×

Approximation properties of certain operator-induced norms on Hilbert spaces. (English) Zbl 1262.41015

Let \(L^2\) be the space of square-integrable functions on a compact set in \(\mathbb{R}^d\) with regard to a probabilistic measure, and let \(\|\cdot\|_{L^2}\) be the corresponding norm. If \(\mathcal{H}\) is a Hilbert space, \(\mathcal{H}\subset L^2\) and \(\Phi\) is a vector \(\Phi=(\Phi_1,\dotsc,\Phi_n)\) of continuous linear functionals \(\Phi_i:\mathcal{H}\to\mathbb{R}\), then one can define a norm on \(\mathcal{H}\) by \(\| f\|_{\mathcal{H}}=\sqrt{\sum_{i=1}^n(\Phi_i(f))^2}\), \(f\in\mathcal{H}\). This norm, a substitute of the norm \(\|\cdot\|_{L^2}\), is very useful for practical reasons. The authors are interested in the problem of determining the order of deviation between the two norms. More exactly, the authors consider the space \(\mathcal{H}=\{f\in L^2\mid f=\sum_{k=1}^{\infty}\sqrt{\sigma_k}\alpha_k\Psi_k, \text{ for some }(\alpha_k)_{k=1}^{\infty}\in l_2\}\), where \((\Psi_k)_{k=1}^{\infty}\) is an orthogonal sequence in \(L^2\) and \(\sigma_1\geq\sigma_2\geq\dotsb>0\) is a sequence which decreases to zero. The inner product is given by \(\langle f,g\rangle_{\mathcal{H}}=\sum_{k=1}^{\infty}\alpha_k\beta_k\), if \(f=\sum_{k=1}^{\infty}\sqrt{\sigma_k}\alpha_k\Psi_k\) and \(g=\sum_{k=1}^{\infty}\sqrt{\sigma_k}\beta_k\Psi_k\). The main result gives an upper bound for the quantity \(R_{\Phi}(\varepsilon)=\sup\{\| f\|_{L^2}^2\,|\,f\in\mathcal{H},\;\| f\|_{\mathcal{H}}^2\leq\varepsilon^2\}\). Then, several applications are discussed, namely, reproducing kernel Hilbert space, random domain sampling, Sobolov kernel and Fourier truncation.

MSC:

41A46 Approximation by arbitrary nonlinear expressions; widths and entropy
46C05 Hilbert and pre-Hilbert spaces: geometry and topology (including spaces with semidefinite inner product)
42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis
47B32 Linear operators in reproducing-kernel Hilbert spaces (including de Branges, de Branges-Rovnyak, and other structured spaces)
15A45 Miscellaneous inequalities involving matrices
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
62F12 Asymptotic properties of parametric estimators
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Adamczak, R., A tail inequality for suprema of unbounded empirical processes with applications to Markov chains, Electronic Journal of Probability, 13, 1000-1034 (2008) · Zbl 1190.60010
[2] Ahlswede, R.; Winter, A., Strong converse for identification via quantum channels, IEEE Transactions on Information Theory, 48, 3, 569-579 (2002) · Zbl 1071.94530
[3] Bartlett, P. L.; Bousquet, O.; Mendelson, S., Local Rademacher complexities, Annals of Statistics, 33, 4, 1497-1537 (2005) · Zbl 1083.62034
[4] R. DeVore, Approximation of functions, in: Proc. Symp. Applied Mathematics, vol. 36, 1986, pp. 1-20.; R. DeVore, Approximation of functions, in: Proc. Symp. Applied Mathematics, vol. 36, 1986, pp. 1-20.
[5] Gamkrelidze, R. V.; Newton, D.; Tikhomirov, V. M., Analysis: Convex Analysis and Approximation Theory (1990), Birkhäuser
[6] Garling, D. J.H., Inequalities: A Journey into Linear Analysis (2007), Cambridge Univ Pr. · Zbl 1135.26014
[7] Koltchinski, V., Local Rademacher complexities and oracle inequalities in risk minimization, Annals of Statistics, 34, 6, 2593-2656 (2006)
[8] Ledoux, M., (The Concentration of Measure Phenomenon. The Concentration of Measure Phenomenon, Mathematical Surveys and Monographs (2001), American Mathematical Society: American Mathematical Society Providence, RI) · Zbl 0995.60002
[9] Mendelson, S., (Geometric Parameters of Kernel Machines. Geometric Parameters of Kernel Machines, Lecture Notes in Computer Science, vol. 2375 (2002)), 29-43 · Zbl 1050.68070
[10] Pinkus, A., \((N\)-Widths in Approximation Theory. \(N\)-Widths in Approximation Theory, Ergebnisse der Mathematik und ihrer Grenzgebiete 3 Folge (1985), Springer)
[11] A. Pinkus, \(N\); A. Pinkus, \(N\)
[12] G. Raskutti, M.J. Wainwright, B. Yu, Minimax-optimal rates for sparse additive models over kernel classes via convex programming, Tech. Rep., Department of Statistics, UC Berkeley, August 2010. http://arxiv.org/abs/1008.3654; G. Raskutti, M.J. Wainwright, B. Yu, Minimax-optimal rates for sparse additive models over kernel classes via convex programming, Tech. Rep., Department of Statistics, UC Berkeley, August 2010. http://arxiv.org/abs/1008.3654 · Zbl 1283.62071
[13] Riesz, F.; Sz.-Nagy, B., Functional Analysis (1990), Dover Publications
[14] Rudelson, M., Random vectors in the isotropic position, Journal of Functional Analysis, 164, 1, 60-72 (1999) · Zbl 0929.46021
[15] J.A. Tropp, User-friendly tail bounds for sums of random matrices. URL: arxiv:1004.4389]http://arxiv.org/abs/1004.4389.; J.A. Tropp, User-friendly tail bounds for sums of random matrices. URL: arxiv:1004.4389]http://arxiv.org/abs/1004.4389.
[16] van de Geer, S. A., Empirical Processes in \(M\)-Estimation (2000), Cambridge University Press
[17] van der Vaart, A. W.; Wellner, J., Weak Convergence and Empirical Processes (1996), Springer-Verlag: Springer-Verlag New York, NY · Zbl 0862.60002
[18] R. Vershynin, Introduction to the non-asymptotic analysis of random matrices. URL: http://www-personal.umich.edu/ romanv/papers/non-asymptotic-rmt-plain.pdf; R. Vershynin, Introduction to the non-asymptotic analysis of random matrices. URL: http://www-personal.umich.edu/ romanv/papers/non-asymptotic-rmt-plain.pdf · Zbl 1291.15088
[19] Wigderson, A.; Xiao, D., Derandomizing the Ahlswede-Winter matrix-valued Chernoff bound using pessimistic estimators, and applications, Theory of Computing, 4, 1, 53-76 (2008) · Zbl 1213.68697
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.