refs.bib

@misc{chazal2013persistencestabilitygeometriccomplexes,
  title         = {Persistence stability for geometric complexes},
  author        = {Frederic Chazal and Vin de Silva and Steve Oudot},
  year          = {2013},
  eprint        = {1207.3885},
  archiveprefix = {arXiv},
  primaryclass  = {math.AT},
  url           = {https://arxiv.org/abs/1207.3885}
}

@book{edelsbrunner2010computational,
  title     = {Computational Topology: {An} Introduction},
  author    = {Edelsbrunner, H. and Harer, J.},
  isbn      = {9780821849255},
  lccn      = {2009028121},
  series    = {Applied Mathematics},
  url       = {https://books.google.com/books?id=MDXa6gFRZuIC},
  year      = {2010},
  publisher = {American Mathematical Society}
}

@article{Borsuk1948,
  author   = {Borsuk, Karol},
  journal  = {Fundamenta Mathematicae},
  keywords = {Topology},
  language = {eng},
  number   = {1},
  pages    = {217-234},
  title    = {On the imbedding of systems of compacta in simplicial complexes},
  url      = {http://eudml.org/doc/213158},
  volume   = {35},
  year     = {1948}
}

@article{leray1945forme,
  title   = {Sur la forme des espaces topologiques et sur les points fixes des repr{\'e}sentations},
  author  = {Leray, Jean},
  journal = {Journal de Math{\'e}matiques Pures et Appliqu{\'e}es},
  volume  = {24},
  pages   = {95--167},
  year    = {1945}
}

@inproceedings{anai2020dtm,
  title        = {{DTM}-based filtrations},
  author       = {Anai, Hirokazu and Chazal, Fr{\'e}d{\'e}ric and Glisse, Marc and Ike, Yuichi and Inakoshi, Hiroya and Tinarrage, Rapha{\"e}l and Umeda, Yuhei},
  booktitle    = {Topological data analysis: the abel symposium 2018},
  pages        = {33--66},
  year         = {2020},
  organization = {Springer},
  doi          = {https://doi.org/10.1007/978-3-030-43408-3_2}
}

@inproceedings{cohen2005stability,
  title     = {Stability of persistence diagrams},
  author    = {Cohen-Steiner, David and Edelsbrunner, Herbert and Harer, John},
  booktitle = {Proceedings of the twenty-first annual symposium on Computational geometry},
  pages     = {263--271},
  year      = {2005}
}

@misc{schnider2024introduction,
  title        = {Introduction to Topological Data Analysis Lecture Notes {FS} 2024},
  author       = {Schnider, Patrick and Weber, Simon},
  howpublished = {\url{https://ti.inf.ethz.ch/ew/courses/TDA24/Script.pdf}}
}

@article{chazal2021introduction,
  title     = {An introduction to topological data analysis: {Fundamental} and practical aspects for data scientists},
  author    = {Chazal, Fr{\'e}d{\'e}ric and Michel, Bertrand},
  journal   = {Frontiers in artificial intelligence},
  volume    = {4},
  pages     = {667963},
  year      = {2021},
  publisher = {Frontiers Media SA}
}

@book{carlsson2021topological,
  title     = {Topological data analysis with applications},
  author    = {Carlsson, Gunnar and Vejdemo-Johansson, Mikael},
  year      = {2021},
  publisher = {Cambridge University Press}
}

@book{boissonnat2018geometric,
  title     = {Geometric and topological inference},
  author    = {Boissonnat, Jean-Daniel and Chazal, Fr{\'e}d{\'e}ric and Yvinec, Mariette},
  volume    = {57},
  year      = {2018},
  publisher = {Cambridge University Press}
}

@misc{hoefgeest2022christoffeldarbouxkerneltopologicaldata,
  title         = {The Christoffel-Darboux kernel for topological data analysis},
  author        = {Pepijn Roos Hoefgeest and Lucas Slot},
  year          = {2022},
  eprint        = {2211.15489},
  archiveprefix = {arXiv},
  primaryclass  = {math.AT},
  url           = {https://arxiv.org/abs/2211.15489}
}

@misc{chazal2013structurestabilitypersistencemodules,
  title         = {The structure and stability of persistence modules},
  author        = {Frederic Chazal and Vin de Silva and Marc Glisse and Steve Oudot},
  year          = {2013},
  eprint        = {1207.3674},
  archiveprefix = {arXiv},
  primaryclass  = {math.AT},
  url           = {https://arxiv.org/abs/1207.3674}
}

@inproceedings{guibas2011witnessed,
  title     = {Witnessed k-distance},
  author    = {Guibas, Leonidas J and M{\'e}rigot, Quentin and Morozov, Dmitriy},
  booktitle = {Proceedings of the twenty-seventh annual symposium on Computational geometry},
  pages     = {57--64},
  year      = {2011}
}

@misc{phillips2015geometricinferencekerneldensity,
  title         = {Geometric Inference on Kernel Density Estimates},
  author        = {Jeff M. Phillips and Bei Wang and Yan Zheng},
  year          = {2015},
  eprint        = {1307.7760},
  archiveprefix = {arXiv},
  primaryclass  = {cs.CG},
  url           = {https://arxiv.org/abs/1307.7760}
}

