1. Learning and Vision: Discriminative Models
Chris Bishop and Paul Viola

2. Part II: Algorithms and Applications
Part I: Fundamentals
Part II: Algorithms and Applications
Support Vector Machines
Face and pedestrian detection
AdaBoost
Faces
Building Fast Classifiers
Trading off speed for accuracy…
Face and object detection
Memory Based Learning
Simard
Moghaddam

3. History Lesson 1950’s Perceptrons are cool
Very simple learning rule, can learn “complex” concepts
Generalized perceptrons are better -- too many weights
1960’s Perceptron’s stink (M+P)
Some simple concepts require exponential # of features
Can’t possibly learn that, right?
1980’s MLP’s are cool (R+M / PDP)
Sort of simple learning rule, can learn anything (?)
Create just the features you need
MLP’s stink
Hard to train : Slow / Local Minima
Perceptron’s are cool

4. Why did we need multi-layer perceptrons?
Problems like this seem to require very complex non-linearities.
Minsky and Papert showed that an exponential number of features is necessary to solve generic problems.

5. Why an exponential number of features?
14th Order???
120 Features
N=21, k=5 --> 65,000 features

6. MLP’s vs. Perceptron MLP’s are hard to train…
Takes a long time (unpredictably long)
Can converge to poor minima
MLP are hard to understand
What are they really doing?
Perceptrons are easy to train…
Type of linear programming. Polynomial time.
One minimum which is global.
Generalized perceptrons are easier to understand.
Polynomial functions.

7. Perceptron Training is Linear Programming
Polynomial time in the number of variables
and in the number of constraints.
What about linearly inseparable?

8. Rebirth of Perceptrons
How to train effectively
Linear Programming (… later quadratic programming)
Though on-line works great too.
How to get so many features inexpensively?!?
Kernel Trick
How to generalize with so many features?
VC dimension. (Or is it regularization?)
Support Vector Machines

9. Lemma 1: Weight vectors are simple
The weight vector lives in a sub-space spanned by the examples…
Dimensionality is determined by the number of examples not the complexity of the space.

10. Lemma 2: Only need to compare examples
This is called the kernel matrix.

11. Simple Kernels yield Complex Features

12. But Kernel Perceptrons Can Generalize Poorly

13. Perceptron Rebirth: Generalization
Too many features … Occam is unhappy
Perhaps we should encourage smoothness?
This not a convergent.
Smoother

14. Linear Program is not unique
The linear program can return any multiple of the correct
weight vector...
Slack variables & Weight prior
- Force the solution toward zero

15. Definition of the Margin
Geometric Margin: Gap between negatives and positives measured perpendicular to a hyperplane
Classifier Margin

16. Require non-zero margin
Allows solutions
with zero margin
Enforces a non-zero
margin between examples
and the decision boundary.

17. Constrained Optimization
Find the smoothest function that separates data
Quadratic Programming (similar to Linear Programming)
Single Minima
Polynomial Time algorithm

18. Constrained Optimization 2

19. SVM: examples

20. SVM: Key Ideas Augment inputs with a very large feature set
Polynomials, etc.
Use Kernel Trick(TM) to do this efficiently
Enforce/Encourage Smoothness with weight penalty
Introduce Margin
Find best solution using Quadratic Programming

21. SVM: Zip Code recognition
Data dimension: 256
Feature Space: 4 th order
roughly 100,000,000 dims

22. The Classical Face Detection Process
Larger
Scale
Smallest
Scale
50,000 Locations/Scales

23. Classifier is Learned from Labeled Data
Training Data
5000 faces
All frontal
108 non faces
Faces are normalized
Scale, translation
Many variations
Across individuals
Illumination
Pose (rotation both in plane and out)
This situation with negative examples is actually quite common… where negative examples are free.

24. Key Properties of Face Detection
Each image contains thousand locs/scales
Faces are rare per image
1000 times as many non-faces as faces
Extremely small # of false positives: 10-6
As I said earlier the classifier is evaluated 50,000
Faces are quite rare…. Perhaps 1 or 2 faces per image
A reasonable goal is to make the false positive rate less than the true positive rate...

