On the power of random information

We study approximation and integration problems and compare the quality of optimal information with the quality of random information. For some problems random information is almost optimal and for some other problems random information is much worse than optimal information. We prove new results and give a short survey of known results.

[1]  UDC,et al.  ON KOLMOGOROV DIAMETERS OF OCTAHEDRA , 2010 .

[2]  Martin Ehler,et al.  Optimal Monte Carlo integration on closed manifolds , 2017, Statistics and Computing.

[3]  E. Novak,et al.  Tractability of Multivariate Problems , 2008 .

[4]  R. DeVore,et al.  Compressed sensing and best k-term approximation , 2008 .

[5]  E. Novak,et al.  The inverse of the star-discrepancy depends linearly on the dimension , 2001 .

[6]  H. Triebel,et al.  Function Spaces in Lipschitz Domains and Optimal Rates of Convergence for Sampling , 2006 .

[7]  Djalil Chafaï,et al.  Interactions between compressed sensing, random matrices, and high dimensional geometry , 2012 .

[8]  Erich Novak,et al.  Linear information versus function evaluations for L2-approximation , 2008, J. Approx. Theory.

[9]  R. Cramer,et al.  Contemporary Cryptology (Advanced Courses in Mathematics - CRM Barcelona) , 2005 .

[10]  Denka Kutzarova,et al.  Stability of low-rank matrix recovery and its connections to Banach space geometry , 2014, ArXiv.

[11]  Henryk Wozniakowski,et al.  On the Power of Standard Information for Weighted Approximation , 2001, Found. Comput. Math..

[12]  David L Donoho,et al.  Compressed sensing , 2006, IEEE Transactions on Information Theory.

[13]  Albert Cohen,et al.  RECOVERY OF FUNCTIONS OF MANY VARIABLES VIA COMPRESSIVE SENSING , 2011 .

[14]  Emmanuel J. Candès,et al.  Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information , 2004, IEEE Transactions on Information Theory.

[15]  Adam Krzyzak,et al.  Nonparametric estimation of a function from noiseless observations at random points , 2017, J. Multivar. Anal..

[16]  Henryk Wozniakowski,et al.  Approximation of infinitely differentiable multivariate functions is intractable , 2009, J. Complex..

[17]  A. G. Sukharev.  Optimal method of constructing best uniform approximations for functions of a certain class , 1978 .

[18]  Henryk Wozniakowski,et al.  On the power of standard information for multivariate approximation in the worst case setting , 2009, J. Approx. Theory.

[19]  Erich Novak,et al.  Random sections of ellipsoids and the power of random information , 2019, Transactions of the American Mathematical Society.

[20]  Elizabeth L. Wilmer,et al.  Markov Chains and Mixing Times , 2008 .

[21]  Hoang Tran,et al.  Polynomial approximation via compressed sensing of high-dimensional functions on lower sets , 2016, Math. Comput..

[22]  Francis R. Bach,et al.  On the Equivalence between Kernel Quadrature Rules and Random Feature Expansions , 2015, J. Mach. Learn. Res..

[23]  Kenji Fukumizu,et al.  Convergence guarantees for kernel-based quadrature rules in misspecified settings , 2016, NIPS.

[24]  Aicke Hinrichs,et al.  Entropy numbers of embeddings of Schatten classes , 2016, 1612.08105.

[25]  David Krieg,et al.  Algorithms and Complexity for some Multivariate Problems , 2019, 1905.01166.

[26]  Henryk Wozniakowski,et al.  When Are Quasi-Monte Carlo Algorithms Efficient for High Dimensional Integrals? , 1998, J. Complex..

[27]  Holger Rauhut,et al.  The Gelfand widths of lp-balls for 0p<=1 , 2010, J. Complex..

[28]  D. Darling.  On a Class of Problems Related to the Random Division of an Interval , 1953 .

[29]  Vladimir N. Temlyakov,et al.  Hyperbolic Cross Approximation , 2016, 1601.03978.

[30]  H. Woxniakowski.  Information-Based Complexity , 1988 .

[31]  Holger Rauhut,et al.  The Gelfand widths of ℓp-balls for 0 , 2010, ArXiv.