Hyper-plane in logistic regression vs linear regression for same number of features

Question

Geometric interpretation of Logistic Regression and Linear regression is considered here.

I was going through Logistic regression and Linear regression. In the optimization equation of both following term is used. $$W^{T}.X$$

W is a vector which holds weights of the hyper-plane.

I realized following about the dimensions of the fitted hyper-plane. Want to confirm it.

Let, d = Number of features for both Logistic Regression and Linear Regression.

Logistic Regression case: Fitted hyper-plane is d-dimensional.

Linear Regression case: Fitted hyper-plane is (d + 1) dimensions.

Example

d = 2

feature 1 : weight, feature 2 : height

Logistic Regression: Its a 2 class classification. y : {obsess, normal)

Linear Regression: y: blood pressure (real value)

Here,

• Logistic Regression will fit a 2-D line.
• Linear Regression will fit a 3-D plane.

Please confirm if this understanding is correct and same happens even in higher dimensions.

Answer 1

Mm, I understand what you want to say, but I think you're slightly wrong with terminology. If you consider a d-dimensional vector space, people usually a hyper-plane (d - 1)-dimensional, because it's a subspace of lower dimensionality. So for your example d = 2 the logistic regression would fit a 1-dimentional line (which would separate 2 classes) and the linear regression would fit a 2-dimentional plane (for every combination of 2 features the prediction would lay on some plane in 3-D space). So the general rule would be logreg -> (d-1)-dimensional hyperplane in d-dimensional vector space, linear regression -> d-dimensional hyperplane in (d+1)-dimensional vector space.