Classification Tamara Berg CSE 595 Words & Pictures

Classification

Tamara BergCSE 595 Words & Pictures

HW2

• Online after class – Due Oct 10, 11:59pm• Use web text descriptions as proxy for class

labels. • Train color attribute classifiers on web

shopping images. • Classify test images as to whether they display

attributes.

Topic Presentations

• First group starts on Tuesday• Audience – please read papers!

Example: Image classification

apple

pear

tomato

cow

dog

horse

input desired output

Slide credit: Svetlana Lazebnik

Slide from Dan Kleinhttp://yann.lecun.com/exdb/mnist/index.html

http://yann.lecun.com/exdb/mnist/index.html

Slide from Dan Klein




Example: Seismic data

Body wave magnitude

Surf

ace

wav

e m

agni

tude

Nuclear explosions

Earthquakes



The basic classification framework

y = f(x)

• Learning: given a training set of labeled examples {(x1,y1), …, (xN,yN)}, estimate the parameters of the prediction function f

• Inference: apply f to a never before seen test example x and output the predicted value y = f(x)

output classification function

input


Some classification methods

106 examples

Nearest neighbor

Shakhnarovich, Viola, Darrell 2003Berg, Berg, Malik 2005…

Neural networks

LeCun, Bottou, Bengio, Haffner 1998Rowley, Baluja, Kanade 1998…

Support Vector Machines and Kernels Conditional Random Fields

McCallum, Freitag, Pereira 2000Kumar, Hebert 2003…

Guyon, VapnikHeisele, Serre, Poggio, 2001…

Slide credit: Antonio Torralba

Example: Training and testing

• Key challenge: generalization to unseen examples

Training set (labels known) Test set (labels unknown)


Slide credit: Dan Klein

Slide from Min-Yen Kan

Classification by Nearest Neighbor

Word vector document classification – here the vector space is illustrated as having 2 dimensions. How many dimensions would the data actually live in?




Classify the test document as the class of the document “nearest” to the query document (use vector similarity to find most similar doc)


Classification by kNN

Classify the test document as the majority class of the k documents “nearest” to the query document. Slide from Min-Yen Kan


What are the features? What’s the training data? Testing data? Parameters?

Classification by kNN

Decision tree classifier

Example problem: decide whether to wait for a table at a restaurant, based on the following attributes:1. Alternate: is there an alternative restaurant nearby?2. Bar: is there a comfortable bar area to wait in?3. Fri/Sat: is today Friday or Saturday?4. Hungry: are we hungry?5. Patrons: number of people in the restaurant (None, Some, Full)6. Price: price range ($, $$, $$$)7. Raining: is it raining outside?8. Reservation: have we made a reservation?9. Type: kind of restaurant (French, Italian, Thai, Burger)10. WaitEstimate: estimated waiting time (0-10, 10-30, 30-60, >60)






Linear classifier

• Find a linear function to separate the classes

f(x) = sgn(w1x1 + w2x2 + … + wDxD) = sgn(w x)


Discriminant Function• It can be arbitrary functions of x, such as:

Nearest Neighbor

Decision Tree

LinearFunctions

( ) Tg b x w x

Slide credit: Jinwei Gu

Linear Discriminant Function• g(x) is a linear function:

( ) Tg b x w x

x1

x2

wT x + b = 0

wT x + b < 0

wT x + b > 0

A hyper-plane in the feature space


denotes +1

denotes -1

x1

• How would you classify these points using a linear discriminant function in order to minimize the error rate?

Linear Discriminant Function

denotes +1

denotes -1

x1

x2

Infinite number of answers!




x1

x2


denotes +1

denotes -1Slide credit: Jinwei Gu



x1

x2


denotes +1


x1

x2• How would you classify these points using a linear discriminant function in order to minimize the error rate?



