The $$n$$n-term Approximation of Periodic Generalized Lévy Processes

Fageot, Julien; Unser, Michaël; Ward, John Paul

doi:10.1007/s10959-018-00877-7

Cited by 11 publications

(20 citation statements)

References 45 publications

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“…Hence, it is not surprising that our results based on the quantization entropy show that impulsive Poisson innovation processes are by far more compressible than heavy-tailed αstable innovation processes; the previous studies on their k-term approximation sort them in the opposite order when the jump distribution in the Poisson innovation is not heavy-tailed. It is interesting to mention that the same ordering of impulsive Poisson and heavy-tailed -stable innovation processes is observed in [19].…”

Section: Introductionsupporting

confidence: 58%

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How Compressible Are Innovation Processes?

Ghourchian

Amini

Gohari

2018

IEEE Trans. Inform. Theory

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The sparsity and compressibility of finite-dimensional signals are of great interest in fields such as compressed sensing. The notion of compressibility is also extended to infinite sequences of i.i.d. or ergodic random variables based on the observed error in their nonlinear k-term approximation. In this work, we use the entropy measure to study the compressibility of continuous-domain innovation processes (alternatively known as white noise). Specifically, we define such a measure as the entropy limit of the doubly quantized (time and amplitude) process. This provides a tool to compare the compressibility of various innovation processes. It also allows us to identify an analogue of the concept of "entropy dimension" which was originally defined by Rényi for random variables. Particular attention is given to stable and impulsive Poisson innovation processes. Here, our results recognize Poisson innovations as the more compressible ones with an entropy measure far below that of stable innovations. While this result departs from the previous knowledge regarding the compressibility of fat-tailed distributions, our entropy measure ranks stable innovations according to their tail decay. Index TermsCompressibility, entropy, impulsive Poisson process, stable innovation, white Lévy noise. Symbol DefinitionSets Caligraphic letters like A, C, D, . . . Real and natural numbers R, NBorel sets in R B(R) or just B Random variablesCapital letters like: A, X, Y, Z, . . . Probability density function (pdf) of (continous) X p X or q X (lower-case letter p)Probability mass function (pmf) of (discrete) X P X (upper-case letter P )Cumulative distribution function (cdf) of X F X X 0 for a given white noise X(t) 1 0 X(t) dt; a random variable Definition 2 (Discrete Random Variable). [23] A random variable X is called discrete if it takes values in a countable alphabet set X ⊂ R. Definition 3 (Discrete-Continuous Random Variable).[23] A random variable X is called discrete-continuous with parameters (p c , P D , Pr {X ∈ D}) if there exists a countable set D, a discrete probability mass function P D , whose support is D, and a pdf p c ∈ AC such that 0 < Pr {X ∈ D} < 1,

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Section: Introductionsupporting

confidence: 58%

“…Proof of (20): First, we find the characteristic function of Y n in terms of the characteristic function of A. From the definition of Y n in (19), we have that Because of the definition of Poisson process in Definition 12, we have that…”

Section: Proof Of Lemmamentioning

confidence: 99%

How Compressible Are Innovation Processes?

Ghourchian

Amini

Gohari

2018

IEEE Trans. Inform. Theory

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“…A process with a strong p -characterization has an almost everywhere asymptotic (with n) pattern for its p -approximation error when a finite rate of significant coefficients is considered (i.e., a fixed-rate analysis). On top of the structure introduced in Definition 5, a relevant scenario to consider is when the limiting function f p,µ (X, r), in (11), is constant (independent of X) µ-almost surely. This can be interpreted as an ergodic property of X with respect to its best-k term p -approximation error, reflecting a typical (almost sure) approximation attribute that is constant for the entire process 3 .…”

Section: Revisiting the P -Approximation Error Analysismentioning

confidence: 99%

“…Let us assume that X has a strong p -characterization (Def. 5) and that its limiting function in (11) is constant µ-almost surely, denoted by (f p,µ (r)) r∈ (0,1] . Assume that d o = f p,µ (ro) for some r o ∈ (0, 1] 4 .…”

Section: Revisiting the P -Approximation Error Analysismentioning

confidence: 99%

“…Various forms of compressibility for a random sequence have been used in signal processing problems, for instance in regression [8], signal reconstruction (in the classical random Gaussian linear measuring setting used in CS) [2,3], and inference-decision [9,10]. Compressibility for random sequences has also been used to analyze continuous-time processes [7,5], for instance, in periodic generalized Lévy processes [11].…”

Section: Introductionmentioning

confidence: 99%

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Compressibility Analysis of Asymptotically Mean Stationary Processes

Silva¹

2021

Preprint

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This work provides new results for the analysis of random sequences in terms of p -compressibility. The results characterize the degree in which a random sequence can be approximated by its best k-sparse version under different rates of significant coefficients (compressibility analysis). In particular, the notion of strong p -characterization is introduced to denote a random sequence that has a well-defined asymptotic limit (sample-wise) of its best k-term approximation error when a fixed rate of significant coefficients is considered (fixed-rate analysis). The main theorem of this work shows that the rich family of asymptotically mean stationary (AMS) processes has a strong p -characterization. Furthermore, we present results that characterize and analyze the p -approximation error function for this family of processes. Adding ergodicity in the analysis of AMS processes, we introduce a theorem demonstrating that the approximation error function is constant and determined in closed-form by the stationary mean of the process. Our results and analyses contribute to the theory and understanding of discrete-time sparse processes and, on the technical side, confirm how instrumental the point-wise ergodic theorem is to determine the compressibility expression of discrete-time processes even when stationarity and ergodicity assumptions are relaxed.

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Compressibility analysis of asymptotically mean stationary processes

Silva

2022

Applied and Computational Harmonic Analysis

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The $$n$$n-term Approximation of Periodic Generalized Lévy Processes

Cited by 11 publications

References 45 publications

How Compressible Are Innovation Processes?

How Compressible Are Innovation Processes?

Compressibility Analysis of Asymptotically Mean Stationary Processes

Compressibility analysis of asymptotically mean stationary processes

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