Pub Date : 2025-04-04DOI: 10.1016/j.jco.2025.101946
Ariel Neufeld , Tuan Anh Nguyen , Sizhou Wu
Neufeld and Wu (2023) [49] developed a multilevel Picard (MLP) algorithm which can approximately solve general semilinear parabolic PDEs with gradient-dependent nonlinearities, allowing also for coefficient functions of the corresponding PDE to be non-constant. By introducing a particular stochastic fixed-point equation (SFPE) motivated by the Feynman-Kac representation and the Bismut-Elworthy-Li formula and identifying the first and second component of the unique fixed-point of the SFPE with the unique viscosity solution of the PDE and its gradient, they proved convergence of their algorithm. However, it remained an open question whether the proposed MLP schema in Neufeld and Wu (2023) [49] does not suffer from the curse of dimensionality. In this paper, we prove that the MLP algorithm in Neufeld and Wu (2023) [49] indeed can overcome the curse of dimensionality, i.e. that its computational complexity only grows polynomially in the dimension and the reciprocal of the accuracy ε, under some suitable assumptions on the nonlinear part of the corresponding PDE.
{"title":"Multilevel Picard approximations overcome the curse of dimensionality in the numerical approximation of general semilinear PDEs with gradient-dependent nonlinearities","authors":"Ariel Neufeld , Tuan Anh Nguyen , Sizhou Wu","doi":"10.1016/j.jco.2025.101946","DOIUrl":"10.1016/j.jco.2025.101946","url":null,"abstract":"<div><div>Neufeld and Wu (2023) <span><span>[49]</span></span> developed a multilevel Picard (MLP) algorithm which can approximately solve <em>general</em> semilinear parabolic PDEs with gradient-dependent nonlinearities, allowing also for coefficient functions of the corresponding PDE to be non-constant. By introducing a particular stochastic fixed-point equation (SFPE) motivated by the Feynman-Kac representation and the Bismut-Elworthy-Li formula and identifying the first and second component of the unique fixed-point of the SFPE with the unique viscosity solution of the PDE and its gradient, they proved convergence of their algorithm. However, it remained an open question whether the proposed MLP schema in Neufeld and Wu (2023) <span><span>[49]</span></span> does not suffer from the curse of dimensionality. In this paper, we prove that the MLP algorithm in Neufeld and Wu (2023) <span><span>[49]</span></span> indeed can overcome the curse of dimensionality, i.e. that its computational complexity only grows polynomially in the dimension <span><math><mi>d</mi><mo>∈</mo><mi>N</mi></math></span> and the reciprocal of the accuracy <em>ε</em>, under some suitable assumptions on the nonlinear part of the corresponding PDE.</div></div>","PeriodicalId":50227,"journal":{"name":"Journal of Complexity","volume":"90 ","pages":"Article 101946"},"PeriodicalIF":1.8,"publicationDate":"2025-04-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144070001","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}
Pub Date : 2025-03-31DOI: 10.1016/j.jco.2025.101944
Salim Bouzebda, Nourelhouda Taachouche
This study presents a comprehensive framework for conditional U-statistics of a general order in the context of Lie group-valued predictors affected by measurement errors. Such situations arise in a variety of modern statistical problems. Our approach is grounded in an abstract harmonic analysis on Lie groups, a setting relatively underexplored in statistical research. In a unified study, we introduce an innovative deconvolution method for conditional U-statistics and investigate its convergence rate and asymptotic distribution for the first time. Furthermore, we explore the application of conditional U-statistics to variables that combine, in a nontrivial way, Euclidean and non-Euclidean elements subject to measurement errors, an area largely uncharted in statistical research. We derive general asymptotic properties, including convergence rates across various modes and the asymptotic distribution. All results are established under fairly general conditions on the underlying models. Additionally, our results are used to derive the asymptotic confidence intervals derived from the asymptotic distribution of the estimator. We also discuss applications of the general approximation results and give new insights into the Kendall rank correlation coefficient and discrimination problems.
