Foundations of Computational Mathematics最新文献

{"title":"Covering Numbers for Deep ReLU Networks with Applications to Function Approximation and Nonparametric Regression","authors":"Weigutian Ou,Helmut Bölcskei","doi":"10.1007/s10208-026-09762-w","DOIUrl":"https://doi.org/10.1007/s10208-026-09762-w","url":null,"abstract":"Abstract Covering numbers of (deep) ReLU networks have been used to characterize approximation-theoretic performance, to upper-bound prediction error in nonparametric regression, and to quantify classification capacity. These results rely on covering number upper bounds obtained via explicit constructions of coverings. Lower bounds on covering numbers do not appear to be available in the literature. The present paper fills this gap by deriving tight (up to multiplicative constants) lower and upper bounds on the metric entropy (i.e., the logarithm of the covering numbers) of fully-connected networks with bounded weights, sparse networks with bounded weights, and fully-connected networks with quantized weights. The tightness of these bounds yields a fundamental understanding of the impact of sparsity, quantization, bounded versus unbounded weights, and network output truncation. Moreover, the bounds allow one to characterize fundamental limits of neural network transformation, including network compression, and lead to sharp upper bounds on the prediction error in nonparametric regression through deep networks. In particular, we remove a $$(log (n))^6$$ ( log ( n ) ) 6 -factor from the best known sample complexity rate for estimating Lipschitz functions via deep networks, thereby establishing optimality. Finally, we identify a systematic relation between optimal nonparametric regression and optimal approximation through deep networks, unifying numerous results in the literature and revealing underlying general principles.","PeriodicalId":55151,"journal":{"name":"Foundations of Computational Mathematics","volume":"109 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2026-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"149021138","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A Nonsmooth Extension of the Brezzi-Rappaz-Raviart Approximation Theorem Via Metric Regularity Techniques and Applications to Nonlinear PDEs","authors":"Jules Berry, Olivier Ley, Francisco J. Silva","doi":"10.1007/s10208-026-09767-5","DOIUrl":"https://doi.org/10.1007/s10208-026-09767-5","url":null,"abstract":"","PeriodicalId":55151,"journal":{"name":"Foundations of Computational Mathematics","volume":"19 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2026-08-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148768975","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Any Nonincreasing Convergence Curves are Simultaneously Possible for GMRES and Weighted GMRES, As Well As for Left and Right Preconditioned GMRES","authors":"Pierre Matalon, Nicole Spillane","doi":"10.1007/s10208-026-09768-4","DOIUrl":"https://doi.org/10.1007/s10208-026-09768-4","url":null,"abstract":"","PeriodicalId":55151,"journal":{"name":"Foundations of Computational Mathematics","volume":"13 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2026-08-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148768979","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Alpha Shapes and Optimal Transport on the Sphere","authors":"Erik Carlsson, Greg DePaul","doi":"10.1007/s10208-026-09765-7","DOIUrl":"https://doi.org/10.1007/s10208-026-09765-7","url":null,"abstract":"In [3], the authors used the Legendre transform to give a tractable method for studying Topological Data Analysis (TDA) in terms of sums of Gaussian kernels. In this paper, we prove a variant for sums of cosine similarity-based kernel functions, which requires considering the more general “ <jats:italic>c</jats:italic> -transform” from optimal transport theory [18]. We then apply these methods to a point cloud arising from a recent breakthrough study, which exhibits a toroidal structure in the brain activity of rats [12]. A key part of this application is that the transport map and transformed density function arising from the theorem replace certain delicate preprocessing steps related to density-based denoising and subsampling.","PeriodicalId":55151,"journal":{"name":"Foundations of Computational Mathematics","volume":"32 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2026-08-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148728732","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Conley–Morse Persistence Barcode: A Homological Signature of Combinatorial Bifurcations","authors":"Tamal K. Dey, Michał Lipiński, Manuel Soriano-Trigueros","doi":"10.1007/s10208-026-09766-6","DOIUrl":"https://doi.org/10.1007/s10208-026-09766-6","url":null,"abstract":"Bifurcation characterizes the qualitative changes in parameterized dynamical systems and is one of the major topics in the field. In this work, we study combinatorial bifurcations within the framework of combinatorial dynamical systems—a young but already well-established theory. We introduce the Conley–Morse persistence barcode, a compact algebraic descriptor of combinatorial bifurcations. This barcode captures structural changes in a dynamical system at the level of Morse decompositions and