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35 results on '"Kunsch, Robert J."'

1. Adaptive and non-adaptive randomized approximation of high-dimensional vectors

Author

Kunsch, Robert J. and Wnuk, Marcin

Subjects
Mathematics - Numerical Analysis, 65C, 65D, 46, 60
Abstract

We study approximation of the embedding $\ell_p^m \hookrightarrow \ell_q^m$, $1 \leq p < q \leq \infty$, based on randomized algorithms that use up to $n$ arbitrary linear functionals as information on a problem instance where $n \ll m$. By analysing adaptive methods we show upper bounds for which the information-based complexity $n$ exhibits only a $(\log\log m)$-dependence. In the case $q < \infty$ we use a multi-sensitivity approach in order to reach optimal polynomial order in $n$ for the Monte Carlo error. We also improve on non-adaptive methods for $q < \infty$ by denoising known algorithms for uniform approximation.

Published
2024

2. Uniform approximation of vectors using adaptive randomized information

Author

Kunsch, Robert J. and Wnuk, Marcin

Subjects
Mathematics - Numerical Analysis, 65C, 65D, 46, 60
Abstract

We study approximation of the embedding $\ell_p^m \rightarrow \ell_{\infty}^m$, $1 \leq p \leq 2$, based on randomized adaptive algorithms that use arbitrary linear functionals as information on a problem instance. We show upper bounds for which the complexity $n$ exhibits only a $(\log\log m)$-dependence. Our results for $p=1$ lead to an example of a gap of order $n$ (up to logarithmic factors) for the error between best adaptive and non-adaptive Monte Carlo methods. This is the largest possible gap for linear problems.

Published
2024

3. Linear Monte Carlo quadrature with optimal confidence intervals

Author

Kunsch, Robert J.

Subjects
Mathematics - Numerical Analysis, 65C05, 65D30, 65J05
Abstract

We study the numerical integration of functions from isotropic Sobolev spaces $W_p^s([0,1]^d)$ using finitely many function evaluations within randomized algorithms, aiming for the smallest possible probabilistic error guarantee $\varepsilon > 0$ at confidence level $1-\delta \in (0,1)$. For spaces consisting of continuous functions, non-linear Monte Carlo methods with optimal confidence properties have already been known, in few cases even linear methods that succeed in that respect. In this paper we promote a new method called stratified control variates (SCV) and by it show that already linear methods achieve optimal probabilistic error rates in the high smoothness regime without the need to adjust algorithmic parameters to the uncertainty $\delta$. We also analyse a version of SCV in the low smoothness regime where $W_p^s([0,1]^d)$ may contain functions with singularities. Here, we observe a polynomial dependence of the error on $\delta^{-1}$ which cannot be avoided for linear methods. This is worse than what is known to be possible using non-linear algorithms where only a logarithmic dependence on $\delta^{-1}$ occurs if we tune in for a specific value of $\delta$., Comment: 22 pages, 2 figures

Published
2023

6. Expected dispersion of uniformly distributed points

Author

Hinrichs, Aicke, Krieg, David, Kunsch, Robert J., and Rudolf, Daniel

Subjects
Mathematics - Probability, Computer Science - Computational Geometry, 62D05 (Primary), 52B55, 65Y20, 68Q25 (Secondary)
Abstract

The dispersion of a point set in $[0,1]^d$ is the volume of the largest axis parallel box inside the unit cube that does not intersect with the point set. We study the expected dispersion with respect to a random set of $n$ points determined by an i.i.d. sequence of uniformly distributed random variables. Depending on the number of points $n$ and the dimension $d$ we provide an upper and lower bound of the expected dispersion. In particular, we show that the minimal number of points required to achieve an expected dispersion less than $\varepsilon\in(0,1)$ depends linearly on the dimension $d$., Comment: 13 pages

Published
2019

7. Optimal confidence for Monte Carlo integration of smooth functions

Author

Kunsch, Robert J. and Rudolf, Daniel

Subjects
Mathematics - Numerical Analysis, 65C05, 65D30, 65J05
Abstract

We study the complexity of approximating integrals of smooth functions at absolute precision $\varepsilon > 0$ with confidence level $1 - \delta \in (0,1)$. The optimal error rate for multivariate functions from classical isotropic Sobolev spaces $W_p^r(G)$ with sufficient smoothness on bounded Lipschitz domains $G \subset \mathbb{R}^d$ is determined. It turns out that the integrability index $p$ has an effect on the influence of the uncertainty $\delta$ in the complexity. In the limiting case $p = 1$ we see that deterministic methods cannot be improved by randomization. In general, higher smoothness reduces the additional effort for diminishing the uncertainty. Finally, we add a discussion about this problem for function spaces with mixed smoothness., Comment: 23 pages