{"title":"Nonparametric conditional U-statistics on Lie groups with measurement errors","authors":"Salim Bouzebda, Nourelhouda Taachouche","doi":"10.1016/j.jco.2025.101944","DOIUrl":"10.1016/j.jco.2025.101944","url":null,"abstract":"<div><div>This study presents a comprehensive framework for conditional <em>U</em>-statistics of a general order in the context of Lie group-valued predictors affected by measurement errors. Such situations arise in a variety of modern statistical problems. Our approach is grounded in an abstract harmonic analysis on Lie groups, a setting relatively underexplored in statistical research. In a unified study, we introduce an innovative deconvolution method for conditional <em>U</em>-statistics and investigate its convergence rate and asymptotic distribution for the first time. Furthermore, we explore the application of conditional <em>U</em>-statistics to variables that combine, in a nontrivial way, Euclidean and non-Euclidean elements subject to measurement errors, an area largely uncharted in statistical research. We derive general asymptotic properties, including convergence rates across various modes and the asymptotic distribution. All results are established under fairly general conditions on the underlying models. Additionally, our results are used to derive the asymptotic confidence intervals derived from the asymptotic distribution of the estimator. We also discuss applications of the general approximation results and give new insights into the Kendall rank correlation coefficient and discrimination problems.</div></div>","PeriodicalId":50227,"journal":{"name":"Journal of Complexity","volume":"89 ","pages":"Article 101944"},"PeriodicalIF":1.8,"publicationDate":"2025-03-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143768448","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-03-04DOI: 10.1016/j.jco.2025.101935
Yang Liu
We consider the integrability of weak mixed first-order derivatives of the integrand and study convergence rates of scrambled digital nets. We show that the generalized Vitali variation with parameter from [Dick and Pillichshammer, 2010] is bounded above by the norm of the weak mixed first-order derivative, where . Consequently, when the weak mixed first-order derivative belongs to for , the variance of the scrambled digital nets estimator convergences at a rate of . Numerical experiments further validate the theoretical results.
考虑被积函数的弱混合一阶导数的Lp可积性，研究了乱置数字网络的收敛速率。我们从[Dick and Pillichshammer， 2010]中证明了参数α∈[12,1]的广义Vitali变分是由弱混合一阶导数的Lp范数所限定的，其中p=23−2α。因此，当1≤p≤2的弱混合一阶导数属于Lp时，加扰数字网络估计量的方差收敛速率为O（N−4+2plogs−1 N）。数值实验进一步验证了理论结果。
{"title":"Integrability of weak mixed first-order derivatives and convergence rates of scrambled digital nets","authors":"Yang Liu","doi":"10.1016/j.jco.2025.101935","DOIUrl":"10.1016/j.jco.2025.101935","url":null,"abstract":"<div><div>We consider the <span><math><msup><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msup></math></span> integrability of weak mixed first-order derivatives of the integrand and study convergence rates of scrambled digital nets. We show that the generalized Vitali variation with parameter <span><math><mi>α</mi><mo>∈</mo><mo>[</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>,</mo><mn>1</mn><mo>]</mo></math></span> from [Dick and Pillichshammer, 2010] is bounded above by the <span><math><msup><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msup></math></span> norm of the weak mixed first-order derivative, where <span><math><mi>p</mi><mo>=</mo><mfrac><mrow><mn>2</mn></mrow><mrow><mn>3</mn><mo>−</mo><mn>2</mn><mi>α</mi></mrow></mfrac></math></span>. Consequently, when the weak mixed first-order derivative belongs to <span><math><msup><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msup></math></span> for <span><math><mn>1</mn><mo>≤</mo><mi>p</mi><mo>≤</mo><mn>2</mn></math></span>, the variance of the scrambled digital nets estimator convergences at a rate of <span><math><mi>O</mi><mo>(</mo><msup><mrow><mi>N</mi></mrow><mrow><mo>−</mo><mn>4</mn><mo>+</mo><mfrac><mrow><mn>2</mn></mrow><mrow><mi>p</mi></mrow></mfrac></mrow></msup><msup><mrow><mi>log</mi></mrow><mrow><mi>s</mi><mo>−</mo><mn>1</mn></mrow></msup><mo></mo><mi>N</mi><mo>)</mo></math></span>. Numerical experiments further validate the theoretical results.</div></div>","PeriodicalId":50227,"journal":{"name":"Journal of Complexity","volume":"89 ","pages":"Article 101935"},"PeriodicalIF":1.8,"publicationDate":"2025-03-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143552355","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}
Pub Date : 2025-02-26DOI: 10.1016/j.jco.2025.101932
Denis Belomestny , John Schoenmakers , Veronika Zorina
We introduce a mesh-type approach for tackling discrete-time, finite-horizon Markov Decision Processes (MDPs) characterized by state and action spaces that are general, encompassing both finite and infinite (yet suitably regular) subsets of Euclidean space. In particular, for bounded state and action spaces, our algorithm achieves a computational complexity that is tractable in the sense of Novak & Woźniakowski [12], and is polynomial in the time horizon. For an unbounded state space the algorithm is “semi-tractable” in the sense that the complexity is proportional to with some dimension independent , to achieve precision ε, and polynomial in the time horizon with linear degree in the underlying dimension. As such, the proposed approach has some flavor of the randomization method by Rust [14] which uses uniform sampling in compact state space. However, the present approach is essentially different due to the inhomogeneous finite horizon setting, which involves general transition distributions over a possibly non-compact state space. To demonstrate the effectiveness of our algorithm, we provide illustrations based on Linear-Quadratic Gaussian (LQG) control problems.
