Why decision boundary differs between multinomial (softmax) and One-vs-Rest Logistic Regression for multiclass classification

Question

Can someone please explain why the decision boundary differs between multinomial (softmax) and One-vs-Rest Logistic Regression for multiclass classification. Example shown below

http://scikit-learn.org/stable/auto_examples/linear_model/plot_logistic_multinomial.html#sphx-glr-auto-examples-linear-model-plot-logistic-multinomial-py

I was under the wrong impression that both would yield the same decision boundary and it was just that the probabilities that softmax gives are normalized and interpretable.

Answer 1

Most likely it is because in One-versus-rest you are training independent binary classifiers using the sigmoid function.

In multinomial logistic regression you use another different function, i.e., the softmax function which forces the outputs to sum to $1$.

Being the functions different, also the boundaries will be different.