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Learning and Vision:Discriminative Models

Chris Bishop and Paul Viola

Viola 2003

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

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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

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

• 1996 Perceptron’s are cool

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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.

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Why an exponential number of features?

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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.

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Perceptron Training is Linear Programming

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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

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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.

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Lemma 2: Only need to compare examples

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Simple Kernels yield Complex Features

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But Kernel Perceptrons CanGeneralize Poorly

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Perceptron Rebirth: Generalization• Too many features … Occam is unhappy

– Perhaps we should encourage smoothness?

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The linear program can return any multiple of the correct weight vector...

Linear Program is not unique

Slack variables & Weight prior - Force the solution toward zero

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Definition of the Margin

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

• Classifier Margin iT

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Require non-zero margin

Allows solutionswith zero margin

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Enforces a non-zeromargin between examplesand the decision boundary.

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Constrained Optimization

• Find the smoothest function that separates data– Quadratic Programming (similar to Linear

Programming)• Single Minima

• Polynomial Time algorithm

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Constrained Optimization 2

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SVM: examples

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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

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SVM: Zip Code recognition

• Data dimension: 256• Feature Space: 4 th order

– roughly 100,000,000 dims

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The Classical Face Detection Process

SmallestScale

LargerScale

50,000 Locations/Scales

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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)

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Key Properties of Face Detection

• 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

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Sung and Poggio

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Rowley, Baluja & Kanade

First Fast System - Low Res to Hi

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Osuna, Freund, and Girosi

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Support Vectors

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P, O, & G: First Pedestrian Work

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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

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AdaBoostWeak

Classifier 1

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Final classifier is linear combination of weak classifiers

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AdaBoost Properties

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AdaBoost: Super Efficient Feature Selector

• Features = Weak Classifiers

• Each round selects the optimal feature given:– Previous selected features– Exponential Loss

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Boosted Face Detection: Image Features

“Rectangle filters”

Similar to Haar wavelets Papageorgiou, et al.

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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

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Example Classifier for Face Detection

ROC curve for 200 feature classifier

A classifier with 200 rectangle features was learned using AdaBoost

95% correct detection on test set with 1 in 14084false positives.

Not quite competitive...

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Building Fast Classifiers

• Given a nested set of classifier hypothesis classes

• Computational Risk Minimization

vs false neg determined by

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Other Fast Classification Work

• Simard

• Rowley (Faces)• Fleuret & Geman (Faces)

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Cascaded Classifier

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• 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)

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Comparison to Other Systems

(94.8)Roth-Yang-Ahuja

94.4Schneiderman-Kanade

89.990.189.286.083.2Rowley-Baluja-Kanade

93.791.891.190.890.190.088.885.278.3Viola-Jones

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False Detections

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Output of Face Detector on Test Images

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Solving other “Face” Tasks

Facial Feature Localization

DemographicAnalysis

Profile Detection

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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

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Feature Localization Features

• Learned features reflect the task

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Profile Detection

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More Results

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Profile Features

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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…

Thanks toAndrew Moore

Viola 2003

1-Nearest Neighbor is an example of…. Instance-based learning

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?

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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 toAndrew Moore

Viola 2003

Nearest Neighbor

Four things make a memory based learner:

1. A distance metricEuclidian

2. How many nearby neighbors to look at?One

3. A weighting function (optional)Unused

4. How to fit with the local points?Just predict the same output as the nearest

neighbor.

Thanks toAndrew Moore

Viola 2003

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.

Thanks toAndrew Moore

Viola 2003

Euclidean Distance Metric

Other Metrics…

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

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Thanks toAndrew Moore

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Notable Distance MetricsThanks toAndrew Moore

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Simard: Tangent Distance

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Simard: Tangent Distance

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FERET Photobook Moghaddam & Pentland (1995)

Thanks toBaback Moghaddam

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Eigenfaces Moghaddam & Pentland (1995)

Normalized Eigenfaces

Thanks toBaback Moghaddam

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Euclidean (Standard) “Eigenfaces” Turk & Pentland (1992) Moghaddam & Pentland (1995)

Projects all the training facesonto a universal eigenspace to “encode” variations (“modes”)via principal components (PCA)

