Distance Metric Learning: A Comprehensive Survey Liu Yang Advisor: Rong Jin May 8th, 2006.

Distance Metric Learning: A Comprehensive Survey

Liu Yang Advisor: Rong Jin

May 8th, 2006

Outline

Introduction Supervised Global Distance Metric Learning Supervised Local Distance Metric Learning Unsupervised Distance Metric Learning Distance Metric Learning based on SVM Kernel Methods for Distance Metrics Learning Conclusions

Introduction

Definition Distance Metric learning is to learn a distance metric for the

input space of data from a given collection of pair of similar/dissimilar points that preserves the distance relation among the training data pairs.

Importance Many machine learning algorithms, heavily rely on the

distance metric for the input data patterns. e.g. kNN A learned metric can significantly improve the performance in

classification, clustering and retrieval tasks: e.g. KNN classifier, spectral clustering, content-based image

retrieval (CBIR).

Contributions of this Survey

Review distance metric learning under different learning conditions supervised learning vs. unsupervised learning learning in a global sense vs. in a local sense distance matrix based on linear kernel vs. nonlinear kernel

Discuss central techniques of distance metric learning K nearest neighbor dimension reduction semidefinite programming kernel learning large margin classification

Supervised Distance Metric Learning

Local

Local Adaptive Distance Metric Learning

Neighborhood Components Analysis

Relevant Component Analysis

Unsupervised Distance Metric Learning Nonlinear embedding

LLE, ISOMAP, Laplacian Eigenmaps

Distance Metric Learning based on SVM

Large Margin Nearest Neighbor Based Distance Metric Learning

Cast Kernel Margin Maximization into a SDP problem

Kernel Methods for Distance Metrics Learning

Kernel Alignment with SDP

Learning with Idealized Kernel

Linear embeddingPCA, MDS

Global Distance Metric Learning by Convex Programming

Outline

Introduction Supervised Global Distance Metric Learning Supervised Local Distance Metric Learning Unsupervised Distance Metric Learning Distance Metric Learning based on SVM Kernel Methods for Distance Metrics Learning

Supervised Global Distance Metric Learning (Xing et al. 2003)

Goal : keep all the data points within the same classes close,

while separating all the data points from different classes.

Formulate as a constrained convex programming problem minimize the distance between the data pairs in S Subject to data pairs in D are well separated

22A

Equivalence constraints: {( , ) | and belong to the same class}

Inequivalence constraints: {( , ) | and belong to different classes},

d ( , ) ( ) ( ), is the distanc

i j i j

i j i j

T m m

A

S x x x x

D x x x x

x y x y x y A x y A S

e metric

Global Distance Metric Learning (Cont’d)

A is positive semi-definite Ensure the negativity and the triangle inequality of the metric

The number of parameters is quadratic in the number of features Difficult to scale to a large number of features

Simplify the computation

2 2

( , ) ( , )

min . . 0, 1m m

i j i j

i j i jA R x x S x x DA A

x x s t A x x

(a)Data Dist. of the original dataset

(b) Data scaled by the global metric

Global Distance Metric Learning: Example I

Keep all the data points within the same classes close Separate all the data points from different classes

Global Distance Metric Learning: Example II

Diagonalize distance metric A can simplify computation, but could lead to disastrous results

(a) Original data (c) Rescaling by learned diagonal A

(b) rescaling by learned full A

(a) Data Dist. of the original dataset

Multimodal data distributions prevent global distance metrics from simultaneously satisfying constraints on within-class compactness and between-class separability.

(b) Data scaled by the global metric

Problems with Global Distance Metric Learning

Outline


Supervised Local Distance Metric Learning


Local Feature Relevance Locally Adaptive Feature Relevance Analysis Local Linear Discriminative Analysis

Neighborhood Components Analysis

Relevant Component Analysis


K Nearest Neighbor Classifier

0

0 0

1 1

0( )0

( ) : nearest neighbors of

, , , , : training examples

1( )

0 . .

