Isye 6740 homework 1

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3.2D.Report.pdf - ISYE 6740 Homework SOHAM GHOSH sghosh315... Doc Preview. Pages 1. Total views 2. Georgia Institute Of Technology. ISYE. ISYE 6740. CorporalTree6409. 7/22/2023. View full document. Students also studied. Ps5.pdf. London School of Economics. ECONOMICS EC1C1. class 1 draw to learn line graph 23.pdf..

1 Clustering. [100 points total. Each part is 25 points.][a-b] Given N data points xn(n = 1, . . . , N), K-means clustering algorithm groups them into K clusters byminimizing the distortion function o. ... ISYE 6740 Homework 1 solved 1 Clustering. [100 points total. Each part is 25 points.]ISYE 6740, Homework 2 100 points + 15 bonus points 1. PCA: Food consumption in European countries [50 points] The data food-consumption.csv contains 16 countries in Europe and their consumption for 20 food items, such as tea, jam, coffee, yogurt, and others. We will perform principal componentView Notes - Lecture_1.pdf from ISYE 6740 at Georgia Institute Of Technology. Lecture 1: Supervised Learning Tuo Zhao Schools of ISYE and CSE, Georgia Tech 2017 Fall from. AI Homework Help. ... Homework 3 Solutions.pdf. Solutions Available. Auckland. BUSMGT 733. homework1_solution.pdf. Solutions Available. Georgia Institute Of Technology. ISYE ...K(x) = 1 √ 2π e − (x 2 1+x 2 2) 2 . Recall in this case, the kernel density estimator (KDE) for a density is given by p(x) = 1 m Xm i=1 1 h K x i − x h where x i are two-dimensional vectors, h > 0 is the kernel bandwidth. Set an appropriate h so you can see the shape of the distribution clearly. Plot of contour plot (like the ones in ...View 7401-HW9sol.pdf from ISYE 7401 at Georgia Institute Of Technology. ISyE 7401, Spring - 2019 Instructor : A.Shapiro Homework # 9 \u0010 \u0011 πi 1. Consider the logistic regression model log 1−πi = β0 +CS 7641 CSE/ISYE 6740 Homework 1 Le Song Deadline: Sep. 26 Monday, 11:55pm • Submit your answers as an electronic copy on T-square. • No unapproved extension of deadline is allowed. Zero credit will be assigned for late submissions. Email request for late submission may not be replied. • For typed answers with LaTeX (recommended) or word processors, extra credits will be given.Homework assignments for ISYE 6740 Computational Data Analysis (Spring 2022) - isye_6740/Canlapan_Inah_HW3.ipynb at main · inahpatrizia/isye_6740Name. Computational Data Analysis: Learning, Mining, and Computation. Listed As. ISYE-6740. Credit Hours. 3. Available to. AN students. Description. …ISYE6740- Homework 1 Solved Image compression using clustering [60 points] In this programming assignment, you are going to apply clustering algorithms for image compression. Your task is implementing K-means for this purpose. It is required you implementing the algorithms yourself rather than calling k-means from a package.ISyE 6740 – Computational Data Analysis Fall 2022 Tentative Syllabus Instructor: Prof. Tuo Zhao, Groseclose 344, Email: [email protected] If you want to contact me via email, please include ISYE6740 in the subject! Time/Location: 11:00am – 12:15pm KAC2447 Office hour: 3pm-4pm MW Location: Groseclose 344 TA: Akshay Shetty, …View ISYE 6740 - (FA22) Syllabus.pdf from ISYE 6740 at Georgia Institute Of Technology. 8/20/2022 ONLINE MASTER OF SCIENCE IN ANALYTICS OMSA 6740 - COMPUTATIONAL D ATA ANALYSIS / MACHINE LEARNING ... (1) You can have up to 10 days of homework extension without penalty. Please email and notify your assigned TA …View ISYE 6501 Homework Week 11-1.pdf from ISYE 6501 at Georgia Institute Of Technology. Question 15.2 In the videos, we saw the "diet problem". (The diet problem is one of the first large-scale1. Implementing EM algorithm for MNIST dataset. Implement the EM algorithm for fitting a Gaussian mixture model for the MNIST dataset. We reduce the dataset to be only two cases, of digits "2" and "6" only. Thus, you will fit GMM with C = 2. Use the data file data.mat or data.dat on Canvas. True […]ISYE-6740 - Computational Data Analytics. Search for a Course: Toggle to Select Spcific Semesters. ... Homework 1-3 are very tough but after Homework 4, the difficult drastically decreases. ... This was a good class for a survey of machine learning methods that goes into more detail than ISYE 6501 and 6040. It felt like a fusion of both classes ...