25. Sung and Poggio

26. Rowley, Baluja & Kanade
First Fast System
- Low Res to Hi

27. Osuna, Freund, and Girosi

28. Support Vectors

29. P, O, & G: First Pedestrian Work
Flaws: very poor model for feature selection.
Little real motivation for features at all. Why not use the original image.

30. On to AdaBoost Given a set of weak classifiers
None much better than random
Iteratively combine classifiers
Form a linear combination
Training error converges to 0 quickly
Test error is related to training margin

31. AdaBoost Freund & Shapire Weak Classifier 1 Weights Increased Weak
Final classifier is
linear combination of weak classifiers

32. AdaBoost Properties

33. AdaBoost: Super Efficient Feature Selector
Features = Weak Classifiers
Each round selects the optimal feature given:
Previous selected features
Exponential Loss

34. Boosted Face Detection: Image Features
“Rectangle filters”
Similar to Haar wavelets
Papageorgiou, et al.
For real problems results are only as good as the features used...
This is the main piece of ad-hoc (or domain) knowledge
Rather than the pixels, we have selected a very large set of simple functions
Sensitive to edges and other critcal features of the image
** At multiple scales
Since the final classifier is a perceptron it is important that the features be non-linear… otherwise the final classifier will be a simple perceptron.
We introduce a threshold to yield binary features
Unique Binary Features

37. Feature Selection For each round of boosting:
Evaluate each rectangle filter on each example
Sort examples by filter values
Select best threshold for each filter (min Z)
Select best filter/threshold (= Feature)
Reweight examples
M filters, T thresholds, N examples, L learning time
O( MT L(MTN) ) Naïve Wrapper Method
O( MN ) Adaboost feature selector

38. Example Classifier for Face Detection
A classifier with 200 rectangle features was learned using AdaBoost
95% correct detection on test set with 1 in 14084
false positives.
Not quite competitive...
ROC curve for 200 feature classifier

39. Building Fast Classifiers
Given a nested set of classifier hypothesis classes
Computational Risk Minimization vs
false
neg
determined by
% False Pos
% Detection 50
In general simple classifiers, while they are more efficient, they are also weaker.
We could define a computational risk hierarchy (in analogy with structural risk minimization)…
A nested set of classifier classes
The training process is reminiscent of boosting…
- previous classifiers reweight the examples used to train subsequent classifiers
The goal of the training process is different
- instead of minimizing errors minimize false positives
FACE
IMAGE
SUB-WINDOW
Classifier 1 F T
NON-FACE
Classifier 3
Classifier 2

40. Other Fast Classification Work
Simard
Rowley (Faces)
Fleuret & Geman (Faces)

41. Cascaded Classifier
50%
20% 2%
IMAGE
SUB-WINDOW
1 Feature
5 Features
20 Features
FACE F F F
NON-FACE
NON-FACE
NON-FACE
A 1 feature classifier achieves 100% detection rate and about 50% false positive rate.
A 5 feature classifier achieves 100% detection rate and 40% false positive rate (20% cumulative)
using data from previous stage.
A 20 feature classifier achieve 100% detection rate with 10% false positive rate (2% cumulative)

42. Comparison to Other Systems
(94.8)
Roth-Yang-Ahuja
94.4
Schneiderman-Kanade
89.9
90.1
89.2
86.0
83.2
Rowley-Baluja-Kanade
93.7
91.8
91.1
90.8
90.0
88.8
85.2
78.3
Viola-Jones
422
167
110 95 78 65 50 31 10
False Detections
Detector

43. Output of Face Detector on Test Images

44. Solving other “Face” Tasks
Profile Detection
Facial Feature Localization
Demographic
Analysis

45. Feature Localization Surprising properties of our framework
The cost of detection is not a function of image size
Just the number of features
Learning automatically focuses attention on key regions
Conclusion: the “feature” detector can include a large contextual region around the feature