1Pr( | ) ( )

( )i

n n

i

i

ix N x

N x x

x y x y

y jy j

o w

j x y jN x

Modified local neighborhood by a distance metric Elongate the distance along the dimensions where

the class labels change rapidly Squeeze the distance along the dimensions that are

almost independent from the class labels

Assumption of KNN Pr(y|x) in the local NN is constant or smooth

However, this is not necessarily true! Near class boundaries Irrelevant dimensions


Local Feature Relevance[J. Friedman,1994]

(x)p(x)dxEf f

i[ | ] (x)p(x|x =z)dx,iE f x z f i

p(x) ( )p(x|x =z) =

p(x') ( ' )i

i

x z

x z

2 2 2 2( ) [( (x)-E ) | ] [( (x)-E( (x) | ) | ] ( [ | ])i i i i iI z E f f x z E f f x z x z Ef E f x z

2 2 2

1

( ) ( ) / ( )p

i i i k kk

r z I z I z

1( , , )mz z z

ix z

ix z

x = z

Assume least-squared estimate for predicting f(x) is

Conditioned at , then the least-squared estimate of f(x)

The improvement in prediction error with knowing

Consider , a measure of relative influence of the ith input variable to the variation of f(x) at is given by

Locally Adaptive Feature Relevance Analysis [C. Domeniconi, 2002]

20

01 0

[ ( | X) ( | x )](X, x )

( | x )

J

j

p j p jr

p j

0x

Use a Chi-squared distance analysis to compute metric for producing a neighborhood, in which

The posterior probabilities are approximately constant Highly adaptive to query locations

Chi-squared distance between the true and estimated posterior at the test point

Use the Chi-squared distance for feature relevance: ---- to tell to which extent the ith dimension can be relied on for predicting p(j| )0x

Local Relevance Measure in ith Dimension

2

i1

[Pr( | ) Pr( | )]r (z) =

Pr( | )

Ji i

ji i

j z j x z

j x z

ir (z)

Pr( | )i ij x z

is a conditional expectation of p(j|x)Pr( | ) (Pr(j|x) | )i i i ij x z E x z

ir (z) 0x

The closer is to p(j|z), the more information the ith dimension provides for predicting p(j|z)

measures the distance between Pr(j|z) and the conditional expectation of Pr(j|x) at location z

Calculate for each point z in the neighborhood of

0i 0 0 1 0 0

01

( )w ( ) , where ( ) (max ( )) ( )

( )

t= 1 or 2, corresponds to linear and quadratic weighting.

tqi

i j j iqt

ll

R xx R x r x r x

R x

q2

i=1

(x,y) = ( )i i iD w x y

Locally Adaptive Feature Relevance Analysis

0

0(x )

1(x ) ( )i i

z N

r r zK

0(x )N is the neighborhood of 0x

A local relevance measure in dimension i

Relative relevance

Weighted distance

Local Linear Discriminative Analysis[T. Hastie et al. 1996]

Sb : the between-class covariance matrix

Sw : the within-class covariance matrix

-1T = Sw Sb. LDA finds principle eigenvectors of matrix

to keep patterns from the same class close

separate patterns from different classes apart

LDA metric : stacking principle eigenvectors of T together

Local Linear Discriminative Analysis

1 1 1 1

2 2 2 2[ I]w w b w wS S S S S

0x

0x

Need local adaptation of the nearest neighbor metric

Initialize as identical matrix Given a testing point , iterate below two steps:

Estimate Sb and Sw based on the local neighbor of measured by Form a local metric behaving like LDA metric

is a small tuning parameter to prevent neighborhoods extending to infinity

Local Sb shows the inconsistency of the class centriods The estimated metric shrinks the neighborhood in directions in which the local class centroids differ to produce a neighborhood in which the class centriod coincide shrinks neighborhoods in directions orthogonal to these local decision boundaries, and elongates them parallel to the boundaries.

Local Linear Discriminative Analysis

Overfitting, Scalability problem, # parameters is quadratic in #features.

