Compressibility of symmetric-&amp;#x03B1;-stable processes

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

doi:10.1109/sampta.2015.7148887

Cited by 4 publications

(4 citation statements)

References 15 publications

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“…Using this characterization along with standard embedding properties of approximation spaces [14,Chapter 7], we derive our result. In particular, (50) together with the aforementioned embedding implies that…”

Section: The Compressibility Of Generalized Lévy Processesmentioning

confidence: 99%

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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: The Compressibility Of Generalized Lévy Processesmentioning

confidence: 99%

“…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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confidence: 95%

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

2019

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“…The decay rate of the best M -term error is known to be directly linked to the Besov regularity [31], [45], which has been quantified for a broad class of Lévy processes [46]- [51]. Hence, the compressibility of Lévy processes has already been characterized using this approach [52], [53] and synthesized in [35,Chapter 6]. In a nutshell, state-of-the-art results show that the best M -term quadratic approximation error of the Brownian motion behaves asymptotically like 1/M 1 ,…”

Section: Gaussian Versus Poisson: Two Extreme Compressibility Behaviorsmentioning

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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“…Specifically, when L is an nth-order ordinary differential operator of the form (1) and w = w α is an SαS innovation, then s α = L −1 w α can be shown to be included in the periodic Besov space B n−1+1/α α,∞ ([−π, π]) with probability one [41]. By invoking the approximation properties of Besov spaces [42], this implies that s α − s α,M L2 = O(M −τ0 ) with τ 0 = n + 1 α − 1 − for any > 0, where s α,M denotes the M -term approximation of s α in a suitable (e.g., wavelet-like) basis.…”

Section: Compressibility Of Sparse Processesmentioning

confidence: 99%

Sampling and (sparse) stochastic processes: A tale of splines and innovation

Unser

2015

2015 International Conference on Sampling Theory and Applications (SampTA)

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Abstract-The commonality between splines and Gaussian or sparse stochastic processes is that they are ruled by the same type of differential equations. Our purpose here is to demonstrate that this has profound implications for the three primary forms of sampling: uniform, nonuniform, and compressed sensing.The connection with classical sampling is that there is a one-to-one correspondence between spline interpolation and the minimum-mean-square-error reconstruction of a Gaussian process from its uniform or nonuniform samples. The caveat, of course, is that the spline type has to be matched to the operator that whitens the process.The connection with compressed sensing is that the nonGaussian processes that are ruled by linear differential equations generally admit a parsimonious representation in a wavelet-like basis. There is also a construction based on splines that yields a wavelet-like basis that is matched to the underlying differential operator. It has been observed that expansions in such bases provide excellent M -term approximations of sparse processes. This property is backed by recent estimates of the local Besov regularity of sparse processes.

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Compressibility of symmetric-α-stable processes

Cited by 4 publications

References 15 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

Sampling and (sparse) stochastic processes: A tale of splines and innovation

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