46. Feature Localization Features
Learned features reflect the task

47. Profile Detection

48. More Results

49. Profile Features

50. Thanks to
Andrew Moore
One-Nearest Neighbor …One nearest neighbor for fitting is described shortly…
Similar to Join The Dots with two Pros and one Con.
PRO: It is easy to implement with multivariate inputs.
CON: It no longer interpolates locally.
PRO: An excellent introduction to instance-based learning…

51. 1-Nearest Neighbor is an example of…. Instance-based learning
Thanks to
Andrew Moore
1-Nearest Neighbor is an example of…. Instance-based learning
x y1
x y2
x y3 .
xn yn
A function approximator that has been around since about 1910.
To make a prediction, search database for similar datapoints, and fit with the local points.
Four things make a memory based learner:
A distance metric
How many nearby neighbors to look at?
A weighting function (optional)
How to fit with the local points?

52. Nearest Neighbor Four things make a memory based learner:
Thanks to
Andrew Moore
Nearest Neighbor
Four things make a memory based learner:
A distance metric Euclidian
How many nearby neighbors to look at? One
A weighting function (optional) Unused
How to fit with the local points? Just predict the same output as the nearest neighbor.

53. Multivariate Distance Metrics
Thanks to
Andrew Moore
Multivariate Distance Metrics
Suppose the input vectors x1, x2, …xn are two dimensional:
x1 = ( x11 , x12 ) , x2 = ( x21 , x22 ) , …xN = ( xN1 , xN2 ).
One can draw the nearest-neighbor regions in input space.
Dist(xi,xj) = (xi1 – xj1)2 + (xi2 – xj2)2
Dist(xi,xj) =(xi1 – xj1)2+(3xi2 – 3xj2)2
The relative scalings in the distance metric affect region shapes.

54. Euclidean Distance Metric
Thanks to
Andrew Moore
Euclidean Distance Metric
Or equivalently,
where
Other Metrics…
Mahalanobis, Rank-based, Correlation-based (Stanfill+Waltz, Maes’ Ringo system…)

55. Notable Distance Metrics
Thanks to
Andrew Moore
Notable Distance Metrics

56. Simard: Tangent Distance

57. Simard: Tangent Distance

58. FERET Photobook Moghaddam & Pentland (1995)
Thanks to
Baback Moghaddam
FERET Photobook Moghaddam & Pentland (1995)

59. Eigenfaces Moghaddam & Pentland (1995)
Thanks to
Baback Moghaddam
Eigenfaces Moghaddam & Pentland (1995)
Normalized Eigenfaces

60. Thanks to
Baback Moghaddam
Euclidean (Standard) “Eigenfaces” Turk & Pentland (1992) Moghaddam & Pentland (1995)
Projects all the training faces
onto a universal eigenspace
to “encode” variations (“modes”)
via principal components (PCA)
Uses inverse-distance
as a similarity measure
for matching & recognition

61. Euclidean Similarity Measures
Thanks to
Baback Moghaddam
Metric (distance-based) Similarity Measures
template-matching, normalized correlation, etc
Disadvantages
Assumes isotropic variation (that all variations are equi-probable)
Can not distinguish incidental changes from the critical ones
Particularly bad for Face Recognition in which so many are incidental!
for example: lighting and expression

62. PCA-Based Density Estimation Moghaddam & Pentland ICCV’95
Thanks to
Baback Moghaddam
Solve for minimal KL divergence residual for the orthogonal subspace:
Perform PCA and factorize into (orthogonal)
Gaussians subspaces:
See Tipping & Bishop (97) for an ML derivation within a more general factor analysis framework (PPCA)

63. dual subspaces for dyads (image pairs)
Thanks to
Baback Moghaddam
Bayesian Face Recognition Moghaddam et al ICPR’96, FG’98, NIPS’99, ICCV’99
Intrapersonal
Extrapersonal
dual subspaces for dyads (image pairs)
Equate “similarity” with posterior on
Moghaddam ICCV’95
PCA-based density estimation