Neighborhood Components Analysis[J. Goldberger et al. 2005]

ix

2

i j

i 2

i k

exp( Ax Ax )Here C { | },

exp( Ax Ax )i j ij

k i

j c c p

n

ii=1

f(A) = p ,

ipi

ijj C

p

NCA learns a Mahalanobis distance metric for the KNN classifier by maximizing the leave-one-out cross validation.

The probability of classifying correctly, weighted counting involving pairwise distance

The expected number of correctly classification points:

RCA [N. Shen et al. 2002]

unlabeled data labeled datachuklet data

^ ^ ^T

j jji ji1 1

1C (x m )(x m ) ,

jnk

j ip

1

^ 2y C x

j^

njji i=1chunklet j : {x } , with mean m

Constructs a Mahalanobis distance metric based on a sum of in-chunklet covariance matrices

Chunklet : data have same but unknown class labels

Sum of in-chunklet covariance matrices for p points in k chunklets:

Apply linear transformation

Information maximization under chunklet constraints[A. Bar-Hillel etal, 2003]

Maximizes the mutual information I(X,Y) Constraints: within-chunklet compactness

T

2

jB

1 1 B

Let B =A A, (*) can be further written into

1max | B | s.t. m , B 0

p

jnk

jij i

x K

2

yj

1 1

yj

1max I(X,Y) s.t. m . (*)

p

m is the transformed mean in the jth chunklet.

K is threshold constant.

jnk

jif F

j i

y K

RCA algorithm applied to synthetic Gaussian data

(a) The fully labeled data set with 3 classes. (b) Same data unlabeled; classes' structure is less evident. (c) The set of chunklets (d) The centered chunklets, and their empirical covariance. (e) The RCA transformation applied to the chunklets. (centered) (f) The original data after applying the RCA transformation.

Outline


Unsupervised Distance Metric Learning

A Unified Framework for Dimension Reduction Solution 1 Solution 2

linear nonlinear

Global PCA, MDS ISOMAP

Local LLE, Laplacian Eigenmap

Most dimension reduction approaches are to learn a distance metric without label information. e.g. PCA

I will present five methods for dimensionality reduction.

Dimensionality Reduction Algorithms

PCA finds the subspace that best preserves the variance of the data.

MDS finds the subspace that best preserves the interpoint distances.

Isomap finds the subspace that best preserves the geodesic interpoint distances. [Tenenbaum et al, 2000].

LLE finds the subspace that best preserves the local linear structure of the data [Roweis and Saul, 2000].

Laplacian Eigenmap finds the subspace that best preserves local neighborhood information in the adjacency graph [M. Belkin and P. Niyogi,2003].

Multidimensional Scaling (MDS)

MDS finds the rank m projection that best preserves the inter-point distance given by matrix D

Converts distances to inner products

Calculate X

Rank m projections Y closet to X

Given the distance matrix among cities, MDS produces this map:

1MDS MDS 2m mY= V ( )

TB= (D)= X X

1MDS MDS 2

MDS MDS[V , ] =eig(B)

X = V ( )

PCA (Principal Component Analysis)

1PCA MDS PCA MDS PCA PCA MDS2V XV , , Y ( ) Y

=Var(X)

PCA PCA[V , ]=eig( )

PCAY = V Xm

PCA finds the subspace that best preserves the data variance.

PCA projection of X with rank m

PCA vs. MDS

In the Euclidean case, MDS only differs from PCA by starting with D and calculating X.

A B

Isometric Feature Mapping (ISOMAP)[Tenenbaum et al, 2000]

Geodesic :the shortest curve on a manifold that connects two points on the manifold

e.g. on a sphere, geodesics are great circles

Geodesic distance: length of the geodesic

Points far apart measured by geodesic dist. appear close measured by Euclidean dist.

ISOMAP

Take a distance matrix as input

Construct a weighted graph G based on neighborhood relations

Estimate pairwise geodesic distance by “a sequence of short hops” on G

Apply MDS to the geodesic distance matrix

Locally Linear Embedding (LLE)[Roweis and Saul, 2000]

LLE finds the subspace that best preserves the local linear structure of the data

Assumption: manifold is locally “linear”

Each sample in the input space is a linearly weighted average of its neighbors.

