Transcript pptx - University of Pittsburgh

CS 1699: Intro to Computer Vision
Bias-Variance Trade-off +
Other Models and Problems
Prof. Adriana Kovashka
University of Pittsburgh
November 3, 2015
Outline
•
•
•
•
Support Vector Machines (review + other uses)
Bias-variance trade-off
Scene recognition: Spatial pyramid matching
Other classifiers
– Decision trees
– Hidden Markov models
• Other problems
– Clustering: agglomerative clustering
– Dimensionality reduction
Lines in R2
x0 , y0 
Let
D
w
a 
w 
c 
 x
x 
 y
ax  cy  b  0
wx b  0
D
Kristen Grauman
ax0  cy0  b
a c
2
2

|w xb|

w
distance from
point to line
Support vector machines
• Want line that maximizes the margin.
xi positive ( yi  1) :
xi  w  b  1
xi negative ( yi  1) :
xi  w  b  1
For support, vectors,
xi  w  b  1
Distance between point
and line:
Support vectors
Margin
| xi  w  b |
|| w ||
For support vectors:
wΤ x  b  1
1
1
2

M


w
w
w
w
w
C. Burges, A Tutorial on Support Vector Machines for Pattern Recognition, Data Mining and Knowledge Discovery, 1998
Finding the maximum margin line
1. Maximize margin 2/||w||
2. Correctly classify all training data points:
xi positive ( yi  1) :
xi  w  b  1
xi negative ( yi  1) :
xi  w  b  1
Quadratic optimization problem:
1 T
Minimize
w w
2
Subject to yi(w·xi+b) ≥ 1
One constraint for each
training point.
Note sign trick.
C. Burges, A Tutorial on Support Vector Machines for Pattern Recognition, Data Mining and Knowledge Discovery, 1998
Finding the maximum margin line
• Solution: w  i  i yi xi
b = yi – w·xi (for any support vector)
• Classification function:
f ( x)  sign (w  x  b)
 sign
 y x  x  b
i i i i
If f(x) < 0, classify as negative, otherwise classify as positive.
• Notice that it relies on an inner product between the test
point x and the support vectors xi
• (Solving the optimization problem also involves
computing the inner products xi · xj between all pairs of
training points)
C. Burges, A Tutorial on Support Vector Machines for Pattern Recognition, Data Mining and Knowledge Discovery, 1998
Nonlinear SVMs
• Datasets that are linearly separable work out great:
x
0
• But what if the dataset is just too hard?
x
0
• We can map it to a higher-dimensional space:
x2
0
Andrew Moore
x
Examples of kernel functions
K ( xi , x j )  xi x j
T

Linear:

Gaussian RBF:
K ( xi ,x j )  exp( 

xi  x j
2
2
2
)
Histogram intersection:
K ( xi , x j )   min( xi (k ), x j (k ))
k
Andrew Moore
Allowing misclassifications
Misclassification
cost
# data samples
Slack variable
The w that minimizes…
Maximize margin
Minimize misclassification
What about multi-class SVMs?
• Unfortunately, there is no “definitive” multi-class SVM
formulation
• In practice, we have to obtain a multi-class SVM by
combining multiple two-class SVMs
• One vs. others
– Training: learn an SVM for each class vs. the others
– Testing: apply each SVM to the test example, and assign it to the
class of the SVM that returns the highest decision value
• One vs. one
– Training: learn an SVM for each pair of classes
– Testing: each learned SVM “votes” for a class to assign to the
test example
Svetlana Lazebnik
Evaluating Classifiers
• Accuracy
– # correctly classified / # all test examples
• Precision/recall
– Precision = # predicted true pos / # predicted pos
– Recall = # predicted true pos / # true pos
• F-measure
= 2PR / (P + R)
• Want evaluation metric to be in some range, e.g. [0 1]
– 0 = worst possible classifier, 1 = best possible classifier
Precision / Recall / F-measure
True positives
(images that contain people)
True negatives
(images that do not contain people)
Predicted positives
(images predicted to contain people)
Predicted negatives
(images predicted not to contain people)
•
•
•
Precision
Recall
F-measure
= 2 / 5 = 0.4
= 2 / 4 = 0.5
= 2*0.4*0.5 / 0.4+0.5 = 0.44
Accuracy: 5 / 10 = 0.5
Support Vector Regression
Regression
Regression is like classification except the labels are real valued
Example applications:
‣ Stock value prediction
‣ Income prediction
‣ CPU power consumption
Subhransu Maji
Regularized Error Function for Regression
In linear regression, we minimize the error function:
N
{ y
n 1
 tn } 
2
n
true value

