Foundations of Computational Mathematics最新文献

{"title":"Stabilizing Decomposition of Multiparameter Persistence Modules","authors":"Håvard Bakke Bjerkevik","doi":"10.1007/s10208-025-09695-w","DOIUrl":"https://doi.org/10.1007/s10208-025-09695-w","url":null,"abstract":"<p>While decomposition of one-parameter persistence modules behaves nicely, as demonstrated by the algebraic stability theorem, decomposition of multiparameter modules is known to be unstable in a certain precise sense. Until now, it has not been clear that there is any way to get around this and build a meaningful stability theory for multiparameter module decomposition. We introduce new tools, in particular <span>(epsilon )</span>-refinements and <span>(epsilon )</span>-erosion neighborhoods, to start building such a theory. We then define the <span>(epsilon )</span>-pruning of a module, which is a new invariant acting like a “refined barcode” that shows great promise to extract features from a module by approximately decomposing it. Our main theorem can be interpreted as a generalization of the algebraic stability theorem to multiparameter modules up to a factor of 2<i>r</i>, where <i>r</i> is the maximal pointwise dimension of one of the modules. Furthermore, we show that the factor 2<i>r</i> is close to optimal. Finally, we discuss the possibility of strengthening the stability theorem for modules that decompose into pointwise low-dimensional summands, and pose a conjecture phrased purely in terms of basic linear algebra and graph theory that seems to capture the difficulty of doing this. We also show that this conjecture is relevant for other areas of multipersistence, like the computational complexity of approximating the interleaving distance, and recent applications of relative homological algebra to multipersistence.</p>","PeriodicalId":55151,"journal":{"name":"Foundations of Computational Mathematics","volume":"48 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2025-01-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143049647","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":"Optimal Regularization for a Data Source","authors":"Oscar Leong, Eliza O’ Reilly, Yong Sheng Soh, Venkat Chandrasekaran","doi":"10.1007/s10208-025-09693-y","DOIUrl":"https://doi.org/10.1007/s10208-025-09693-y","url":null,"abstract":"<p>In optimization-based approaches to inverse problems and to statistical estimation, it is common to augment criteria that enforce data fidelity with a regularizer that promotes desired structural properties in the solution. The choice of a suitable regularizer is typically driven by a combination of prior domain information and computational considerations. Convex regularizers are attractive computationally but they are limited in the types of structure they can promote. On the other hand, nonconvex regularizers are more flexible in the forms of structure they can promote and they have showcased strong empirical performance in some applications, but they come with the computational challenge of solving the associated optimization problems. In this paper, we seek a systematic understanding of the power and the limitations of convex regularization by investigating the following questions: Given a distribution, what is the optimal regularizer for data drawn from the distribution? What properties of a data source govern whether the optimal regularizer is convex? We address these questions for the class of regularizers specified by functionals that are continuous, positively homogeneous, and positive away from the origin. We say that a regularizer is optimal for a data distribution if the Gibbs density with energy given by the regularizer maximizes the population likelihood (or equivalently, minimizes cross-entropy loss) over all regularizer-induced Gibbs densities. As the regularizers we consider are in one-to-one correspondence with star bodies, we leverage dual Brunn-Minkowski theory to show that a radial function derived from a data distribution is akin to a “computational sufficient statistic” as it is the key quantity for identifying optimal regularizers and for assessing the amenability of a data source to convex regularization. Using tools such as <span>(Gamma )</span>-convergence from variational analysis, we show that our results are robust in the sense that the optimal regularizers for a sample drawn from a distribution converge to their population counterparts as the sample size grows large. Finally, we give generalization guarantees for various families of star bodies that recover previous results for polyhedral regularizers (i.e., dictionary learning) and lead to new ones for a variety of classes of star bodies. </p>","PeriodicalId":55151,"journal":{"name":"Foundations of Computational Mathematics","volume":"48 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2025-01-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143049667","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":"Sharp Bounds for Max-sliced Wasserstein Distances","authors":"March T. Boedihardjo","doi":"10.1007/s10208-025-09690-1","DOIUrl":"https://doi.org/10.1007/s10208-025-09690-1","url":null,"abstract":"<p>We obtain essentially matching upper and lower bounds for the expected max-sliced 1-Wasserstein distance between a probability measure on a separable Hilbert space and its empirical distribution from <i>n</i> samples. By proving a Banach space version of this result, we also obtain an upper bound, that is sharp up to a log factor, for the expected max-sliced 2-Wasserstein distance between a symmetric probability measure <span>(mu )</span> on a Euclidean space and its symmetrized empirical distribution in terms of the operator norm of the covariance matrix of <span>(mu )</span> and the diameter of the support of <span>(mu )</span>.