provides a characterization of the nature of observed transitions in terms of the Conley index. The construction of the Conley–Morse persistence barcode builds upon ideas from topological persistence. Specifically, we consider a persistence module obtained from the Conley index of invariant sets indexed over a poset. Using gentle algebras, we prove that this module decomposes into simple intervals (bars) and compute them by adapting the zigzag persistence algorithm to our purpose.","PeriodicalId":55151,"journal":{"name":"Foundations of Computational Mathematics","volume":"32 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2026-08-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148690838","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Generalised Hardness of Approximation and the SCI Hierarchy –On Determining the Boundaries of Training Algorithms in AI","authors":"Luca Eva Gazdag, Alexander Bastounis, Anders C. Hansen","doi":"10.1007/s10208-026-09764-8","DOIUrl":"https://doi.org/10.1007/s10208-026-09764-8","url":null,"abstract":"Generalised hardness of approximation (GHA) is the phenomenon that one can easily compute an &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;jats:tex-math&gt;$$epsilon $$&lt;/jats:tex-math&gt; &lt;mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;mml:mi&gt;ϵ&lt;/mml:mi&gt; &lt;/mml:math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt; -approximation to a solution of a computational problem for &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;jats:tex-math&gt;$$epsilon&gt; epsilon _1 &gt; 0$$&lt;/jats:tex-math&gt; &lt;mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;mml:mrow&gt; &lt;mml:mi&gt;ϵ&lt;/mml:mi&gt; &lt;mml:mo&gt;&gt;&lt;/mml:mo&gt; &lt;mml:msub&gt; &lt;mml:mi&gt;ϵ&lt;/mml:mi&gt; &lt;mml:mn&gt;1&lt;/mml:mn&gt; &lt;/mml:msub&gt; &lt;mml:mo&gt;&gt;&lt;/mml:mo&gt; &lt;mml:mn&gt;0&lt;/mml:mn&gt; &lt;/mml:mrow&gt; &lt;/mml:math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt; , but for &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;jats:tex-math&gt;$$epsilon &lt; epsilon _1$$&lt;/jats:tex-math&gt; &lt;mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;mml:mrow&gt; &lt;mml:mi&gt;ϵ&lt;/mml:mi&gt; &lt;mml:mo&gt;&lt;&lt;/mml:mo&gt; &lt;mml:msub&gt; &lt;mml:mi&gt;ϵ&lt;/mml:mi&gt; &lt;mml:mn&gt;1&lt;/mml:mn&gt; &lt;/mml:msub&gt; &lt;/mml:mrow&gt; &lt;/mml:math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt; (the approximation threshold) it suddenly becomes hard, for example, non-computable or intractable (non-polynomial time). In this paper we demonstrate the phenomenon that GHA happens when using AI techniques for solving inverse problems, namely training neural networks (NNs) to optimally perform on the training data. In particular, for any non-zero underdetermined linear inverse problem the following phase transition can occur: For a certain family of training sets &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;jats:tex-math&gt;$$Omega $$&lt;/jats:tex-math&gt; &lt;mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;mml:mi&gt;Ω&lt;/mml:mi&gt; &lt;/mml:math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt; , one can prove the existence of optimal NNs for solving the inverse problem for each &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;jats:tex-math&gt;$$mathcal {T}in Omega $$&lt;/jats:tex-math&gt; &lt;mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;mml:mrow&gt; &lt;mml:mi&gt;T&lt;/mml:mi&gt; &lt;mml:mo&gt;∈&lt;/mml:mo&gt; &lt;mml:mi&gt;Ω&lt;/mml:mi&gt; &lt;/mml:mrow&gt; &lt;/mml:math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt; , however, these optimal neural networks can only be computed to a certain accuracy &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;jats:tex-math&gt;$$epsilon _1 &gt; 0$$&lt;/jats:tex-math&gt; &lt;mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;mml:mrow&gt; &lt;mml:msub&gt; &lt;mml:mi&gt;ϵ&lt;/mml:mi&gt; &lt;mml:mn&gt;1&lt;/mml:mn&gt; &lt;/mml:msub&gt; &lt;mml:mo&gt;&gt;&lt;/mml:mo&gt; &lt;mml:mn&gt;0&lt;/mml:mn&gt; &lt;/mml:mrow&gt; &lt;/mml:math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt; . Below the approximation threshold &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;jats:tex-math&gt;$$epsilon _1$$&lt;/jats:tex-math&gt; &lt;mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;mml:msub&gt; &lt;mml:mi&gt;ϵ&lt;/mml:mi&gt; &lt;mml:mn&gt;1&lt;/mml:mn&gt; &lt;/mml:msub&gt; &lt;/mml:math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt; , not only does it become intractable to compute the NNs, it becomes impossible regardless of computing power","PeriodicalId":55151,"journal":{"name":"Foundations of Computational Mathematics","volume":"26 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2026-07-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148682207","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Validated Enclosure of Renormalization Fixed Points Via Chebyshev Series and the DFT","authors":"Maxime Breden, Jorge Gonzalez, J. D. Mireles James","doi":"10.1007/s10208-026-09763-9","DOIUrl":"https://doi.org/10.1007/s10208-026-09763-9","url":null,"abstract":"","PeriodicalId":55151,"journal":{"name":"Foundations of Computational Mathematics","volume":"27 