Journal of Complexity最新文献

{"title":"On the expected number of real roots of polynomials and exponential sums","authors":"Gregorio Malajovich","doi":"10.1016/j.jco.2022.101720","DOIUrl":"https://doi.org/10.1016/j.jco.2022.101720","url":null,"abstract":"<div><p><span>The expected number of real projective roots of orthogonally invariant random homogeneous real polynomial systems is known to be equal to the square root of the Bézout number. A similar result is known for random multi-homogeneous systems, invariant through a product of </span>orthogonal groups. In this note, those results are generalized to certain families of sparse polynomial systems, with no orthogonal invariance assumed.</p></div>","PeriodicalId":50227,"journal":{"name":"Journal of Complexity","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49882303","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Consistency of randomized integration methods","authors":"Julian Hofstadler,&nbsp;Daniel Rudolf","doi":"10.1016/j.jco.2023.101740","DOIUrl":"https://doi.org/10.1016/j.jco.2023.101740","url":null,"abstract":"<div><p>We prove that a class of randomized integration methods, including averages based on <span><math><mo>(</mo><mi>t</mi><mo>,</mo><mi>d</mi><mo>)</mo></math></span><span>-sequences, Latin hypercube<span><span> sampling, Frolov points as well as Cranley-Patterson rotations, consistently estimates expectations of integrable functions<span>. Consistency here refers to convergence in mean and/or convergence in probability of the estimator to the integral of interest. Moreover, we suggest median modified methods and show for </span></span>integrands in </span></span><span><math><msup><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msup></math></span> with <span><math><mi>p</mi><mo>&gt;</mo><mn>1</mn></math></span><span> consistency in terms of almost sure convergence.</span></p></div>","PeriodicalId":50227,"journal":{"name":"Journal of Complexity","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49882274","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Improved bounds on the gain coefficients for digital nets in prime power base","authors":"Takashi Goda ,&nbsp;Kosuke Suzuki","doi":"10.1016/j.jco.2022.101722","DOIUrl":"https://doi.org/10.1016/j.jco.2022.101722","url":null,"abstract":"<div><p><span>We study randomized<span> quasi-Monte Carlo integration by scrambled nets. The scrambled net quadrature has long gained its popularity because it is an unbiased estimator of the true integral, allows for a practical error estimation, achieves a high order decay of the variance for smooth functions, and works even for </span></span><span><math><msup><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msup></math></span>-functions with any <span><math><mi>p</mi><mo>≥</mo><mn>1</mn></math></span>. The variance of the scrambled net quadrature for <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>-functions can be evaluated through the set of the so-called <em>gain coefficients.</em></p><p>In this paper, based on the system of Walsh functions and the concept of dual nets, we provide improved upper bounds on the gain coefficients for digital nets in general prime power base. Our results explain the known bound by Owen (1997) for Faure sequences, the recently improved bound by Pan and Owen (2022) for digital nets in base 2 (including Sobol' sequences as a special case), and their finding that all the nonzero gain coefficients for digital nets in base 2 must be powers of two, all in a unified way.</p></div>","PeriodicalId":50227,"journal":{"name":"Journal of Complexity","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49882300","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Average case tractability of non-homogeneous tensor product problems with the absolute error criterion","authors":"Guiqiao Xu","doi":"10.1016/j.jco.2023.101743","DOIUrl":"https://doi.org/10.1016/j.jco.2023.101743","url":null,"abstract":"<div><p>We study average case tractability of non-homogeneous tensor product problems with the absolute error criterion. We consider algorithms that use finitely many evaluations of arbitrary linear functionals<span>. For general non-homogeneous tensor product problems, we obtain the matching necessary and sufficient conditions for strong polynomial tractability in terms of the one-dimensional eigenvalues. We give some examples to show that strong polynomial tractability is not equivalent to polynomial tractability, and