{"title":"Expected multivolumes of random amoebas","authors":"Turgay Bayraktar , Ali Ulaş Özgür Kişisel","doi":"10.1016/j.jco.2023.101802","DOIUrl":"https://doi.org/10.1016/j.jco.2023.101802","url":null,"abstract":"<div><p>We compute the expected multivolume of the amoeba of a random half dimensional complete intersection in <span><math><msup><mrow><mi>CP</mi></mrow><mrow><mn>2</mn><mi>n</mi></mrow></msup></math></span>. We also give a relative generalization of our result to the toric case.</p></div>","PeriodicalId":50227,"journal":{"name":"Journal of Complexity","volume":"80 ","pages":"Article 101802"},"PeriodicalIF":1.7,"publicationDate":"2023-09-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49886730","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 strongly monotonic polygonal Euler scheme","authors":"Tim Johnston, Sotirios Sabanis","doi":"10.1016/j.jco.2023.101801","DOIUrl":"https://doi.org/10.1016/j.jco.2023.101801","url":null,"abstract":"<div><p>In recent years tamed schemes have become an important technique for simulating SDEs and SPDEs whose continuous coefficients display superlinear growth. The taming method involves curbing the growth of the coefficients as a function of stepsize, but so far has not been adapted to preserve the monotonicity of the coefficients. This has arisen as an issue in <span>[4]</span>, where the lack of a strongly monotonic tamed scheme forces strong conditions on the setting. In this article we give a novel and explicit method for truncating monotonic functions in separable real Hilbert spaces, and show how this can be used to define a polygonal (tamed) Euler scheme on finite dimensional space, preserving the monotonicity of the drift coefficient, and converging to the true solution at the same rate as the classical Euler scheme for Lipschitz coefficients. Our construction is the first explicit method for truncating monotone functions we are aware of, and the first in infinite dimensions.</p></div>","PeriodicalId":50227,"journal":{"name":"Journal of Complexity","volume":"80 ","pages":"Article 101801"},"PeriodicalIF":1.7,"publicationDate":"2023-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49887910","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 power of standard information for tractability for L∞ approximation of periodic functions in the worst case setting","authors":"Jiaxin Geng, Heping Wang","doi":"10.1016/j.jco.2023.101790","DOIUrl":"https://doi.org/10.1016/j.jco.2023.101790","url":null,"abstract":"<div><p>We study multivariate approximation of periodic functions in the worst case setting with the error measured in the <span><math><msub><mrow><mi>L</mi></mrow><mrow><mo>∞</mo></mrow></msub></math></span> norm. We consider algorithms that use standard information <span><math><msup><mrow><mi>Λ</mi></mrow><mrow><mi>std</mi></mrow></msup></math></span> consisting of function values or general linear information <span><math><msup><mrow><mi>Λ</mi></mrow><mrow><mi>all</mi></mrow></msup></math></span> consisting of arbitrary continuous linear functionals. We investigate equivalences of various notions of algebraic and exponential tractability for <span><math><msup><mrow><mi>Λ</mi></mrow><mrow><mi>std</mi></mrow></msup></math></span> and <span><math><msup><mrow><mi>Λ</mi></mrow><mrow><mi>all</mi></mrow></msup></math></span> under the absolute or normalized error criterion, and show that the power of <span><math><msup><mrow><mi>Λ</mi></mrow><mrow><mi>std</mi></mrow></msup></math></span> is the same as the one of <span><math><msup><mrow><mi>Λ</mi></mrow><mrow><mi>all</mi></mrow></msup></math></span> for various notions of algebraic and exponential tractability. Our results can be applied to weighted Korobov spaces and Korobov spaces with exponential weights. This gives a special solution to Open Problem 145 as posed by Novak and Woźniakowski (2012) <span>[40]</span>.</p></div>","PeriodicalId":50227,"journal":{"name":"Journal of Complexity","volume":"80 ","pages":"Article 101790"},"PeriodicalIF":1.7,"publicationDate":"2023-08-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49887908","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":"Erich Novak","doi":"10.1016/j.jco.2023.101792","DOIUrl":"https://doi.org/10.1016/j.jco.2023.101792","url":null,"abstract":"","PeriodicalId":50227,"journal":{"name":"Journal of Complexity","volume":"79 ","pages":"Article 101792"},"PeriodicalIF":1.7,"publicationDate":"2023-08-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49877024","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":"Bypassing the quadrature exactness assumption of hyperinterpolation on the sphere","authors":"Congpei An , Hao-Ning Wu","doi":"10.1016/j.jco.2023.101789","DOIUrl":"https://doi.org/10.1016/j.jco.2023.101789","url":null,"abstract":"<div><p>This paper focuses on the approximation of continuous functions on the unit sphere by spherical polynomials of degree <em>n</em> via hyperinterpolation. Hyperinterpolation of degree <em>n</em> is a discrete approximation of the <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>-orthogonal projection of the same degree with its Fourier coefficients evaluated by a positive-weight quadrature