Uses inverse-distanceas a similarity measurefor matching & recognition

U

Thanks toBaback Moghaddam

Viola 2003

• 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

Euclidean Similarity Measures

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Thanks toBaback Moghaddam

Viola 2003

PCA-Based Density Estimation Moghaddam & Pentland ICCV’95

Perform PCA and factorize into (orthogonal)Gaussians subspaces:

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

Solve for minimal KL divergence residual for the orthogonal subspace:

Thanks toBaback Moghaddam

Viola 2003

Bayesian Face Recognition Moghaddam et al ICPR’96, FG’98, NIPS’99, ICCV’99

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Thanks toBaback Moghaddam

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Intra-Extra (Dual) Subspaces

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Extra

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Thanks toBaback Moghaddam

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Intra-Extra Subspace Geometry

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

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1I

1

Thanks toBaback Moghaddam

Viola 2003

• 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

Bayesian Similarity Measure

)|()( I PSML

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!

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Thanks toBaback Moghaddam

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FERET Identification: Pre-Test

Bayesian (Intra-Extra) Standard (Eigenfaces)

Thanks toBaback Moghaddam

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Official 1996 FERET Test

Bayesian (Intra-Extra) Standard (Eigenfaces)

Thanks toBaback Moghaddam

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One-Nearest Neighbor

Objection:

That noise-fitting is really objectionable.

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

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

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k-Nearest Neighbor

Four things make a memory based learner:

1. A distance metricEuclidian

2. How many nearby neighbors to look at? k

3. A weighting function (optional)Unused

4. How to fit with the local points? Just predict the average output among the k nearest neighbors.

Thanks toAndrew Moore

Viola 2003

k-Nearest Neighbor (here k=9)

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?

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.

Thanks toAndrew Moore

Viola 2003

Kernel Regression

Four things make a memory based learner:

1. A distance metricScaled Euclidian

2. How many nearby neighbors to look at? All of them

3. A weighting function (optional) wi = exp(-D(xi, query)2 / Kw

2)

Nearby points to the query are weighted strongly, far points weakly. The KW parameter is the Kernel Width. Very important.

4. How to fit with the local points?Predict the weighted average of the outputs:

predict = Σwiyi / Σwi

Thanks toAndrew Moore

Viola 2003

Kernel Regression in Pictures

Take this dataset…

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

Thanks toAndrew Moore

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Varying the Query

xq = 150 xq = 395

Thanks toAndrew Moore

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Varying the kernel width

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

As Kwinfinity, the prediction tends to the global average.

xq = 310

KW = 50 (see the double arrow at top of diagram)

xq = 310 (the same)

KW = 100

xq = 310 (the same)

KW = 150

Thanks toAndrew Moore

Viola 2003

Kernel Regression Predictions

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

As Kwinfinity, the prediction tends to the global average.

KW=10 KW=20 KW=80

Thanks toAndrew Moore

Viola 2003

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.

KW=1/32 of x-axis width.

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.

Thanks toAndrew Moore

Viola 2003

Weighting functions

Let

d=D(xi,xquery)/KW

Then here are some commonly used weighting functions…

(we use a Gaussian)

Thanks toAndrew Moore

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Kernel Regression can look bad

KW = Best.

Clearly not capturing the simple structure of the data.. Note the complete failure to extrapolate at edges.

KW = Best.

Also much too local. Why wouldn’t increasing Kw help? Because then it would all be “smeared”.

KW = Best.

Three noisy linear segments. But best kernel regression gives poor gradients.

Time to try something more powerful…

Thanks toAndrew Moore

Viola 2003

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….

Thanks toAndrew Moore

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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?

Thanks toAndrew Moore

Viola 2003

Locally Weighted Regression

Four things make a memory-based learner:1. A distance metric

Scaled Euclidian2. How many nearby neighbors to look at?

All of them3. 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.

4. How to fit with the local points?1. First form a local linear model. Find the β that minimizes the locally weighted sum of squared

residuals:

2

1

2

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xβyβ argmin

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kk

Tkkw

Then predict ypredict=βT xquery

Thanks toAndrew Moore

Viola 2003

How LWR works

Linear regression not flexible but trains like lightning.

Locally weighted regression is very flexible and fast to train.

Query

Thanks toAndrew Moore

Viola 2003

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?

Thanks toAndrew Moore

Viola 2003

Features, Features, Features

• In almost every case:

Good Features beat Good Learning

Learning beats No Learning

• Critical classifier ratio:

• AdaBoost >> SVM

complexity

quality