Given a radius $R>0$ and positive integers $m\le n$, I want to generate a random analytic function $f: B_R(0 \in \mathbb R^m) \rightarrow \mathbb R^n$ (where $B_R(0 \in \mathbb R^m) = \{ x \in \mathbb R^m | \|x \|_2 < R \}$) such that $f$ is a smooth embedding, i.e. the differential $\text{d}f$ is of full rank on $B_R(0 \in \mathbb R^m)$.
Is there a reasonably easy method to do this or some prior research done on this question? I've done some quick Google searches with keywords like 'random analytic function', 'random analytic embedding', 'random parametric surface' and they didn't return satisfactory results.
A simple way to achieve this would be to perturb the canonical embedding $i_{\mathbb R^m \hookrightarrow \mathbb R^n}: \mathbb R^m \hookrightarrow \mathbb R^n$ with an analytic function with suitably small coefficients that disallow determinants from vanishing. More precisely, letting $$f = (f_1, \cdots, f_n) = i_{\mathbb R^m \hookrightarrow \mathbb R^n} + (\tilde f_1, \cdots \tilde f_n)$$ such that $\tilde f_i$ are $m$-variable power series with small coefficients bounded below some $\epsilon>0$ would ensure that at least one of the sub-determinants of $\text{d}f$ are $>0$. However, this would restrict the possible classes of functions quite heavily (intuitively, the image $\text{im}(f)$ would look like a flat sheet). If possible, I'd like to choose each $f_i$ from a large class, while the relationships between $f_i$ ensure that $\text d f$ is of full rank.
This question is in data science context, so I eventually need an algorithm to do this.