(10 points) Now choose ` 1 distance (or Manhattan distance) between images (recall the definition from "Clustering" lecture)). Repeat the steps above. Repeat the steps above. Again construct a similarity graph with vertices corresponding to the images, and tune the threshold so that each node has at least 100 neighbors.Computational Data Analy - 29323 - ISYE 6740 - PAN. Associated Term: Spring 2021. Levels: Graduate Semester, Undergraduate Semester. Georgia Tech-Atlanta * Campus. Lecture* Schedule Type. Partially at a Distance (BOR) Instructional Method. Learning Objectives: Canvas Course Description. Required Materials: Technical Requirements:View homework4.pdf from CSE 6740 at Georgia Institute Of Technology. ISYE 6740, Summer 2021, Homework 4 100 points + 3 bonus points 1. Comparing classifiers. (65 points) In lectures, we learnChong Zhang Homework 1 ISyE 6740 August 27, 2019 1 Since the distortion function: J = ∑ N n =1 ∑ K k =1 r nk || x n-μ k || 2 and the dissimilarity function: || x n-μ k || 2 if we want to find the values for {r nk} and {μ k } to minimize J. We can perform a iteration of two steps: first,minimize J with respect to the {r nk}, keeping the ...View HW3.report.pdf from CSE 6740 at Georgia Institute Of Technology. ISYE 6740 Homework 3 Total 100 points. 1. Basic optimization. (30 points.) Consider a simplified logistic regression problem.ISyE 7406: Data Mining and Statistical Learning (Due: 01/30/2020) Homework Assignment #2 Instructor: Yajun, Mei Name: Chen-Yang(Jim), Liu GTID: 90345**** Problem 1: R exercise for linear Regression (50 points) (a) Introduction The dataset in this problem is from faraway library in R. This dataset contains 252 observations with 18 features. We will use this dataset to perform linear regression.I took ISyE 6501 in the spring and am taking ISyE 6740 this semester. 6501 was definitely a good intro to 6740, but 6740 is a step up. ... We've had a number of mathematical proofs in the homework (not my forté) and a decent amount of Python (or Matlab) programming. The homeworks have been well thought out to reinforce the mathematical basis ...ISYE 6740 Homework 6 Fall 2020 Total 100 points. Shasha Liao . 1. AdaBoost. (30 points) Consider the following dataset, plotting in the following figure. The first two coordinates represent the value of two features, and the last coordinate is the binary label of the data.ISYE/CSE 6740 Homework 1 August 30, 2019 • Submit your answers as an electronic copy on Canvas. • No unapproved extension of deadline is allowed. Zero credit will be assigned for late submissions. Email request for late submission may not be replied. • For typed answers with LaTeX (recommended) or word processors, extra credits (10 …When it comes to purchasing heavy machinery such as Kubota equipment, negotiating with the dealer can be a crucial step in getting the best deal possible. Before entering into nego...Computational Data Analy - 29323 - ISYE 6740 - PAN. Associated Term: Spring 2021. Levels: Graduate Semester, Undergraduate Semester. Georgia Tech-Atlanta * Campus. Lecture* Schedule Type. Partially at a Distance (BOR) Instructional Method. Learning Objectives: Canvas Course Description. Required Materials: Technical Requirements:CSE/ISYE 6740 Homework 1. Submit your answers as an electronic copy. No unapproved extension of deadline is allowed. Zero credit will be assigned for late submissions. Email request for late submission may not be replied. For typed answers with LaTeX (recommended) or word processors, extra credits will be given.ISYE 6740, Spring 2024, Homework 4 100 points 1. Optimization (35 points). Consider a simplified logistic regression problem. Given m training samples (xi, yi), i = 1,... , m. The data xi ∈ R 2 , and yi ∈ { 0 , 1 }. To fit a logistic regression model for classification, we solve the following optimization problem, where θ ∈ R is a ...About. Homework assignments for ISYE 6740 Computational Data Analysis (Spring 2022) MIT license. Activity. 