Topological Representation Research Articles

Stacked Topological Preserving Dynamic Brain Networks Representation and Classification.

In recent years, numerous studies have adopted rs-fMRI to construct dynamic functional connectivity networks (DFCNs) and applied them to the diagnosis of brain diseases, such as epilepsy and schizophrenia. Compared with the static brain networks, the DFCNs have a natural advantage in reflecting the process of brain activity due to the time information contained in it. However, most of the current methods for constructing DFCNs fail to aggregate the brain topology structure and temporal variation of the functional architecture associated with brain regions, and often ignore the inherent multi-dimensional feature representation of DFCNs for classification. In order to address these issues, we propose a novel DFCNs construction and representation method and apply it to brain disease diagnosis. Specifically, we fuse the blood oxygen level dependent (BOLD) signal and interactions between brain regions to distinguish the brain topology within each time domain and across different time domains, by embedding block structure in the adjacency matrix. After that, a sparse tensor decomposition method with sparse local structure preserving regularization is developed to extract DFCNs features from a multi-dimensional perspective. Finally, the kernel discriminant analysis is employed to provide the decision result. We validate the proposed method on epilepsy and schizophrenia identification tasks, respectively. The experimental results show that the proposed method outperforms several state-of-the-art methods in the diagnosis of brain diseases.

An Effective Topological Representation and Dimensionality Reduction Approach for Multi-Material Structural Topology Optimization

Abstract Topology optimization is among the most effective tools for innovative and lightweight structural designs. Multi-material design is considered to achieve better structural performance than single-material design. To significantly reduce the design space dimensionality and facilitate the optimization of multi-material structural design problems, this study proposes an effective topological representation and dimensionality reduction approach based on the material-field series expansion (MFSE) model. In the proposed method, a specified number of material phases is described within a single material field with a piecewise Heaviside projection function. The topology optimization problem is solved by determining the optimal MFSE coefficients. Owing to the single material-field topological description and series expansion, the number of design variables is independent of the finite element mesh and the number of material phases. In terms of dimensionality reduction, the proposed method outperformed all reported state-of-the-art algorithms for multi-material topology optimization. The validity and universality of the proposed method are illustrated in two- and three-dimensional numerical examples.

ROADMAP FOR INTEROPERABLE 3D DATA MODELS IN OGC APIS AND OTHER DATA EXCHANGE APPROACHES

Abstract. Many data exchange standards for 3D spatial data applications exist, ranging from the general Geography Markup Language (GML) underpinning CityGML to specific models for business application domains, such as BuildingSMART Industry Foundation Classes (BIM/IFC). There are a number of different approaches to modelling 3D objects, and in general the geometry aspects of these can be readily understood in the context of the visualisation needs of different applications. The topology, or relationships between elements of these objects, on the other hand is either not directly supported by such geometry models or implemented in different ways by different standards. We discuss limitations of existing standards for describing topological relationships in particular. In some cases topology information is embedded in geometry objects using identifiers for vertices, edges and faces, but in general there is scope to develop a standardised model for describing alternatives for topology and 3D geometry representations. A limited set of such models allows for interoperability via transformations between different representations. The ISO 19107 Spatial Schema provides an adequate conceptual model for these concerns, so we present the argument that a profile of this comprehensive model be defined for the limited set of such representation options required for Smart Cities and other similar applications.

CTTE: Customized Travel Time Estimation via Mobile Crowdsensing

Estimating the origin-destination travel time is a fundamental problem in many location-based services for vehicles, e.g., ride-hailing, vehicle dispatching, and route planning. Recent work has made significant progress to accuracy, but they largely rely on GPS trajectories which are too coarse to model many personalized driving behaviors, e.g., differentiating novice and veteran drivers. In this paper, we propose Customized Travel Time Estimation (CTTE) that fuses GPS trajectories, smartphone inertial data, and road network within a deep recurrent neural network. It constructs a road link traffic database with topology representation, speed statistics, and query distribution. It also calibrates inertial readings, estimates the arbitrary phone’s pose in car, and detects multiple aggressive driving events (e.g., bump judders, sharp turns, sharp slopes, frequent lane shifts, overspeeds, and sudden brakes). Finally, we demonstrate our solution on two typical transportation problems, i.e., predicting traffic speed at holistic level and estimating customized travel time at personal level, within a multi-task learning structure. Experiments on two large-scale real-world traffic datasets from DiDi platform show our effectiveness compared with the state-of-the-art.

