A rate-distortion theory for permutation spaces

We investigate the lossy compression of the permutation space by analyzing the trade-off between the size of a source code and the distortion with respect to either Kendall tau distance or ℓ
₁
distance of the inversion vectors. For both distortion measures, we characterize the rate-distortion functions and provide explicit code designs that achieve them. Finally, we provide bounds on the higher order terms in the codebook size when the distortion levels lead to degenerate code rates (0 or 1).