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Persistent Homology Captures the Generalization of Neural Networks Without A Validation Set [article]

Asier Gutiérrez-Fandiño, David Pérez-Fernández, Jordi Armengol-Estapé, Marta Villegas
2021 arXiv   pre-print
In this work, we suggest studying the training of neural networks with Algebraic Topology, specifically Persistent Homology (PH).  ...  Results show that the PH diagram distance between consecutive neural network states correlates with the validation accuracy, implying that the generalization error of a neural network could be intrinsically  ...  Watanabe and H. Yamana. Topological measurement of deep neural networks using persistent homology. In ISAIM, 2020. [30] S. Watanabe and H. Yamana.  ... 
arXiv:2106.00012v1 fatcat:hnp2myzmivgu5omqptwg66m7ru

Shapley Homology: Topological Analysis of Sample Influence for Neural Networks [article]

Kaixuan Zhang, Qinglong Wang, Xue Liu, C. Lee Giles
2019 arXiv   pre-print
Our empirical studies show that when using the 0-dimensional homology, on neighboring graphs, samples with higher influence scores have more impact on the accuracy of neural networks for determining the  ...  In this work, we study the influence of a sample on determining the intrinsic topological features of its underlying manifold.  ...  In a more general case, it is easy to show by the homology theory (Schapira, 2001 ) that once all subcomplex gets Betti number β K = 0 for some K, then for any other k-dimensional homology with k > K,  ... 
arXiv:1910.06509v1 fatcat:iufvubjivjfztlbabmh6ftdicy

TopologyNet: Topology based deep convolutional and multi-task neural networks for biomolecular property predictions

Zixuan Cang, Guo-Wei Wei, Roland L. Dunbrack
2017 PLoS Computational Biology  
We introduce topology, i.e., element specific persistent homology (ESPH), to untangle geometric complexity and biological complexity.  ...  We further integrate ESPH and convolutional neural networks to construct a multichannel topological neural network (TopologyNet) for the predictions of protein-ligand binding affinities and protein stability  ...  In this framework, element-specific persistent homology reduces geometric and biological complexities and provides a sufficient and structured low level representation for neural networks.  ... 
doi:10.1371/journal.pcbi.1005690 pmid:28749969 pmcid:PMC5549771 fatcat:ntcs3edkyreunor2lm6uamzrzu

Characterizing and Measuring the Similarity of Neural Networks with Persistent Homology [article]

David Pérez-Fernández and Asier Gutiérrez-Fandiño and Jordi Armengol-Estapé and Marta Villegas
2021 arXiv   pre-print
In this work, we observe that neural networks can be represented as abstract simplicial complex and analyzed using their topological 'fingerprints' via Persistent Homology (PH).  ...  Characterizing the structural properties of neural networks is crucial yet poorly understood, and there are no well-established similarity measures between networks.  ...  They show that the path homology of these networks is non-trivial in higher dimensions and depends on the number and size of the network layers.  ... 
arXiv:2101.07752v3 fatcat:rv42imvhira6lgade6chfa7rpe

A Survey of Topological Machine Learning Methods

Felix Hensel, Michael Moor, Bastian Rieck
2021 Frontiers in Artificial Intelligence  
such as deep neural networks.  ...  Next to their applications in the aforementioned areas, TDA methods have also proven to be effective in supporting, enhancing, and augmenting both classical machine learning and deep learning models.  ...  ., 2014) is comprised of two sub-networks, a generator and a discriminator.  ... 
doi:10.3389/frai.2021.681108 pmid:34124648 pmcid:PMC8187791 fatcat:r3d6yjc5hbf53n3rqwxiv54pxq

Characterizing the Shape of Activation Space in Deep Neural Networks [article]

Thomas Gebhart, Paul Schrater, Alan Hylton
2019 arXiv   pre-print
We introduce a method for computing persistent homology over the graphical activation structure of neural networks, which provides access to the task-relevant substructures activated throughout the network  ...  The representations learned by deep neural networks are difficult to interpret in part due to their large parameter space and the complexities introduced by their multi-layer structure.  ...  Figure 1 . 1 The neural network filtration and persistent subgraph reconstruction process.  ... 
arXiv:1901.09496v2 fatcat:a5cyzqstrbhfhlfl45u6762zfy

Shapley Homology: Topological Analysis of Sample Influence for Neural Networks

Kaixuan Zhang, Qinglong Wang, Xue Liu, C. Lee Giles
2020 Neural Computation  
Empirical studies show that when the zero-dimensional Shapley homology is used on neighboring graphs, samples with higher influence scores have a greater impact on the accuracy of neural networks that  ...  In this work, we study the influence of a sample on determining the intrinsic topological features of its underlying manifold.  ...  Acknowledgments We gratefully acknowledge useful comments from the referees and Christopher Griffin.  ... 
doi:10.1162/neco_a_01289 pmid:32433903 fatcat:k2mic7xs2ne4hnuy4f5frwjeh4

The importance of the whole: topological data analysis for the network neuroscientist [article]