64. Intra-Extra (Dual) Subspaces
Thanks to
Baback Moghaddam
Intra-Extra (Dual) Subspaces
specs
light
mouth
smile
Intra
Extra
Note that the Extra and Standard subspaces are qualitatively the same (eg. both have beard-encoders, not present in the Intra space). In fact, their 1st eigenvector is almost exactly the same (bangs in the forehead)
Standard
PCA

65. Intra-Extra Subspace Geometry
Thanks to
Baback Moghaddam
The lower-order eigenvectors/eigenvalues allow us to “visualize” these two distributions and to judge their (relative) size and orientation at least.
The key thing is that we’re effectively talking about two “pencils” which only intersect near the origin (of delta space). Discriminating between these two subspaces (a binary classification task) is often much easier than discriminating between N faces (N-ary classification). In fact, not only easier, but very often much more effective.
Two “pancake” subspaces with different orientations intersecting near the origin. If each is in fact Gaussian, then the optimal discriminant is hyperquadratic

66. Bayesian Similarity Measure
Thanks to
Baback Moghaddam
Bayesian (MAP) Similarity
priors can be adjusted to reflect operational settings or used for Bayesian fusion (evidential “belief” from another level of inference)
Likelihood (ML) Similarity
Intra-only (ML) recognition is only slightly inferior to MAP (by few %). Therefore, if you had to pick only one subspace to work in, you should pick Intra – and not standard eigenfaces!

67. FERET Identification: Pre-Test
Thanks to
Baback Moghaddam
Bayesian (Intra-Extra)
Standard (Eigenfaces)

68. Official 1996 FERET Test Bayesian (Intra-Extra) Standard (Eigenfaces)
Thanks to
Baback Moghaddam
Bayesian (Intra-Extra)
Standard (Eigenfaces)

69. ..let’s leave distance metrics for now, and go back to….
Thanks to
Andrew Moore
One-Nearest Neighbor
Objection:
That noise-fitting is really objectionable.
What’s the most obvious way of dealing with it?

70. k-Nearest Neighbor Four things make a memory based learner:
Thanks to
Andrew Moore
k-Nearest Neighbor
Four things make a memory based learner:
A distance metric Euclidian
How many nearby neighbors to look at? k
A weighting function (optional) Unused
How to fit with the local points? Just predict the average output among the k nearest neighbors.

71. k-Nearest Neighbor (here k=9)
Thanks to
Andrew Moore
k-Nearest Neighbor (here k=9)
A magnificent job of noise-smoothing. Three cheers for 9-nearest-neighbor.
But the lack of gradients and the jerkiness isn’t good.
Appalling behavior! Loses all the detail that join-the-dots and 1-nearest-neighbor gave us, yet smears the ends.
Fits much less of the noise, captures trends. But still, frankly, pathetic compared with linear regression.
K-nearest neighbor for function fitting smoothes away noise, but there are clear deficiencies.
What can we do about all the discontinuities that k-NN gives us?

72. Kernel Regression Four things make a memory based learner:
Thanks to
Andrew Moore
Kernel Regression
Four things make a memory based learner:
A distance metric Scaled Euclidian
How many nearby neighbors to look at? All of them
A weighting function (optional) wi = exp(-D(xi, query)2 / Kw2)
Nearby points to the query are weighted strongly, far points weakly. The KW parameter is the Kernel Width. Very important.
How to fit with the local points? Predict the weighted average of the outputs:
predict = Σwiyi / Σwi

73. Kernel Regression in Pictures
Thanks to
Andrew Moore
Kernel Regression in Pictures
Take this dataset…
..and do a kernel prediction with xq (query) = 310, Kw = 50.