Published
2018

8. Solvable Integration Problems and Optimal Sample Size Selection

Author

Kunsch, Robert J., Novak, Erich, and Rudolf, Daniel

Subjects
Mathematics - Numerical Analysis, Statistics - Computation, 65C05, 62D99, 65Y20, 62L12
Abstract

We compute the integral of a function or the expectation of a random variable with minimal cost and use, for our new algorithm and for upper bounds of the complexity, i.i.d. samples. Under certain assumptions it is possible to select a sample size based on a variance estimation, or -- more generally -- based on an estimation of a (central absolute) $p$-moment. That way one can guarantee a small absolute error with high probability, the problem is thus called solvable. The expected cost of the method depends on the $p$-moment of the random variable, which can be arbitrarily large. In order to prove the optimality of our algorithm we also provide lower bounds. These bounds apply not only to methods based on i.i.d. samples but also to general randomized algorithms. They show that -- up to constants -- the cost of the algorithm is optimal in terms of accuracy, confidence level, and norm of the particular input random variable. Since the considered classes of random variables or integrands are very large, the worst case cost would be infinite. Nevertheless one can define adaptive stopping rules such that for each input the expected cost is finite. We contrast these positive results with examples of integration problems that are not solvable., Comment: 38 pages, to appear in the Journal of Complexity

Published
2018

9. The Difficulty of Monte Carlo Approximation of Multivariate Monotone Functions

Author

Kunsch, Robert J.

Subjects
Mathematics - Numerical Analysis, 41A25, 41A46, 41A63, 65C05
Abstract

We study the $L_1$-approximation of $d$-variate monotone functions based on information from $n$ function evaluations. It is known that this problem suffers from the curse of dimensionality in the deterministic setting, that is, the number $n(\varepsilon,d)$ of function evaluations needed in order to approximate an unknown monotone function within a given error threshold $\varepsilon$ grows at least exponentially in $d$. This is not the case in the randomized setting (Monte Carlo setting) where the complexity $n(\varepsilon,d)$ grows exponentially in $\sqrt{d}$ (modulo logarithmic terms) only. An algorithm exhibiting this complexity is presented. Still, the problem remains difficult as best known methods are deterministic if $\varepsilon$ is comparably small, namely $\varepsilon \preceq 1/\sqrt{d}$. This inherent difficulty is confirmed by lower complexity bounds which reveal a joint $(\varepsilon,d)$-dependency and from which we deduce that the problem is not weakly tractable., Comment: Based on a chapter of the author's PhD thesis arXiv:1704.08213

Published
2018

10. Breaking the Curse for Uniform Approximation in Hilbert Spaces via Monte Carlo Methods

Author

Kunsch, Robert J.

Subjects
Mathematics - Numerical Analysis, 41A25, 41A45, 41A63, 41A65, 65C05, 65J05
Abstract

We study the $L_{\infty}$-approximation of $d$-variate functions from Hilbert spaces via linear functionals as information. It is a common phenomenon in tractability studies that unweighted problems (with each dimension being equally important) suffer from the curse of dimensionality in the deterministic setting, that is, the number $n(\varepsilon,d)$ of information needed in order to solve a problem to within a given accuracy $\varepsilon > 0$ grows exponentially in $d$. We show that for certain approximation problems in periodic tensor product spaces, in particular Korobov spaces with smoothness $r > 1/2$, switching to the randomized setting can break the curse of dimensionality, now having polynomial tractability, namely $n(\varepsilon,d) \preceq \varepsilon^{-2} \, d \, (1 + \log d)$. Similar benefits of Monte Carlo methods in terms of tractability have only been known for integration problems so far., Comment: 1 figure. Based on a chapter of the author's PhD thesis arXiv:1704.08213, yet with improved results in some settings

