🍩 Database of Original & Non-Theoretical Uses of Topology

(found 21 matches in 0.002776s)

  1. Simplicial Neural Networks (2020)

    Stefania Ebli, Michaël Defferrard, Gard Spreemann
    Abstract We present simplicial neural networks (SNNs), a generalization of graph neural networks to data that live on a class of topological spaces called simplicial complexes. These are natural multi-dimensional extensions of graphs that encode not only pairwise relationships but also higher-order interactions between vertices - allowing us to consider richer data, including vector fields and \$n\$-fold collaboration networks. We define an appropriate notion of convolution that we leverage to construct the desired convolutional neural networks. We test the SNNs on the task of imputing missing data on coauthorship complexes.

  2. A Topological Framework for Deep Learning (2020)

    Mustafa Hajij, Kyle Istvan
    Abstract We utilize classical facts from topology to show that the classification problem in machine learning is always solvable under very mild conditions. Furthermore, we show that a softmax classification network acts on an input topological space by a finite sequence of topological moves to achieve the classification task. Moreover, given a training dataset, we show how topological formalism can be used to suggest the appropriate architectural choices for neural networks designed to be trained as classifiers on the data. Finally, we show how the architecture of a neural network cannot be chosen independently from the shape of the underlying data. To demonstrate these results, we provide example datasets and show how they are acted upon by neural nets from this topological perspective.

  3. CCF-GNN: A Unified Model Aggregating Appearance, Microenvironment, and Topology for Pathology Image Classification (2023)

    Hongxiao Wang, Gang Huang, Zhuo Zhao, Liang Cheng, Anna Juncker-Jensen, Máté Levente Nagy, Xin Lu, Xiangliang Zhang, Danny Z. Chen
    Abstract Pathology images contain rich information of cell appearance, microenvironment, and topology features for cancer analysis and diagnosis. Among such features, topology becomes increasingly important in analysis for cancer immunotherapy. By analyzing geometric and hierarchically structured cell distribution topology, oncologists can identify densely-packed and cancer-relevant cell communities (CCs) for making decisions. Compared to commonly-used pixel-level Convolution Neural Network (CNN) features and cell-instance-level Graph Neural Network (GNN) features, CC topology features are at a higher level of granularity and geometry. However, topological features have not been well exploited by recent deep learning (DL) methods for pathology image classification due to lack of effective topological descriptors for cell distribution and gathering patterns. In this paper, inspired by clinical practice, we analyze and classify pathology images by comprehensively learning cell appearance, microenvironment, and topology in a fine-to-coarse manner. To describe and exploit topology, we design Cell Community Forest (CCF), a novel graph that represents the hierarchical formulation process of big-sparse CCs from small-dense CCs. Using CCF as a new geometric topological descriptor of tumor cells in pathology images, we propose CCF-GNN, a GNN model that successively aggregates heterogeneous features (e.g., appearance, microenvironment) from cell-instance-level, cell-community-level, into image-level for pathology image classification. Extensive cross-validation experiments show that our method significantly outperforms alternative methods on H&E-stained; immunofluorescence images for disease grading tasks with multiple cancer types. Our proposed CCF-GNN establishes a new topological data analysis (TDA) based method, which facilitates integrating multi-level heterogeneous features of point clouds (e.g., for cells) into a unified DL framework.

  4. Connectivity in fMRI: Blind Spots and Breakthroughs (2018)

    Victor Solo, Jean-Baptiste Poline, Martin A. Lindquist, Sean L. Simpson, F. DuBois Bowman, Moo K. Chung, Ben Cassidy
    Abstract In recent years, driven by scientific and clinical concerns, there has been an increased interest in the analysis of functional brain networks. The goal of these analyses is to better understand how brain regions interact, how this depends upon experimental conditions and behavioral measures and how anomalies (disease) can be recognized. In this work we provide, firstly, a brief review of some of the main existing methods of functional brain network analysis. But rather than compare them, as a traditional review would do, instead, we draw attention to their significant limitations and blind spots. Then, secondly, relevant experts, sketch a number of emerging methods, which can break through these limitations. In particular we discuss five such methods. The first two, stochastic block models and exponential random graph models, provide an inferential basis for network analysis lacking in the exploratory graph analysis methods. The other three address: network comparison via persistent homology, time-varying connectivity that distinguishes sample fluctuations from neural fluctuations and, network system identification that draws inferential strength from temporal autocorrelation.