7 stars. 1 watching. 1 fork. Report repository. Releases. No …View homework1 (1).docx from FINC FINC-420 at The University of Tennessee, Knoxville. ISYE 6740 Summer 2023 Homework 1 (100 points) In this homework, the superscript of a symbol xi denotes the indexISYE 6740 is CDA right? I certainly wouldn’t call it an easier course. It’s only homework and no exams so I guess maybe in terms of grading, but the content is quite difficult, and I found the homework challenging and time consuming. I also found the lectures really varied in quality, some homework questions you could solve with lecture ...University: Georgia Institute of Technology. Info. Download. AI Quiz. Homework 6 random forest question 6:01 pm isye 6740 hw6 in import numpy as np import csv from sklearn import tree import matplotlib.pyplot as plt from sklearn.ISYE 6740, Spring 2024, Homework 4 100 points 1. Optimization (35 points). Consider a simplified logistic regression problem. Given m training samples (xi, yi), i = 1,... , m. The data xi ∈ R 2 , and yi ∈ { 0 , 1 }. To fit a logistic regression model for classification, we solve the following optimization problem, where θ ∈ R is a ...View ISyE6740-Fall2022-Xie-Syllabus.pdf from ISYE 6740 at Georgia Institute Of Technology. 8/20/2022 ONLINE MASTER OF SCIENCE IN ANALYTICS ISYE 6740 - COMPUTATIONAL DATA ANALYSIS / MACHINE LEARNING. ... Homework 1, release 08/23, Due 09/06 Week 2 08/29-09/02 Spectral Clustering We will discuss another type of clustering algorithm: ...View Homework Help - homework4 from ISYE 6740 at Georgia Institute Of Technology. Spring 2017 ISYE 6740 Computational Data Analysis: Homework 4 1 ISYE 6740 Computational Data Analysis: HomeworkGiven N data points xn(n = 1,...,N), K-medoids clustering algorithm groups them into K clusters by minimizing the distortion function ), where D(x,y) is a distance measure between two vectors x and y in same size (in case of K-means, D(x,y) = kx − yk2), µk is the center of k-th cluster; and rnk = 1 if xn belongs to the k-th cluster and rnk ...q, p 6= q. Write down the the Bayes classifier f : X → Y for binary classification Y ∈ {−1, +1}. Simplify the. (a) Suppose the class conditional distribution is a Gaussian. Based on the general loss function in problem. 3.1, write the Bayes classifier as f (X) = sign (h (X)) and simplify h as much as possible.CS 7641 CSE/ISYE 6740 Homework 2 Solutions October 11, 2016 1 EM for Mixture of Gaussians. Mixture of K Gaussians is represented as. p(x) = ∑ K. k= πkN (x|μk, Σk), (1) …Brief Solution of HW#1 of ISyE 7406 General Peer Grader Guidance. When grading your peers, you will not only learn how to improve your future homework submissions but you will also gain deeper understanding of the concepts in the assignments. When assigning scores, consider the responses to the questions given your understanding of the problem ...ISYE 6740 Summer 2021 Homework 1 (100 points + 2 bonus points) 1 Image compression using clustering [60 points] In this programming assignment, you are going to apply clustering algorithms for image compression. Your task is implementing K-means for this purpose.ISYE 6740 Summer 2022 Homework 1 Concept Questions. What's the main difference between supervised and unsupervised learning? Supervised learning uses labelled historic data to predict or classify new data based on labels associated with historic data. The learning algorithm learns from training datasets iteratively, adjusting to improve the ...ISYE 6740 Homework 2 solved K-medoids In class, we learned that the basic K-means works in Euclidean space for computing distance between data points as well as for updating centroids by arithmetic mean. Sometimes, however, the dataset may work ... nk = 1 if k = arg minj D(xn, µj), and rContribute to hsharifi7/ISYE-6740 development by creating an account on GitHub.If you’re trying to help a student with math homework and questions involving slope come up, you might need a refresher on learning