# Affine point matching in general dimensions

Fix a positive integer $$d$$ and consider the $$d$$-dimensional Euclidean space $$\mathbb{R}^d$$. Let $$S$$ and $$T$$ finite subsets of $$\mathbb{R}^d$$ of the same size $$n$$. Under the assumption that $$S$$ and $$T$$ are affinely equivalent i.e. there exists an affine transformation $$\alpha:\mathbb{R}^d\rightarrow \mathbb{R}^d$$ such that $$\alpha(S)=T$$; the problem of reconstructing this affine transformation based on the unlabelled sets $$S$$ and $$T$$ is called the affine matching problem. A thirty years old paper titled 'Affine matching problem' due to Spinzak and Werman presents an algorithm which does the job with time complexity $$O(n^{d-2}\text{log}n)$$.

Question: Has there been developed a more efficient algorithm where $$d$$ is not necessarily equal to $$2$$ or $$3$$?

I have tried searching on Google scholar prior to asking this question. The search gives me results specific to $$d=2$$ or $$d=3$$ and not for the general case. I am not a computational scientist and any help would be appreciated.

PS- By unlabelled sets, I mean, for $$S=\{s_1,\ldots,s_n\}$$ and $$T=\{t_1,\ldots ,t_n\}$$, it is not necesarry that $$\alpha(s_i)=t_i$$, $$\forall i\in\{1,\ldots,n\}$$.