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代写数学考试 ECE498AC Midterm代考math 密码学离散数学

2018-10-17 08:00 星期三 所属: 数学代写代考,北美/加拿大/英国靠谱的数学作业代写机构 浏览:1077

代写数学考试 youwill receive a non-transferrable 2% bonus for using LATEX ,There are 2 problems over 3 math代写

 

Bonus Problem Set 4

 

  • Thisis an open book/Internet homework, but you should not interact with your classmates
  • This is a bonus problem set with a single harddeadline
  • Youwill receive a non-transferrable 2% bonus for using LATEX
  • There are 2 problems over 3
  • Theproblems are assigned the same weight for the overall
  • Due Monday April 29, 2019 11:59pm EST (Monday May 6, 2019 for DistanceLearning Students)
  • Pleaseturn in your solution using the LATEX template as the first page
代写数学考试
代写数学考试

K-means clustering with the Euclidean distance inherently assumes that each pair of clusters is linearly separable, which may not be the case in practice. In this problem you will derive a strategy for dealing with this limitation that we did not discuss in class. Specifically, you will show that like so many other algorithms we have discussed in class, K-means can be “kernelized.” In the following,

we consider a dataset {xi}N .

  • Let zij≜ 1 xi j , where j denotes the jth  Show that the rule for updating the

jth cluster center mj given this cluster assignment can be expressed as

 

N

mj = αijxi (1)

i=1

by expressing αij as a function of the zij’s.

  • Giventwo points x1 and x2, show that ∥x1 − x2∥2 can be computed using only linear com-

binations of inner products.

  • Given the results of the previous parts, show how to compute ∥ximj∥2 using only(linear

combinations of) inner products between the data points {xi}N .

  • Describehow to use the results from the previous parts to “kernelize” the K-means clustering algorithm described in

 

In this problem we consider the scenario seen in class, where x is drawn uniformly on [−1, 1] and y = sin(πx), for which we are given N = 2 training samples. Here, we will consider an alternative approach to fitting a line to the data based on Tikhonov regularization. Specifically, we let

 

 

y = y1 y2

A = 1 x1

1 x2

] θ = [ b

] (2)

 

We will then consider Tikhonov regularized least squares estimators of the form

θˆ ≜ (AA ΓΓ)1Ay. (3)

  • Howshould we set Γ to reduce this estimator to fitting a constant function (i.e., finding an h(x) of the form h(x) = b)? (Hint: For the purposes of this problem, it is sufficient to set Γ in a way that just makes a 0. To make a = 0 exactly requires setting Γ in a way that makes the matrix AA ΓΓ singular, but note that this does not mean that the regularized least-squares optimization problem cannot be solved; you must just use a different formula than the one in (3).
  • Howshould we set Γ to reduce this estimator to fitting a line of the form h(x) = ax b that passes through the observed data points (x1, y1) and (x2, y2)?
  • (Optional)Play around and see if you can find a (diagonal) matrix Γ that results in a smaller risk than either of the two approaches we discussed in  You will need to do this numeri- cally using Python or MATLAB. Report the Γ that gives you the best results. (You can restrict your search to diagonal Γ to simplify this.)
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