Published
2017

11. Monte Carlo Methods for Uniform Approximation on Periodic Sobolev Spaces with Mixed Smoothness

Author

Byrenheid, Glenn, Kunsch, Robert J., and Nguyen, Van Kien

Subjects
Mathematics - Numerical Analysis, 41A25, 41A46, 41A63, 41A65, 65C05, 65J05
Abstract

We consider the order of convergence for linear and nonlinear Monte Carlo approximation of compact embeddings from Sobolev spaces of dominating mixed smoothness defined on the torus $\mathbb{T}^d$ into the space $L_{\infty}(\mathbb{T}^d)$ via methods that use arbitrary linear information. These cases are interesting because we can gain a speedup of up to $1/2$ in the main rate compared to the worst case approximation. In doing so we determine the rate for some cases that have been left open by Fang and Duan.

Published
2017
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12. High-Dimensional Function Approximation: Breaking the Curse with Monte Carlo Methods

Author

Kunsch, Robert J.

Subjects
Mathematics - Numerical Analysis, 41A17, 41A25, 41A46, 41A63, 41A65, 65C05, 65J05
Abstract

In this dissertation we study the tractability of the information-based complexity $n(\varepsilon,d)$ for $d$-variate function approximation problems. In the deterministic setting for many unweighted problems the curse of dimensionality holds, that means, for some fixed error tolerance $\varepsilon>0$ the complexity $n(\varepsilon,d)$ grows exponentially in $d$. For integration problems one can usually break the curse with the standard Monte Carlo method. For function approximation problems, however, similar effects of randomization have been unknown so far. The thesis contains results on three more or less stand-alone topics. For an extended five page abstract, see the section "Introduction and Results". Chapter 2 is concerned with lower bounds for the Monte Carlo error for general linear problems via Bernstein numbers. This technique is applied to the $L_{\infty}$-approximation of certain classes of $C^{\infty}$-functions, where it turns out that randomization does not affect the tractability classification of the problem. Chapter 3 studies the $L_{\infty}$-approximation of functions from Hilbert spaces with methods that may use arbitrary linear functionals as information. For certain classes of periodic functions from unweighted periodic tensor product spaces, in particular Korobov spaces, we observe the curse of dimensionality in the deterministic setting, while with randomized methods we achieve polynomial tractability. Chapter 4 deals with the $L_1$-approximation of monotone functions via function values. It is known that this problem suffers from the curse in the deterministic setting. An improved lower bound shows that the problem is still intractable in the randomized setting. However, Monte Carlo breaks the curse, in detail, for any fixed error tolerance $\varepsilon>0$ the complexity $n(\varepsilon,d)$ grows exponentially in $\sqrt{d}$ only., Comment: This is the author's submitted PhD thesis, still in the referee process

Published
2017

20. Sparse Deterministic and Stochastic Channels: Identification of Spreading Functions and Covariances

Author

Kaplan, Alihan, Lee, Dae Gwan, Pfander, Götz E., Pohl, Volker, Benedetto, John J., Series Editor, Czaja, Wojciech, Series Editor, Okoudjou, Kasso, Series Editor, Aldroubi, Akram, Editorial Board Member, Cochran, Douglas, Editorial Board Member, Feichtinger, Hans G., Editorial Board Member, Heil, Christopher, Editorial Board Member, Jaffard, Stéphane, Editorial Board Member, Kutyniok, Gitta, Editorial Board Member, Maggioni, Mauro, Editorial Board Member, Shen, Zuowei, Editorial Board Member, Strohmer, Thomas, Editorial Board Member, Wang, Yang, Editorial Board Member, Rauhut, Holger, editor, and Kunsch, Robert J., editor

Published
2022
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21. Proof Methods for Robust Low-Rank Matrix Recovery

Author

Fuchs, Tim, Gross, David, Jung, Peter, Krahmer, Felix, Kueng, Richard, Stöger, Dominik, Benedetto, John J., Series Editor, Czaja, Wojciech, Series Editor, Okoudjou, Kasso, Series Editor, Aldroubi, Akram, Editorial Board Member, Cochran, Douglas, Editorial Board Member, Feichtinger, Hans G., Editorial Board Member, Heil, Christopher, Editorial Board Member, Jaffard, Stéphane, Editorial Board Member, Kutyniok, Gitta, Editorial Board Member, Maggioni, Mauro, Editorial Board Member, Shen, Zuowei, Editorial Board Member, Strohmer, Thomas, Editorial Board Member, Wang, Yang, Editorial Board Member, Rauhut, Holger, editor, and Kunsch, Robert J., editor

