Finding value of theta in linear classification

Question

I have an examing coming up, and I'm practicing with exams from previous years. However, the answers to the questions are not provided unfortunately.

I'm currently doing the question below, and the intuitive answer I get is 2. However, this seems too easy. Am I correct?

kate-melnykova · Accepted Answer · 2020-12-12 19:35:30Z

1

Since it is an exam question, I will give you hints and leading questions instead of answers.

Note that $x^T w$ must be well-defined, and $x$ is 2-dimensional. What is the dimension of $w$?

How is bias incorporated into the model?

The margin satisfies $\widehat{y} = 0.$ What is the equation of the margin (as a function of x and bias)?

P.S. I expect a typo/vagueness in this question.

answered Dec 12, 2020 at 19:35

kate-melnykova

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$\begingroup$ The dimension of 𝑤 are also 2-dimensional, no? 𝑤 = [2, 2] $\endgroup$
– Adnos
Commented Dec 12, 2020 at 21:44
1

$\begingroup$ Let's give it a shot. The margin line passes through $x = (1, 4).$ For this $x$, $x^T w$ is "indecisive", and, therefore, it should be zero. Does your answer fit? $\endgroup$
– kate-melnykova
Commented Dec 13, 2020 at 0:47
$\begingroup$ After going through the lecture slides, I finally ended up with 𝑤1 = 1 and 𝑤2 = -(1/3). Is this correct? $\endgroup$
– Adnos
Commented Dec 14, 2020 at 20:40
$\begingroup$ The equation of the line is $-2-2x_1 + x_2=0.$ First, I make sure that positives are above the line and negatives are below the line. Secondly, we need to choose the appropriate scaling, e.g., $-4-4x_1+2x_2=0$ would be the equation for the same line but would lead to a different $w$. I use the margins for that. And I got a different answer. $\endgroup$
– kate-melnykova
Commented Dec 15, 2020 at 0:00
$\begingroup$ However, my margin line has a bias, and in the example, there is no bias. I am very confused about it. $\endgroup$
– kate-melnykova
Commented Dec 15, 2020 at 0:00

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