</p>","PeriodicalId":55151,"journal":{"name":"Foundations of Computational Mathematics","volume":"22 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2025-01-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143020600","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":"Active Manifolds, Stratifications, and Convergence to Local Minima in Nonsmooth Optimization","authors":"Damek Davis, Dmitriy Drusvyatskiy, Liwei Jiang","doi":"10.1007/s10208-025-09691-0","DOIUrl":"https://doi.org/10.1007/s10208-025-09691-0","url":null,"abstract":"<p>In this work, we develop new regularity conditions in nonsmooth analysis that parallel the stratification conditions of Whitney, Kuo, and Verdier. They quantify how subgradients interact with a certain “active manifold” that captures the nonsmooth activity of the function. Based on these new conditions, we show that several subgradient-based methods converge only to local minimizers when applied to generic Lipschitz and subdifferentially regular functions that are definable in an o-minimal structure. At a high level, our argument is appealingly transparent: we interpret the nonsmooth dynamics as an approximate Riemannian gradient method on the active manifold. As a by-product, we extend the stochastic processes techniques of Pemantle.</p>","PeriodicalId":55151,"journal":{"name":"Foundations of Computational Mathematics","volume":"74 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2025-01-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143020602","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":"Optimal Convergence Rates for the Spectrum of the Graph Laplacian on Poisson Point Clouds","authors":"Scott Armstrong, Raghavendra Venkatraman","doi":"10.1007/s10208-025-09696-9","DOIUrl":"https://doi.org/10.1007/s10208-025-09696-9","url":null,"abstract":"<p>We prove optimal convergence rates for eigenvalues and eigenvectors of the graph Laplacian on Poisson point clouds. Our results are valid down to the critical percolation threshold, yielding error estimates for relatively sparse graphs.</p>","PeriodicalId":55151,"journal":{"name":"Foundations of Computational Mathematics","volume":"20 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2025-01-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143020601","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":"Accuracy Controlled Schemes for the Eigenvalue Problem of the Radiative Transfer Equation","authors":"Wolfgang Dahmen, Olga Mula","doi":"10.1007/s10208-025-09694-x","DOIUrl":"https://doi.org/10.1007/s10208-025-09694-x","url":null,"abstract":"<p>The criticality problem in nuclear engineering asks for the principal eigenpair of a Boltzmann operator describing neutron transport in a reactor core. Being able to reliably design, and control such reactors requires assessing these quantities within quantifiable accuracy tolerances. In this paper, we propose a paradigm that deviates from the common practice of approximately solving the corresponding spectral problem with a fixed, presumably sufficiently fine discretization. Instead, the present approach is based on first contriving iterative schemes, formulated in function space, that are shown to converge at a quantitative rate without assuming any a priori excess regularity properties, and that exploit only properties of the optical parameters in the underlying radiative transfer model. We develop the analytical and numerical tools for approximately realizing each iteration step within judiciously chosen accuracy tolerances, verified by a posteriori estimates, so as to still warrant quantifiable convergence to the exact eigenpair. This is carried out in full first for a Newton scheme. Since this is only locally convergent we analyze in addition the convergence of a power iteration in function space to produce sufficiently accurate initial guesses. Here we have to deal with intrinsic difficulties posed by compact but unsymmetric operators preventing standard arguments used in the finite dimensional case. Our main point is that we can avoid any condition on an initial guess to be already in a small neighborhood of the exact solution. We close with a discussion of remaining intrinsic obstructions to a certifiable numerical implementation, mainly related to not knowing the gap between the principal eigenvalue and the next smaller one in modulus. </p>","PeriodicalId":55151,"journal":{"name":"Foundations of Computational Mathematics","volume":"20 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2025-01-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142991448","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 Index for Multivalued Maps on Finite Topological Spaces","authors":"Jonathan Barmak, Marian Mrozek, Thomas Wanner","doi":"10.1007/s10208-024-09685-4","DOIUrl":"https://doi.org/10.1007/s10208-024-09685-4","url":null,"abstract":"<p>We develop Conley’s theory for multivalued maps on finite topological spaces. More precisely, for discrete-time dynamical systems generated by the iteration of a multivalued map which satisfies appropriate regularity conditions, we establish the notions of isolated invariant sets and index pairs, and use them to introduce a well-defined Conley index. In addition, we verify some of its fundamental properties such as the Ważewski property and continuation.