2
w
2
predicted value
An example of Є-insensitive error function:
E  0
E | f ( x)  y | 
for
| f ( x)  y | 
otherwise
Use the Є-insensitive error function:
N
 2
C  E ( y ( xn )  t n )  w
2
n 1
Adapted from Huan Liu
Slack Variables for Regression
For a target point to lie
inside the tube:
yn   tn  yn  
Introduce slack variables
to allow points to lie
outside the tube:
Subject to:
tn  y ( xn )   n
n  0
tn  y ( xn )   n
 0
Adapted from Huan Liu

n
Minimize:
N
1 2
C  ( n   )  w
2
n 1

n
Next time:
Support Vector Ranking
Outline
•
•
•
•
Support Vector Machines (review + other uses)
Bias-variance trade-off
Scene recognition: Spatial pyramid matching
Other classifiers
– Decision trees
– Hidden Markov models
• Other problems
– Clustering: agglomerative clustering
– Dimensionality reduction
Generalization
Training set (labels known)
Test set (labels
unknown)
• How well does a learned model generalize from
the data it was trained on to a new test set?
Slide credit: L. Lazebnik
Generalization
• Components of generalization error
– Bias: how much the average model over all training sets differs
from the true model
• Error due to inaccurate assumptions/simplifications made by
the model
– Variance: how much models estimated from different training
sets differ from each other
• Underfitting: model is too “simple” to represent all the
relevant class characteristics
– High bias and low variance
– High training error and high test error
• Overfitting: model is too “complex” and fits irrelevant
characteristics (noise) in the data
– Low bias and high variance
– Low training error and high test error
Slide credit: L. Lazebnik
Bias-Variance Trade-off
• Models with too few
parameters are
inaccurate because of a
large bias (not enough
flexibility).
• Models with too many
parameters are
inaccurate because of a
large variance (too much
sensitivity to the sample).
Slide credit: D. Hoiem
Fitting a model
Is this a good fit?
Figures from Bishop
With more training data
Figures from Bishop
Bias-variance tradeoff
Overfitting
Error
Underfitting
Test error
Training error
High Bias
Low Variance
Complexity
Low Bias
High Variance
Slide credit: D. Hoiem
Bias-variance tradeoff
Test Error
Few training examples
High Bias
Low Variance
Many training examples
Complexity
Low Bias
High Variance
Slide credit: D. Hoiem
Choosing the trade-off
• Need validation set
• Validation set is separate from the test set
Error
Test error
Training error
High Bias
Low Variance
Complexity
Low Bias
High Variance
Slide credit: D. Hoiem
Effect of Training Size
Error
Fixed prediction model
Testing
Generalization Error
Training
Number of Training Examples
Adapted from D. Hoiem
How to reduce variance?
• Choose a simpler classifier
• Use fewer features
• Get more training data
• Regularize the parameters
Slide credit: D. Hoiem
Regularization
Figures from Bishop
Characteristics of vision learning problems
• Lots of continuous features
– Spatial pyramid may have ~15,000 features
• Imbalanced classes
– Often limited positive examples, practically infinite
negative examples
• Difficult prediction tasks
• Recently, massive training sets became available
– If we have a massive training set, we want classifiers
with low bias (high variance is ok) and reasonably
efficient training
Adapted from D. Hoiem
Remember…
• No free lunch: machine learning
algorithms are tools
• Three kinds of error
– Inherent: unavoidable
– Bias: due to over-simplifications
– Variance: due to inability to perfectly estimate parameters
