Metric Entropy of Integration Operators and Small Ball Probabilities for the Brownian Sheet

Dunker, Thomas; Linde, Werner; Kühn, Thomas; Lifshits, Mikhail

doi:10.1006/jath.1999.3354

Cited by 20 publications

(13 citation statements)

References 15 publications

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“…It is known (see Belinsky (1998) and Dunker et al (1999)) that e n (ū : The entropy numbers of the generating operator u, as calculated by T. Kühn (private communication), are e n (u) ≈ n 1/α−1/γ−1 (log n) −β/γ , which agrees with (7.12) from the point of view of Theorem 1.3. However, the connection between the entropy and small deviations completely breaks down in the critical case (7.13).…”

Section: Riemann-liouville Processessupporting

confidence: 74%

Small deviations of stable processes and entropy of the associated random operators

Аурзада¹,

Lifshits²,

Linde³

2009

Bernoulli

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We investigate the relation between the small deviation problem for a symmetric α-stable random vector in a Banach space and the metric entropy properties of the operator generating it. This generalizes former results due to Li and Linde and to Aurzada. It is shown that this problem is related to the study of the entropy numbers of a certain random operator. In some cases, an interesting gap appears between the entropy of the original operator and that of the random operator generated by it. This phenomenon is studied thoroughly for diagonal operators. Basic ingredients here are techniques related to random partitions of the integers. The main result concerning metric entropy and small deviations allows us to determine or provide new estimates for the small deviation rate for several symmetric α-stable random processes, including unbounded Riemann-Liouville processes, weighted Riemann-Liouville processes and the (d-dimensional) α-stable sheet. This is an electronic reprint of the original article published by the ISI/BS in Bernoulli, 2009, Vol. 15, No. 4, 1305-1334. This reprint differs from the original in pagination and typographic detail. 1350-7265 c 2009 ISI/BS 1306 F. Aurzada, M. Lifshits and W. LindeIn this case, we say that X is generated by the operator u. This approach is very useful for investigating symmetric α-stable random processes with paths in E ′ . We refer to Section 5 of Li and Linde (2004) or Section 7.1.1 below for a discussion of how all natural examples of SαS processes fit into this framework. For example, if u from L p [0, 1] to L α [0, 1] is defined by (uf )(t) :=

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Section: Riemann-liouville Processessupporting

confidence: 74%

Small deviations of stable processes and entropy of the associated random operators

Аурзада¹,

Lifshits²,

Linde³

2009

Bernoulli

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“…where > 0 and X m is an m-dimensional integrated Brownian sheet (a description of integrated Brownian sheet is given in Section 8.4.2). Required bounds on this small ball probability are then obtained using the results from Dunker et al [13,Theorem 6] and Chen and Li [10,Theorem 1.2]. Specifically, we show that there exist positive constants C m and m depending only on m such that log P sup…”

Section: Risk Analysis For Equally Spaced Lattice Designsmentioning

confidence: 80%

“…where N ( , D m , • 2 ) is the -covering number of D m under the L 2 metric. Our proof of ( 13) is based on establishing a connection between the metric entropy of D m and the small ball probability of integrated Brownian sheet (using ideas and results from Blei et al [3], Gao [18], Li and Linde [22], and Artstein et al [2]) and then deriving the small ball probability of integrated Brownian sheet (using ideas and results from Dunker et al [13] and Chen and Li [10]).…”

Section: Contributionmentioning

confidence: 99%

“…For these reasons, for the integrated Brownian sheet X m , we can restate Theorem 8.7 as follows: for α > 0 and β ∈ R, We now turn our attention to the small ball probability of X m . It is known that the covariance operator of the Brownian sheet B m is T m T * m (see, e.g., Dunker et al [13,Section 3]). Thus, by applying Theorem 8.8, we can obtain…”

Section: Proof Of Theorem 53mentioning

confidence: 99%

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MARS via LASSO

Ki¹,

Fang²,

Guntuboyina³

2021

Preprint

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MARS is a popular method for nonparametric regression introduced by Friedman in 1991. MARS fits simple nonlinear and non-additive functions to regression data. We propose and study a natural LASSO variant of the MARS method. Our method is based on least squares estimation over a convex class of functions obtained by considering infinite-dimensional linear combinations of functions in the MARS basis and imposing a variation based complexity constraint. We show that our estimator can be computed via finite-dimensional convex optimization and that it is naturally connected to nonparametric function estimation techniques based on smoothness constraints. Under a simple design assumption, we prove that our estimator achieves a rate of convergence that depends only logarithmically on dimension and thus avoids the usual curse of dimensionality to some extent. We implement our method with a cross-validation scheme for the selection of the involved tuning parameter and show that it has favorable performance compared to the usual MARS method in simulation and real data settings.

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“…By Corollary 3.1, it would suffice to determine the asymptotic behavior of the average Kolmogorov numbers. The best bounds known up to now are established in [3]. Compare [9] and [13] for more information and related results.…”

Section: Gaussian Markov Processesmentioning

confidence: 97%

Relations between Classical, Average, and Probabilistic Kolmogorov Widths

Creutzig

2002

Journal of Complexity

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Let m be a centered Gaussian Radon measure on a Banach space E, and let H m ı E be its reproducing kernel Hilbert space with unit ball K m . We prove that for the m-average widths d (a) n (E, m) of E and the classical Kolmogorov widthsn (E, m) as well. Furthermore, we show that for the probabilistic widths dfor some universal constant c 1 > 0 and for all d < d 0 . These results are applied to find concrete estimates in some specific settings. © 2002 Elsevier Science (USA)

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Metric Entropy of Integration Operators and Small Ball Probabilities for the Brownian Sheet

Cited by 20 publications

References 15 publications

Small deviations of stable processes and entropy of the associated random operators

Small deviations of stable processes and entropy of the associated random operators

MARS via LASSO

Relations between Classical, Average, and Probabilistic Kolmogorov Widths

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