@misc{buchet2014efficientrobustpersistenthomology,
  title         = {Efficient and Robust Persistent Homology for Measures},
  author        = {Mickael Buchet and Frederic Chazal and Steve Y. Oudot and Donald R. Sheehy},
  year          = {2014},
  eprint        = {1306.0039},
  archiveprefix = {arXiv},
  primaryclass  = {cs.CG},
  url           = {https://arxiv.org/abs/1306.0039}
}

@article{fractalfract8120731,
  author         = {Lee, Minhyeok and Lee, Soyeon},
  title          = {Persistent Homology Analysis of {AI-Generated} Fractal Patterns: {A} Mathematical Framework for Evaluating Geometric Authenticity},
  journal        = {Fractal and Fractional},
  volume         = {8},
  year           = {2024},
  number         = {12},
  article-number = {731},
  url            = {https://www.mdpi.com/2504-3110/8/12/731},
  issn           = {2504-3110},
  abstract       = {We present a mathematical framework for analyzing fractal patterns in AI-generated images using persistent homology. Given a text-to-image mapping M:T→I, we demonstrate that the persistent homology groups Hk(t) of sublevel set filtrations {f−1((−∞,t])}t∈R characterize multi-scale geometric structures, where f:M(p)→R is the grayscale intensity function of a generated image. The primary challenge lies in quantifying self-similarity in scales, which we address by analyzing birth–death pairs (bi,di) in the persistence diagram PD(M(p)). Our contribution extends beyond applying the stability theorem to AI-generated fractals; we establish how the self-similarity inherent in fractal patterns manifests in the persistence diagrams of generated images. We validate our approach using the Stable Diffusion 3.5 model for four fractal categories: ferns, trees, spirals, and crystals. An analysis of guidance scale effects γ∈[4.0,8.0] reveals monotonic relationships between model parameters and topological features. Stability testing confirms robustness under noise perturbations η≤0.2, with feature count variations Δμf<0.5. Our framework provides a foundation for enhancing generative models and evaluating their geometric fidelity in fractal pattern synthesis.},
  doi            = {10.3390/fractalfract8120731}
}

@book{moise2013geometric,
  title     = {Geometric topology in dimensions 2 and 3},
  author    = {Moise, Edwin E},
  volume    = {47},
  year      = {2013},
  publisher = {Springer Science \& Business Media}
}

@article{genton2001classes,
  title   = {Classes of kernels for machine learning: {A} statistics perspective},
  author  = {Genton, Marc G},
  journal = {Journal of machine learning research},
  volume  = {2},
  number  = {Dec},
  pages   = {299--312},
  year    = {2001}
}

@misc{cagliari2010finitenessrankinvariantsmultidimensional,
  title         = {Finiteness of rank invariants of multidimensional persistent homology groups},
  author        = {Francesca Cagliari and Claudia Landi},
  year          = {2010},
  eprint        = {1001.0358},
  archiveprefix = {arXiv},
  primaryclass  = {math.AT},
  url           = {https://arxiv.org/abs/1001.0358}
}

@book{grafakos2008classical,
  title     = {Classical fourier analysis},
  author    = {Grafakos, Loukas and others},
  volume    = {2},
  year      = {2008},
  publisher = {Springer}
}

@book{weaver2018lipschitz,
  title     = {Lipschitz algebras},
  author    = {Weaver, Nik},
  year      = {2018},
  publisher = {World Scientific}
}

@book{lee2003smooth,
  title     = {Smooth manifolds},
  author    = {Lee, John M},
  year      = {2003},
  publisher = {Springer}
}

@book{milnor1963morse,
  title     = {Morse theory},
  author    = {Milnor, John Willard},
  number    = {51},
  year      = {1963},
  publisher = {Princeton university press}
}

@article{mclachlan1999mahalanobis,
  title   = {Mahalanobis distance},
  author  = {McLachlan, Goeffrey J},
  journal = {Resonance},
  volume  = {4},
  number  = {6},
  pages   = {20--26},
  year    = {1999}
}

@book{matsumoto2002introduction,
  title     = {An introduction to Morse theory},
  author    = {Matsumoto, Yukio},
  volume    = {208},
  year      = {2002},
  publisher = {American Mathematical Soc.}
}

@phdthesis{schmahl2023topics,
  title  = {Topics in Persistent Homology: {From} {Morse} Theory for Minimal Surfaces to Efficient Computation of Image Persistence},
  author = {Schmahl, Maximilian},
  year   = {2023},
  school = {Ruprecht Karl University of Heidelberg}
}

@article{cohen2009extending,
  title     = {Extending persistence using {P}oincar{\'e} and {L}efschetz duality},
  author    = {Cohen-Steiner, David and Edelsbrunner, Herbert and Harer, John},
  journal   = {Foundations of Computational Mathematics},
  volume    = {9},
  number    = {1},
  pages     = {79--103},
  year      = {2009},
  publisher = {Springer}
}

@book{hatcher2005algebraic,
  title     = {Algebraic topology},
  author    = {Hatcher, Allen},
  year      = {2002},
  publisher = {Cambridge: Cambridge University Press}
}

@book{munkres2000topology,
  title     = {Topology},
  author    = {Munkres, James R},
  year      = {2000},
  publisher = {Prentice Hall},
  edition   = {2nd}
}

@article{ott2005prevalence,
  title   = {Prevalence},
  author  = {Ott, William and Yorke, James},
  journal = {Bulletin of the American Mathematical Society},
  volume  = {42},
  number  = {3},
  pages   = {263--290},
  year    = {2005}
}