# Modified sign function VC dimenson

If we have $$f:\mathbb{R} \rightarrow \{\pm 1\}$$, and $$\mathcal{F}$$ and $$\mathcal{F}'$$, what are the VC dimensions of

$$\mathcal{F} = \{sign(\prod_{i=1}^n (x-\theta_i), \forall a_i \in \mathbb{R} \}$$

$$\mathcal{F}' = \cup_{n=1}^n \mathcal{F}$$

I think VC dimesion of $$\mathcal{F}$$ is $$n$$ and $$\mathcal{F}'$$ is infinity

For $$\mathcal{F}$$, we know that it is in 1D, and expanding the polynimals in makes it possible to separate the points with a large polynomial

For $$\mathcal{F}'$$ taking the union will add at least one in each increment, and there are infinite functions. Thus infinity.

Do I have the right approach?

• If you can prove your claims, then you have the right approach. Have you tried that? – Yuval Filmus Apr 30 '20 at 7:23