Published
2022
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22. Angular Scattering Function Estimation Using Deep Neural Networks

Author

Song, Yi, Caire, Giuseppe, Benedetto, John J., Series Editor, Czaja, Wojciech, Series Editor, Okoudjou, Kasso, Series Editor, Aldroubi, Akram, Editorial Board Member, Cochran, Douglas, Editorial Board Member, Feichtinger, Hans G., Editorial Board Member, Heil, Christopher, Editorial Board Member, Jaffard, Stéphane, Editorial Board Member, Kutyniok, Gitta, Editorial Board Member, Maggioni, Mauro, Editorial Board Member, Shen, Zuowei, Editorial Board Member, Strohmer, Thomas, Editorial Board Member, Wang, Yang, Editorial Board Member, Rauhut, Holger, editor, and Kunsch, Robert J., editor

Published
2022
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23. Recovery Under Side Constraints

Author

Ardah, Khaled, Haardt, Martin, Liu, Tianyi, Matter, Frederic, Pesavento, Marius, Pfetsch, Marc E., Benedetto, John J., Series Editor, Czaja, Wojciech, Series Editor, Okoudjou, Kasso, Series Editor, Aldroubi, Akram, Editorial Board Member, Cochran, Douglas, Editorial Board Member, Feichtinger, Hans G., Editorial Board Member, Heil, Christopher, Editorial Board Member, Jaffard, Stéphane, Editorial Board Member, Kutyniok, Gitta, Editorial Board Member, Maggioni, Mauro, Editorial Board Member, Shen, Zuowei, Editorial Board Member, Strohmer, Thomas, Editorial Board Member, Wang, Yang, Editorial Board Member, Rauhut, Holger, editor, and Kunsch, Robert J., editor

Published
2022
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24. Atmospheric Radar Imaging Improvements Using Compressed Sensing and MIMO

Author

Chau, Jorge Luis, Urco, Juan Miguel, Weber, Tobias, Aweda, Jeremy Olaore, Benedetto, John J., Series Editor, Czaja, Wojciech, Series Editor, Okoudjou, Kasso, Series Editor, Aldroubi, Akram, Editorial Board Member, Cochran, Douglas, Editorial Board Member, Feichtinger, Hans G., Editorial Board Member, Heil, Christopher, Editorial Board Member, Jaffard, Stéphane, Editorial Board Member, Kutyniok, Gitta, Editorial Board Member, Maggioni, Mauro, Editorial Board Member, Shen, Zuowei, Editorial Board Member, Strohmer, Thomas, Editorial Board Member, Wang, Yang, Editorial Board Member, Rauhut, Holger, editor, and Kunsch, Robert J., editor

Published
2022
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25. Information Theory and Recovery Algorithms for Data Fusion in Earth Observation

Author

Fornasier, Massimo, Hong, Danfeng, Kramer, Gerhard, Palzer, Lars, Rauchensteiner, Michael, Zhu, Xiao Xiang, Benedetto, John J., Series Editor, Czaja, Wojciech, Series Editor, Okoudjou, Kasso, Series Editor, Aldroubi, Akram, Editorial Board Member, Cochran, Douglas, Editorial Board Member, Feichtinger, Hans G., Editorial Board Member, Heil, Christopher, Editorial Board Member, Jaffard, Stéphane, Editorial Board Member, Kutyniok, Gitta, Editorial Board Member, Maggioni, Mauro, Editorial Board Member, Shen, Zuowei, Editorial Board Member, Strohmer, Thomas, Editorial Board Member, Wang, Yang, Editorial Board Member, Rauhut, Holger, editor, and Kunsch, Robert J., editor

Published
2022
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26. Compressive Sensing and Neural Networks from a Statistical Learning Perspective