{"title":"Weighted mesh algorithms for general Markov decision processes: Convergence and tractability","authors":"Denis Belomestny , John Schoenmakers , Veronika Zorina","doi":"10.1016/j.jco.2025.101932","DOIUrl":"10.1016/j.jco.2025.101932","url":null,"abstract":"<div><div>We introduce a mesh-type approach for tackling discrete-time, finite-horizon Markov Decision Processes (MDPs) characterized by state and action spaces that are general, encompassing both finite and infinite (yet suitably regular) subsets of Euclidean space. In particular, for bounded state and action spaces, our algorithm achieves a computational complexity that is tractable in the sense of Novak & Woźniakowski <span><span>[12]</span></span>, and is polynomial in the time horizon. For an unbounded state space the algorithm is “semi-tractable” in the sense that the complexity is proportional to <span><math><msup><mrow><mi>ε</mi></mrow><mrow><mo>−</mo><mi>c</mi></mrow></msup></math></span> with some dimension independent <span><math><mi>c</mi><mo>≥</mo><mn>2</mn></math></span>, to achieve precision <em>ε</em>, and polynomial in the time horizon with linear degree in the underlying dimension. As such, the proposed approach has some flavor of the randomization method by Rust <span><span>[14]</span></span> which uses uniform sampling in compact state space. However, the present approach is essentially different due to the inhomogeneous finite horizon setting, which involves general transition distributions over a possibly non-compact state space. To demonstrate the effectiveness of our algorithm, we provide illustrations based on Linear-Quadratic Gaussian (LQG) control problems.</div></div>","PeriodicalId":50227,"journal":{"name":"Journal of Complexity","volume":"88 ","pages":"Article 101932"},"PeriodicalIF":1.8,"publicationDate":"2025-02-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143520997","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-02-25DOI: 10.1016/j.jco.2025.101934
Alexander Demin , Joris van der Hoeven
Consider a sparse polynomial in several variables given explicitly as a sum of non-zero terms with coefficients in an effective field. In this paper, we present several algorithms for factoring such polynomials and related tasks (such as gcd computation, square-free factorization, content-free factorization, and root extraction). Our methods are all based on sparse interpolation, but follow two main lines of attack: iteration on the number of variables and more direct reductions to the univariate or bivariate case. We present detailed probabilistic complexity bounds in terms of the complexity of sparse interpolation and evaluation.
