On the Characterization of $\ell _{p}$ -Compressible Ergodic Sequences

This work offers a necessary and sufficient condition for a stationary and ergodic process to be lp-compressible in the sense proposed by Amini, Unser and Marvasti [“Compressibility of deterministic and random infinity sequences,” IEEE Trans. Signal Process., vol. 59, no. 11, pp. 5193-5201, 2011, Def. 6]. The condition reduces to check that the p-moment of the invariant distribution of the process is well defined, which contextualizes and extends the result presented by Gribonval, Cevher and Davies in [“Compressible distributions for high-dimensional statistics,” IEEE Trans. Inf. Theory, vol. 58, no. 8, pp. 5016-5034, 2012, Prop. 1]. Furthermore, for the scenario of non-ℓ_p-compressible ergodic sequences, we provide a closed-form expression for the best k-term relative approximation error (in the ℓ_p-norm sense) when only a fraction (rate) of the most significant sequence coefficients are kept as the sequence-length tends to infinity. We analyze basic properties of this rate-approximation error curve, which is again a function of the invariant measure of the process. Revisiting the case of i.i.d. sequences, we completely identify the family of lp-compressible processes, which reduces to look at a polynomial order decay (heavy-tail) property of the distribution.