High-rate transform coding: how high is high, and does it matter?

Karhunen-Loeve transforms (KLT's) are the optimal orthogonal transforms for transform coding of Gaussian sources. This well-known fact is usually established with approximations from high-resolution quantization theory. How high does the rate have to be for these approximations to be accurate? The minimum rate allocated to any component should be at least about one bit. (The average rate per component may be much higher.) Does the rate actually have to be high for the KLT to be optimal? No, the KLT is optimal more generally. Two new, simple proofs of this fact are described. They rely on a scale invariance property, but not on high-resolution approximations or properties of optimal fixed-rate quantization.