A good projection should best preserve this geometric locality property

LLE W: a linear representation of every data point by its neighbors Choose W by minimized the reconstruction error

Calculate a neighborhood preserving mapping Y, by minimizing the reconstruction error

Y is given by the eigenvectors of the m lowest nonzero eigenvalues of matrix

2n

i iji=1 1

ij i ij j i1

minimizing x W

s.t. W 1, x ; W 0 if x is not a neighbor of x

K

ijj

n

j

x

* *i

W1

(Y)= y , where W arg min (W)K

ij iji

W y

T(I-W) (I-W)

Laplacian Eigenmap finds the subspace that best preserves local neighborhood information in adjacency graph

Graph Laplacian: Given a graph G with weight matrix W D is a diagonal matrix with L =D –W is the graph Laplacian

Detailed steps: Construct adjacency graph G. Weight the edges: Generalized eigen-decomposition of Embedding : eigenvectors with top m nonzero eigenvalues

Laplacian Eigenmap [M. Belkin and P. Niyogi,2003]

ii jij

D W

Lf= DfijW 1, if nodes i and j are connected, and 0 otw.

A Unified Framework for Dimension Reduction Algorithms

All use an eigendecomposition to obtain a lower-dimensional embedding of data lying on a non-linear manifold. Normalize affinity matrix

The embedding of has two alternative solutions Solution 1 : (MDS & Isomap)

is the best approximation of in the squared error sense.

Solution 2 : (LLE & Laplacian Eigenmap)

i it tiy with y = v

i it t ite with e = v

^

t t(H) the m largest positive eigenvalues and eigenvectors veig

ix

i je ,e^

Hij

ijj

H ^

H H

Outline


Distance Metric Learning based on SVM

Large Margin Nearest Neighbor Based Distance Metric Learning Objective Function Reformulation as SDP

Cast Kernel Margin Maximization into a SDP Problem Maximum Margin Cast into SDP problem Apply to Hard Margin and Soft Margin

After training k=3 target neighbors lie within a smaller radius differently labeled inputs lie outside this smaller radius with a margin of at least one unit distance.

Large Margin Nearest Neighbor Based Distance Metric Learning[K. Weinberger et al., 2006]

Learns a Mahanalobis distance metric in the kNN classification setting by SDP, that Enforces the k-nearest neighbors belong to the same class examples from different classes are separated by a large margin

Large Margin Nearest Neighbor Based Distance Metric Learning

Cost function:

Penalize large distances between each input and its target neighbors

The hinge loss is incurred by differently labeled inputs whose distances do not exceed the distance from input to any of its target neighbors by one absolute unit of distance -> do not threaten to invade each other’s neighborhoods

2 2 2

ij i j ij i j i l 22 2ij ijl

(L) = L(x -x ) C (1 )[1 L(x -x ) L(x -x ) ]

[ ] max(z,0) denotes the standard hinge loss and the constant C > 0.

ily

z

ij i j

ij j i

y {0,1} indicate whether or not the label y and y match

{0,1} indicate whether x is a target neighbor of x

ix

Reformulation as SDP

Ti j i j

M

T Ti j i j i l i l

The resulting SDP is :

min (x x ) M(x x ) C (1 )

. . (x x ) M(x x ) (x x ) M(x x ) 1

0,M =0

ij ij il ijlij ijl

ijl

ijl

y

s t

2

i j i j i j2Let L(x -x ) (x -x ) M(x -x ), and introducing slack variable T

ijl

Cast Kernel Margin Maximization into a SDP Problem [G. R. G. Lanckriet et al, 2004]

Maximum margin : the decision boundary has the maximum minimum distance from the closest training point. Hard Margin: linearly separable Soft Margin: nonlinearly separable

The performance measure, generalized from dual solution of different maximizing margin problem

T T T, (K) max 2 ( ( ) ) : 0, y 0

with 0 on the training data w.r.t K. G is Gram matrix.

