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Venkat Anantharam - One of the best experts on this subject based on the ideXlab platform.
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universal algorithms building a case for Pointwise Convergence
Allerton Conference on Communication Control and Computing, 2012Co-Authors: Narayana Santhanam, Venkat AnantharamAbstract:We consider algorithms for prediction, compression and entropy estimation in a universal setup. In each case, we estimate some function of an unknown distribution p over the set of natural numbers, using only n observations generated i.i.d. from p. While p is unknown, it belongs to a known collection P of possible models. When the supports of distributions in P are uniformly bounded, consistent algorithms exist for each of the problems. Namely, the Convergence of the estimate to the true value can be bounded by a function depending only on the sample size, n, and not on the underlying distribution p. However, when the supports of distributions in P are not uniformly bounded, a more natural approach involves algorithms that are Pointwise consistent, namely, the Convergence to the true value is at a rate that depends on both n and the underlying (unknown) distribution p. The obvious practical difficulty with Pointwise Convergence is that the asymptotic consistency of the algorithm may indicate nothing about the performance of the algorithm for any fixed sample size, since the underlying distribution is unknown. In this paper, we first note that for many complex model classes P, we can still circumvent the above practical difficulty with Pointwise Convergence. Secondly, we take here a preliminary step towards characterizing a broad framework establishing the hierarchy of difficulty of problems involving Pointwise Convergence. We look for connections among the following problems which we define for a Pointwise Convergence scenario: (i) predicting good upper bounds on the next unseen sample, (ii) weak universal compression, and (iii) entropy estimation. We construct counter-examples to show that no two properties above imply the third.
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Allerton Conference - Universal algorithms: Building a case for Pointwise Convergence
2012 50th Annual Allerton Conference on Communication Control and Computing (Allerton), 2012Co-Authors: Narayana Santhanam, Venkat AnantharamAbstract:We consider algorithms for prediction, compression and entropy estimation in a universal setup. In each case, we estimate some function of an unknown distribution p over the set of natural numbers, using only n observations generated i.i.d. from p. While p is unknown, it belongs to a known collection P of possible models. When the supports of distributions in P are uniformly bounded, consistent algorithms exist for each of the problems. Namely, the Convergence of the estimate to the true value can be bounded by a function depending only on the sample size, n, and not on the underlying distribution p. However, when the supports of distributions in P are not uniformly bounded, a more natural approach involves algorithms that are Pointwise consistent, namely, the Convergence to the true value is at a rate that depends on both n and the underlying (unknown) distribution p. The obvious practical difficulty with Pointwise Convergence is that the asymptotic consistency of the algorithm may indicate nothing about the performance of the algorithm for any fixed sample size, since the underlying distribution is unknown. In this paper, we first note that for many complex model classes P, we can still circumvent the above practical difficulty with Pointwise Convergence. Secondly, we take here a preliminary step towards characterizing a broad framework establishing the hierarchy of difficulty of problems involving Pointwise Convergence. We look for connections among the following problems which we define for a Pointwise Convergence scenario: (i) predicting good upper bounds on the next unseen sample, (ii) weak universal compression, and (iii) entropy estimation. We construct counter-examples to show that no two properties above imply the third.
Narayana Santhanam - One of the best experts on this subject based on the ideXlab platform.
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universal algorithms building a case for Pointwise Convergence
Allerton Conference on Communication Control and Computing, 2012Co-Authors: Narayana Santhanam, Venkat AnantharamAbstract:We consider algorithms for prediction, compression and entropy estimation in a universal setup. In each case, we estimate some function of an unknown distribution p over the set of natural numbers, using only n observations generated i.i.d. from p. While p is unknown, it belongs to a known collection P of possible models. When the supports of distributions in P are uniformly bounded, consistent algorithms exist for each of the problems. Namely, the Convergence of the estimate to the true value can be bounded by a function depending only on the sample size, n, and not on the underlying distribution p. However, when the supports of distributions in P are not uniformly bounded, a more natural approach involves algorithms that are Pointwise consistent, namely, the Convergence to the true value is at a rate that depends on both n and the underlying (unknown) distribution p. The obvious practical difficulty with Pointwise Convergence is that the asymptotic consistency of the algorithm may indicate nothing about the performance of the algorithm for any fixed sample size, since the underlying distribution is unknown. In this paper, we first note that for many complex model classes P, we can still circumvent the above practical difficulty with Pointwise Convergence. Secondly, we take here a preliminary step towards characterizing a broad framework establishing the hierarchy of difficulty of problems involving Pointwise Convergence. We look for connections among the following problems which we define for a Pointwise Convergence scenario: (i) predicting good upper bounds on the next unseen sample, (ii) weak universal compression, and (iii) entropy estimation. We construct counter-examples to show that no two properties above imply the third.
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Allerton Conference - Universal algorithms: Building a case for Pointwise Convergence
2012 50th Annual Allerton Conference on Communication Control and Computing (Allerton), 2012Co-Authors: Narayana Santhanam, Venkat AnantharamAbstract:We consider algorithms for prediction, compression and entropy estimation in a universal setup. In each case, we estimate some function of an unknown distribution p over the set of natural numbers, using only n observations generated i.i.d. from p. While p is unknown, it belongs to a known collection P of possible models. When the supports of distributions in P are uniformly bounded, consistent algorithms exist for each of the problems. Namely, the Convergence of the estimate to the true value can be bounded by a function depending only on the sample size, n, and not on the underlying distribution p. However, when the supports of distributions in P are not uniformly bounded, a more natural approach involves algorithms that are Pointwise consistent, namely, the Convergence to the true value is at a rate that depends on both n and the underlying (unknown) distribution p. The obvious practical difficulty with Pointwise Convergence is that the asymptotic consistency of the algorithm may indicate nothing about the performance of the algorithm for any fixed sample size, since the underlying distribution is unknown. In this paper, we first note that for many complex model classes P, we can still circumvent the above practical difficulty with Pointwise Convergence. Secondly, we take here a preliminary step towards characterizing a broad framework establishing the hierarchy of difficulty of problems involving Pointwise Convergence. We look for connections among the following problems which we define for a Pointwise Convergence scenario: (i) predicting good upper bounds on the next unseen sample, (ii) weak universal compression, and (iii) entropy estimation. We construct counter-examples to show that no two properties above imply the third.