{"title":"Factoring sparse polynomials fast","authors":"Alexander Demin , Joris van der Hoeven","doi":"10.1016/j.jco.2025.101934","DOIUrl":"10.1016/j.jco.2025.101934","url":null,"abstract":"<div><div>Consider a sparse polynomial in several variables given explicitly as a sum of non-zero terms with coefficients in an effective field. In this paper, we present several algorithms for factoring such polynomials and related tasks (such as gcd computation, square-free factorization, content-free factorization, and root extraction). Our methods are all based on sparse interpolation, but follow two main lines of attack: iteration on the number of variables and more direct reductions to the univariate or bivariate case. We present detailed probabilistic complexity bounds in terms of the complexity of sparse interpolation and evaluation.</div></div>","PeriodicalId":50227,"journal":{"name":"Journal of Complexity","volume":"88 ","pages":"Article 101934"},"PeriodicalIF":1.8,"publicationDate":"2025-02-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143529702","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}
Pub Date : 2025-02-20DOI: 10.1016/j.jco.2025.101933
Ben Adcock , Nick Dexter , Sebastian Moraga
Infinite-dimensional, holomorphic functions have been studied in detail over the last several decades, due to their relevance to parametric differential equations and computational uncertainty quantification. The approximation of such functions from finitely-many samples is of particular interest, due to the practical importance of constructing surrogate models to complex mathematical models of physical processes. In a previous work, [5] we studied the approximation of so-called Banach-valued, -holomorphic functions on the infinite-dimensional hypercube from m (potentially adaptive) samples. In particular, we derived lower bounds for the adaptive m-widths for classes of such functions, which showed that certain algebraic rates of the form are the best possible regardless of the sampling-recovery pair. In this work, we continue this investigation by focusing on the practical case where the samples are pointwise evaluations drawn identically and independently from the underlying probability measure for the problem. Specifically, for Hilbert-valued -holomorphic functions, we show that the same rates can be achieved (up to a small polylogarithmic or algebraic factor) for tensor-product Jacobi measures. Our reconstruction maps are based on least squares and compressed sensing procedures using the corresponding orthonormal Jacobi polynomials. In doing so, we strengthen and generalize past work that has derived weaker nonuniform guarantees for the uniform and Chebyshev measures (and corresponding polynomials) only. We also extend various best s-term polynomial approximation error bounds to arbitrary Jacobi polynomial expansions. Overall, we demonstrate that i.i.d. pointwise samples drawn from an underlying probability measure are near-optimal for the recovery of infinite-dimensional, holomorphic functions.
{"title":"Optimal approximation of infinite-dimensional holomorphic functions II: Recovery from i.i.d. pointwise samples","authors":"Ben Adcock , Nick Dexter , Sebastian Moraga","doi":"10.1016/j.jco.2025.101933","DOIUrl":"10.1016/j.jco.2025.101933","url":null,"abstract":"<div><div>Infinite-dimensional, holomorphic functions have been studied in detail over the last several decades, due to their relevance to parametric differential equations and computational uncertainty quantification. The approximation of such functions from finitely-many samples is of particular interest, due to the practical importance of constructing surrogate models to complex mathematical models of physical processes. In a previous work, <span><span>[5]</span></span> we studied the approximation of so-called Banach-valued, <span><math><mo>(</mo><mi>b</mi><mo>,</mo><mi>ε</mi><mo>)</mo></math></span>-holomorphic functions on the infinite-dimensional hypercube <span><math><msup><mrow><mo>[</mo><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>]</mo></mrow><mrow><mi>N</mi></mrow></msup></math></span> from <em>m</em> (potentially adaptive) samples. In particular, we derived lower bounds for the adaptive <em>m</em>-widths for classes of such functions, which showed that certain algebraic rates of the form <span><math><msup><mrow><mi>m</mi></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn><mo>−</mo><mn>1</mn><mo>/</mo><mi>p</mi></mrow></msup></math></span> are the best possible regardless of the sampling-recovery pair. In this work, we continue this investigation by focusing on the practical case where the samples are pointwise evaluations drawn identically and independently from the underlying probability measure for the problem. Specifically, for Hilbert-valued <span><math><mo>(</mo><mi>b</mi><mo>,</mo><mi>ε</mi><mo>)</mo></math></span>-holomorphic functions, we show that the same rates can be achieved (up to a small polylogarithmic or algebraic factor) for tensor-product Jacobi measures. Our reconstruction maps are based on least squares and compressed sensing procedures using the corresponding orthonormal Jacobi polynomials. In doing so, we strengthen and generalize past work that has derived weaker nonuniform guarantees for the uniform and Chebyshev measures (and corresponding polynomials) only. We also extend various best <em>s</em>-term polynomial approximation error bounds to arbitrary Jacobi polynomial expansions. Overall, we demonstrate that i.i.d. pointwise samples drawn from an underlying probability measure are near-optimal for the recovery of infinite-dimensional, holomorphic functions.</div></div>","PeriodicalId":50227,"journal":{"name":"Journal of Complexity","volume":"89 ","pages":"Article 101933"},"PeriodicalIF":1.8,"publicationDate":"2025-02-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143552356","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}
Pub Date : 2025-02-19DOI: 10.1016/j.jco.2025.101931
Aoli Yang , Jun Fan , Dao-Hong Xiang
The pursuit of individualized treatment rules in precision medicine has generated significant interest due to its potential to optimize clinical outcomes for patients with diverse treatment responses. One approach that has gained attention is outcome weighted learning, which is tailored to estimate optimal individualized treatment rules by leveraging each patient's unique characteristics under a weighted classification framework. However, traditional offline learning algorithms, which process all available data at once, face limitations when applied to high-dimensional electronic health records data due to its sheer volume. Additionally, the dynamic nature of precision medicine requires that learning algorithms can effectively handle streaming data that arrives in a sequential manner. To overcome these challenges, we present a novel framework that combines outcome weighted learning with online gradient descent algorithms, aiming to enhance precision medicine practices. Our framework provides a comprehensive analysis of the learning theory associated with online outcome weighted learning algorithms, taking into account general classification loss functions. We establish the convergence of these algorithms for the first time, providing explicit convergence rates while assuming polynomially decaying step sizes, with (or without) a regularization term. Our findings present a non-trivial extension of online classification to online outcome weighted learning, contributing to the theoretical foundations of learning algorithms tailored for processing streaming input-output-reward type data.