Cw e G K I C

Cast into SDP Problem

2 tr 2 tr , trK =0min (K ) s.t. trace(K)=c. Here (K ) =w (K )S Sw w

Hard Margin

1-norm soft margin

2-norm soft margin

tr tr ,0 trK =0min (K ) s.t. trace(K)=c. Here (K ) =w (K )w w

1 tr 1 tr C,0 trK =0min (K ) s.t. trace(K)=c. Here (K ) =w (K )S Sw w

K,t, , ,

tr

T T

min

. . trace(K)=c, K =0, 0, 0,

G(K ) y 0

( y) t-2C

trn

t

s t

I e

e e

,K =0min (K) . . trace(K) = cCw s t

Outline


Kernel Methods for Distance Metrics Learning

Learning a good kernel is equivalent to distance metric learning

Kernel Alignment

Kernel Alignment with SDP

Learning with Idealized Kernel

Ideal Kernel The Idealized Kernel

Kernel Alignment [N. Cristianini,2001]

T^ 1 F

1 2

1 1 F

K , yyA(S, k ,k ) , y { 1}

K ,K

m

m

A measure of similarity between two kernel functions or between a kernel and a target function

The inner product between two kernel matrices based on kernel k1 and k2.

The alignment of K1 and K2 w.r.t S:

Measure the degree of agreement between a kernel and a given learning task.

1 2 1 i j 2 i jF, 1

K ,K K (x , x )K (x , x )n

i j

^

1 2 F1 2

1 1 2 2F F

K ,KA(S, k ,k )

K ,K K ,K

Kernel Alignment with SDP [G. R. G. Lanckriet et al, 2004]

Optimizing the alignment between a set of labels and a kernel matrix using SDP in a transductive setting.

Optimizing an objective function over the training data block -> automatic tuning of testing data block

Introduce A with , this reduces toT

trFA,K

T

n

max K , yy

A K. . trace(A) 1, K =0, =0.

K Is t

^T

1K

max A( ,K , yy ) s.t. K =0, trace(K) =1S

tr tr,t

ij i j tr tTtr,t

K KK= , where K (x ), (x ) , i, j =1, , n n .

K K

T K K =A and trace(A) 1

Learning with Idealized Kernel[J. T. Kwok and I.W. Tsang,2003]

Idealize a given kernel by making it more similar to the ideal kernel matrix.

Ideal kernel:

Idealized kernel:

The alignment of will be greater than k, if are the number of positive and negative samples.

Under the original distance metric M:

i j*i j

i j

1, y(x ) y(x )k (x , x )

0, y(x ) y(x )

~

*k = k + k2

*

2 2

K,K

n n

~

k

2~ ~ ~ ij i j

2ij i j

d y =yK K 2K

d y yii jj ij

T 2 Ti j i j ij i j i j k(x , x ) = x Mx , M =0; d (x - x ) M(x - x )

,n n

iji j i j

2 TS2B, ,

(x ,x ) (x ,x )

~ ~2 2ij ij i j

~2 2ij ij i j

1 1min B , where B= AA

2

, (x , x ). . , 0, 0,

, (x , x )

ij D ijS DS D

ij

ij

ij

CC

N N

d d Ds t

d d S

Idealized kernel

We modify Search for a matrix A under which

different classes : pulled apart by an amount of at least same class :getting close together.

Introduce slack variables for error tolerance

2~ij i j2

ij 2ij i j

d y = yd

d y y

2~T T

ij i j i j(x - x ) A A(x - x )d

Conclusions

A comprehensive review, covers: Supervised distance metric learning Unsupervised distance metric learning Maximum margin based distance metric learning approaches Kernel methods towards distance metrics

Challenge: Unsupervised distance metric learning. Going local in a principle manner. Learn an explicit nonlinear distance metric in the local sense. Efficiency issue.

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Distance Metric Learning: A Comprehensive Survey Liu Yang Advisor: Rong Jin May 8th, 2006.

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