🍩 Database of Original & NonTheoretical Uses of Topology
(found 19 matches in 0.003418s)


Exploring the Geometry and Topology of Neural Network Loss Landscapes (2022)
Stefan Horoi, Jessie Huang, Bastian Rieck, Guillaume Lajoie, Guy Wolf, Smita KrishnaswamyAbstract
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 nonlinear dimensionality reduction of the jump and retrain trajectories via PHATE, a trajectory and manifoldpreserving 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. 
Topological Regularization for Dense Prediction (2021)
Deqing Fu, Bradley J. NelsonAbstract
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. 
TDANet: Fusion of Persistent Homology and Deep Learning Features for COVID19 Detection From Chest XRay Images (2021)
Mustafa Hajij, Ghada Zamzmi, Fawwaz BataynehAbstract
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 TDANet, a novel ensemble network that fuses topological and deep features for the purpose of enhancing model generalizability and accuracy. We apply the proposed TDANet to a critical application, which is the automated detection of COVID19 from CXR images. Experimental results showed that the proposed network achieved excellent performance and suggested the applicability of our method in practice. 
Determining Structural Properties of Artificial Neural Networks Using Algebraic Topology (2021)
David Pérez Fernández, Asier GutiérrezFandiño, Jordi ArmengolEstapé, Marta VillegasAbstract
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 subproblems (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. 
Simplicial Neural Networks (2020)
Stefania Ebli, Michaël Defferrard, Gard SpreemannAbstract
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 multidimensional extensions of graphs that encode not only pairwise relationships but also higherorder 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. 
PersGNN: Applying Topological Data Analysis and Geometric Deep Learning to StructureBased Protein Function Prediction (2020)
Nicolas Swenson, Aditi S. Krishnapriyan, Aydin Buluc, Dmitriy Morozov, Katherine YelickAbstract
Understanding protein structurefunction 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 structurebased prediction schemes. We present PersGNN  an endtoend 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. 
Cell Complex Neural Networks (2020)
Mustafa Hajij, Kyle Istvan, Ghada ZamzamiAbstract
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 networktype computations on cell complexes. Furthermore, we introduce intercellular 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. 
Graph Filtration Learning (2020)
Christoph Hofer, Florian Graf, Bastian Rieck, Marc Niethammer, Roland KwittAbstract
We propose an approach to learning with graphstructured data in the problem domain of graph classification. In particular, we present a novel type of readout operation to aggregate node features into a graphlevel representation. To this end, we leverage persistent homology computed via a realvalued, 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. 
Can Neural Networks Learn Persistent Homology Features? (2020)
Guido Montúfar, Nina Otter, Yuguang WangAbstract
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 oneparameter families of spaces associated with data, and persistence diagrams describe the lifetime of topological invariants, such as connected components or holes, across the oneparameter 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. 
PINet: A Deep Learning Approach to Extract Topological Persistence Images (2020)
Anirudh Som, Hongjun Choi, Karthikeyan Natesan Ramamurthy, Matthew Buman, Pavan TuragaAbstract
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 realworld data, such as viewpoint, 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 onestep approach to generate PIs directly from the input data. We design two separate convolutional neural network architectures, one designed to take in multivariate time series signals as input and another that accepts multichannel images as input. We call these networks Signal PINet 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 PINet architectures on two applications: human activity recognition using triaxial 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/PINet. 
Topologically Densified Distributions (2020)
Christoph Hofer, Florian Graf, Marc Niethammer, Roland KwittAbstract
We study regularization in the context of small samplesize learning with overparametrized 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. 
Classification of Skin Lesions by Topological Data Analysis Alongside With Neural Network (2020)
Naiereh Elyasi, Mehdi Hosseini MoghadamAbstract
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. 
A Topological Framework for Deep Learning (2020)
Mustafa Hajij, Kyle IstvanAbstract
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. 
AtomSpecific Persistent Homology and Its Application to Protein Flexibility Analysis (2020)
David Bramer, GuoWei WeiAbstract
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 Bfactor prediction. This work introduces atomspecific 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 atomspecific topological representation of individual atomic properties in a molecule. Atomspecific topological features are integrated with various machine learning algorithms, including gradient boosting trees and convolutional neural network for protein thermal fluctuation analysis and Bfactor prediction. Extensive numerical results indicate the proposed method provides a powerful topological tool for analyzing and predicting localized information in complex macromolecules. 
TopoResNet: A Hybrid Deep Learning Architecture and Its Application to Skin Lesion Classification (2019)
YuMin Chung, ChuanShen Hu, Austin Lawson, Clifford Smyth 
The Importance of the Whole: Topological Data Analysis for the Network Neuroscientist (2019)
Ann E. Sizemore, Jennifer E. PhillipsCremins, Robert Ghrist, Danielle S. BassettAbstract
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 algebrotopological 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. 
Connectivity in fMRI: Blind Spots and Breakthroughs (2018)
Victor Solo, JeanBaptiste Poline, Martin A. Lindquist, Sean L. Simpson, F. DuBois Bowman, Moo K. Chung, Ben CassidyAbstract
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, timevarying connectivity that distinguishes sample fluctuations from neural fluctuations and, network system identification that draws inferential strength from temporal autocorrelation. 
Representability of Algebraic Topology for Biomolecules in Machine Learning Based Scoring and Virtual Screening (2018)
Zixuan Cang, Lin Mu, GuoWei WeiAbstract
This work introduces a number of algebraic topology approaches, including multicomponent persistent homology, multilevel persistent homology, and electrostatic persistence for the representation, characterization, and description of small molecules and biomolecular complexes. In contrast to the conventional persistent homology, multicomponent persistent homology retains critical chemical and biological information during the topological simplification of biomolecular geometric complexity. Multilevel persistent homology enables a tailored topological description of inter and/or intramolecular 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 knearest neighbors, ensemble of trees, and deep convolutional neural networks, to manifest their descriptive and predictive powers for proteinligand binding analysis and virtual screening of small molecules. Extensive numerical experiments involving 4,414 proteinligand complexes from the PDBBind database and 128,374 ligandtarget and decoytarget 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 proteinligand binding affinity prediction and liganddecoy discrimination.