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Learning and Vision: Discriminative Models

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Learning and Vision:Discriminative Models

Chris Bishop and Paul Viola

- 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

- 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?

- Some simple concepts require exponential # of features
- 1980’s MLP’s are cool (R+M / PDP)
- Sort of simple learning rule, can learn anything (?)
- Create just the features you need

- 1990 MLP’s stink
- Hard to train : Slow / Local Minima

- 1996 Perceptron’s are cool

- 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.

14th Order???

120 Features

N=21, k=5 --> 65,000 features

- 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.

What about linearly inseparable?

Polynomial time in the number of variables

and in the number of constraints.

Support Vector Machines

- 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?)

- 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.

- Too many features … Occam is unhappy
- Perhaps we should encourage smoothness?

Smoother

The linear program can return any multiple of the correct

weight vector...

Slack variables & Weight prior

- Force the solution toward zero

- Geometric Margin: Gap between negatives and positives measured perpendicular to a hyperplane
- Classifier Margin

Allows solutions

with zero margin

Enforces a non-zero

margin between examples

and the decision boundary.

- Find the smoothest function that separates data
- Quadratic Programming (similar to Linear Programming)
- Single Minima
- Polynomial Time algorithm

- Quadratic Programming (similar to Linear Programming)

- 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

- Data dimension: 256
- Feature Space: 4 th order
- roughly 100,000,000 dims

Larger

Scale

Smallest

Scale

50,000 Locations/Scales

- Training Data
- 5000 faces
- All frontal

- 108 non faces
- Faces are normalized
- Scale, translation

- 5000 faces
- Many variations
- Across individuals
- Illumination
- Pose (rotation both in plane and out)

- Each image contains 10 - 50 thousand locs/scales
- Faces are rare 0 - 50 per image
- 1000 times as many non-faces as faces

- Extremely small # of false positives: 10-6

First Fast System

- Low Res to Hi

- 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

Weak

Classifier 1

Weights

Increased

Weak

Classifier 2

Weak

classifier 3

Final classifier is

linear combination of weak classifiers

Freund & Shapire

- Features = Weak Classifiers
- Each round selects the optimal feature given:
- Previous selected features
- Exponential Loss

“Rectangle filters”

Similar to Haar wavelets

Papageorgiou, et al.

Unique Binary Features

- 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

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

% False Pos

0

50

vs

false

neg

determined by

50 100

% Detection

T

T

T

T

IMAGE

SUB-WINDOW

Classifier 2

Classifier 3

FACE

Classifier 1

F

F

F

F

NON-FACE

NON-FACE

NON-FACE

NON-FACE

- Given a nested set of classifier hypothesis classes
- Computational Risk Minimization

- Simard
- Rowley (Faces)
- Fleuret & Geman (Faces)

- 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)

50%

20%

2%

IMAGE

SUB-WINDOW

5 Features

20 Features

FACE

1 Feature

F

F

F

NON-FACE

NON-FACE

NON-FACE

10

31

50

65

78

95

110

167

422

Viola-Jones

78.3

85.2

88.8

90.0

90.1

90.8

91.1

91.8

93.7

Rowley-Baluja-Kanade

83.2

86.0

89.2

90.1

89.9

Schneiderman-Kanade

94.4

Roth-Yang-Ahuja

(94.8)

False Detections

Detector

Profile Detection

Facial Feature Localization

Demographic

Analysis

- 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

- The cost of detection is not a function of image size
- Conclusion: the “feature” detector can include a large contextual region around the feature

- Learned features reflect the task

Thanks to

Andrew Moore

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…

Thanks to

Andrew Moore

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?

x1y1

x2 y2

x3 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.

Thanks to

Andrew Moore

Four things make a memory based learner:

- A distance metricEuclidian
- 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.

Thanks to

Andrew Moore

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.

The relative scalings in the distance metric affect region shapes.

Thanks to

Andrew Moore

Other Metrics…

- Mahalanobis, Rank-based, Correlation-based (Stanfill+Waltz, Maes’ Ringo system…)

Or equivalently,

where

Thanks to

Andrew Moore

Thanks to

Baback Moghaddam

Normalized Eigenfaces

Thanks to

Baback Moghaddam

Thanks to

Baback Moghaddam

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

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

PCA-Based Density Estimation Moghaddam & Pentland ICCV’95

Perform PCA and factorize into (orthogonal)

Gaussians subspaces:

Solve for minimal KL divergence residual for the orthogonal subspace:

Thanks to

Baback Moghaddam

See Tipping & Bishop (97) for an ML derivation within a more general factor analysis framework (PPCA)

Intrapersonal

Extrapersonal

dual subspaces for dyads (image pairs)

Equate “similarity” with posterior on

PCA-based density estimation

Moghaddam ICCV’95

Thanks to

Baback Moghaddam

smile

smile

smile

light

specs

mouth

specs

smile

Intra

Extra

Standard

PCA

Thanks to

Baback Moghaddam

Intra-Extra (Dual) Subspaces

Intra-Extra Subspace Geometry

Thanks to

Baback Moghaddam

Two “pancake” subspaces with different orientations intersecting near the origin. If each is in fact Gaussian, then the optimal discriminant is hyperquadratic

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!

FERET Identification: Pre-Test

Thanks to

Baback Moghaddam

Bayesian (Intra-Extra)

Standard (Eigenfaces)

Official 1996 FERET Test

Thanks to

Baback Moghaddam

Bayesian (Intra-Extra)

Standard (Eigenfaces)

..let’s leave distance metrics for now, and go back to….

Thanks to

Andrew Moore

Objection:

That noise-fitting is really objectionable.

What’s the most obvious way of dealing with it?

Thanks to

Andrew Moore

Four things make a memory based learner:

- A distance metricEuclidian
- 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.

Thanks to

Andrew Moore

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?

Thanks to

Andrew Moore

Four things make a memory based learner:

- A distance metricScaled 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

Thanks to

Andrew Moore

Take this dataset…

..and do a kernel prediction with xq (query) = 310, Kw = 50.

Thanks to

Andrew Moore

xq= 150

xq= 395

Thanks to

Andrew Moore

Increasing the kernel width Kw means further away points get an opportunity to influence you.

As Kwinfinity, the prediction tends to the global average.

Thanks to

Andrew Moore

Increasing the kernel width Kw means further away points get an opportunity to influence you.

As Kwinfinity, the prediction tends to the global average.

Thanks to

Andrew Moore

Choosing a good Kw is important. Not just for Kernel Regression, but for all the locally weighted learners we’re about to see.

Thanks to

Andrew Moore

Let

d=D(xi,xquery)/KW

Then here are some commonly used weighting functions…

(we use a Gaussian)

Thanks to

Andrew Moore

Time to try something more powerful…

Thanks to

Andrew Moore

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….

Thanks to

Andrew Moore

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?

Thanks to

Andrew Moore

Four things make a memory-based learner:

- A distance metricScaled 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 Kwparameter 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=βTxquery

Thanks to

Andrew Moore

Query

Linear regression not flexible but trains like lightning.

Locally weighted regression is very flexible and fast to train.

Thanks to

Andrew Moore

- In almost every case:
Good Features beat Good Learning

Learning beats No Learning

- Critical classifier ratio:
- AdaBoost >> SVM