Journal of Complexity最新文献

{"title":"Nonasymptotic analysis of robust regression with modified Huber's loss","authors":"Hongzhi Tong","doi":"10.1016/j.jco.2023.101744","DOIUrl":"https://doi.org/10.1016/j.jco.2023.101744","url":null,"abstract":"<div><p><span>To achieve robustness against the outliers or heavy-tailed sampling distribution, we consider an Ivanov regularized empirical risk minimization scheme associated with a modified Huber's loss for nonparametric regression in reproducing kernel </span>Hilbert space<span>. By tuning the scaling and regularization parameters in accordance with the sample size, we develop nonasymptotic concentration results for such an adaptive estimator. Specifically, we establish the best convergence rates for prediction error when the conditional distribution satisfies a weak moment condition.</span></p></div>","PeriodicalId":50227,"journal":{"name":"Journal of Complexity","volume":"76 ","pages":"Article 101744"},"PeriodicalIF":1.7,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49882302","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":"On oracle factoring of integers","authors":"Andrzej Dąbrowski ,&nbsp;Jacek Pomykała ,&nbsp;Igor E. Shparlinski","doi":"10.1016/j.jco.2023.101741","DOIUrl":"https://doi.org/10.1016/j.jco.2023.101741","url":null,"abstract":"<div><p><span>We present an oracle factorisation algorithm, which in polynomial deterministic time, finds a nontrivial factor of almost all positive integers </span><em>n</em><span> based on the knowledge of the number of points on certain elliptic curves<span> in residue rings modulo </span></span><em>n</em>.</p></div>","PeriodicalId":50227,"journal":{"name":"Journal of Complexity","volume":"76 ","pages":"Article 101741"},"PeriodicalIF":1.7,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49882275","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":"Bit-complexity of classical solutions of linear evolutionary systems of partial differential equations","authors":"Ivan Koswara ,&nbsp;Gleb Pogudin ,&nbsp;Svetlana Selivanova ,&nbsp;Martin Ziegler","doi":"10.1016/j.jco.2022.101727","DOIUrl":"https://doi.org/10.1016/j.jco.2022.101727","url":null,"abstract":"<div><p><span>We study the bit-complexity intrinsic to solving the initial-value and (several types of) boundary-value problems for linear evolutionary systems of partial differential equations (PDEs), based on the Computable Analysis approach. Our algorithms are guaranteed to compute classical solutions to such problems approximately up to error </span><span><math><mn>1</mn><mo>/</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>n</mi></mrow></msup></math></span>, so that <em>n</em> corresponds to the number of reliable bits of the output; bit-cost is measured with respect to <em>n</em><span><span>. Computational Complexity Theory allows us to prove in a rigorous sense that PDEs with </span>constant coefficients are algorithmically ‘easier’ than general ones. Indeed, solutions to the latter are shown (under natural assumptions) computable using a polynomial number of memory bits, and we prove that the complexity class </span><span>PSPACE</span> is in general optimal; while the case of constant coefficients can be solved in #<span>P</span>—also essentially optimally so: the Heat Equation ‘requires’ <span><math><msub><mrow><mi>#</mi><mtext>P</mtext></mrow><mrow><mn>1</mn></mrow></msub></math></span><span>. Our algorithms raise difference schemes to exponential powers, efficiently: we compute any desired entry of such a power in #P, provided that the underlying exponential-sized matrices are circulant of constant bandwidth. Exponentially powering modular two-band circulant matrices is established even feasible in </span><span>P</span><span>; and under additional conditions, also the solution to certain linear PDEs<span> becomes polynomial time computable.</span></span></p></div>","PeriodicalId":50227,"journal":{"name":"Journal of Complexity","volume":"76 ","pages":"Article 101727"},"PeriodicalIF":1.7,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49882262","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":"Adaptive iterative hard thresholding for low-rank matrix recovery and rank-one measurements","authors":"Yu Xia ,&nbsp;Likai Zhou","doi":"10.1016/j.jco.2022.101725","DOIUrl":"https://doi.org/10.1016/j.jco.2022.101725","url":null,"abstract":"<div><p>In low-rank matrix recovery, many kinds of measurements fail to meet the standard restricted isometry property (RIP), such as rank-one measurements, that is, <span><math><msub><mrow><mo>[</mo><mi>A</mi><mo>(</mo><mi>X</mi><mo>)</mo><mo>]</mo></mrow><mrow><mi>i</mi></mrow></msub><mo>=</mo><mo>〈</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>,</mo><mi>X</mi><mo>〉</mo></math></span> with <span><math><mtext>rank</mtext><mo>(</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>)</mo><mo>=</mo><mn>1</mn></math></span>, <span><math><mi>i</mi><mo>=</mo><mn>1</mn><mo>,</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>,</mo><mi>m</mi></math></span><span>. Historical iterative hard thresholding sequence for low-rank matrix recovery and rank-one measurements was taken as </span><span><math><msup><mrow><mi>X</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mo>=</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>s</mi></mrow></msub><mo>(</mo><msup><mrow><mi>X</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>−</mo><msub><mrow><mi>μ</mi></mrow><mrow><mi>n</mi></mrow></msub><msub><mrow><mi>P</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>(</mo><msup><mrow><mi>A</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mtext>sign</mtext><mo>(</mo><mi>A</mi><mo>(</mo><msup><mrow><mi>X</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>)</mo><mo>−</mo><mi>y</mi><mo>)</mo><mo>)</mo><mo>)</mo></math></span>, which introduced the “tail” and “head” approximations <span><math><msub><mrow><mi>P</mi></mrow><mrow><mi>s</mi></mrow></msub></math></span> and <span><math><msub><mrow><mi>P</mi></mrow><mrow><mi>t</mi></mrow></msub></math></span>, respectively. In this paper, we remove the term <span><math><msub><mrow><mi>P</mi></mrow><mrow><mi>t</mi></mrow></msub></math></span><span> and provide a new iterative hard thresholding algorithm with adaptive step size (abbreviated as AIHT). The linear convergence analysis and stability results on AIHT are established under the </span><span><math><msub><mrow><mi>ℓ</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>/</mo><msub><mrow><mi>ℓ</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span><span>-RIP. Particularly, we discuss the rank-one Gaussian measurements under the tight upper and lower bounds on </span><span><math><mi>E</mi><msub><mrow><mo>‖</mo><mi>A</mi><mo>(</mo><mi>X</mi><mo>)</mo><mo>‖</mo></mrow><mrow><mn>1</mn></mrow></msub></math></span>, and provide better convergence rate and sampling complexity. Besides, several empirical experiments are provided to show that AIHT performs better than the historical rank-one iterative hard thresholding method.</p></div>","PeriodicalId":50227,"journal":{"name":"Journal of Complexity","volume":"76 ","pages":"Article 101725"},"PeriodicalIF":1.7,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49882272","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":"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":"76 ","pages":"Article 101720"},"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":"76 ","pages":"Article 101740"},"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":"76 ","pages":"Article 101722"},"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":"76 ","pages":"Article 101743"},"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":"76 ","pages":"Article 101721"},"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":"76 ","pages":"Article 101742"},"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}