Which one is the best?

denotes +1


Large Margin Linear Classifier

“safe zone”• The linear discriminant

function (classifier) with the maximum margin is the best

Margin is defined as the width that the boundary could be increased by before hitting a data point

Why it is the best? strong generalization ability

Margin

x1

x2

Linear SVMSlide credit: Jinwei Gu

Large Margin Linear Classifier

x1

x2 Margin

wT x + b = 0

wT x + b = -1w

T x + b = 1

x+

x+

x-

Support Vectors


Solving the Optimization Problem

SV

( ) T Ti i

i

g b b

x w x x x

The linear discriminant function is:

Notice it relies on a dot product between the test point x and the support vectors xi


Linear separability


Non-linear SVMs: Feature Space General idea: the original input space can be mapped to

some higher-dimensional feature space where the training set is separable:

Φ: x → φ(x)

This slide is courtesy of www.iro.umontreal.ca/~pift6080/documents/papers/svm_tutorial.ppt

Nonlinear SVMs: The Kernel Trick With this mapping, our discriminant function is now:

SV

( ) ( ) ( ) ( )T Ti i

i

g b b

x w x x x

No need to know this mapping explicitly, because we only use the dot product of feature vectors in both the training and test.

A kernel function is defined as a function that corresponds to a dot product of two feature vectors in some expanded feature space:

( , ) ( ) ( )Ti j i jK x x x x


Nonlinear SVMs: The Kernel Trick

Linear kernel:

2

2( , ) exp( )

2i j

i jK

x x

x x

( , ) Ti j i jK x x x x

( , ) (1 )T pi j i jK x x x x

0 1( , ) tanh( )Ti j i jK x x x x

Examples of commonly-used kernel functions:

Polynomial kernel:

Gaussian (Radial-Basis Function (RBF) ) kernel:

Sigmoid:


Support Vector Machine: Algorithm

• 1. Choose a kernel function

• 2. Choose a value for C

• 3. Solve the quadratic programming problem (many software packages available)

• 4. Construct the discriminant function from the support vectors


Some Issues• Choice of kernel - Gaussian or polynomial kernel is default - if ineffective, more elaborate kernels are needed - domain experts can give assistance in formulating appropriate similarity

measures

• Choice of kernel parameters - e.g. σ in Gaussian kernel - σ is the distance between closest points with different classifications - In the absence of reliable criteria, applications rely on the use of a

validation set or cross-validation to set such parameters.

This slide is courtesy of www.iro.umontreal.ca/~pift6080/documents/papers/svm_tutorial.ppt Slide credit: Jinwei Gu

Summary: Support Vector Machine

• 1. Large Margin Classifier – Better generalization ability & less over-fitting

• 2. The Kernel Trick– Map data points to higher dimensional space in

order to make them linearly separable.– Since only dot product is used, we do not need to

represent the mapping explicitly.


• A simple algorithm for learning robust classifiers– Freund & Shapire, 1995– Friedman, Hastie, Tibshhirani, 1998

• Provides efficient algorithm for sparse visual feature selection– Tieu & Viola, 2000– Viola & Jones, 2003

• Easy to implement, doesn’t require external optimization tools.

Boosting


• Defines a classifier using an additive model:

Boosting

Strong classifier

Weak classifier

WeightFeaturesvector


• Defines a classifier using an additive model:

• We need to define a family of weak classifiers

Boosting

Strong classifier

Weak classifier

WeightFeaturesvector

from a family of weak classifiers


Adaboost


Each data point has

a class label:

wt =1and a weight:

+1 ( )

-1 ( )yt =

Boosting• It is a sequential procedure:

xt=1

xt=2

xt


Toy exampleWeak learners from the family of lines

h => p(error) = 0.5 it is at chance

Each data point has

a class label:

wt =1and a weight:

+1 ( )

-1 ( )yt =


Toy example

This one seems to be the best

Each data point has

a class label:

wt =1and a weight:

+1 ( )

-1 ( )yt =

This is a ‘weak classifier’: It performs slightly better than chance.Slide credit: Antonio Torralba

Toy example

We set a new problem for which the previous weak classifier performs at chance again

Each data point has

a class label:

wt wt exp{-yt Ht}

We update the weights:

+1 ( )

-1 ( )yt =


Toy example


Each data point has

a class label:

wt wt exp{-yt Ht}


+1 ( )

-1 ( )yt =


Toy example


Each data point has

a class label:

wt wt exp{-yt Ht}


+1 ( )

-1 ( )yt =


Toy example


Each data point has

a class label:

wt wt exp{-yt Ht}


+1 ( )

-1 ( )yt =


Toy example

The strong (nonlinear) classifier is built as the combination of all the weak (linear) classifiers.

f1 f2

f3

f4


Adaboost


Documents

Classification Tamara Berg CSE 595 Words & Pictures