Ann E. Sizemore, Jennifer Phillips-Cremins, Robert Ghrist, Danielle S. Bassett
2018 arXiv   pre-print
Finally we suggest avenues for future work and highlight new advances in mathematics that appear ready for use in revealing the architecture and function of neural systems.  ...  The application of network techniques to the analysis of neural data has greatly improved our ability to quantify and describe these rich interacting systems.  ...  Acknowledgments The authors would like to thank Richard Betzel, Harvey Huang, and Sunnia Chen for helpful discussions. D.S.B. and A.E.S. acknowledge support from the John D. and Catherine T.  ... 
arXiv:1806.05167v1 fatcat:3n7penfjxjdobdognbhsdmw42m

The importance of the whole: Topological data analysis for the network neuroscientist

Ann E. Sizemore, Jennifer E. Phillips-Cremins, Robert Ghrist, Danielle S. Bassett
2018 Network Neuroscience  
Finally, we suggest avenues for future work and highlight new advances in mathematics ready for use in neural systems.  ...  Data analysis techniques from network science have fundamentally improved our understanding of neural systems and the complex behaviors that they support.  ...  ACKNOWLEDGMENTS The authors would like to thank Richard Betzel, Harvey Huang, and Sunnia Chen for helpful discussions.  ... 
doi:10.1162/netn_a_00073 pmid:31410372 pmcid:PMC6663305 fatcat:svlizgxp4ne75hydwxtqibyn7e

Why Topology for Machine Learning and Knowledge Extraction?

Massimo Ferri
2018 Machine Learning and Knowledge Extraction  
Data has shape, and shape is the domain of geometry and in particular of its "free" part, called topology. The aim of this paper is twofold.  ...  Such interactions can be beneficial for both the generation of novel theoretical tools and finding cutting-edge practical applications.  ...  The funding sponsor had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, and in the decision to publish the results.  ... 
doi:10.3390/make1010006 dblp:journals/make/Ferri19 fatcat:fqhscd3tv5c63emycx46m3hmru

Atom-specific persistent homology and its application to protein flexibility analysis

David Bramer, Guo-Wei Wei
2019 Computational and Mathematical Biophysics  
Recently, persistent homology has had tremendous success in biomolecular data analysis.  ...  Atom-specific topological features are integrated with various machine learning algorithms, including gradient boosting trees and convolutional neural network for protein thermal fluctuation analysis and  ...  Acknowledgment: This work was supported in part by NSF Grants DMS-1721024, DMS-1761320, and IIS1900473 and NIH grant GM126189.  ... 
doi:10.1515/cmb-2020-0001 fatcat:qjgklbghwrhw5j5ftqbb2h6zgy

Simplicial complexes and complex systems

Vsevolod Salnikov, Daniele Cassese, Renaud Lambiotte
2018 European journal of physics  
The methods, based on notion of simplicial complexes, generalise standard network tools by naturally allowing for many-body interactions and providing results robust under continuous deformations of the  ...  Topological data analysis provides a set of tools to characterise the shape of data, in terms of the presence of holes or cavities between the points.  ...  In that case, each clique of size d in the hypergraph leads to the a simplex of dimension d − 1, as well as all its intrinsic simplicies.  ... 
doi:10.1088/1361-6404/aae790 fatcat:npivtygmurhgpgi3mbx3t42wyu

Dive into Layers: Neural Network Capacity Bounding using Algebraic Geometry [article]

Ji Yang and Lu Sang and Daniel Cremers
2021 arXiv   pre-print
To mathematically prove this, we borrow a tool in topological algebra: Betti numbers to measure the topological geometric complexity of input data and the neural network.  ...  By characterizing the expressive capacity of a neural network with its topological complexity, we conduct a thorough analysis and show that the network's expressive capacity is limited by the scale of  ...  Homological Structure of Neural Networks In this section, we will analyze the homology structure on each layer of a neural network.  ... 
arXiv:2109.01461v2 fatcat:o6hpr7ynjzc5vjkuf2icvp6tum

Topology of deep neural networks [article]

Gregory Naitzat, Andrey Zhitnikov, Lek-Heng Lim
2020 arXiv   pre-print
We performed extensive experiments on the persistent homology of a wide range of point cloud data sets, both real and simulated.  ...  , i.e., with perfect accuracy on training set and near-zero generalization error (≈ 0.01%).  ...  ), and the University of Chicago (Chicago-Vienna Faculty Grant and Eckhardt Faculty Fund).  ... 
arXiv:2004.06093v1 fatcat:xybs7clxtvgizdpmkoqttizgj4

Persistent Homology-Based Topological Analysis on the Gestalt Patterns during Human Brain Cognition Process

Zaisheng Liu, Fei Ni, Rongpeng Li, Honggang Zhang, Chang Liu, Jiefang Zhang, Songyun Xie, Siti Anom Ahmad
2021 Journal of Healthcare Engineering  
In general, this paper evaluates and quantifies cognitively related neural correlates by persistent homology features of EEG signals, which provides an approach to realizing the digitization of neural  ...  Specifically, we try to extract the physiologically meaningful features of the brain responding to different contours and shapes in images in Gestalt cognitive tests by combining persistent homology analysis  ...  Introduction In recent years, with the development of neural networks, researchers are committed to explaining the intrinsic nature of human consciousness generation and artificial intelligence (AI).  ... 
doi:10.1155/2021/2334332 pmid:34760139 pmcid:PMC8575602 fatcat:m3fzfka4ybdnfmt5k63glh7qxq
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