from limited data
• Try simple classifiers first
• Better to have smart features and simple classifiers
than simple features and smart classifiers
• Use increasingly powerful classifiers with more
training data (bias-variance tradeoff)
Adapted from D. Hoiem
Outline
•
•
•
•
Support Vector Machines (review + other uses)
Bias-variance trade-off
Scene recognition: Spatial pyramid matching
Other classifiers
– Decision trees
– Hidden Markov models
• Other problems
– Clustering: agglomerative clustering
– Dimensionality reduction
Beyond Bags of Features: Spatial Pyramid Matching
for Recognizing Natural Scene Categories
CVPR 2006
Svetlana Lazebnik ([email protected])
Beckman Institute, University of Illinois at Urbana-Champaign
Cordelia Schmid ([email protected])
INRIA Rhône-Alpes, France
Jean Ponce ([email protected])
Ecole Normale Supérieure, France
http://www-cvr.ai.uiuc.edu/ponce_grp
Bags of words
Slide credit: L. Lazebnik
Bag-of-features steps
1.
2.
3.
4.
Extract local features
Learn “visual vocabulary” using clustering
Quantize local features using visual vocabulary
Represent images by frequencies of “visual words”
Slide credit: L. Lazebnik
Local feature extraction
…
Slide credit: Josef Sivic
Learning the visual vocabulary
…
Slide credit: Josef Sivic
Learning the visual vocabulary
…
Clustering
Slide credit: Josef Sivic
Learning the visual vocabulary
…
Visual vocabulary
Clustering
Slide credit: Josef Sivic
Image categorization with bag of words
Training
1.
2.
Extract bag-of-words representation
Train classifier on labeled examples using histogram values as features
Testing
1.
2.
3.
4.
Extract keypoints/descriptors
Quantize into visual words using the clusters computed at training time
Compute visual word histogram
Compute label using classifier
Slide credit: D. Hoiem
What about spatial layout?
All of these images have the same color histogram
Slide credit: D. Hoiem
Spatial pyramid
Compute histogram in each spatial bin
Slide credit: D. Hoiem
Spatial pyramid
[Lazebnik et al. CVPR 2006]
Slide credit: D. Hoiem
Pyramid matching
Indyk & Thaper (2003), Grauman & Darrell (2005)
Matching using pyramid and histogram intersection for some particular visual word:
xi
xj
Original images
Feature histograms:
Level 3
Level 2
Level 1
Level 0
Total
K( xweight
i , xj ) (value of pyramid match kernel):
Adapted from L. Lazebnik
Scene category dataset
Fei-Fei & Perona (2005), Oliva & Torralba (2001)
http://www-cvr.ai.uiuc.edu/ponce_grp/data
Multi-class classification results (100 training images per class)
Fei-Fei & Perona: 65.2%
Slide credit: L. Lazebnik
Scene category confusions
Difficult indoor images
kitchen
living room
bedroom
Slide credit: L. Lazebnik
Caltech101 dataset
Fei-Fei et al. (2004)
http://www.vision.caltech.edu/Image_Datasets/Caltech101/Caltech101.html
Multi-class classification results (30 training images per class)
Slide credit: L. Lazebnik
Outline
•
•
•
•
Support Vector Machines (review + other uses)
Bias-variance trade-off
Scene recognition: Spatial pyramid matching
Other classifiers
– Decision trees
– Hidden Markov models
• Other problems
– Clustering: agglomerative clustering
– Dimensionality reduction
Decision tree classifier
Example problem: decide whether to wait for a table at a
restaurant, based on the following attributes:
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
Alternate: is there an alternative restaurant nearby?
Bar: is there a comfortable bar area to wait in?
Fri/Sat: is today Friday or Saturday?