rule that exactly integrates all spherical polynomials of degree at most 2<em>n</em>. This paper aims to bypass this quadrature exactness assumption by replacing it with the Marcinkiewicz–Zygmund property proposed in a previous paper. Consequently, hyperinterpolation can be constructed by a positive-weight quadrature rule (not necessarily with quadrature exactness). This scheme is referred to as <em>unfettered hyperinterpolation</em>. This paper provides a reasonable error estimate for unfettered hyperinterpolation. The error estimate generally consists of two terms: a term representing the error estimate of the original hyperinterpolation of full quadrature exactness and another introduced as compensation for the loss of exactness degrees. A guide to controlling the newly introduced term in practice is provided. In particular, if the quadrature points form a quasi-Monte Carlo (QMC) design, then there is a refined error estimate. Numerical experiments verify the error estimates and the practical guide.</p></div>","PeriodicalId":50227,"journal":{"name":"Journal of Complexity","volume":"80 ","pages":"Article 101789"},"PeriodicalIF":1.7,"publicationDate":"2023-08-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49887909","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":"Convergence of the Gauss-Newton method for convex composite optimization problems under majorant condition on Riemannian manifolds","authors":"Qamrul Hasan Ansari , Moin Uddin , Jen-Chih Yao","doi":"10.1016/j.jco.2023.101788","DOIUrl":"10.1016/j.jco.2023.101788","url":null,"abstract":"<div><p>In this paper, we consider convex composite optimization problems on Riemannian manifolds, and discuss the semi-local convergence of the Gauss-Newton method with quasi-regular initial point and under the majorant condition. As special cases, we also discuss the convergence of the sequence generated by the Gauss-Newton method under Lipschitz-type condition, or under <em>γ</em>-condition.</p></div>","PeriodicalId":50227,"journal":{"name":"Journal of Complexity","volume":"80 ","pages":"Article 101788"},"PeriodicalIF":1.7,"publicationDate":"2023-08-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42426919","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":"The minimal radius of Galerkin information for the problem of numerical differentiation","authors":"S.G. Solodky, S.A. Stasyuk","doi":"10.1016/j.jco.2023.101787","DOIUrl":"10.1016/j.jco.2023.101787","url":null,"abstract":"<div><p>The problem of numerical differentiation<span> for periodic functions with finite smoothness is investigated. For multivariate functions<span>, different variants of the truncation method are constructed and their approximation properties are obtained. Based on these results, sharp bounds (in the power scale) of the minimal radius of Galerkin information for the problem under study are found.</span></span></p></div>","PeriodicalId":50227,"journal":{"name":"Journal of Complexity","volume":"81 ","pages":"Article 101787"},"PeriodicalIF":1.7,"publicationDate":"2023-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42869659","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":"Sampling numbers of smoothness classes via ℓ1-minimization","authors":"Thomas Jahn , Tino Ullrich , Felix Voigtlaender","doi":"10.1016/j.jco.2023.101786","DOIUrl":"https://doi.org/10.1016/j.jco.2023.101786","url":null,"abstract":"<div><p>Using techniques developed recently in the field of compressed sensing we prove new upper bounds for general (nonlinear) sampling numbers of (quasi-)Banach smoothness spaces in <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span><span>. In particular, we show that in relevant cases such as mixed and isotropic weighted Wiener classes or Sobolev spaces with mixed smoothness, sampling numbers in </span><span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> can be upper bounded by best <em>n</em><span>-term trigonometric widths in </span><span><math><msup><mrow><mi>L</mi></mrow><mrow><mo>∞</mo></mrow></msup></math></span>. We describe a recovery procedure from <em>m</em> function values based on <span><math><msup><mrow><mi>ℓ</mi></mrow><mrow><mn>1</mn></mrow></msup></math></span>-minimization (basis pursuit denoising). With this method, a significant gain in the rate of convergence compared to recently developed linear recovery methods is achieved. In this deterministic worst-case setting we see an additional speed-up of <span><math><msup><mrow><mi>m</mi></mrow><mrow><mo>−</mo><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup></math></span> (up to log factors) compared to linear methods in case of weighted Wiener spaces. For their quasi-Banach counterparts even arbitrary polynomial speed-up is possible. Surprisingly, our approach allows to recover mixed smoothness Sobolev functions belonging to <span><math><msubsup><mrow><mi>S</mi></mrow><mrow><mi>p</mi></mrow><mrow><mi>r</mi></mrow></msubsup><mi>W</mi><mo>(</mo><msup><mrow><mi>T</mi></mrow><mrow><mi>d</mi></mrow></msup><mo>)</mo></math></span> on the <em>d</em>-torus with a logarithmically better rate of convergence than any linear method can achieve when <span><math><mn>1</mn><mo><</mo><mi>p</mi><mo><</mo><mn>2</mn></math></span> and <em>d</em> is large. This effect is not present for isotropic Sobolev spaces.