I agree with the cubic time and quadratic space is a big limitation for now and I'm looking for ways to make them linear (or close to linear).
I figured as much :-) because using n and m for vector and linear map dimensions is actually the older, more established convention.
> I'm looking for ways to make them linear (or close to linear)
The holy grail in AI research right now. I imagine you're looking, or have looked, at mapping models to frequency space to make complexity O(n log n). Take a look at the work the Hazy Research folks have been doing at Stanford -- if you haven't already.