  5. PersGNN: Applying Topological Data Analysis and Geometric Deep Learning to Structure-Based Protein Function Prediction (2020)

    Nicolas Swenson, Aditi S. Krishnapriyan, Aydin Buluc, Dmitriy Morozov, Katherine Yelick
    Abstract Understanding protein structure-function relationships is a key challenge in computational biology, with applications across the biotechnology and pharmaceutical industries. While it is known that protein structure directly impacts protein function, many functional prediction tasks use only protein sequence. In this work, we isolate protein structure to make functional annotations for proteins in the Protein Data Bank in order to study the expressiveness of different structure-based prediction schemes. We present PersGNN - an end-to-end trainable deep learning model that combines graph representation learning with topological data analysis to capture a complex set of both local and global structural features. While variations of these techniques have been successfully applied to proteins before, we demonstrate that our hybridized approach, PersGNN, outperforms either method on its own as well as a baseline neural network that learns from the same information. PersGNN achieves a 9.3% boost in area under the precision recall curve (AUPR) compared to the best individual model, as well as high F1 scores across different gene ontology categories, indicating the transferability of this approach.

  7. Cell Complex Neural Networks (2020)

    Mustafa Hajij, Kyle Istvan, Ghada Zamzami
    Abstract Cell complexes are topological spaces constructed from simple blocks called cells. They generalize graphs, simplicial complexes, and polyhedral complexes that form important domains for practical applications. We propose a general, combinatorial, and unifying construction for performing neural network-type computations on cell complexes. Furthermore, we introduce inter-cellular message passing schemes, message passing schemes on cell complexes that take the topology of the underlying space into account. In particular, our method generalizes many of the most popular types of graph neural networks.

  8. Representability of Algebraic Topology for Biomolecules in Machine Learning Based Scoring and Virtual Screening (2018)

    Zixuan Cang, Lin Mu, Guo-Wei Wei
    Abstract This work introduces a number of algebraic topology approaches, including multi-component persistent homology, multi-level persistent homology, and electrostatic persistence for the representation, characterization, and description of small molecules and biomolecular complexes. In contrast to the conventional persistent homology, multi-component persistent homology retains critical chemical and biological information during the topological simplification of biomolecular geometric complexity. Multi-level persistent homology enables a tailored topological description of inter- and/or intra-molecular interactions of interest. Electrostatic persistence incorporates partial charge information into topological invariants. These topological methods are paired with Wasserstein distance to characterize similarities between molecules and are further integrated with a variety of machine learning algorithms, including k-nearest neighbors, ensemble of trees, and deep convolutional neural networks, to manifest their descriptive and predictive powers for protein-ligand binding analysis and virtual screening of small molecules. Extensive numerical experiments involving 4,414 protein-ligand complexes from the PDBBind database and 128,374 ligand-target and decoy-target pairs in the DUD database are performed to test respectively the scoring power and the discriminatory power of the proposed topological learning strategies. It is demonstrated that the present topological learning outperforms other existing methods in protein-ligand binding affinity prediction and ligand-decoy discrimination.

  9. Graph Filtration Learning (2020)

    Christoph Hofer, Florian Graf, Bastian Rieck, Marc Niethammer, Roland Kwitt
    Abstract We propose an approach to learning with graph-structured data in the problem domain of graph classification. In particular, we present a novel type of readout operation to aggregate node features into a graph-level representation. To this end, we leverage persistent homology computed via a real-valued, learnable, filter function. We establish the theoretical foundation for differentiating through the persistent homology computation. Empirically, we show that this type of readout operation compares favorably to previous techniques, especially when the graph connectivity structure is informative for the learning problem.

