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According to Polynomial Regression concept, high order terms in a model such as $x_1^2, \,\, x_2^3, \,\, x_1^2x_2$ are replaced with new features. In this way, the model equation is converted to linear form.

For example, I have model equation like this: $w_0 + w_1x_1^2 + w_2x_2^2$

It is a circular classification line which separates first class from second class in a binary classification problem. However, I want to solve this problem by using linear equation. To accomplish this, I decided to use polynomial regression, and I replace high order terms with new features like this: $w_0 + w_1x_3 + w_2x_4$

This operation is also mentioned in Andrew Ng machine learning course. In other words, it is correct and non-problematic approach.

What I am wondering is what is the difference between polynomial regression and feature transformation ?

Feature transformation performs same thing. All non-linear terms are converted to linear terms by using transformation functions.

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    $\begingroup$ If you use the same numeric values in the actual fitting code, changing their names will not affect the regression. If the numeric values are different, the fitted weights will be different. I am not sure I understand your question correctly. $\endgroup$
    – James Phillips
    Commented Jul 1, 2019 at 12:09
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    $\begingroup$ Many "feature transformations" are not expressible as polynomials. $\endgroup$
    – whuber ♦
    Commented Jul 1, 2019 at 13:13

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