Persistent Homology and the Upper Box Dimension

被引：10
|
作者：
Schweinhart, Benjamin [1 ]
机构：
[1] Ohio State Univ, Dept Math, 231 W 18th Ave, Columbus, OH 43210 USA
来源：
DISCRETE & COMPUTATIONAL GEOMETRY | 2021年 / 65卷 / 02期
关键词：
Persistent homology; Upper box dimension; Fractal geometry; Metric geometry; Extremal combinatorics; MINIMAL SPANNING-TREES; NUMBERS;
D O I：
10.1007/s00454-019-00145-3
中图分类号：
TP301 [理论、方法];
学科分类号：
081202 ;
摘要：
We introduce a fractal dimension for a metric space defined in terms of the persistent homology of extremal subsets of that space. We exhibit hypotheses under which this dimension is comparable to the upper box dimension; in particular, the dimensions coincide for subsets of R2 whose upper box dimension exceeds 1.5. These results are related to extremal questions about the number of persistent homology intervals of a set of n points in a metric space.
引用
收藏
页码：331 / 364
页数：34
相关论文
共 50 条
  • [1] Persistent Homology and the Upper Box Dimension
    Benjamin Schweinhart
    Discrete & Computational Geometry, 2021, 65 : 331 - 364
  • [2] A Fractal Dimension for Measures via Persistent Homology
    Adams, Henry
    Aminian, Manuchehr
    Farnell, Elin
    Kirby, Michael
    Mirth, Joshua
    Neville, Rachel
    Peterson, Chris
    Shonkwiler, Clayton
    TOPOLOGICAL DATA ANALYSIS, ABEL SYMPOSIUM 2018, 2020, 15 : 1 - 31
  • [3] Fractal dimension and the persistent homology of random geometric complexes
    Schweinhart, Benjamin
    ADVANCES IN MATHEMATICS, 2020, 372
  • [4] Fractal dimension estimation with persistent homology: A comparative study
    Jaquette, Jonathan
    Schweinhart, Benjamin
    COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION, 2020, 84 (84):
  • [5] Intrinsic Dimension, Persistent Homology and Generalization in Neural Networks
    Birdal, Tolga
    Lou, Aaron
    Guibas, Leonidas
    Simsekli, Umut
    ADVANCES IN NEURAL INFORMATION PROCESSING SYSTEMS 34 (NEURIPS 2021), 2021, 34
  • [6] The minimal spanning tree and the upper box dimension
    Kozma, G
    Lotker, Z
    Stupp, G
    PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY, 2006, 134 (04) : 1183 - 1187
  • [7] Hausdorff dimension and upper box dimension of a class of homogeneous Moran sets
    Liu, Shishuang
    Li, Yanzhe
    Zong, Wenqi
    An, Chengshuai
    DYNAMICAL SYSTEMS-AN INTERNATIONAL JOURNAL, 2024, 39 (03): : 449 - 460
  • [8] Hausdorff and upper box dimension estimate of hyperbolic recurrent sets
    Mrinal Kanti Roychowdhury
    Israel Journal of Mathematics, 2014, 201 : 507 - 523
  • [9] HAUSDORFF AND UPPER BOX DIMENSION ESTIMATE OF HYPERBOLIC RECURRENT SETS
    Roychowdhury, Mrinal Kanti
    ISRAEL JOURNAL OF MATHEMATICS, 2014, 201 (02) : 507 - 523
  • [10] Going beyond persistent homology using persistent homology
    Immonen, Johanna
    Souza, Amauri H.
    Garg, Vikas
    ADVANCES IN NEURAL INFORMATION PROCESSING SYSTEMS 36 (NEURIPS 2023), 2023,
12345