polynomial tractability is not equivalent to quasi-polynomial tractability. But for non-homogeneous tensor product problems with decreasing eigenvalues, we prove that strong polynomial tractability is always equivalent to polynomial tractability, and strong polynomial tractability is even equivalent to quasi-polynomial tractability when the one-dimensional largest eigenvalues are less than one. In particular, we find an example that quasi-polynomial tractability with the absolute error criterion is not equivalent to that with the normalized error criterion even if all the one-dimensional largest eigenvalues are one. Finally we consider a special class of non-homogeneous tensor product problems with improved monotonicity condition of the eigenvalues.</span></p></div>","PeriodicalId":50227,"journal":{"name":"Journal of Complexity","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49882304","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A note on the CBC-DBD construction of lattice rules with general positive weights","authors":"Peter Kritzer","doi":"10.1016/j.jco.2022.101721","DOIUrl":"https://doi.org/10.1016/j.jco.2022.101721","url":null,"abstract":"<div><p><span>Lattice rules are among the most prominently studied quasi-Monte Carlo methods to approximate multivariate integrals. A rank-1 lattice rule for an </span><em>s</em>-dimensional integral is specified by its <span><em>generating vector</em></span> <span><math><mi>z</mi><mo>∈</mo><msup><mrow><mi>Z</mi></mrow><mrow><mi>s</mi></mrow></msup></math></span> and its number of points <em>N</em>. While there are many results on the existence of “good” rank-1 lattice rules, there are no explicit constructions of good generating vectors for dimensions <span><math><mi>s</mi><mo>≥</mo><mn>3</mn></math></span>. Therefore one resorts to computer search algorithms. In a recent paper by Ebert et al. in the Journal of Complexity, we showed a component-by-component digit-by-digit (CBC-DBD) construction for good generating vectors for integration of functions in weighted Korobov classes equipped with product weights. Here, we generalize this result to arbitrary positive weights, answering an open question from the paper of Ebert et al. We include a section on how the algorithm can be implemented in the case of POD weights, implying that the CBC-DBD construction is competitive with the classical CBC construction.</p></div>","PeriodicalId":50227,"journal":{"name":"Journal of Complexity","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49882301","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Low-energy points on the sphere and the real projective plane","authors":"Carlos Beltrán,&nbsp;Ujué Etayo,&nbsp;Pedro R. López-Gómez","doi":"10.1016/j.jco.2023.101742","DOIUrl":"https://doi.org/10.1016/j.jco.2023.101742","url":null,"abstract":"<div><p>We present a generalization of a family of points on <span><math><msup><mrow><mi>S</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>, the Diamond ensemble, containing collections of <em>N</em> points on <span><math><msup><mrow><mi>S</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> with very small logarithmic energy for all <span><math><mi>N</mi><mo>∈</mo><mi>N</mi></math></span>. We extend this construction to the real projective plane <span><math><msup><mrow><mi>RP</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> and we obtain upper and lower bounds with explicit constants for the Green and logarithmic energy on this last space.</p></div>","PeriodicalId":50227,"journal":{"name":"Journal of Complexity","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49882297","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Changes of the Editorial Board","authors":"Josef Dick (The senior editor of the journal),&nbsp;Aicke Hinrichs (The senior editor of the journal),&nbsp;Erich Novak (The senior editor of the journal),&nbsp;Klaus Ritter (The senior editor of the journal),&nbsp;Grzegorz Wasilkowski (The senior editor of the journal),&nbsp;Henryk Woźniakowski (The senior editor of the journal)","doi":"10.1016/j.jco.2022.101728","DOIUrl":"https://doi.org/10.1016/j.jco.2022.101728","url":null,"abstract":"","PeriodicalId":50227,"journal":{"name":"Journal of Complexity","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2023-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50186379","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Central Limit Theorem for (t,s)-sequences in base 2","authors":"Mordechay