Homework 4 Support Vector Machines

Machine Learning 
Homework 4, Version 1

This homework contains 3 questions. The last two questions require programming. Question 3 requires
the SVM implementation from Question 2. The maximum number of points is 100 plus 20 bonus points.
1 Question 1 – Support Vector Machines (20 points)
1.1 Linear case (10 points)
Consider training a linear SVM on linearly separable dataset consisting of n points. Let m be the number of
support vectors obtained by training on the entire set. Show that the LOOCV error is bounded above by m
n
.
Hint: Consider two cases: (1) removing a support vector data point and (2) removing a non-support
vector data point.
1.2 General case (10 points)
Now consider the same problem as above. But instead of using a linear SVM, we will use a general kernel.
Assuming that the data is linearly separable in the high dimensional feature space corresponding to the
kernel, does the bound in previous section still hold? Explain why or why not.
2 Question 2 – Implementation of SVMs (35pts + 10 bonus pts)
In this problem, you will implement SVMs using quadratic programming. Quadratic programs refer to
optimization problems in which the objective function is quadratic and the constraints are linear. Quadratic
programs are well studied in optimization literature, and there are efficient solvers. Many Machine Learning
algorithms are reduced to solving quadratic programs. In this question, you will use the quadratic program
solver of Matlab to optimize the dual objective of a kernel SVM.
The dual objective of kernel SVM can be written as:
maximize
α
Xn
j=1
αj −
1
2
Xn
i=1
Xn
j=1
yiαiyjαjk(xi
, xj ) (1)
s.t. Xn
j=1
yjαj = 0 (2)
0 ≤ αj ≤ C ∀j. (3)
1. (10 points) Write the SVM dual objective as a quadratic program. Look at the quadprog function of
Matlab, and write down what H,f, A, b, Aeq, beq, lb, ub are.
2. Use quadratic programming to optimize the dual SVM objective. In Matlab, you can use the function
quadprog.
3. Write a program to compute w and b of the primal from α of the dual. You only need to do this for
linear kernel.
1
4. (10 points) Set C = 0.1, train an SVM with linear kernel using trD, trLb in q2 1 data.mat (in
Matlab, load the data using load q2 1 data.mat). Test the obtained SVM on valD, valLb,
and report the accuracy, the objective value of SVM, the number of support vectors, and the confusion
matrix.
5. (5 points) Repeat the above question with C = 10.
6. (10 points + 10 Bonus)
For this question, you will use multiple binary kernel SVMs to do Crowd Image Classification(Task
is same as the one in HW3). This data has 4 classes. Train your multiclass SVM classifier for
these 4 classes and compete in an in-class Kaggle competition: https://www.kaggle.com/
c/cse512hw4.
Training, Test and Validation data can be downloaded from the Kaggle page https://www.kaggle.
com/c/cse512hw4. Use Training date for training your SVM classifier. You can use the Validation
data for hyperparameter tuning. Submit the predictions on the test data to the Kaggle page.
We have already computed feature vectors for you. Each feature vector has 512 features. For reference,
we also provide the jpeg images from which the feature vectors were extracted, but you are not required
to use them. For multi-class classification, you can use one-versus-one or one-versus-rest approaches.
You’re not allowed to use any other classifiers for this submission. Report the best accuracy and the
approach, the kernel, the parameters you used to achieve that.
We will maintain a leader board, and the top three entries at the end of the competition (assignment
due date) will receive 10 bonus points. Any submission that rises to top three after the assignment
deadline is not eligible for bonus points. The ranking will be based on the Categorization accuracy
(percentage of correct label).
To prevent exploiting test data, you are allowed to make a maximum of 3 submissions per 24 hours.
Your submission will be evaluated immediately and the leader board will be updated.
For this question, you don’t need to have the highest accuracy to earn full points. However, you might
loose all or some points if your performance is much lower than a certain threshold. The threshold will
be determined by us, based on what we believe to be the minimum value that a correct implementation
should achieves.
3 Question 3 – SVM for object detection (45 pts + 10 bonus pts)
In this question, you will train an SVM and use it for detecting human upper bodies in your favorite TV
series The Big Bang Theory. You must use your SVM implementation from Question 2.