Author

Behboodi, Arash, Rauhut, Holger, Schnoor, Ekkehard, Benedetto, John J., Series Editor, Czaja, Wojciech, Series Editor, Okoudjou, Kasso, Series Editor, Aldroubi, Akram, Editorial Board Member, Cochran, Douglas, Editorial Board Member, Feichtinger, Hans G., Editorial Board Member, Heil, Christopher, Editorial Board Member, Jaffard, Stéphane, Editorial Board Member, Kutyniok, Gitta, Editorial Board Member, Maggioni, Mauro, Editorial Board Member, Shen, Zuowei, Editorial Board Member, Strohmer, Thomas, Editorial Board Member, Wang, Yang, Editorial Board Member, Rauhut, Holger, editor, and Kunsch, Robert J., editor

Published
2022
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27. Unbiasing in Iterative Reconstruction Algorithms for Discrete Compressed Sensing

Author

Fischer, Robert F. H., Sippel, Carmen, Benedetto, John J., Series Editor, Czaja, Wojciech, Series Editor, Okoudjou, Kasso, Series Editor, Aldroubi, Akram, Editorial Board Member, Cochran, Douglas, Editorial Board Member, Feichtinger, Hans G., Editorial Board Member, Heil, Christopher, Editorial Board Member, Jaffard, Stéphane, Editorial Board Member, Kutyniok, Gitta, Editorial Board Member, Maggioni, Mauro, Editorial Board Member, Shen, Zuowei, Editorial Board Member, Strohmer, Thomas, Editorial Board Member, Wang, Yang, Editorial Board Member, Rauhut, Holger, editor, and Kunsch, Robert J., editor

Published
2022
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28. Analysis of Sparse Recovery Algorithms via the Replica Method

Author

Bereyhi, Ali, Müller, Ralf R., Schulz-Baldes, Hermann, Benedetto, John J., Series Editor, Czaja, Wojciech, Series Editor, Okoudjou, Kasso, Series Editor, Aldroubi, Akram, Editorial Board Member, Cochran, Douglas, Editorial Board Member, Feichtinger, Hans G., Editorial Board Member, Heil, Christopher, Editorial Board Member, Jaffard, Stéphane, Editorial Board Member, Kutyniok, Gitta, Editorial Board Member, Maggioni, Mauro, Editorial Board Member, Shen, Zuowei, Editorial Board Member, Strohmer, Thomas, Editorial Board Member, Wang, Yang, Editorial Board Member, Rauhut, Holger, editor, and Kunsch, Robert J., editor

Published
2022
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29. New Challenges in Covariance Estimation: Multiple Structures and Coarse Quantization

Author

Maly, Johannes, Yang, Tianyu, Dirksen, Sjoerd, Rauhut, Holger, Caire, Giuseppe, Benedetto, John J., Series Editor, Czaja, Wojciech, Series Editor, Okoudjou, Kasso, Series Editor, Aldroubi, Akram, Editorial Board Member, Cochran, Douglas, Editorial Board Member, Feichtinger, Hans G., Editorial Board Member, Heil, Christopher, Editorial Board Member, Jaffard, Stéphane, Editorial Board Member, Kutyniok, Gitta, Editorial Board Member, Maggioni, Mauro, Editorial Board Member, Shen, Zuowei, Editorial Board Member, Strohmer, Thomas, Editorial Board Member, Wang, Yang, Editorial Board Member, Rauhut, Holger, editor, and Kunsch, Robert J., editor

Published
2022
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30. Compressed Sensing in the Spherical Near-Field to Far-Field Transformation

Author

Culotta-López, Cosme, Bangun, Arya, Mathar, Rudolf, Heberling, Dirk, Benedetto, John J., Series Editor, Czaja, Wojciech, Series Editor, Okoudjou, Kasso, Series Editor, Aldroubi, Akram, Editorial Board Member, Cochran, Douglas, Editorial Board Member, Feichtinger, Hans G., Editorial Board Member, Heil, Christopher, Editorial Board Member, Jaffard, Stéphane, Editorial Board Member, Kutyniok, Gitta, Editorial Board Member, Maggioni, Mauro, Editorial Board Member, Shen, Zuowei, Editorial Board Member, Strohmer, Thomas, Editorial Board Member, Wang, Yang, Editorial Board Member, Rauhut, Holger, editor, and Kunsch, Robert J., editor

Published
2022
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31. Hierarchical Compressed Sensing

