UNIVERSITY COLLEGE LONDON
Engineering Sciences代写 Assignment Release Date: 2nd October 2019Assignment Hand-in Date: 16th October 2019 at 11.55amFormat: Problems
Faculty of Engineering Sciences
Department of Computer Science
COMP0036: LSA – Assignment 2
Dr. Dariush Hosseini (dariush.hosseini@ucl.ac.uk)
Overview Engineering Sciences代写
- Assignment Release Date: 2nd October2019
- AssignmentHand-in Date: 16th October 2019 at 55am
- Format:Problems
Guidelines
- You should answer all THREE
- Note that not all questions carry equal
- You should submit your final report as a pdf using Turnitin accessed via themodule’s Moodle page.
- Within your report you should begin each question on a newpage.
- You should preface your report with a single page containing, on twolines:
- The module code:‘COMP0036’
- The assignment title: ‘LSA – Assignment2’
- Your report should be neat andlegible.Engineering Sciences代写
You are strongly advised to use LATEX to format the report, however this is not a hard requirement, and you may submit a report formatted with the aid of another system, or even a handwritten report, provided that you have converted it to a pdf (see above).
- Please attempt to express your answers as succinctly as succinctly as possible.
- Please note that if your answer to a question or sub-question is illegible orincomprehen- sible to the marker then you will receive no marks for that question or sub-question.Engineering Sciences代写
- Please remember to detail your working, and state clearly any assumptions which you make.
- Failure to adhere to any of the guidelines may result in question-specific deduction of marks. If warranted these deductions may be punitive.
- Assumean unlabelled dataset,
, with sample mean, 
, and an orthonormal basis set
,where d < m.
We have investigated the ‘Projected Variance Maximisation’ approach to PCA in which we are interested in finding the d-dimensional subspace spanned by
for which the sum of the sample variance of the data projected onto this subspace is max-imised.
This leads to a formulation of the PCA problem as:
(1)
However there are several other approaches to PCA. One is the ‘Reconstruction Error Minimisation’ approach.
Here we are interested in finding the d-dimensional subspace spanned by .u[j]Σd
which minimises the reconstruction error, i.e.:(2)
(a) [6 marks]
Show that problems (1) and (2) are equivalent.
(b) [4 marks]
In each analysis we were careful to centre our input data by effectively subtracting off the mean. Why is it important to centre the data in this way?Engineering Sciences代写
(c) [6 marks]
Show that we can re-write the objective of problem (1) as follows, and provide an expression for the matrix S:
(3)
(d) [4 marks]
If we were to replace S with the sample correlation matrix and then proceed to perform PCA with this objective, under what circumstances would this form of PCA differ from the covariance matrix version?
- Assume a set of unlabelledpoints,
, which we wish to separate into k clusters.Engineering Sciences代写
We wish to use a ‘Spherical’ Gaussian Mixture Model (GMM) for this clustering task. Here we assume that each point has an unknown, latent, cluster assignment associated with it, z ∈ {1, …, k}, which is the outcome of a multinomial random variable, Z:
Furthermore, contingent on the cluster assignment, z, we assume that each point, x, is the outcome of a Gaussian random variable, X :
Where:
µ[j] ∈ Rd is the mean associated with cluster j, and;
Σ[j] = sI is the covariance associated with each cluster j. Here s is a constant, and I denotes the d × d identity matrix.
(a) [6 marks]
Give an expression for the log likelihood of the dataset, {x(i)}n
in terms of s and{µ[j], π[j]}k=1.Engineering Sciences代写
(b) [5 marks]
We wish to maximise this expression subject to the usual constraints using the EM algorithm.
Give an expression for the responsibilities, γi[j], in this case.
(c) [8 marks]
Explain why, as s → 0, the solution we generate from the spherical GMM EM clustering algorithm will tend towards the solution we would generate from the k-means clustering algorithm.Engineering Sciences代写
(d) [3 marks]
Describe and explain the form of the boundary which discriminates between clusters in this case.
(e) [3 marks]
Now consider the more usual GMM EM clustering algorithm, in which we assume that the covariances, Σ[j] > 0, are no longer constrained to take the isotropic form which we assumed earlier and are in general distinct ∀j.
Describe and explain the form of the boundary which discriminates between clusters in this case in general.
- (a) [4marks]
Explain the importance of the Representer Theorem?
(b) [3 marks]
Consider the A1-regularised Logistic regression optimisation problem:
Here:
represents a set of training data, where x ∈ Rm are input attributes,
while y ∈ {0, 1} is the output label;
w ∈ Rm is the weight vector of the linear discriminant which we seek, and;
λ > 0 is some constant.
Is this problem susceptible to the kernel trick? Explain.
(c) [4 marks]
Consider the following clustering problem:
Here:
n i=1represents a set of unlabelled points, where x ∈ Rm are input attributes;
{µ[j] ∈ Rm}krepresent the k cluster centroids, and;
{ρi[j] ∈ {0, 1}}n,k represent a set of assignment variables.
Let us seek to perform this optimisation using the k-means algorithm.
Is the M-step of this procedure susceptible to the kernel trick? Explain.
(d) [4 marks]
Assuming 2-dimensional input attributes, x = [x1, x2]T , what kernel function, κ, is associated with the following feature map, φ (express your answer in terms of vector dot products):
(e) [5 marks]
Assuming 1-dimensional input attributes, x, what feature map, φ : x ›→ φ(x), is associated with the following kernel, κ:
κ(x(i), x(j)) = exp .−σ2“x(i) − x(j)“2Σ
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