To detect human upper bodies in images, we need a classifier that can distinguish between upper-body
image patches from non-upper-body patches. To train such a classifier, we can use SVMs. The training data
is typically a set of images with bounding boxes of the upper bodies. Positive training examples are image
patches extracted at the annotated locations. A negative training example can be any image patch that does
not significantly overlap with the annotated upper bodies. Thus there are potentially many more negative
training examples than positive training examples. Due to memory limitation, it will not be possible to use
all negative training examples at the same time. In this question, you will implement hard-negative mining
to find hardest negative examples and iteratively train an SVM.
3.1 Data
Training images are provided in the subdirectory trainIms. The annotated locations of the upper bodies
are given in trainAnno.mat. This file contains a cell structure ubAnno; ubAnno{i} is the annotated
locations of the upper bodies in the i
th image. ubAnno{i} is 4×k matrix, where each column corresponds
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to an upper body. The rows encode the left, top, right, bottom coordinates of the upper bodies (the origin of
the image coordinate is at the top left corner).
Images for validation and test are given in valIms, testIms respectively. The annotation file for
test images is not released.
3.2 External library
Raw image intensity values are not robust features for classification. In this question, we will use Histogram
of Oriented Gradient (HOG) as image features. HOG uses the gradient information instead of intensities,
and this is more robust to changes in color and illumination conditions. See [1] for more information about
HOG, but it is not required for this assignment.
To use HOG, you will need to install an VL FEAT: http://www.vlfeat.org. This is an excellent
cross-platform library for computer vision and machine learning. However, in this homework, you are only
allowed to use the HOG calculation and visualization function vl hog. In fact, you should not call vl hog
directly. Use the supplied helper functions instead; they will call vl hog.
3.3 Helper functions
To help you, a number of utility functions and classes are provided. The most important functions are in
HW4 Utils.m.
1. Run HW4 Utils.demo1 to see how to read and display upper body annotation
2. Run HW4 Utils.demo2 to display image patches and HOG feature images. Compare HOG features
for positive and negative examples, can you see why HOG would be useful for detect upper bodies?
3. Use HW4 Utils.getPosAndRandomNeg() to get initial training and validation data. Positive
instances are HOG features extracted at the locations of upper bodies. Negative instances are HOG
features at random locations of the images. The data used in Question 3 is actually generated using
this function.
4. Use HW4 Utils.detect to run the sliding window detector. This returns a list of locations and
SVM scores. This function can be used for detecting upper bodies in an image. It can also be used to
find hardest negative examples in an image.
5. Use HW4 Utils.cmpFeat to compute HOG feature vector for an image patch.
6. Use HW4 Utils.genRsltFile to generate result file.
7. Use HW4 Utils.cmpAP to compute the Average Precision for the result file.
8. Use HW4 Utils.rectOverlap to compute the overlap between two rectangular regions. The
overlap is defined as the area of the intersection over the area of the union. A returned detection region
is considered correct (true positive) if there is an annotated upper body such that the overlap between
the two boxes is more than 0.5.
9. Some useful Matlab functions to work with images are: imread, imwrite, imshow, rgb2gray, imresize.
3.4 What to implement?
1. (15 points) Use the training data in HW4 Utils.getPosAndRandomNeg() to train an SVM classifier. Use this classifier to generate a result file (use HW4 Utils.genRsltFile) for validation
data. Use HW4 Utils.cmpAP to compute the AP and plot the precision recall curve. Submit your
AP and precision recall curve (on validation data).
3
Algorithm 1 Hard negative mining algorithm
P osD ← all annotated upper bodies
NegD ← random image patches
(w, b) ← trainSVM(P osD, NegD)
for iter = 1, 2, · · · do
A ← All non support vectors in NegD.
B ← Hardest negative examples . Run UB detection and find negative patches that
. violate the SVM margin constraint the most
NegD ← (NegD \ A) ∪ B.
(w, b) ← trainSVM(P osD, NegD)
end for
2. Implement hard negative mining algorithm given in Algorithm 2. Positive training data and random
negative training data can be generated using HW4 Utils.getPosAndRandomNeg(). At each
iteration, you should remove negative examples that do not correspond to support vectors from the
negative set. Use the function HW4 Utils.detect on train images to identify hardest negative
examples and include them in the negative training set. Use HW4 Utils.cmpFeat to compute
HOG feature vectors.