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2026-07-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148408868","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"The Likelihood Correspondence","authors":"Thomas Kahle, Hal Schenck, Bernd Sturmfels, Maximilian Wiesmann","doi":"10.1007/s10208-026-09755-9","DOIUrl":"https://doi.org/10.1007/s10208-026-09755-9","url":null,"abstract":"An arrangement of hypersurfaces in projective space is strict normal crossing (SNC) if and only if its Euler discriminant is nonzero. We study the critical loci of arbitrary Laurent monomials in the equations of the smooth hypersurfaces. The family of these loci forms an irreducible variety in the product of two projective spaces, known in algebraic statistics as the likelihood correspondence and in particle physics as the scattering correspondence. We establish an explicit determinantal representation for the minimal generators of the bihomogeneous prime ideal that defines this variety.","PeriodicalId":55151,"journal":{"name":"Foundations of Computational Mathematics","volume":"34 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2026-07-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148408869","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Interior-Point Methods on Manifolds: Theory and Applications","authors":"Hiroshi Hirai, Harold Nieuwboer, Michael Walter","doi":"10.1007/s10208-026-09756-8","DOIUrl":"https://doi.org/10.1007/s10208-026-09756-8","url":null,"abstract":"Interior-point methods offer a highly versatile framework for convex optimization that is effective in theory and practice. A key notion in their theory is that of a self-concordant barrier. We give a suitable generalization of self-concordance to Riemannian manifolds and show that it gives the same structural results and guarantees as in the Euclidean setting, in particular local quadratic convergence of Newton’s method. We analyze a path-following method for optimizing compatible objectives over a convex domain for which one has a self-concordant barrier, and obtain the standard complexity guarantees as in the Euclidean setting. We provide general constructions of barriers, and show that on the space of positive-definite matrices and other symmetric spaces, the squared distance to a point is self-concordant. To demonstrate the versatility of our framework, we give algorithms with state-of-the-art complexity guarantees for the general class of scaling and non-commutative optimization problems, which have been of much recent interest, and we provide the first algorithms for efficiently finding high-precision solutions for computing minimal enclosing balls and geometric medians in nonpositive curvature.","PeriodicalId":55151,"journal":{"name":"Foundations of Computational Mathematics","volume":"16 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2026-07-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148408870","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"First-Order Optimality Conditions for Non-Commutative Optimization Problems","authors":"Mateus Araújo, Igor Klep, Andrew J. P. Garner, Tamás Vértesi, Miguel Navascués","doi":"10.1007/s10208-026-09761-x","DOIUrl":"https://doi.org/10.1007/s10208-026-09761-x","url":null,"abstract":"We consider the problem of optimizing the state average of a polynomial of non-commuting variables, over all states and operators satisfying a number of polynomial constraints, and over all Hilbert spaces where such states and operators are defined. Such non-commutative polynomial optimization (NPO) problems are routinely solved through hierarchies of semidefinite programming (SDP) relaxations. By formulating the general NPO problem in Lagrangian terms, we heuristically derive first-order optimality conditions via small variations in the problem variables. Although the derivation is not rigorous, it gives rise to two types of optimality conditions—state and operator—which are rigorously analyzed in the paper. Both types of conditions can be enforced through additional positive semidefinite constraints in the SDP hierarchies. State optimality conditions are shown to be satisfied by all NPO problems. For NPO problems with optimal solutions (such as, e.g., Archimedean ones) they allow enforcing a new type of constraints: namely, restricting the optimization over states to the set of common ground states of an arbitrary number of operators. Operator optimality conditions are the non-commutative analogs of the Karush–Kuhn–Tucker (KKT) conditions, which are known to hold in many classical optimization problems. In this regard, we prove that a weak form of operator optimality holds for all NPO problems; stronger versions require the problem constraints to satisfy some qualification criterion, just like in the classical case (e.g.: Mangasarian–Fromovitz constraint qualification). We test the power of the new optimality conditions by computing local properties of ground states of many-body spin systems and the maximum quantum violation of Bell inequalities.","PeriodicalId":55151,"journal":{"name":"Foundations of Computational Mathematics","volume":"216 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2026-07-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148408867","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}