74. Thanks to
Andrew Moore
Varying the Query
xq = 150
xq = 395

75. Varying the kernel width
Thanks to
Andrew Moore
Varying the kernel width
xq = 310
KW = 50 (see the double arrow at top of diagram)
xq = 310 (the same)
KW = 100
KW = 150
Increasing the kernel width Kw means further away points get an opportunity to influence you.
As Kwinfinity, the prediction tends to the global average.

76. Kernel Regression Predictions
Thanks to
Andrew Moore
Kernel Regression Predictions
KW=10
KW=20
KW=80
Increasing the kernel width Kw means further away points get an opportunity to influence you.
As Kwinfinity, the prediction tends to the global average.

77. Kernel Regression on our test cases
Thanks to
Andrew Moore
Kernel Regression on our test cases
KW=1/32 of x-axis width.
It’s nice to see a smooth curve at last. But rather bumpy. If Kw gets any higher, the fit is poor.
Quite splendid. Well done, kernel regression. The author needed to choose the right KW to achieve this.
KW=1/16 axis width.
Nice and smooth, but are the bumps justified, or is this overfitting?
Choosing a good Kw is important. Not just for Kernel Regression, but for all the locally weighted learners we’re about to see.

78. Weighting functions Let d=D(xi,xquery)/KW
Thanks to
Andrew Moore
Weighting functions
Let
d=D(xi,xquery)/KW
Then here are some commonly used weighting functions…
(we use a Gaussian)

79. Kernel Regression can look bad
Thanks to
Andrew Moore
Kernel Regression can look bad
KW = Best.
Clearly not capturing the simple structure of the data.. Note the complete failure to extrapolate at edges.
Also much too local. Why wouldn’t increasing Kw help? Because then it would all be “smeared”.
Three noisy linear segments. But best kernel regression gives poor gradients.
Time to try something more powerful…

80. Locally Weighted Regression
Thanks to
Andrew Moore
Locally Weighted Regression
Kernel Regression:
Take a very very conservative function approximator called AVERAGING. Locally weight it.
Locally Weighted Regression:
Take a conservative function approximator called LINEAR REGRESSION. Locally weight it.
Let’s Review Linear Regression….

81. Unweighted Linear Regression
Thanks to
Andrew Moore
Unweighted Linear Regression
You’re lying asleep in bed. Then Nature wakes you.
YOU: “Oh. Hello, Nature!”
NATURE: “I have a coefficient β in mind. I took a bunch of real numbers called x1, x2 ..xN thus: x1=3.1,x2=2, …xN=4.5.
For each of them (k=1,2,..N), I generated yk= βxk+εk
where εk is a Gaussian (i.e. Normal) random variable with mean 0 and standard deviation σ. The εk’s were generated independently of each other.
Here are the resulting yi’s: y1=5.1 , y2=4.2 , …yN=10.2”
You: “Uh-huh.”
Nature: “So what do you reckon β is then, eh?”
WHAT IS YOUR RESPONSE?

82. Locally Weighted Regression
Thanks to
Andrew Moore
Locally Weighted Regression
Four things make a memory-based learner:
A distance metric Scaled Euclidian
How many nearby neighbors to look at? All of them
A weighting function (optional) wk = exp(-D(xk, xquery)2 / Kw2) Nearby points to the query are weighted strongly, far points weakly. The Kw parameter is the Kernel Width.
How to fit with the local points?
First form a local linear model. Find the β that minimizes the locally weighted sum of squared residuals:
Then predict ypredict=βT xquery

83. How LWR works Query Thanks to Andrew Moore
Linear regression not flexible but trains like lightning.
Locally weighted regression is very flexible and fast to train.

84. LWR on our test cases KW = 1/16 of x-axis width.
Thanks to
Andrew Moore
LWR on our test cases
KW = 1/16 of x-axis width.
KW = 1/32 of x-axis width.
KW = 1/8 of x-axis width.
Nicer and smoother, but even now, are the bumps justified, or is this overfitting?

85. Features, Features, Features
In almost every case:
Good Features beat Good Learning
Learning beats No Learning
Critical classifier ratio:
AdaBoost >> SVM
This is not to say that a