Field
Randomised, and the guarantee that changes kind
Every other claim on this site is deterministic: the residual is below a bound or it is not. A randomised low-rank approximation has an error bound that holds with a probability, and the seed changes the answer — so the figures draw a band rather than a line. What sketching buys is the same quality for fewer passes over the data, which is a statement about cost. On a matrix with no structure to find it buys nothing, and neither does anything else.
A bound that holds with probability
Every other guarantee on this site is deterministic. The randomised low-rank approximation offers one that holds with a probability, the seed changes the answer, and the honest figure is a band rather than a line.
The dimension does not appear
A random projection preserves the lengths of a set of vectors to within a distortion that depends on how many vectors there are and not on how many coordinates each one has. That is the fact the whole field rests on, and it is genuinely surprising.
Randomisation does not create structure
On a matrix whose singular values are all equal, a rank-ten randomised approximation has error 1.0 — and so does the optimal deterministic one. Neither achieved anything, and only one of them is usually sold with the implication that it might.