¯ ¯ ¯ ¯ ¯ £ C ² \ E M D = L 1 i f : \ E M D - a N a t u r a l E x t e n s i o n t o L 1

The Earth Mover’s Distance Complexity Zoo General ground distance: Orlin 88 normalized 1D histograms Werman, Peleg and Rosenfeld 85 L 1 O ( N 3 l o g N ) O ( N )

The Earth Mover’s Distance Complexity Zoo normalized 1D cyclic histograms Pele and Werman 08 (Werman, Peleg, Melter, and Kong 86) L 1 O ( N )

The Earth Mover’s Distance Complexity Zoo Manhattan grids Ling and Okada 07 O ( N 2 l o g N ( D + l o g N ) ) L 1

What about N-dimensional histograms with a cyclic dimensions? The Earth Mover’s Distance Complexity Zoo

The Earth Mover’s Distance Complexity Zoo general histograms Gudmundsson, Klein, Knauer and Smid 07 L 1 p 1 p 2 p 3 p 4 p 5 p 6 p 7 p 8 l 2 l 1 l ` 2 O ( N 2 l o g 2 D ¡ 1 N )

The Earth Mover’s Distance Complexity Zoo 1D linear/cyclic histograms Pele and Werman 08 m i n ( L 1 ; 2 ) O ( N ) f 1 ; 2 ; : : : ; ¢ g

The Earth Mover’s Distance Complexity Zoo general histograms is the number of edges with cost 1 Pele and Werman 08 m i n ( L 1 ; 2 ) f 1 ; 2 ; : : : ; ¢ g D K O ( N 2 K l o g ( N K ) )

The Earth Mover’s Distance Complexity Zoo Any thresholded distance Pele and Werman 09 O ( N 2 l o g N ( K + l o g N ) ) number of edges with cost different from the threshold K = O ( l o g N ) O ( N 2 l o g 2 N )

Thresholded Distances EMD with a thresholded ground distance is not an approximation of EMD. It has better performance.

The Flow Network Transformation Original Network Simplified Network

The Flow Network Transformation Flowing the Monge sequence (if ground distance is a metric, zero-cost edges are a Monge sequence)

The Flow Network Transformation Removing Empty Bins and their edges

The Flow Network Transformation We actually finished here….

Combining Algorithms EMD algorithms can be combined. For example : L 1

Combining Algorithms EMD algorithms can be combined. For example, thresholded : L 1

The Earth Mover’s Distance Approximations

Charikar 02, Indyk and Thaper 03 – approximated EMD on by embedding it into the norm. Time complexity: Distortion (in expectation): The Earth Mover’s Distance Approximations L 1 f 1 ; : : : ; ¢ g d O ( T N d l o g ¢ ) O ( d l o g ¢ )

The Earth Mover’s Distance Approximations Grauman and Darrell 05 – Pyramid Match Kernel (PMK) same as Indyk and Thaper, replacing with histogram intersection . PMK approximates EMD with partial matching. PMK is a mercer kernel. Time complexity & distortion – same as Indyk and Thaper (proved in Grauman and Darrell 07). L 1

The Earth Mover’s Distance Approximations Lazebnik, Schmid and Ponce 06 – used PMK in the spatial domain (SPM).

The Earth Mover’s Distance Approximations level 0 + + + + + + + + + + + + + + + + + + + + + + + + x 1 / 8 x 1 / 4 x 1 / 2 + + + level 1 level 2 + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +

The Earth Mover’s Distance Approximations Shirdhonkar and Jacobs 08 - approximated EMD using the sum of absolute values of the weighted wavelet coefficients of the difference histogram. D i f f e r e n c e P Q Wavelet Transform j=0 j=1 Absolute values + £ 2 ¡ 2 £ 0 £ 2 ¡ 2 £ 1

The Earth Mover’s Distance Approximations Khot and Naor 06 – any embedding of the EMD over the d-dimensional Hamming cube into must incur a distortion of . Andoni, Indyk and Krauthgamer 08 - for sets with cardinalities upper bounded by a parameter the distortion reduces to . Naor and Schechtman 07 - any embedding of the EMD over must incur a distortion of . L 1 ( d ) s O ( l o g s l o g d ) f 0 ; 1 ; : : : ; ¢ g 2 ( p l o g ¢ )

Robust Distances Very high distances outliers same difference.

Robust Distances With colors, the natural choice. 0 0 20 40 60 80 100 120 CIEDE200 Distance from Blue

Robust Distances ( b l u e ; r e d ) = 5 6 ¢E 0 0 ( b l u e ; y e l l o w ) = 1 0 2 ¢E 0 0

Robust Distances - Exponent Usually a negative exponent is used: ² L e t d ( a ; b ) b e a d i s t a n c e m e a s u r e b e t w e e n t w o f e a t u r e s - a ; b . ² T h e n e g a t i v e e x p o n e n t d i s t a n c e i s : d e ( a ; b ) = 1 ¡ e ¡ d ( a ; b ) ¾

Robust Distances - Exponent Exponent is used because (Ruzon and Tomasi 01) : robust, smooth, monotonic, and a metric Input is always discrete anyway …

Robust Distances - Thresholded ² L e t d ( a ; b ) b e a d i s t a n c e m e a s u r e b e t w e e n t w o f e a t u r e s - a ; b . ² T h e t h r e s h o l d e d d i s t a n c e w i t h a t h r e s h o l d o f t > 0 i s : d t ( a ; b ) = m i n ( d ( a ; b ) ; t ) .