Hints: (1) a negative example should not have significant overlap with any annotated upper body. You
can experiment with different threshold but 0.3 is a good starting point. (2) make sure you normalize
the feature vectors for new negative examples. (3) you should compute the objective value at each
iteration; the objective values should not decrease.
3. (15 points) Run the negative mining for 10 iterations. Assume your computer is not so powerful and so
you cannot add more than 1000 new negative training examples at each iteration. Record the objective
values (on train data) and the APs (on validation data) through the iterations. Plot the objective values.
Plot the APs.
4. (15 points) For this question, you will need to generate a result file for test data using the function
HW4 Utils.genRsltFile. You will need to submit this file on https://docs.google.
com/forms/d/e/1FAIpQLSeqg5Gb3bS1omsEn6zdpHYn7AeD68c7IheA8qKZ0VcuuSabLQ/
viewform?usp=sf_link to receive the AP on test data. Report the AP in your answer file. Important Note: You MUST use your Stony Brook ID as the name of your submission file, i.e.,
your SBU ID.mat (e.g., 012345679.mat). Your submission will not be evaluated if you don’t use
your SBU ID. For this question, you don’t need to have the highest AP to earn full marks. However,
you might loose all or some points if your performance is much lower than a certain threshold. The
threshold will be determined by us, based on what we believe to be the minimum value that a correct
implementation should achieve.
5. (10 bonus points) Your submitted result file for test data will be automatically entered in a competition
for fame. We will maintain a leader board (https://docs.google.com/spreadsheets/d/
1Deg3NNjZrwwZOVfZAs1zZTApcdKO_P8Xg6H7ZJcCu9s/edit?usp=sharing) and the top
three entries at the end of the competition (due date) will receive 10 bonus points. The ranking is based
on AP.
You can submit the result as frequent as you want. However, the evaluation server will only evaluate all
submissions two times a day, at 12:00pm and 10:00pm. The system only keeps the recent submission
file, and your new submission will override the previous ones. Therefore, you have two chances a day
to evaluate your method. The leader board will be updated in 30 minutes after every evaluation cycle.
4
You are allowed to use any feature types for this part of the homework. For example, you can use
different parameter settings for HOG feature computation. You can even combine multiple HOG
features. You can also append HOG features with geometric features (e.g., think about the locations
of the upper body). You are allowed to perform different types of feature normalization (e.g, L1,
L2). You can use both training and validation data to train your classifier. You are allowed to use
SVMs, Ridge Regression, Lasso Regression, or any technique that we have covered. You can run hard
negative mining algorithm for as many iterations as you want, and the number of negative examples
added at each iteration is not limited by 1000. You are not allowed to use Deep Learning features.
4 What to submit?
4.1 Blackboard submission
You will need to submit both your code and your answers to questions on Blackboard. Put the answer file and
your code in a folder named: SUBID FirstName LastName (e.g., 10947XXXX lionel messi). Zip this folder
and submit the zip file on Blackboard. Your submission must be a zip file, i.e, SUBID FirstName LastName.zip.
The answer file should be named: answers.pdf. The first page of the answers.pdf should be the filled cover
page at the end of this homework. The remaining of the answer file should contain:
1. Answers to Question 1 and 2
2. Answers to Question 3.4, including the requested plots.
4.2 Prediction submission
For Question 2.6, you must submit a .csv file to get the accuracy through Kaggle (https://www.
kaggle.com/c/cse512hw4. A submission file should contain two columns: Id(same as Image name)
and Class. The file should contain a header and have the following format.
Id, Class
1, 1
2, 10
... ...
A sample submission file is available from the competition site and our handout.
For Questions 3.4.4, 3.4.5, you must submit a mat file. A submission file can be automatically generated
by HW4 Utils.genRsltFile. You need to upload this result file on our evaluation server (https://
docs.google.com/forms/d/e/1FAIpQLSeqg5Gb3bS1omsEn6zdpHYn7AeD68c7IheA8qKZ0VcuuSabLQ/
viewform?usp=sf_link ) to obtain AP.