Hungry: are we hungry?
Patrons: number of people in the restaurant (None, Some, Full)
Price: price range ($, $$, $$$)
Raining: is it raining outside?
Reservation: have we made a reservation?
Type: kind of restaurant (French, Italian, Thai, Burger)
WaitEstimate: estimated waiting time (0-10, 10-30, 30-60, >60)
Slide credit: L. Lazebnik
Decision tree classifier
Slide credit: L. Lazebnik
Decision tree classifier
Slide credit: L. Lazebnik
Sequence Labeling Problem
• Unlike most computer vision problems, many
NLP problems can viewed as sequence labeling.
• Each token in a sequence is assigned a label.
• Labels of tokens are dependent on the labels of
other tokens in the sequence, particularly their
neighbors (not i.i.d).
foo
Adapted from Ray Mooney
bar
blam
zonk
zonk
bar
blam
Markov Model / Markov Chain
• A finite state machine with probabilistic
state transitions.
• Makes Markov assumption that next state
only depends on the current state and
independent of previous history.
Ray Mooney
Sample Markov Model for POS
0.05
0.1
Noun
Det
0.5
0.95
0.9
stop
Verb
0.05
0.25
0.1
PropNoun
0.4
0.8
0.1
0.5
0.1
start
Ray Mooney
0.25
Sample Markov Model for POS
0.05
0.1
Noun
Det
0.5
0.95
0.9
stop
Verb
0.05
0.25
0.1
PropNoun
0.4
0.8
0.1
0.5
0.25
0.1
start
Adapted from Ray Mooney
P(PropNoun Verb Det Noun) = 0.4*0.8*0.25*0.95*0.1=0.0076
“Mary
ate
the cake.”
Hidden Markov Model
• Probabilistic generative model for sequences.
• Assume an underlying set of hidden (unobserved)
states in which the model can be (e.g. parts of
speech).
• Assume probabilistic transitions between states over
time (e.g. transition from POS to another POS as
sequence is generated).
• Assume a probabilistic generation of tokens from
states (e.g. words generated for each POS).
Ray Mooney
Sample HMM for POS
0.05
the
a the
a
the a the
that
Det
0.1
cat
dog
car bed
pen apple
0.95
Noun
0.5
0.9
0.05
0.1
0.4
0.1
Ray Mooney
0.25
Verb
0.8
0.1
PropNoun
0.5
start
Tom
John Mary
Alice
Jerry
0.25
bit
ate saw
played
hit gave
stop
Outline
•
•
•
•
Support Vector Machines (review + other uses)
Bias-variance trade-off
Scene recognition: Spatial pyramid matching
Other classifiers
– Decision trees
– Hidden Markov models
• Other problems
– Clustering: agglomerative clustering
– Dimensionality reduction
Slide credit: D. Hoiem
Clustering Strategies
• K-means
– Iteratively re-assign points to the nearest cluster
center
• Mean-shift clustering
– Estimate modes
• Graph cuts
– Split the nodes in a graph based on assigned links with
similarity weights
• Agglomerative clustering
– Start with each point as its own cluster and iteratively
merge the closest clusters
Agglomerative clustering
Agglomerative clustering
Agglomerative clustering
Agglomerative clustering
Agglomerative clustering
Agglomerative clustering
How to define cluster similarity?
- Average distance between points, maximum
distance, minimum distance
How many clusters?
distance
- Clustering creates a dendrogram (a tree)
- Threshold based on max number of clusters
or based on distance between merges
Adapted from J. Hays
Why do we cluster?
• Summarizing data
– Look at large amounts of data
– Represent a large continuous vector with the cluster number
• Counting
– Histograms of texture, color, SIFT vectors
• Segmentation
– Separate the image into different regions
• Prediction
– Images in the same cluster may have the same labels
Slide credit: J. Hays, D. Hoiem
Slide credit: D. Hoiem
Dimensionality Reduction
Figure from Genevieve Patterson, IJCV 2014