  10. Can Neural Networks Learn Persistent Homology Features? (2020)

    Guido Montúfar, Nina Otter, Yuguang Wang
    Abstract Topological data analysis uses tools from topology -- the mathematical area that studies shapes -- to create representations of data. In particular, in persistent homology, one studies one-parameter families of spaces associated with data, and persistence diagrams describe the lifetime of topological invariants, such as connected components or holes, across the one-parameter family. In many applications, one is interested in working with features associated with persistence diagrams rather than the diagrams themselves. In our work, we explore the possibility of learning several types of features extracted from persistence diagrams using neural networks.

  11. PI-Net: A Deep Learning Approach to Extract Topological Persistence Images (2020)

    Anirudh Som, Hongjun Choi, Karthikeyan Natesan Ramamurthy, Matthew Buman, Pavan Turaga
    Abstract Topological features such as persistence diagrams and their functional approximations like persistence images (PIs) have been showing substantial promise for machine learning and computer vision applications. This is greatly attributed to the robustness topological representations provide against different types of physical nuisance variables seen in real-world data, such as view-point, illumination, and more. However, key bottlenecks to their large scale adoption are computational expenditure and difficulty incorporating them in a differentiable architecture. We take an important step in this paper to mitigate these bottlenecks by proposing a novel one-step approach to generate PIs directly from the input data. We design two separate convolutional neural network architectures, one designed to take in multi-variate time series signals as input and another that accepts multi-channel images as input. We call these networks Signal PI-Net and Image PINet respectively. To the best of our knowledge, we are the first to propose the use of deep learning for computing topological features directly from data. We explore the use of the proposed PI-Net architectures on two applications: human activity recognition using tri-axial accelerometer sensor data and image classification. We demonstrate the ease of fusion of PIs in supervised deep learning architectures and speed up of several orders of magnitude for extracting PIs from data. Our code is available at https://github.com/anirudhsom/PI-Net.

  12. Topologically Densified Distributions (2020)

    Christoph Hofer, Florian Graf, Marc Niethammer, Roland Kwitt
    Abstract We study regularization in the context of small sample-size learning with over-parametrized neural networks. Specifically, we shift focus from architectural properties, such as norms on the network weights, to properties of the internal representations before a linear classifier. Specifically, we impose a topological constraint on samples drawn from the probability measure induced in that space. This provably leads to mass concentration effects around the representations of training instances, i.e., a property beneficial for generalization. By leveraging previous work to impose topological constrains in a neural network setting, we provide empirical evidence (across various vision benchmarks) to support our claim for better generalization.

  13. Classification of Skin Lesions by Topological Data Analysis Alongside With Neural Network (2020)

    Naiereh Elyasi, Mehdi Hosseini Moghadam
    Abstract In this paper we use TDA mapper alongside with deep convolutional neural networks in the classification of 7 major skin diseases. First we apply kepler mapper with neural network as one of its filter steps to classify the dataset HAM10000. Mapper visualizes the classification result by a simplicial complex, where neural network can not do this alone, but as a filter step neural network helps to classify data better. Furthermore we apply TDA mapper and persistent homology to understand the weights of layers of mobilenet network in different training epochs of HAM10000. Also we use persistent diagrams to visualize the results of analysis of layers of mobilenet network.

  14. Topological Regularization for Dense Prediction (2021)

    Deqing Fu, Bradley J. Nelson
    Abstract Dense prediction tasks such as depth perception and semantic segmentation are important applications in computer vision that have a concrete topological description in terms of partitioning an image into connected components or estimating a function with a small number of local extrema corresponding to objects in the image. We develop a form of topological regularization based on persistent homology that can be used in dense prediction tasks with these topological descriptions. Experimental results show that the output topology can also appear in the internal activations of trained neural networks which allows for a novel use of topological regularization to the internal states of neural networks during training, reducing the computational cost of the regularization. We demonstrate that this topological regularization of internal activations leads to improved convergence and test benchmarks on several problems and architectures.

