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

2. 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

3. 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).

4. 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

5. Global Distance Metric Learning by Convex Programming
Local Adaptive Distance
Metric Learning
Supervised
Distance Metric Learning
Relevant Component Analysis
Local
Neighborhood Components Analysis
Linear embedding
PCA, MDS
Unsupervised
Distance Metric Learning
Nonlinear embedding
LLE, ISOMAP, Laplacian Eigenmaps
Large Margin Nearest Neighbor
Based Distance Metric Learning
Distance Metric Learning
based on SVM
Cast Kernel Margin
Maximization into a SDP problem
Kernel Alignment with SDP
Kernel Methods for
Distance Metrics Learning
Learning with Idealized Kernel

6. 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

7. 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

8. 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

9. Global Distance Metric Learning: Example I
(b) Data scaled by the global metric
Data Dist. of the original dataset
Keep all the data points within the same classes close
Separate all the data points from different classes

10. Global Distance Metric Learning: Example II
Original data
(b) rescaling by learned full A
(c) Rescaling by learned diagonal A
Diagonalize distance metric A can simplify computation, but could lead to disastrous results

11. Problems with Global Distance Metric Learning
Data Dist. of the original dataset
(b) Data scaled by the global metric
Multimodal data distributions prevent global distance metrics from simultaneously satisfying constraints on within-class compactness and between-class separability.

12. 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

13. Supervised Local Distance Metric Learning
Local Adaptive Distance Metric Learning
Local Feature Relevance
Locally Adaptive Feature Relevance Analysis
Local Linear Discriminative Analysis
Neighborhood Components Analysis
Relevant Component Analysis

14. Local Adaptive Distance Metric Learning
K Nearest Neighbor Classifier

15. Local Adaptive Distance Metric Learning
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
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

16. Local Feature Relevance [J. Friedman,1994]
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

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

18. Local Relevance Measure in ith Dimension
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
is a conditional expectation of p(j|x)
The closer is to p(j|z), the more information the ith dimension provides for predicting p(j|z)

19. Locally Adaptive Feature Relevance Analysis
A local relevance measure in dimension i
Relative relevance
Weighted distance
is the neighborhood of

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

21. Discriminative Analysis
Local Linear
Discriminative Analysis
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

22. Local Linear Discriminative Analysis
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.

23. Neighborhood Components Analysis [J. Goldberger et al. 2005]
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:
Overfitting, Scalability problem, # parameters is quadratic in #features.

24. RCA [N. Shen et al. 2002]
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
unlabeled data
chuklet data
labeled data

25. Information maximization under chunklet constraints [A
Information maximization under chunklet constraints [A. Bar-Hillel etal, 2003]
Maximizes the mutual information I(X,Y)
Constraints: within-chunklet compactness

26. 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.

27. 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

28. Unsupervised Distance Metric Learning
Most dimension reduction approaches are to learn a distance metric without label information. e.g. PCA
I will present five methods for dimensionality reduction.
A Unified Framework for Dimension Reduction
Solution 1
Solution 2
linear
nonlinear
Global
PCA, MDS
ISOMAP
Local
LLE, Laplacian Eigenmap

29. 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].

30. 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:

31. PCA (Principal Component Analysis)
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.

32. 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. A B

33. 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

34. 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

35. 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

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

37. 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)

38. 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

39. 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

40. 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
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.

41. 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

42. Reformulation as SDP

43. Cast Kernel Margin Maximization into a SDP Problem [G. R. G
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

44. Cast into SDP Problem Hard Margin 1-norm soft margin

45. 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

46. 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

47. Kernel Alignment [N. Cristianini,2001]
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.

48. 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 to

49. 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:

50. 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

51. 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.