B. Levin","doi":"10.1016/j.jco.2022.101699","DOIUrl":"https://doi.org/10.1016/j.jco.2022.101699","url":null,"abstract":"<div><p>Let <span><math><msub><mrow><mo>(</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo></mrow><mrow><mi>n</mi><mo>≥</mo><mn>0</mn></mrow></msub></math></span> be a digital <span><math><mo>(</mo><mi>t</mi><mo>,</mo><mi>s</mi><mo>)</mo></math></span>-sequence in base 2, <span><math><msub><mrow><mi>P</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>=</mo><msubsup><mrow><mo>(</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo></mrow><mrow><mi>n</mi><mo>=</mo><mn>0</mn></mrow><mrow><msup><mrow><mn>2</mn></mrow><mrow><mi>m</mi></mrow></msup><mo>−</mo><mn>1</mn></mrow></msubsup></math></span>, and let <span><math><mi>D</mi><mo>(</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>,</mo><mi>Y</mi><mo>)</mo></math></span> be the local discrepancy of <span><math><msub><mrow><mi>P</mi></mrow><mrow><mi>m</mi></mrow></msub></math></span>. Let <span><math><mi>T</mi><mo>⊕</mo><mi>Y</mi></math></span> be the digital addition of <em>T</em> and <em>Y</em>, and let<span><span><span><math><msub><mrow><mi>M</mi></mrow><mrow><mi>s</mi><mo>,</mo><mi>p</mi></mrow></msub><mo>(</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>)</mo><mo>=</mo><msup><mrow><mo>(</mo><munder><mo>∫</mo><mrow><msup><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></mrow><mrow><mn>2</mn><mi>s</mi></mrow></msup></mrow></munder><mo>|</mo><mi>D</mi><mo>(</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>⊕</mo><mi>T</mi><mo>,</mo><mi>Y</mi><mo>)</mo><msup><mrow><mo>|</mo></mrow><mrow><mi>p</mi></mrow></msup><mi>d</mi><mi>T</mi><mi>d</mi><mi>Y</mi><mo>)</mo></mrow><mrow><mn>1</mn><mo>/</mo><mi>p</mi></mrow></msup><mo>.</mo></math></span></span></span> In this paper, we prove that <span><math><mi>D</mi><mo>(</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>⊕</mo><mi>T</mi><mo>,</mo><mi>Y</mi><mo>)</mo><mo>/</mo><msub><mrow><mi>M</mi></mrow><mrow><mi>s</mi><mo>,</mo><mn>2</mn></mrow></msub><mo>(</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>)</mo></math></span><span> weakly converges to the standard Gaussian distribution for </span><span><math><mi>m</mi><mo>→</mo><mo>∞</mo></math></span>, where <span><math><mi>T</mi><mo>,</mo><mi>Y</mi></math></span> are uniformly distributed random variables in <span><math><msup><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></mrow><mrow><mi>s</mi></mrow></msup></math></span>. In addition, we prove that<span><span><span><math><msub><mrow><mi>M</mi></mrow><mrow><mi>s</mi><mo>,</mo><mi>p</mi></mrow></msub><mo>(</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>)</mo><mo>/</mo><msub><mrow><mi>M</mi></mrow><mrow><mi>s</mi><mo>,</mo><mn>2</mn></mrow></msub><mo>(</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>)</mo><mo>→</mo><mfrac><mrow><mn>1</mn></mrow><mrow><msqrt><mrow><mn>2</mn><mi>π</mi></mrow></msqrt></mrow></mfrac><munderover><mo>∫</mo><mrow><mo>−</mo><mo>∞</mo></mrow><mrow><mo>∞</","PeriodicalId":50227,"journal":{"name":"Journal of Complexity","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2023-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50186376","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"An unfeasibility view of neural network learning","authors":"Joos Heintz ,&nbsp;Luis Miguel Pardo ,&nbsp;Enrique Carlos Segura ,&nbsp;Hvara Ocar ,&nbsp;Andrés Rojas Paredes","doi":"10.1016/j.jco.2022.101710","DOIUrl":"10.1016/j.jco.2022.101710","url":null,"abstract":"<div><p>We define the notion of a continuously differentiable perfect learning algorithm for multilayer neural network architectures and show that such algorithms do not exist provided that the length of the data set exceeds the number of involved parameters and the activation functions are logistic, tanh or sin.</p></div>","PeriodicalId":50227,"journal":{"name":"Journal of Complexity","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2023-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48585253","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Best Paper Award of the Journal of Complexity","authors":"Erich Novak","doi":"10.1016/j.jco.2022.101731","DOIUrl":"https://doi.org/10.1016/j.jco.2022.101731","url":null,"abstract":"","PeriodicalId":50227,"journal":{"name":"Journal of Complexity","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2023-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50168429","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}