</p></div>","PeriodicalId":50227,"journal":{"name":"Journal of Complexity","volume":"79 ","pages":"Article 101786"},"PeriodicalIF":1.7,"publicationDate":"2023-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49877022","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":"Random-prime–fixed-vector randomised lattice-based algorithm for high-dimensional integration","authors":"Frances Y. Kuo , Dirk Nuyens , Laurence Wilkes","doi":"10.1016/j.jco.2023.101785","DOIUrl":"https://doi.org/10.1016/j.jco.2023.101785","url":null,"abstract":"<div><p>We show that a very simple randomised algorithm for numerical integration can produce a near optimal rate of convergence for integrals of functions in the <em>d</em><span>-dimensional weighted Korobov space. This algorithm uses a lattice<span> rule with a fixed generating vector and the only random element is the choice of the number of function evaluations. For a given computational budget </span></span><em>n</em> of a maximum allowed number of function evaluations, we uniformly pick a prime <em>p</em> in the range <span><math><mi>n</mi><mo>/</mo><mn>2</mn><mo><</mo><mi>p</mi><mo>≤</mo><mi>n</mi></math></span>. We show error bounds for the randomised error, which is defined as the worst case expected error, of the form <span><math><mi>O</mi><mo>(</mo><msup><mrow><mi>n</mi></mrow><mrow><mo>−</mo><mi>α</mi><mo>−</mo><mn>1</mn><mo>/</mo><mn>2</mn><mo>+</mo><mi>δ</mi></mrow></msup><mo>)</mo></math></span>, with <span><math><mi>δ</mi><mo>></mo><mn>0</mn></math></span>, for a Korobov space with smoothness <span><math><mi>α</mi><mo>></mo><mn>1</mn><mo>/</mo><mn>2</mn></math></span> and general weights. The implied constant in the bound is dimension-independent given the usual conditions on the weights. We present an algorithm that can construct suitable generating vectors <em>offline</em> ahead of time at cost <span><math><mi>O</mi><mo>(</mo><mi>d</mi><msup><mrow><mi>n</mi></mrow><mrow><mn>4</mn></mrow></msup><mo>/</mo><mi>ln</mi><mo></mo><mi>n</mi><mo>)</mo></math></span> when the weight parameters defining the Korobov spaces are so-called product weights. For this case, numerical experiments confirm our theory that the new randomised algorithm achieves the near optimal rate of the randomised error.</p></div>","PeriodicalId":50227,"journal":{"name":"Journal of Complexity","volume":"79 ","pages":"Article 101785"},"PeriodicalIF":1.7,"publicationDate":"2023-08-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49877023","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}
Philipp Grohs , Shokhrukh Ibragimov , Arnulf Jentzen , Sarah Koppensteiner
{"title":"Lower bounds for artificial neural network approximations: A proof that shallow neural networks fail to overcome the curse of dimensionality","authors":"Philipp Grohs , Shokhrukh Ibragimov , Arnulf Jentzen , Sarah Koppensteiner","doi":"10.1016/j.jco.2023.101746","DOIUrl":"https://doi.org/10.1016/j.jco.2023.101746","url":null,"abstract":"<div><p><span>Artificial neural networks<span> (ANNs) have become a very powerful tool in the approximation of high-dimensional functions. Especially, deep ANNs, consisting of a large number of hidden layers, have been very successfully used in a series of practical relevant computational problems involving high-dimensional input data ranging from classification tasks in supervised learning to optimal decision problems in reinforcement learning. There are also a number of mathematical results in the scientific literature which study the approximation capacities of ANNs in the context of high-dimensional target functions. In particular, there are a series of mathematical results in the scientific literature which show that sufficiently deep ANNs have the capacity to overcome the curse of dimensionality in the approximation of certain target function classes in the sense that the number of parameters of the approximating ANNs grows at most polynomially in the dimension </span></span><span><math><mi>d</mi><mo>∈</mo><mi>N</mi></math></span> of the target functions under considerations. In the proofs of several of such high-dimensional approximation results it is crucial that the involved ANNs are sufficiently deep and consist a sufficiently large number of hidden layers which grows in the dimension of the considered target functions. It is the topic of this work to look a bit more detailed to the deepness of the involved ANNs in the approximation of high-dimensional target functions. In particular, the main result of this work proves that there exists a concretely specified sequence of functions which can be approximated without the curse of dimensionality by sufficiently deep ANNs but which cannot be approximated without the curse of dimensionality if the involved ANNs are shallow or not deep enough.</p></div>","PeriodicalId":50227,"journal":{"name":"Journal of Complexity","volume":"77 ","pages":"Article 101746"},"PeriodicalIF":1.7,"publicationDate":"2023-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50200237","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}