# Confusion regarding the criteria for defining a ML model as a linear model

I am confused about the criteria which determines whether a model is linear or not. As far as I understand, the following statements are equivalent :

1. A model is linear
2. Output class label is a linear combination of model parameters w
3. Decision boundary is linear [a hyperplane]

If the above understanding is correct, then how is a single layer perceptron with a non linear activation like sigmoid a linear model? From this post , I understand that it fulfils the 3rd statement that the decision boundary will be linear but the output is clearly not a linear combination of w.

Can anyone please clarify this? Am stuck on this for a while now, any help would be deeply appreciated!

• "Output class label is a linear combination of model parameters w" I don't really understand what this is supposed to mean. Could you give an example of a model that fulfills this criterion? Feb 1 at 10:44
• @Firebug what I meant was that the discriminant function would be linear in w ie in a linear model, prediction y = wTx where w is weight vector and x is the input vector
– TarS
Feb 1 at 10:58

In statistics, we call a model linear when the outcome is a linear combination of the parameters, meaning that you can write $$\hat y_i = \sum_j x_{ij}\hat\beta_j$$ (second definition). We could have something like $$x_{2j} = \left(x_{1j}\right)^2$$ without breaking that, which is how polynomial regression qualifies as a linear model, even though the model could be squaring or cubing terms, leading to a highly nonlinear plot.
If you then throw an activation function on there to get $$\hat y_j = \dfrac{1}{1 + \exp\left(-\sum_j x_{ij}\hat\beta_j\right)}$$, then $$\hat y$$ no longer qualifies as being a linear combination of the $$\beta$$ parameters, and statistics would not consider this a linear model, with the caveat that the nonlinearity can be expressed in a way that qualifies it as a generalized linear model (which some people will consider linear enough to count as somewhat of an honorary linear model).