﻿ 加拿大数学代写 ECE 6254代写 - 作业代写, 数学代写

# 加拿大数学代写 ECE 6254代写

2022-09-09 09:31 星期五 所属： 作业代写 浏览：51 ## Assigned April 22, 2019 – Revised April 22, 2019

### Bonus Problem Set 4  加拿大数学代写

• This is 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

### Problem 1   加拿大数学代写

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 strategyfor 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 • Let , where Cj denotes the jth  Show that the rule for updating thejth cluster center mj given this cluster assignment can be expressed as

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

• Giventwo points x1 and x2, show that ||x1− x2||22 can be computed using only linear com-binations of inner products.
• Giventhe results of the previous parts, show how to compute ||ximj||22  using only (linear combinations of ) inner products between the data points {xi}Ni=1.
• Describehow to use the results from the previous parts to “kernelize” the K-means clustering algorithm described in class.

### Problem 2

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

θˆ ≜ (AA ΓΓ)1Ay. (3)   加拿大数学代写

(a)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).

(b)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)?   加拿大数学代写

(c)(Optional)Play around and see if you can find a (diagonal) matrix Γ that results in a smallerrisk 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.) 