how to calculate this important measurement. Rea...1 ISyE 6740 - Spring 2021 Project Final Report Predicting Wine Scores: What makes for a good wine? Team # 147, Group E, Team AlphaGo Team Members: Jin Jang (GT ID #: 903619645), Preprocessing and Non-Linear Model Fitting Samuel Martin (GT ID #: 903657034), EDA and Linear Model Fitting Jacob Rodriguez (GT ID #: 903619768), Problem Statement and Feature EngineeringCS 7641 CSE/ISYE 6740 Homework 4 Prakash, Fall 2021 Deadline: 12/02, 11:59 pm • Submit your answers as an electronic copy on Gradescope. • No unapproved extension of deadline is allowed. Late submission will lead to 0 credit. • For typed answers with LaTeX (recommended) or word processors, extra credits will be given (= 5 points). If you handwrite, try to be clear as much as possible. View HW2_ISYE_6740.pdf from ISYE 6740 at Georgia Institute Of Technology. Amogh Badugu Email: [email protected] Course: ISYE 6740 Homework - # 1 GT ID: 903687830 Date: September 28, 2021 Problem ISyE 6740 - Spring 2021 Project Proposal Team Member Names: Kyle Thayer, Priscilla Mariani, Vikram Ramanujam Project Title: Predictive Weather Analytics Using First-Order Data Problem Statement Despite the common joke that meteorologists forecast the weather by flipping a coin, weather forecasting is actually a very complex and analytical process. . The unstable and chaotic nature of weather ...Spring 2017 ISYE 6740 Computational Data Analysis: Homework 3 1 ISYE 6740 Computational Data Analysis: Homework 3 Due: Sunday March 26, 2017, 11:59pm 100 Points Total Version 1.0 Instruction: Please write a report to answer the questions and include the plots in the report. You can write the code in R, Python or Matlab and submit …View Homework 2 Solutions.pdf from BUSMGT 733 at Auckland. ISYE 6740 Summer 2022 Homework 2 (100 points + 10 bonus points) 1. Conceptual questions [15 points]. 1. (5 points) Please prove the firstISYE 6501 Intro to Analytics Modeling (Spring 2020) OMSCentral: ... Homework 1: Using a public API to pull data, visualizing it, SQLite, D3 intro, and Python Flask intro (a good gauge to see if ...Ankitcodinghub ISYE6740 Homework 3 Solved. 1. Order of faces using ISOMAP. This question aims to reproduce the ISOMAP algorithm results in the original paper for ISOMAP, J.B. Tenenbaum, V. de Silva, and J.C. Langford, Science 290 (2000) 2319-2323 that we have also seen in the lecture as an exercise (isn't this exciting to go through the process of generating results for a high-impact ...View homework2.pdf from CSE 6740 at Georgia Institute Of Technology. ISYE 6740, Summer 2020, Homework 2 Prof. Yao Xie 1. Order of faces using ISOMAP (30 points) The objective of this question is toWhen you need security to protect your business, hiring a security vendor will be an important task. You can’t afford to make a mistake in this hiring decision, so do your homework...1. First, given a set of images for each person, we generate the so-called eigenface using. these images. The procedure to obtain eigenface is explained as follows. Given n. images of the same person denoted by x1, . . . , xn. Each image originally is a matrix. We vectorize each image to form the vector xi ∈ R. p.January 19 2020 ISyE 6740 Homework 1 Solution Eucledian and Manhattan distances when the K values were large, it cannot be strictly generalized. I could not infer any pattern for the dependence of time on K in the implementation using inf-distance metric. 4. Effect of initial centroids Although there were minor changes in the pixel composition of the output …Given N data points xn(n = 1,…,N), K-medoids clustering algorithm groups them into K clusters by minimizing the distortion function ), where D(x,y) is a distance measure between two vectors x and y in same size (in case of K-means, D(x,y) = kx − yk2), µk is the center of k-th cluster; and rnk = 1 if xn belongs to the k-th cluster and rnk ...hsharifi7 / ISYE-6740 Public. Notifications. Fork 9. Star 16. Projects. Security. Insights. Contribute to hsharifi7/ISYE-6740 development by creating an account on GitHub.ISYE 6740 Homework 6 solution