</p>","PeriodicalId":55151,"journal":{"name":"Foundations of Computational Mathematics","volume":"70 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2024-12-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142797022","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":"Generalized Pseudospectral Shattering and Inverse-Free Matrix Pencil Diagonalization","authors":"James Demmel, Ioana Dumitriu, Ryan Schneider","doi":"10.1007/s10208-024-09682-7","DOIUrl":"https://doi.org/10.1007/s10208-024-09682-7","url":null,"abstract":"<p>We present a randomized, inverse-free algorithm for producing an approximate diagonalization of any <span>(n times n)</span> matrix pencil (<i>A</i>, <i>B</i>). The bulk of the algorithm rests on a randomized divide-and-conquer eigensolver for the generalized eigenvalue problem originally proposed by Ballard, Demmel and Dumitriu (Technical Report 2010). We demonstrate that this divide-and-conquer approach can be formulated to succeed with high probability provided the input pencil is sufficiently well-behaved, which is accomplished by generalizing the recent pseudospectral shattering work of Banks, Garza-Vargas, Kulkarni and Srivastava (Foundations of Computational Mathematics 2023). In particular, we show that perturbing and scaling (<i>A</i>, <i>B</i>) regularizes its pseudospectra, allowing divide-and-conquer to run over a simple random grid and in turn producing an accurate diagonalization of (<i>A</i>, <i>B</i>) in the backward error sense. The main result of the paper states the existence of a randomized algorithm that with high probability (and in exact arithmetic) produces invertible <i>S</i>, <i>T</i> and diagonal <i>D</i> such that <span>(||A - SDT^{-1}||_2 le varepsilon )</span> and <span>(||B - ST^{-1}||_2 le varepsilon )</span> in at most <span>(O left( log ^2 left( frac{n}{varepsilon } right) T_{text {MM}}(n) right) )</span> operations, where <span>(T_{text {MM}}(n))</span> is the asymptotic complexity of matrix multiplication. This not only provides a new set of guarantees for highly parallel generalized eigenvalue solvers but also establishes nearly matrix multiplication time as an upper bound on the complexity of inverse-free, exact-arithmetic matrix pencil diagonalization.</p>","PeriodicalId":55151,"journal":{"name":"Foundations of Computational Mathematics","volume":"47 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2024-12-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142796989","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":"Locally-Verifiable Sufficient Conditions for Exactness of the Hierarchical B-spline Discrete de Rham Complex in $$mathbb {R}^n$$","authors":"Kendrick Shepherd, Deepesh Toshniwal","doi":"10.1007/s10208-024-09659-6","DOIUrl":"https://doi.org/10.1007/s10208-024-09659-6","url":null,"abstract":"<p>Given a domain <span>(Omega subset mathbb {R}^n)</span>, the de Rham complex of differential forms arises naturally in the study of problems in electromagnetism and fluid mechanics defined on <span>(Omega )</span>, and its discretization helps build stable numerical methods for such problems. For constructing such stable methods, one critical requirement is ensuring that the discrete subcomplex is cohomologically equivalent to the continuous complex. When <span>(Omega )</span> is a hypercube, we thus require that the discrete subcomplex be exact. Focusing on such <span>(Omega )</span>, we theoretically analyze the discrete de Rham complex built from hierarchical B-spline differential forms, i.e., the discrete differential forms are smooth splines and support adaptive refinements—these properties are key to enabling accurate and efficient numerical simulations. We provide locally-verifiable sufficient conditions that ensure that the discrete spline complex is exact. Numerical tests are presented to support the theoretical results, and the examples discussed include complexes that satisfy our prescribed conditions as well as those that violate them.</p>","PeriodicalId":55151,"journal":{"name":"Foundations of Computational Mathematics","volume":"82 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2024-12-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142776456","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":"Constrained and Unconstrained Stable Discrete Minimizations for p-Robust Local Reconstructions in Vertex Patches in the de Rham Complex","authors":"Théophile Chaumont-Frelet, Martin Vohralík","doi":"10.1007/s10208-024-09674-7","DOIUrl":"https://doi.org/10.1007/s10208-024-09674-7","url":null,"abstract":"<p>We analyze constrained and unconstrained minimization problems on patches of tetrahedra sharing a common vertex with discontinuous piecewise polynomial data of degree <i>p</i>. We show that the discrete minimizers in the spaces of piecewise polynomials of degree <i>p</i> conforming in the <span>(H^1)</span>, <span>({varvec{H}}(textbf{curl}))</span>, or <span>({varvec{H}}({text {div}}))</span> spaces are as good as the minimizers in these entire (infinite-dimensional) Sobolev spaces, up to a constant that is independent of <i>p</i>. These results are useful in the analysis and design of finite element methods, namely for devising stable local commuting projectors and establishing local-best–global-best equivalences in a priori analysis and in the context of a posteriori error estimation. Unconstrained minimization in <span>(H^1)</span> and constrained minimization in <span>({varvec{H}}({text {div}}))</span> have been previously treated in the literature. Along with improvement of the results in the <span>(H^1)</span> and <span>({varvec{H}}({text {div}}))</span> cases, our key contribution is the treatment of the <span>({varvec{H}}(textbf{curl}))</span> framework. This enables us to cover the whole de Rham diagram in three space dimensions in a single setting.</p>","PeriodicalId":55151,"journal":{"name":"Foundations of Computational Mathematics","volume":"113 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2024-11-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142713198","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}