arXiv - MATH - Metric Geometry最新文献_第3页

Approximating Metric Magnitude of Point Sets 近似点集的度量大小

arXiv - MATH - Metric Geometry Pub Date : 2024-09-06 DOI: arxiv-2409.04411

Rayna Andreeva, James Ward, Primoz Skraba, Jie Gao, Rik Sarkar

{"title":"Approximating Metric Magnitude of Point Sets","authors":"Rayna Andreeva, James Ward, Primoz Skraba, Jie Gao, Rik Sarkar","doi":"arxiv-2409.04411","DOIUrl":"https://doi.org/arxiv-2409.04411","url":null,"abstract":"Metric magnitude is a measure of the \"size\" of point clouds with many\u0000desirable geometric properties. It has been adapted to various mathematical\u0000contexts and recent work suggests that it can enhance machine learning and\u0000optimization algorithms. But its usability is limited due to the computational\u0000cost when the dataset is large or when the computation must be carried out\u0000repeatedly (e.g. in model training). In this paper, we study the magnitude\u0000computation problem, and show efficient ways of approximating it. We show that\u0000it can be cast as a convex optimization problem, but not as a submodular\u0000optimization. The paper describes two new algorithms - an iterative\u0000approximation algorithm that converges fast and is accurate, and a subset\u0000selection method that makes the computation even faster. It has been previously\u0000proposed that magnitude of model sequences generated during stochastic gradient\u0000descent is correlated to generalization gap. Extension of this result using our\u0000more scalable algorithms shows that longer sequences in fact bear higher\u0000correlations. We also describe new applications of magnitude in machine\u0000learning - as an effective regularizer for neural network training, and as a\u0000novel clustering criterion.","PeriodicalId":501444,"journal":{"name":"arXiv - MATH - Metric Geometry","volume":"33 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142187843","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Dynamical self-similarity, $L^{q}$-dimensions and Furstenberg slicing in $mathbb{R}^d$ $mathbb{R}^d$中的动态自相似性、$L^{q}$维数和弗斯滕伯格切片法

arXiv - MATH - Metric Geometry Pub Date : 2024-09-06 DOI: arxiv-2409.04608

Emilio Corso, Pablo Shmerkin

引用次数: 0

Principal frequency of clamped plates on RCD(0,N) spaces: sharpness, rigidity and stability RCD(0,N) 空间上夹板的主频：锐度、刚度和稳定性

arXiv - MATH - Metric Geometry Pub Date : 2024-09-06 DOI: arxiv-2409.04337

Alexandru Kristály, Andrea Mondino

{"title":"Principal frequency of clamped plates on RCD(0,N) spaces: sharpness, rigidity and stability","authors":"Alexandru Kristály, Andrea Mondino","doi":"arxiv-2409.04337","DOIUrl":"https://doi.org/arxiv-2409.04337","url":null,"abstract":"We study fine properties of the principal frequency of clamped plates in the\u0000(possibly singular) setting of metric measure spaces verifying the RCD(0,N)\u0000condition, i.e., infinitesimally Hilbertian spaces with non-negative Ricci\u0000curvature and dimension bounded above by N>1 in the synthetic sense. The\u0000initial conjecture -- an isoperimetric inequality for the principal frequency\u0000of clamped plates -- has been formulated in 1877 by Lord Rayleigh in the\u0000Euclidean case and solved affirmatively in dimensions 2 and 3 by Ashbaugh and\u0000Benguria [Duke Math. J., 1995] and Nadirashvili [Arch. Rat. Mech. Anal., 1995].\u0000The main contribution of the present work is a new isoperimetric inequality for\u0000the principal frequency of clamped plates in RCD(0,N) spaces whenever N is\u0000close enough to 2 or 3. The inequality contains the so-called ``asymptotic\u0000volume ratio\", and turns out to be sharp under the subharmonicity of the\u0000distance function, a condition satisfied in metric measure cones. In addition,\u0000rigidity (i.e., equality in the isoperimetric inequality) and stability results\u0000are established in terms of the cone structure of the RCD(0,N) space as well as\u0000the shape of the eigenfunction for the principal frequency, given by means of\u0000Bessel functions. These results are new even for Riemannian manifolds with\u0000non-negative Ricci curvature. We discuss examples of both smooth and non-smooth\u0000spaces where the results can be applied.","PeriodicalId":501444,"journal":{"name":"arXiv - MATH - Metric Geometry","volume":"59 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142187844","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Minimal displacement set for weakly systolic complexes 弱收缩复合体的最小位移集

arXiv - MATH - Metric Geometry Pub Date : 2024-09-05 DOI: arxiv-2409.03850

Ioana-Claudia Lazar

引用次数: 0

Non-smooth paths having unit speed with respect to the Kobayashi metric 相对于小林公设具有单位速度的非光滑路径

arXiv - MATH - Metric Geometry Pub Date : 2024-09-05 DOI: arxiv-2409.03709

Gautam Bharali, Rumpa Masanta

引用次数: 0

On variants of the Furstenberg set problem 关于弗斯滕伯格集合问题的变体

arXiv - MATH - Metric Geometry Pub Date : 2024-09-05 DOI: arxiv-2409.03678

Jonathan M. Fraser