Binary logistic regression with only positive training examples - does that even make sense?

Question

^{(I have learned about polynomial linear regression, logistic regression, and neural networks.)}

I have a binary logistic regression problem. I need to classify things to be true or false. What makes this hard for me to understand is that I only have y=1 training examples. Is it possible to get a decision boundary for this data set?

I simplified the problem by only thinking about 2 of my features.

I experimented a little bit by placing fake y=0 training examples everywhere.

Gradient descent with the real and fake training sets seems to fit the data ok. It is a bit underfit, taking into account that this is a degree 6 curve and regularization is turned off.

There has to be a better way to approach this problem. Heck, I could even calculate the convex hull! (Not sure how well that will work in 15+ dimensions.) Any advice?

Answer 1

Do you have a test set with $y = 0$ examples? If not, it will be pretty hard to evaluate your classifier!

Anyway, logistic regression isn't the greatest tool for this, but there are other neat techniques that were designed for problems like this, usually falling under the umbrella term of one-class classification. I think the most commonly used is the one-class support vector machine.