Spring 2021. Total 100 points + 10 bonus points. House price dataset (25 points). The HOUSES dataset contains a collection of recent real estate listings in San Luis Obispo county and around it.Computational Data Analy - 29323 - ISYE 6740 - PAN. Associated Term: Spring 2021. Levels: Graduate Semester, Undergraduate Semester. Georgia Tech-Atlanta * Campus. Lecture* Schedule Type. Partially at a Distance (BOR) Instructional Method. Learning Objectives: Canvas Course Description. Required Materials: Technical Requirements:1 Clustering. [100 points total. Each part is 25 points.] [a-b] Given N data points xn(n = 1, . . . , N), K-means clustering algorithm groups them into K clusters by minimizing the distortion function over {r nk, µk} J = X N n=1 X K k=1 r nkkx n − µ k k 2, where r nk = 1 if xn belongs to the k-th cluster and r nk = 0 otherwise.Crosslisted with ISYE 6740. Credit not awarded for both CSE 6740 and CS 4641/7641/ISYE 6740. Data Recovery. ... The midterm was extremely difficult, with questions not similar to the homework at all. The majority of the students struggled but he couldn't care less. Read all 8 reviews. Course Chat. Chat with other students in CSE 6740. Schedule ...ISYE 6740 Homework 6 solution Spring 2021Total 100 points + 10 bonus points.1. House price dataset (25 points).The HOUSES dataset contains a collection of recent real estate listings in San Luis Obispo county and around it. The dataset is provided in RealEstate.csv. You may use \one-hot-keying" to expand the categorical variables.1 O NLINE M ASTER OF S CIENCE IN A NALYTICS ISYE/CSE 6740 – C OMPUTATIONAL D ATA A NALYSIS / M ACHINE L EARNING I T ENTATIVE S YLLABUS (S UBJECT TO CHANGE), S UMMER 2020 H. Milton Stewart School of Industrial and Systems Engineering Georgia Institute of Technology P ROFESSOR : Yao Xie; [email protected] Professor Office Hour: Wed 9-9:30pm.1/10/2021 5 track of that and make adjustments on your CANVAS due date. Homework extension requests should be made before the original homework due date. (2) If you have already used the above 10 days of homework extension, and if you submit the homework late: one day late the grade will be discount to 75% of your total, two days late the grade will discount to 50% of your total, three days ...View homework6.pdf from ISYE 6740 at Georgia Institute Of Technology. ISYE 6740 Homework 6 Fall 2020 Total 100 points. 1. AdaBoost. (30 points) Consider the following dataset, plotting in theHomework is good because it gives students a chance to practice and internalize information presented during classroom lessons. It also encourages parents to get involved in the st...Given N data points xn(n = 1,…,N), K-medoids clustering algorithm groups them into K clusters by minimizing the distortion function ), where D(x,y) is a distance measure between two vectors x and y in same size (in case of K-means, D(x,y) = kx − yk2), µk is the center of k-th cluster; and rnk = 1 if xn belongs to the k-th cluster and rnk ...1 Image compression using clustering In this programming assignment, you are going to apply clustering algorithms for image compression. Your task is implementing K-means for this purpose. It is required you implementing the algorithms yourself rather than calling k-means from a package. However, it is ok to use standard packages such as file i/o, linear …Feb 22, 2024 · To fit a logistic regression model for classification, we solve the following optimization problem, where θ∈Ris a parameter we aim to find: maxθ ℓ(θ), (1) where the log-likelhood function ℓ(θ) = m X. −1)θTxi . 1. (10 points) Show step-by-step mathematical derivation for the gradient of the cost function ℓ(θ) in (1). 2. Studying ISYE 6402 Time Series Analysis at Georgia Institute of Technology? On Studocu you will find assignments, lecture notes and much more for ISYE 6402 Georgia ... ISYE 6402 Homework 4. Assignments None. Lecture notes. Date Rating. year. Ratings. ISYE 6402 Module 1 Case Studies-1. 26 pages 2023/2024 None. 2023/2024 None. Save. M4-spectral ...Contribute to hsharifi7/ISYE-6740 development by creating an account on GitHub.

