Compressibility Measures for Affinely Singular Random Vectors

Charusaie, Mohammad-Amin; Amini, Arash; Rini, Stefano

doi:10.1109/tit.2022.3174623

Cited by 2 publications

(4 citation statements)

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“…Particularly, we study the marginal and joint probability measure of two examples of DCE-ARMA. We show that the existence of atomic components in the excitation noise induces affine singularities [33] and Cantor-type singularities in probability measure of samples.…”

Section: Singularity In Arma Processesmentioning

confidence: 93%

“…Proof: We prove this theorem using the following steps: (i) we show that a truncation ξ n p−q+1 of the excitation noise has an affinely singular probability distribution, (ii) using step (i) we show that the joint random vector [X p ; ξ n p−q+1 ] of a truncation of the ARMA process and its excitation noise is affinely singular, (iii) using the previous steps and linear recursive relation in ARMA processes we show that each truncation X n of the ARMA process has affinely singular distribution, (iv) we show that if the excitation noise is absolutely continuous, then the truncation X n is absolutely continuous, and (v) we use Lemma [33,Lemma 7] and Hankel property of the linear recursion of the ARMA process to show that the singularity dimensions of the truncation X n concentrates around nd(ξ i ) for large n.…”

Section: Appendix B Proof Of Theoremmentioning

confidence: 98%

“…Due to the linear mixture of the white noise samples, these components form affine subsets. The class of vectors with with singularities over affine sets was the topic of our previous publication [33]. We refer to this class of vectors as Affinely Singular Random Vectors (Affinely Singular Random Vectors (ASRV)).…”

Section: Stochastic Processesmentioning

confidence: 99%

“…V , where Q V and P V are unitary matrices and Σ V is a diagonal matrix with rank( Φ −1 Θ ŨV ) non-zero terms. Next, we use [33,Lemma 10] that shows that the result of a multiplication of a fullrow rank matrix and an absolutely continuous RV is another absolutely continuous RV. In fact, Σ V P † V is a concatenation of a full-row rank matrix and null rows.…”

Section: Appendix C Proof Of Theoremmentioning

confidence: 99%

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On the Compressibility of Affinely Singular Random Vectors

Charusaie

Amini

Rini

2020

2020 IEEE International Symposium on Information Theory (ISIT)

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Rényi Information Dimension (RID) plays a central role in quantifying the compressibility of random variables with singularities in their distribution, encompassing and extending beyond the class of sparse sources. The RID, from a high perspective, presents the average number of bits that is needed for coding the i.i.d. samples of a random variable with high precision. There are two main extensions of the RID for stochastic processes: information dimension rate (IDR) and block information dimension (BID). In addition, a more recent approach towards the compressibility of stochastic processes revolves around the concept of ϵ-achievable compression rates, which treat a random process as the limiting point of finite-dimensional random vectors and apply the compressed sensing tools on these random variables. While there is limited knowledge about the interplay of the the BID, the IDR, and ϵ-achievable compression rates, the value of IDR and BID themselves are known only for very specific types of processes, namely i.i.d. sequences (i.e., discretedomain white noise) and moving-average (MA) processes. This paper investigates the IDR and BID of discrete-time Auto-Regressive Moving-Average (ARMA) processes in general, and their relations with ϵ-achievable compression rates when the excitation noise has a discrete-continuous measure. To elaborate, this paper shows that the RID and ϵ-achievable compression rates of this type of processes are equal to that of their excitation noise. In other words, the samples of such ARMA processes can be compressed as much as their sparse excitation noise, although the samples themselves are by no means sparse. The results of this paper can be used to evaluate the compressibility of various types of locally correlated data with finite-or infinite-memory as they are often modelled via ARMA processes.

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Section: Singularity In Arma Processesmentioning

confidence: 93%

Section: Appendix B Proof Of Theoremmentioning

confidence: 98%

Section: Stochastic Processesmentioning

confidence: 99%

Section: Appendix C Proof Of Theoremmentioning

confidence: 99%

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