How Compressible Are Innovation Processes?

Ghourchian, Hamid; Amini, Arash; Gohari, Amin

doi:10.1109/tit.2018.2822660

Cited by 8 publications

(19 citation statements)

References 26 publications

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“…This is achieved by compound-Poisson processes and also by Laplace processes, that are therefore the sparsest processes. These results are coherent with recent works of H. Ghourchian et al in their recent works on the entropy of sparse processes [26]. Here, the authors have shown that, among SαS and compound Poisson white noises, the less sparse is the Gaussian white noise, and the sparsest are the compound Poisson white noises.…”

Section: Discussion and Examplessupporting

confidence: 93%

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

2019

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In this paper, we study the compressibility of random processes and fields, called generalized Lévy processes, that are solutions of stochastic differential equations driven by d-dimensional periodic Lévy white noises. Our results are based on the estimation of the Besov regularity of Lévy white noises and generalized Lévy processes. We show in particular that non-Gaussian generalized Lévy processes are more compressible in a wavelet basis than the corresponding Gaussian processes, in the sense that their n-term approximation errors decay faster. We quantify this compressibility in terms of the Blumenthal-Getoor indices of the underlying Lévy white noise. * This work is an extension of the conference paper [50].

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Section: Discussion and Examplessupporting

confidence: 93%

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

2019

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“…Linear In the second framework, one can quantify the compressibility of a Lévy process in the information theoretic sense through the entropy of the underlying Lévy white noise, as in [54]. These two frameworks are complementary and based on totally different tools, but they are consistent and lead to the same compressibility hierarchy.…”

Section: Brownian Motion Compound Poissonmentioning

confidence: 99%

Wavelet Compressibility of Compound Poisson Processes

Aziznejad¹,

Fageot²

2020

Preprint

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In this paper, we characterize the wavelet compressibility of compound Poisson processes. To that end, we expand a given compound Poisson process over the Haar wavelet basis and analyse its asymptotic approximation properties. By considering only the nonzero wavelet coefficients up to a given scale, what we call the sparse approximation, we exploit the extreme sparsity of the wavelet expansion that derives from the piecewiseconstant nature of compound Poisson processes. More precisely, we provide nearly-tight lower and upper bounds for the mean L2-sparse approximation error of compound Poisson processes. Using these bounds, we then prove that the sparse approximation error has a sub-exponential and super-polynomial asymptotic behavior. We illustrate these theoretical results with numerical simulations on compound Poisson processes. In particular, we highlight the remarkable ability of wavelet-based dictionaries in achieving highly compressible approximations of compound Poisson processes.

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“…For continuous-domain innovation processes (i.e., continuous-domain white noise), the notions of RID and RDE are defined in [10] by vanishingly fine quantization of the time axis and the amplitude range.…”

Section: Related Workmentioning

confidence: 99%

“…It is shown that when the distortion tends to zero, the limiting value of the RDF is closely related to the differential entropy (if it exists) [3]. The compressibility notations are not limited to discrete and continuous sources: there has been some recent efforts to define this notion for sequences [4], [5], [6] and random processes [7], [8], [9], [10].…”

Section: Introductionmentioning

confidence: 99%

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Compressibility Measures for Affinely Singular Random Vectors

Charusaie,

Amini,

Rini

2020

Preprint

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There are several ways to measure the compressibility of a random measure; they include the general rate-distortion curve, as well as more specific notions such as Renyi information dimension (RID), and dimensional-rate bias (DRB). The RID parameter indicates the concentration of the measure around lower-dimensional subsets of the space while the DRB parameter specifies the compressibility of the distribution over these lowerdimensional subsets. While the evaluation of such compressibility parameters is well-studied for continuous and discrete measures (e.g., the DRB is closely related to the entropy and differential entropy in discrete and continuous cases, respectively), the case of discrete-continuous measures is quite subtle. In this paper, we focus on a class of multi-dimensional random measures that have singularities on affine lower-dimensional subsets. These cases are of interest when working with linear transformation of component-wise independent discrete-continuous random variables. Here, we evaluate the RID and DRB for such probability measures. We further provide an upper-bound for the RID of multi-dimensional random measures that are obtained by Lipschitz functions of component-wise independent discretecontinuous random variables (x). The upper-bound is shown to be achievable when the Lipschitz function is Ax, where A satisfies spark(A) = rank(A) + 1 (e.g., Vandermonde matrices). The above results in the case of discrete-domain moving-average processes with non-Gaussian excitation noise allow us to evaluate the block-average RID and to find a relationship between this parameter and other existing compressibility measures.

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How Compressible Are Innovation Processes?

Cited by 8 publications

References 26 publications

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

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

Wavelet Compressibility of Compound Poisson Processes

Compressibility Measures for Affinely Singular Random Vectors

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