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How Given N data points xn(n = 1,…,N), K-medoids clustering algorithm groups them into K clusters by minimizing the distortion function ), where D(x,y) is a distance measure between two vectors x and y in same size (in case of K-means, D(x,y) = kx − yk2), µk is the center of k-th cluster; and rnk = 1 if xn belongs to the k-th cluster and rnk ...Math homework can sometimes feel like an insurmountable challenge. From complex equations to confusing word problems, it’s easy to get overwhelmed. However, with the right techniqu...View homework1 (1).docx from FINC FINC-420 at The University of Tennessee, Knoxville. ISYE 6740 Summer 2023 Homework 1 (100 points) In this homework, the superscript of a symbol xi denotes the index

When CSE/ISYE 6740 Homework 4 George Lan March 19, 2020 • Submit your answers as an electronic copy on T-square. • No unapproved extension of deadline is allowed. Zero credit will be assigned for late submissions. Email request for late submission may not be replied. • For typed answers with LaTeX (recommended) or word processors, extra credits will be given.Document ISYE6740_2024_HW1.pdf, Subject Computer Science, from Georgia Institute Of Technology, Length: 10 pages, Preview: Computational Data Analytics ISYE 6740 Homework 1 Arjun Mishra - 903230877 OVERVIEW In this assignment, we are given aStudying ISYE 7406 Data Mining&Stat Learn at Georgia Institute of Technology? On Studocu you will find 20 assignments, coursework, lecture notes, practice materials ... HW5 - homework 5 sample sol. 18 pages 2021/2022 None. 2021/2022 None. Save. 7406HW03 - homework 3. 2 pages 2023/2024 None. 2023/2024 None. Save. HW3 report - HW 3 SOLUTION.…

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keke wyatt 2022 ISYE 6740, Summer 2023, Homework 3. 100 points + 10 bonus points. Prof. Yao Xie 1. Conceptual questions. [10 points] For the EM algorithm for GMM, please show how to use the Bayes rule to drive τ ki in a closed-form expression. 2. Optimization. [20 points] Consider a simplified logistic regression problem. Given m training samples (xi, yi), i ...ISYE 6740 HW1.pdf - Homework 1 1. Concept Questions 1.... Doc Preview. Pages 1. Identified Q&As 5. Solutions available. Total views 6. Georgia Institute Of Technology. ISYE. ISYE 6740. DoctorPower11599. 1/31/2024. View full document. Students also studied. Week 3.pdf. Georgia Institute Of Technology. ISYE 6740. Week 6.pdf. first street livermorehitman holla girlfriend cinnamon instagram View Notes - Lecture_1.pdf from ISYE 6740 at Georgia Institute Of Technology. Lecture 1: Supervised Learning Tuo Zhao Schools of ISYE and CSE, Georgia Tech 2017 Fall from. AI Homework Help. ... Homework 3 Solutions.pdf. Solutions Available. Auckland. BUSMGT 733. homework1_solution.pdf. Solutions Available. Georgia Institute Of Technology. ISYE ...View homework3.pdf from ISYE 6740 at Georgia Institute Of Technology. ISYE 6740, Fall 2021, Homework 3 100 points Prof. Yao Xie 1. Conceptual questions. [20 points] 1. (10 points) Based on the how much does suicide hotline pay per hourdurwin hickman lotteryfirst assembly of god longview tx View Homework 2 Solutions.pdf from BUSMGT 733 at Auckland. ISYE 6740 Summer 2022 Homework 2 (100 points + 10 bonus points) 1. Conceptual questions [15 points]. 1. (5 points) Please prove the first how to sell old playboy magazines View Habibe_Tommy_HW1_report-2.pdf from ISYE 6740 at Georgia Institute Of Technology. ISYE 6740 Fall 2021 Homework 1 (100 points + 2 bonus points) 1 - Conception questions [30 points] Please provideIn this programming assignment, you are going to apply clustering algorithms for image compression. Your task is implementing K-means for this purpose. It is required you implementing the algorithms yourself rather than calling k-means from a package. However, it is ok to use standard packages such as file i/o, linear algebra, and visualization. Formatting instruction […] federal indictment list missouri 2023funeral home in dalton georgiahawkeye football on radio View 3.2A.Report.pdf from ISYE 6740 at Georgia Institute Of Technology. ISYE 6740 Homework SOHAM GHOSH [email protected] 2.1 X ERN %Y'befo,l L ki ool e R %O(iz'(v, " o pot] o) = 2 toq 1€)AL )iyl AI Homework HelpEnhanced Document Preview: ISYE 6740, Fall 2022, Homework 4 100 points + 5 bonus points. SVM (40 points). 1 (10 points). Explain why can we set the margin c = 1 to derive the SVM formulation? 2. (10 points). Using the Lagrangian dual formulation, show that the weight vector can be represented as w= n iyixi. i=1, where i 0 are the dual variables.