Author

Eisert, Jens, Flinth, Axel, Groß, Benedikt, Roth, Ingo, Wunder, Gerhard, Benedetto, John J., Series Editor, Czaja, Wojciech, Series Editor, Okoudjou, Kasso, Series Editor, Aldroubi, Akram, Editorial Board Member, Cochran, Douglas, Editorial Board Member, Feichtinger, Hans G., Editorial Board Member, Heil, Christopher, Editorial Board Member, Jaffard, Stéphane, Editorial Board Member, Kutyniok, Gitta, Editorial Board Member, Maggioni, Mauro, Editorial Board Member, Shen, Zuowei, Editorial Board Member, Strohmer, Thomas, Editorial Board Member, Wang, Yang, Editorial Board Member, Rauhut, Holger, editor, and Kunsch, Robert J., editor

Published
2022
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32. Active Channel Sparsification: Realizing Frequency-Division Duplexing Massive MIMO with Minimal Overhead

Author

Khalilsarai, Mahdi Barzegar, Haghighatshoar, Saeid, Yi, Xinping, Caire, Giuseppe, Wunder, Gerhard, Benedetto, John J., Series Editor, Czaja, Wojciech, Series Editor, Okoudjou, Kasso, Series Editor, Aldroubi, Akram, Editorial Board Member, Cochran, Douglas, Editorial Board Member, Feichtinger, Hans G., Editorial Board Member, Heil, Christopher, Editorial Board Member, Jaffard, Stéphane, Editorial Board Member, Kutyniok, Gitta, Editorial Board Member, Maggioni, Mauro, Editorial Board Member, Shen, Zuowei, Editorial Board Member, Strohmer, Thomas, Editorial Board Member, Wang, Yang, Editorial Board Member, Rauhut, Holger, editor, and Kunsch, Robert J., editor

Published
2022
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33. Fast Radio Propagation Prediction with Deep Learning

Author

Levie, Ron, Yapar, Çağkan, Caire, Giuseppe, Kutyniok, Gitta, Benedetto, John J., Series Editor, Czaja, Wojciech, Series Editor, Okoudjou, Kasso, Series Editor, Aldroubi, Akram, Editorial Board Member, Cochran, Douglas, Editorial Board Member, Feichtinger, Hans G., Editorial Board Member, Heil, Christopher, Editorial Board Member, Jaffard, Stéphane, Editorial Board Member, Kutyniok, Gitta, Editorial Board Member, Maggioni, Mauro, Editorial Board Member, Shen, Zuowei, Editorial Board Member, Strohmer, Thomas, Editorial Board Member, Wang, Yang, Editorial Board Member, Rauhut, Holger, editor, and Kunsch, Robert J., editor

Published
2022
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34. Sparse Recovery of Sound Fields Using Measurements from Moving Microphones

Author

Katzberg, Fabrice, Mertins, Alfred, Benedetto, John J., Series Editor, Czaja, Wojciech, Series Editor, Okoudjou, Kasso, Series Editor, Aldroubi, Akram, Editorial Board Member, Cochran, Douglas, Editorial Board Member, Feichtinger, Hans G., Editorial Board Member, Heil, Christopher, Editorial Board Member, Jaffard, Stéphane, Editorial Board Member, Kutyniok, Gitta, Editorial Board Member, Maggioni, Mauro, Editorial Board Member, Shen, Zuowei, Editorial Board Member, Strohmer, Thomas, Editorial Board Member, Wang, Yang, Editorial Board Member, Rauhut, Holger, editor, and Kunsch, Robert J., editor

Published
2022
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35. Over-the-Air Computation for Distributed Machine Learning and Consensus in Large Wireless Networks

Author

Frey, Matthias, Bjelaković, Igor, Stańczak, Sławomir, Benedetto, John J., Series Editor, Czaja, Wojciech, Series Editor, Okoudjou, Kasso, Series Editor, Aldroubi, Akram, Editorial Board Member, Cochran, Douglas, Editorial Board Member, Feichtinger, Hans G., Editorial Board Member, Heil, Christopher, Editorial Board Member, Jaffard, Stéphane, Editorial Board Member, Kutyniok, Gitta, Editorial Board Member, Maggioni, Mauro, Editorial Board Member, Shen, Zuowei, Editorial Board Member, Strohmer, Thomas, Editorial Board Member, Wang, Yang, Editorial Board Member, Rauhut, Holger, editor, and Kunsch, Robert J., editor

Published
2022
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