  15. TDA-Net: Fusion of Persistent Homology and Deep Learning Features for COVID-19 Detection From Chest X-Ray Images (2021)

    Mustafa Hajij, Ghada Zamzmi, Fawwaz Batayneh
    Abstract Topological Data Analysis (TDA) has emerged recently as a robust tool to extract and compare the structure of datasets. TDA identifies features in data (e.g., connected components and holes) and assigns a quantitative measure to these features. Several studies reported that topological features extracted by TDA tools provide unique information about the data, discover new insights, and determine which feature is more related to the outcome. On the other hand, the overwhelming success of deep neural networks in learning patterns and relationships has been proven on various data applications including images. To capture the characteristics of both worlds, we propose TDA-Net, a novel ensemble network that fuses topological and deep features for the purpose of enhancing model generalizability and accuracy. We apply the proposed TDA-Net to a critical application, which is the automated detection of COVID-19 from CXR images. Experimental results showed that the proposed network achieved excellent performance and suggested the applicability of our method in practice.

  16. Emotion Recognition in Talking-Face Videos Using Persistent Entropy and Neural Networks (2022)

    Eduardo Paluzo-Hidalgo, Rocio Gonzalez-Diaz, Guillermo Aguirre-Carrazana, Eduardo Paluzo-Hidalgo, Rocio Gonzalez-Diaz, Guillermo Aguirre-Carrazana
    Abstract \textlessabstract\textgreater\textlessp\textgreaterThe automatic recognition of a person's emotional state has become a very active research field that involves scientists specialized in different areas such as artificial intelligence, computer vision, or psychology, among others. Our main objective in this work is to develop a novel approach, using persistent entropy and neural networks as main tools, to recognise and classify emotions from talking-face videos. Specifically, we combine audio-signal and image-sequence information to compute a \textlessitalic\textgreatertopology signature\textless/italic\textgreater (a 9-dimensional vector) for each video. We prove that small changes in the video produce small changes in the signature, ensuring the stability of the method. These topological signatures are used to feed a neural network to distinguish between the following emotions: calm, happy, sad, angry, fearful, disgust, and surprised. The results reached are promising and competitive, beating the performances achieved in other state-of-the-art works found in the literature.\textless/p\textgreater\textless/abstract\textgreater

  17. Determining Structural Properties of Artificial Neural Networks Using Algebraic Topology (2021)

    David Pérez Fernández, Asier Gutiérrez-Fandiño, Jordi Armengol-Estapé, Marta Villegas
    Abstract Artificial Neural Networks (ANNs) are widely used for approximating complex functions. The process that is usually followed to define the most appropriate architecture for an ANN given a specific function is mostly empirical. Once this architecture has been defined, weights are usually optimized according to the error function. On the other hand, we observe that ANNs can be represented as graphs and their topological 'fingerprints' can be obtained using Persistent Homology (PH). In this paper, we describe a proposal focused on designing more principled architecture search procedures. To do this, different architectures for solving problems related to a heterogeneous set of datasets have been analyzed. The results of the evaluation corroborate that PH effectively characterizes the ANN invariants: when ANN density (layers and neurons) or sample feeding order is the only difference, PH topological invariants appear; in the opposite direction in different sub-problems (i.e. different labels), PH varies. This approach based on topological analysis helps towards the goal of designing more principled architecture search procedures and having a better understanding of ANNs.

  18. Exploring the Geometry and Topology of Neural Network Loss Landscapes (2022)

    Stefan Horoi, Jessie Huang, Bastian Rieck, Guillaume Lajoie, Guy Wolf, Smita Krishnaswamy
    Abstract Recent work has established clear links between the generalization performance of trained neural networks and the geometry of their loss landscape near the local minima to which they converge. This suggests that qualitative and quantitative examination of the loss landscape geometry could yield insights about neural network generalization performance during training. To this end, researchers have proposed visualizing the loss landscape through the use of simple dimensionality reduction techniques. However, such visualization methods have been limited by their linear nature and only capture features in one or two dimensions, thus restricting sampling of the loss landscape to lines or planes. Here, we expand and improve upon these in three ways. First, we present a novel “jump and retrain” procedure for sampling relevant portions of the loss landscape. We show that the resulting sampled data holds more meaningful information about the network’s ability to generalize. Next, we show that non-linear dimensionality reduction of the jump and retrain trajectories via PHATE, a trajectory and manifold-preserving method, allows us to visualize differences between networks that are generalizing well vs poorly. Finally, we combine PHATE trajectories with a computational homology characterization to quantify trajectory differences.