{"title":"Online outcome weighted learning with general loss functions","authors":"Aoli Yang , Jun Fan , Dao-Hong Xiang","doi":"10.1016/j.jco.2025.101931","DOIUrl":"10.1016/j.jco.2025.101931","url":null,"abstract":"<div><div>The pursuit of individualized treatment rules in precision medicine has generated significant interest due to its potential to optimize clinical outcomes for patients with diverse treatment responses. One approach that has gained attention is outcome weighted learning, which is tailored to estimate optimal individualized treatment rules by leveraging each patient's unique characteristics under a weighted classification framework. However, traditional offline learning algorithms, which process all available data at once, face limitations when applied to high-dimensional electronic health records data due to its sheer volume. Additionally, the dynamic nature of precision medicine requires that learning algorithms can effectively handle streaming data that arrives in a sequential manner. To overcome these challenges, we present a novel framework that combines outcome weighted learning with online gradient descent algorithms, aiming to enhance precision medicine practices. Our framework provides a comprehensive analysis of the learning theory associated with online outcome weighted learning algorithms, taking into account general classification loss functions. We establish the convergence of these algorithms for the first time, providing explicit convergence rates while assuming polynomially decaying step sizes, with (or without) a regularization term. Our findings present a non-trivial extension of online classification to online outcome weighted learning, contributing to the theoretical foundations of learning algorithms tailored for processing streaming input-output-reward type data.</div></div>","PeriodicalId":50227,"journal":{"name":"Journal of Complexity","volume":"88 ","pages":"Article 101931"},"PeriodicalIF":1.8,"publicationDate":"2025-02-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143474840","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}
We are given a value-oracle for a d-dimensional function f that satisfies the conditions of Miranda's theorem, and therefore has a root. Our goal is to compute an approximate root using a number of evaluations that is polynomial in the number of accuracy digits. For this is always possible using the bisection method, but for this is impossible in general.
We show that, if and f satisfies a single monotonicity condition, then the number of required evaluations is polynomial in the accuracy. The same holds if and f satisfies some particular monotonicity conditions. We show that, if and f satisfies a single monotonicity condition, then the number of required evaluations is polynomial in the accuracy. The same holds if and f satisfies some particular monotonicity conditions. In contrast, if even two of these monotonicity conditions are missing, then the required number of evaluations might be exponential.
As an example application, we show that approximate roots of monotone functions can be used for approximate envy-free cake-cutting.
