# Bound on covering number for overparametrized manifold

I am trying to wrap my head around the following problem:

I have $$p$$ real parameters $$\boldsymbol{\theta} \in \Theta = [0, 2\pi)^p$$ that parametrize functions $$f(\boldsymbol{\theta}) \in f(\Theta)$$ via a mapping that is Lipschitz in the following sense: $$\lVert f(\boldsymbol{\theta}) - f(\boldsymbol{\theta}')\rVert_{\infty} \leq L \lVert \boldsymbol{\theta} - \boldsymbol{\theta}'\rVert_1.$$ I now want to compute the covering number of the manifold $$f(\Theta)$$, which is equivalent to counting how many elements an $$\epsilon$$-net for $$f(\Theta)$$ has. I can get an upper bound by first constructing an $$\epsilon$$-netfor the set of parameters $$\Theta$$ that has $$\mathcal{N}(\Theta, \lVert \cdot \rVert_1, \epsilon) \sim \left(\frac{2\pi}{\epsilon}\right)^p$$ elements, and then can use this together with the Lipschitz property to infer that the maximal number of points to construct an $$\epsilon$$-net for the set of parametrized functions, $$f(\Theta)$$, is upper bounded as $$\mathcal{N}(f(\Theta), \lVert \cdot \rVert_{\infty}, \epsilon) \lesssim \left(\frac{2\pi L}{\epsilon}\right)^p.$$

But I also have the information that the dimension of the manifold $$f(\Theta)$$ is bounded independently of $$p$$, say $$\dim(f(\Theta)) \leq q$$. Can I use this knowledge to improve my bound on the covering number for $$f(\Theta)$$?

I would hope for something along the lines of $$\mathcal{N}(f(\Theta), \lVert \cdot \rVert_{\infty}, \epsilon) \overset{?}{\lesssim} \left(\frac{2\pi L}{\epsilon}\right)^{\min(p, q)},$$ eventually with some sort of pre-factor, but I do not know how to prove this. Of the things I was thinking about to solve this, the following is the only one I'm left with that seems sensible to me:

Effectively, we can always find $$q$$ independent parameters to parametrize $$f(\Theta)$$ given the bound on the dimension. Can I then use this information to pick a subset of the parameters that I denote $$\boldsymbol{\vartheta}$$ (e.g. the first $$q$$ parameters) and make the rest of the parameters a function of these parameters via the implicit function theorem, i.e. $$\boldsymbol{\theta} = \boldsymbol{\theta}(\boldsymbol{\vartheta})\colon [0,2\pi)^q \to [0,2\pi)^p$$. If I could show that this mapping is $$l$$-Lipschitz, i.e. $$\lVert \boldsymbol{\theta}(\boldsymbol{\vartheta})-\boldsymbol{\theta}(\boldsymbol{\vartheta}')\rVert_1 \leq l \lVert \boldsymbol{\vartheta} - \boldsymbol{\vartheta}'\rVert_1,$$ I could use this together with the formula for the generalized arc length (see here) to upper bound the volume of the set of parameters $$\boldsymbol{\theta}([0,2\pi)^q) \subseteq \Theta$$ $$\operatorname{Vol}(\boldsymbol{\theta}([0,2\pi)^q)) \leq \sqrt{1+l^2}(2\pi)^q,$$ which should allow me to relate a covering of $$[0,2\pi)^q$$ to a covering of $$f(\Theta)$$.