  19. Atom-Specific Persistent Homology and Its Application to Protein Flexibility Analysis (2020)

    David Bramer, Guo-Wei Wei
    Abstract Recently, persistent homology has had tremendous success in biomolecular data analysis. It works by examining the topological relationship or connectivity of a group of atoms in a molecule at a variety of scales, then rendering a family of topological representations of the molecule. However, persistent homology is rarely employed for the analysis of atomic properties, such as biomolecular flexibility analysis or B-factor prediction. This work introduces atom-specific persistent homology to provide a local atomic level representation of a molecule via a global topological tool. This is achieved through the construction of a pair of conjugated sets of atoms and corresponding conjugated simplicial complexes, as well as conjugated topological spaces. The difference between the topological invariants of the pair of conjugated sets is measured by Bottleneck and Wasserstein metrics and leads to an atom-specific topological representation of individual atomic properties in a molecule. 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 B-factor prediction. Extensive numerical results indicate the proposed method provides a powerful topological tool for analyzing and predicting localized information in complex macromolecules.

  20. The Importance of the Whole: Topological Data Analysis for the Network Neuroscientist (2019)

    Ann E. Sizemore, Jennifer E. Phillips-Cremins, Robert Ghrist, Danielle S. Bassett
    Abstract Data analysis techniques from network science have fundamentally improved our understanding of neural systems and the complex behaviors that they support. Yet the restriction of network techniques to the study of pairwise interactions prevents us from taking into account intrinsic topological features such as cavities that may be crucial for system function. To detect and quantify these topological features, we must turn to algebro-topological methods that encode data as a simplicial complex built from sets of interacting nodes called simplices. We then use the relations between simplices to expose cavities within the complex, thereby summarizing its topological features. Here we provide an introduction to persistent homology, a fundamental method from applied topology that builds a global descriptor of system structure by chronicling the evolution of cavities as we move through a combinatorial object such as a weighted network. We detail the mathematics and perform demonstrative calculations on the mouse structural connectome, synapses in C. elegans, and genomic interaction data. Finally, we suggest avenues for future work and highlight new advances in mathematics ready for use in neural systems., For the network neuroscientist, this exposition aims to communicate both the mathematics and the advantages of using tools from applied topology for the study of neural systems. Using data from the mouse connectome, electrical and chemical synapses in C. elegans, and chromatin interaction data, we offer example computations and applications to further demonstrate the power of topological data analysis in neuroscience. Finally, we expose the reader to novel developments in applied topology and relate these developments to current questions and methodological difficulties in network neuroscience.

  21. Novel Production Prediction Model of Gasoline Production Processes for Energy Saving and Economic Increasing Based on AM-GRU Integrating the UMAP Algorithm (2023)

    Jintao Liu, Liangchao Chen, Wei Xu, Mingfei Feng, Yongming Han, Tao Xia, Zhiqiang Geng
    Abstract Gasoline, as an extremely important petroleum product, is of great significance to ensure people's living standards and maintain national energy security. In the actual gasoline industrial production environment, the point information collected by industrial devices usually has the characteristics of high dimension, high noise and time series because of the instability of manual operation and equipment operation. Therefore, it is difficult to use the traditional method to predict and optimize gasoline production. In this paper, a novel production prediction model using an attention mechanism (AM) based gated recurrent unit (GRU) (AM-GRU) integrating the uniform manifold approximation and projection (UMAP) is proposed. The data collected in the industrial plant are processed by the box plot to remove the data outside the quartile. Then, the UMAP is used to remove the strong correlation between the data, which can improve the running speed and the performance of the AM-GRU. Compared with the existing time series data prediction method, the superiority of the AM-GRU is verified based on University of California Irvine (UCI) benchmark datasets. Finally, the production prediction model of actual complex gasoline production processes for energy saving and economic increasing based on the proposed method is built. The experiment results show that compared with other time series data prediction models, the proposed model has better stability and higher accuracy with reaching 0.4171, 0.9969, 0.2538 and 0.5038 in terms of the mean squared error, the average absolute accuracy, the mean squared error and the root mean square error. Moreover, according to the optimal scheme of the raw material, the inefficiency production points can be expected to increase about 0.69 tons of the gasoline yield and between about \$645.1 and \$925.6 of economic benefits of industrial production.