{"title":"Computing approximate roots of monotone functions","authors":"Alexandros Hollender , Chester Lawrence, Erel Segal-Halevi","doi":"10.1016/j.jco.2025.101930","DOIUrl":"10.1016/j.jco.2025.101930","url":null,"abstract":"<div><div>We are given a value-oracle for a <em>d</em>-dimensional function <em>f</em> that satisfies the conditions of Miranda's theorem, and therefore has a root. Our goal is to compute an approximate root using a number of evaluations that is polynomial in the number of accuracy digits. For <span><math><mi>d</mi><mo>=</mo><mn>1</mn></math></span> this is always possible using the bisection method, but for <span><math><mi>d</mi><mo>≥</mo><mn>2</mn></math></span> this is impossible in general.</div><div>We show that, if <span><math><mi>d</mi><mo>=</mo><mn>2</mn></math></span> and <em>f</em> satisfies a single monotonicity condition, then the number of required evaluations is polynomial in the accuracy. The same holds if <span><math><mi>d</mi><mo>≥</mo><mn>3</mn></math></span> and <em>f</em> satisfies some particular <span><math><msup><mrow><mi>d</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>−</mo><mi>d</mi></math></span> monotonicity conditions. We show that, if <span><math><mi>d</mi><mo>=</mo><mn>2</mn></math></span> and <em>f</em> satisfies a single monotonicity condition, then the number of required evaluations is polynomial in the accuracy. The same holds if <span><math><mi>d</mi><mo>≥</mo><mn>3</mn></math></span> and <em>f</em> satisfies some particular <span><math><msup><mrow><mi>d</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>−</mo><mi>d</mi></math></span> monotonicity conditions. In contrast, if even two of these monotonicity conditions are missing, then the required number of evaluations might be exponential.</div><div>As an example application, we show that approximate roots of monotone functions can be used for approximate envy-free cake-cutting.</div></div>","PeriodicalId":50227,"journal":{"name":"Journal of Complexity","volume":"88 ","pages":"Article 101930"},"PeriodicalIF":1.8,"publicationDate":"2025-01-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143157479","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-01-16DOI: 10.1016/j.jco.2025.101929
Michael Maller
In previous work we defined a computational saddle transition problem which arises in the dynamics of diffeomorphisms of the 2−dimensional torus, and proved this problem is in Oracle NP, working in a model of computation appropriate for Turing machine computations on problems defined over the real numbers. In this note we report further work on these problems, studying orbit descriptions represented as finite words in periodic points. We show these Orbit Word Problems are again in Oracle NP, in our model. Our methods also reveal structures in the set of realized orbit words, suggesting further applications in complexity.
{"title":"On the complexity of orbit word problems","authors":"Michael Maller","doi":"10.1016/j.jco.2025.101929","DOIUrl":"10.1016/j.jco.2025.101929","url":null,"abstract":"<div><div>In previous work we defined a computational saddle transition problem which arises in the dynamics of diffeomorphisms of the 2−dimensional torus, and proved this problem is in Oracle <strong>NP</strong>, working in a model of computation appropriate for Turing machine computations on problems defined over the real numbers. In this note we report further work on these problems, studying orbit descriptions represented as finite words in periodic points. We show these Orbit Word Problems are again in Oracle <strong>NP</strong>, in our model. Our methods also reveal structures in the set of realized orbit words, suggesting further applications in complexity.</div></div>","PeriodicalId":50227,"journal":{"name":"Journal of Complexity","volume":"88 ","pages":"Article 101929"},"PeriodicalIF":1.8,"publicationDate":"2025-01-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143157480","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}
Pub Date : 2024-12-16DOI: 10.1016/j.jco.2024.101922
Joris van der Hoeven, Grégoire Lecerf
Consider a sparse multivariate polynomial f with integer coefficients. Assume that f is represented as a “modular black box polynomial”, e.g. via an algorithm to evaluate f at arbitrary integer points, modulo arbitrary positive integers. The problem of sparse interpolation is to recover f in its usual sparse representation, as a sum of coefficients times monomials. For the first time we present a quasi-optimal algorithm for this task in term of the product of the number of terms of f by the maximum of the bit-size of the terms of f.
{"title":"Fast interpolation of multivariate polynomials with sparse exponents","authors":"Joris van der Hoeven, Grégoire Lecerf","doi":"10.1016/j.jco.2024.101922","DOIUrl":"10.1016/j.jco.2024.101922","url":null,"abstract":"<div><div>Consider a sparse multivariate polynomial <em>f</em> with integer coefficients. Assume that <em>f</em> is represented as a “modular black box polynomial”, e.g. via an algorithm to evaluate <em>f</em> at arbitrary integer points, modulo arbitrary positive integers. The problem of sparse interpolation is to recover <em>f</em> in its usual sparse representation, as a sum of coefficients times monomials. For the first time we present a quasi-optimal algorithm for this task in term of the product of the number of terms of <em>f</em> by the maximum of the bit-size of the terms of <em>f</em>.</div></div>","PeriodicalId":50227,"journal":{"name":"Journal of Complexity","volume":"87 ","pages":"Article 101922"},"PeriodicalIF":1.8,"publicationDate":"2024-12-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143180614","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}
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