Yes the span, so if we discover that a set of points is actually all in the span of a line, that line is a kind of fact or pattern about the points. So probably there is an equivalent in higher dimensions. I am seeing there is a problem whereby we introduce our own bias in creating our model.

So the extra dimensions are unnecessary? I just realised that there could be some situations in which non orthogonal dimensions are preferable. I can’t exactly think of them. Doesn’t it suggest a pattern in data if a mapping is found that reduces the dimension? Like I picture from linear algebra 101 finding a line that everything is a multiple of so one dimension would do and that line is a ‘pattern’? Sorry I’m high.

So is that saying that there is a kind of linear transformation happening between some space (the reality? Our personal model?) and the embedding space? I don’t know what embedding space is and I shouldn’t be here but you are saying interesting things.

Isotropic? Not isomorphic? Please elaborate