Thresholded Distances Thresholded metrics are also metrics (Pele and Werman ICCV 2009). Better results. Pele and Werman ICCV 2009 algorithm computes EMD with thresholded ground distances much faster. Thresholded distance corresponds to sparse similarities matrix -> faster QC / QF computation. A i j = 1 ¡ D i j m a x i j ( D i j )

Thresholded Distances Thresholded vs. exponent : Fast computation of cross-bin distances with a thresholded ground distance. Exponent changes small distances – can be a problem (e.g. color differences).

Thresholded Distances Color distance should be thresholded (robust). 0 50 100 150 distance from blue ¢ E 0 0 m i n ( ¢ E 0 0 ; 1 0 ) 1 0 ( 1 ¡ e ¡ ¢ E 0 0 4 ) 1 0 ( 1 ¡ e ¡ ¢ E 0 0 5 )

Thresholded Distances Exponent changes small distances ¢ E 0 0 m i n ( ¢ E 0 0 ; 1 0 ) 1 0 ( 1 ¡ e ¡ ¢ E 0 0 4 ) 1 0 ( 1 ¡ e ¡ ¢ E 0 0 5 ) 0 5 10 distance from blue

A Ground Distance for SIFT d R = j j ( x i ; y i ) ¡ ( x j ; y j ) j j 2 + m i n ( j o i ¡ o j j ; M ¡ j o i ¡ o j j ) d T = m i n ( d R ; T ) T h e g r o u n d d i s t a n c e b e t w e e n t w o S I F T b i n s ( x i ; y i ; o i ) a n d ( x j ; y j ; o j ) :

A Ground Distance for Color Image T h e g r o u n d d i s t a n c e s b e t w e e n t w o L A B i m a g e b i n s ( x i ; y i ; L i ; a i ; b i ) a n d ( x j ; y j ; L j ; a j ; b j ) w e u s e a r e : d c T = m i n ( ( j j ( x i ; y i ) ¡ ( x j ; y j ) j j 2 ) + ¢ 0 0 ( ( L i ; a i ; b i ) ; ( L j ; a j ; b j ) ) ; T )

Perceptual Color Differences Euclidean distance on L*a*b* space is widely considered as perceptual uniform. 0 100 200 300 distance from blue j j L a b j j 2

Perceptual Color Differences 0 20 40 60 adist2 distance from blue Purples before blues j j L a b j j 2

Perceptual Color Differences on L*a*b* space is better. Luo, Cui and Rigg 01. Sharma, Wu and Dalal 05. distance from blue 0 5 10 15 20 ¢ E 0 0 ¢ E 0 0

Perceptual Color Differences on L*a*b* space is better. ¢ E 0 0 j j L a b j j 2 ¢ E 0 0

Perceptual Color Differences 0 50 100 150 distance from blue on L*a*b* space is better. ¢ E 0 0 ¢ E 0 0

Perceptual Color Differences on L*a*b* space is better. But still has major problems. Color distance should be thresholded (robust). ( b l u e ; r e d ) = 5 6 ¢E 0 0 ( b l u e ; y e l l o w ) = 1 0 2 ¢E 0 0 ¢ E 0 0

Perceptual Color Differences Color distance should be saturated (robust). 0 50 100 150 distance from blue ¢ E 0 0 m i n ( ¢ E 0 0 ; 1 0 ) 1 0 ( 1 ¡ e ¡ ¢ E 0 0 4 ) 1 0 ( 1 ¡ e ¡ ¢ E 0 0 5 )

Perceptual Color Descriptors 11 basic color terms. Berlin and Kay 69. white red green yellow blue brown purple pink orange grey black

Perceptual Color Descriptors 11 basic color terms. Berlin and Kay 69.

Perceptual Color Descriptors 11 basic color terms. Berlin and Kay 69. Image copyright by Eric Rolph. Taken from: http://upload.wikimedia.org/wikipedia/commons/5/5c/Double-alaskan-rainbow.jpg

Perceptual Color Descriptors How to give each pixel an “11-colors” description ?

Perceptual Color Descriptors Learning Color Names from Real-World Images, J. van de Weijer, C. Schmid, J. Verbeek CVPR 2007.

Perceptual Color Descriptors Learning Color Names from Real-World Images, J. van de Weijer, C. Schmid, J. Verbeek CVPR 2007. chip-based real-world

Perceptual Color Descriptors Learning Color Names from Real-World Images, J. van de Weijer, C. Schmid, J. Verbeek CVPR 2007. For each color returns a probability distribution over the 11 basic colors. white red green yellow blue brown purple pink orange grey black

Perceptual Color Descriptors Applying Color Names to Image Description, J. van de Weijer, C. Schmid ICIP 2007. Outperformed state of the art color descriptors.

Perceptual Color Descriptors Using illumination invariants – black, gray and white are the same. “ Too much invariance” happens in other cases ( Local features and kernels for classification of texture and object categories: An in-depth study - Zhang, Marszalek, Lazebnik and Schmid. IJCV 2007, Learning the discriminative power-invariance trade-off - Varma and Ray. ICCV 2007 ). To conclude: Don’t solve imaginary problems.

Perceptual Color Descriptors This method is still not perfect. 11 color vector for purple (255,0,255) is: In real world images there are no such over-saturated colors. 0 0 0 0 0 0 0 0 0 1 0

Open Questions EMD variant that reduces the effect of large bins. Learning the ground distance for EMD. Learning the similarity matrix and normalization factor for QC.

Hands-On Code Example http://www.cs.huji.ac.il/~ofirpele/FastEMD/code/ http://www.cs.huji.ac.il/~ofirpele/QC/code/

Tutorial: Or “Ofir Pele” http://www.cs.huji.ac.il/~ofirpele/DFML_ECCV2010_tutorial/

ECCV2010: distance function and metric learning part 1

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ECCV2010: distance function and metric learning part 1

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