Shobu Shiraki - One of the best experts on this subject based on the ideXlab platform.
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Pointwise Convergence along restricted directions for the fractional Schrödinger equation
Journal of Fourier Analysis and Applications, 2020Co-Authors: Shobu ShirakiAbstract:We consider the Pointwise Convergence problem for the solution of Schrodinger-type equations along directions determined by a given compact subset of the real line. This problem contains Carleson’s problem as the simplest case and was studied in general by Cho et al. We extend their result from the case of the classical Schrodinger equation to a class of equations which includes the fractional Schrodinger equations. To achieve this, we significantly simplify their proof by completely avoiding a time localization argument.
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Pointwise Convergence along a tangential curve for the fractional Schr\"odinger equation
arXiv: Analysis of PDEs, 2020Co-Authors: Chu-hee Cho, Shobu ShirakiAbstract:In this paper we study the Pointwise Convergence problem along a tangential curve for the fractional Schrodinger equations in one spatial dimension and estimate the capacitary dimension of the divergence set. We extend a prior paper by Lee and the first author for the classical Schrodinger equation, which in itself contains a result due to Lee, Vargas and the first author, to the fractional Schrodinger equation. The proof is based on a decomposition argument without time localization, which has recently been introduced by the second author.
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Pointwise Convergence along restricted directions for the fractional Schr\"odinger equation
arXiv: Analysis of PDEs, 2019Co-Authors: Shobu ShirakiAbstract:We consider the Pointwise Convergence problem for the solution of Schrodinger-type equations along directions determined by a given compact subset of the real line. This problem contains Carleson's problem as the most simple case and was studied in general by Cho--Lee--Vargas. We extend their result from the case of the classical Schrodinger equation to a class of equations which includes the fractional Schrodinger equations. To achieve this, we significantly simplify their proof by completely avoiding a time localization argument.
Michael T Lacey - One of the best experts on this subject based on the ideXlab platform.
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Pointwise Convergence of vector-valued Fourier series
Mathematische Annalen, 2013Co-Authors: Tuomas Hytonen, Michael T LaceyAbstract:We prove a vector-valued version of Carleson’s theorem: let \(Y=[X,H]_\theta \) be a complex interpolation space between an unconditionality of martingale differences (UMD) space \(X\) and a Hilbert space \(H\). For \(p\in (1,\infty )\) and \(f\in L^p(\mathbb T ;Y)\), the partial sums of the Fourier series of \(f\) converge to \(f\) Pointwise almost everywhere. Apparently, all known examples of UMD spaces are of this intermediate form \(Y=[X,H]_\theta \). In particular, we answer affirmatively a question of Rubio de Francia on the Pointwise Convergence of Fourier series of Schatten class valued functions.
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Pointwise Convergence of vector valued fourier series
arXiv: Functional Analysis, 2012Co-Authors: Tuomas Hytonen, Michael T LaceyAbstract:We prove a vector-valued version of Carleson's theorem: Let Y=[X,H]_t be a complex interpolation space between a UMD space X and a Hilbert space H. For p\in(1,\infty) and f\in L^p(T;Y), the partial sums of the Fourier series of f converge to f Pointwise almost everywhere. Apparently, all known examples of UMD spaces are of this intermediate form Y=[X,H]_t. In particular, we answer affirmatively a question of Rubio de Francia on the Pointwise Convergence of Fourier series of Schatten class valued functions.
Sanghyuk Lee - One of the best experts on this subject based on the ideXlab platform.
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Dimension of divergence sets for the Pointwise Convergence of the Schrödinger equation
Journal of Mathematical Analysis and Applications, 2014Co-Authors: Chu-hee Cho, Sanghyuk LeeAbstract:Abstract In this note, we consider the Pointwise Convergence along curves for the Schrodinger equation and obtain estimates for the capacitary dimension of divergence sets which extend our previous result in [6] .
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Problems on Pointwise Convergence of Solutions to the Schrödinger Equation
Journal of Fourier Analysis and Applications, 2012Co-Authors: Chu-hee Cho, Sanghyuk Lee, Ana VargasAbstract:In this paper we consider several variants of the Pointwise Convergence problem for the Schrodinger equation, which generalize the previously known results.
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Pointwise Convergence of solutions to Schrödinger equations
arXiv: Classical Analysis and ODEs, 2007Co-Authors: Sanghyuk LeeAbstract:We study Pointwise Convergence of the solutions to Schr\"odinger equations with initial datum $f\in H^s(\mathbb R^n)$. The conjecture is that the solution $e^{it\Delta}f$ converges to $f$ almost everywhere for all $f\in H^s(\mathbb R^n)$ if and only if $s\ge 1/4$. The conjecture is known true for one spatial dimension and the Convergence when $s>1/2$ was verified for $n\ge 2$. Recently, concrete progresses have been made in $\mathbb R^2$ for some $s 1/2-1/{24}$.