SphereMap: Dynamic Multi-Layer Graph Structure for Rapid Safety-Aware UAV Planning

A flexible topological representation consisting of a two-layer graph structure built on-board an Unmanned Aerial Vehicle (UAV) by continuously filling the free space of an occupancy map with intersecting spheres is proposed in this \paper{}. Most state-of-the-art planning methods find the shortest paths while keeping the UAV at a pre-defined distance from obstacles. Planning over the proposed structure reaches this pre-defined distance only when necessary, maintaining a safer distance otherwise, while also being orders of magnitude faster than other state-of-the-art methods. Furthermore, we demonstrate how this graph representation can be converted into a lightweight shareable topological-volumetric map of the environment, which enables decentralized multi-robot cooperation. The proposed approach was successfully validated in several kilometers of real subterranean environments, such as caves, devastated industrial buildings, and in the harsh and complex setting of the final event of the DARPA SubT Challenge, which aims to mimic the conditions of real search and rescue missions as closely as possible, and where our approach achieved the \nth{2} place in the virtual track.

Predicting miRNA-disease associations based on multi-view information fusion.

MicroRNAs (miRNAs) play an important role in various biological processes and their abnormal expression could lead to the occurrence of diseases. Exploring the potential relationships between miRNAs and diseases can contribute to the diagnosis and treatment of complex diseases. The increasing databases storing miRNA and disease information provide opportunities to develop computational methods for discovering unobserved disease-related miRNAs, but there are still some challenges in how to effectively learn and fuse information from multi-source data. In this study, we propose a multi-view information fusion based method for miRNA-disease association (MDA)prediction, named MVIFMDA. Firstly, multiple heterogeneous networks are constructed by combining the known MDAs and different similarities of miRNAs and diseases based on multi-source information. Secondly, the topology features of miRNAs and diseases are obtained by using the graph convolutional network to each heterogeneous network view, respectively. Moreover, we design the attention strategy at the topology representation level to adaptively fuse representations including different structural information. Meanwhile, we learn the attribute representations of miRNAs and diseases from their similarity attribute views with convolutional neural networks, respectively. Finally, the complicated associations between miRNAs and diseases are reconstructed by applying a bilinear decoder to the combined features, which combine topology and attribute representations. Experimental results on the public dataset demonstrate that our proposed model consistently outperforms baseline methods. The case studies further show the ability of the MVIFMDA model for inferring underlying associations between miRNAs and diseases.

Topology optimization of thin-walled structures with directional straight stiffeners

Continuum topology optimization is a powerful method to find the optimal design of structures. However, it usually leads to complex nonregular configurations of stiffeners for thin-walled structures, which are difficult to manufacture using traditional production techniques. Therefore, this study proposes a novel topology representation and optimization method for thin-walled structures with directional straight stiffeners using the material-field series expansion and multifield superposition. The material field function with an anisotropic correlation is introduced to describe the straight stiffeners in a given direction, and the entire stiffeners’ topology is represented by the superposition of multiple material fields. Moreover, the stiffener's direction can be designed because of the inherent merit of separating the material field and the finite element mesh. The proposed method connects the characteristics of regular shape and topology of directional straight stiffeners into the optimization model without introducing additional manufacturing constraints. The optimization model is effectively solved using a gradient-based optimization algorithm incorporating the design sensitivity information. Several numerical examples are presented to demonstrate that the proposed method can successfully obtain the optimized topology while being easy to manufacture in stiffened structures design problems.

Formation of Machine Learning Features Based on the Construction of Tropical Functions

One of the main methods of computational topology and topological data analysis is persistent homology, which combines geometric and topological information about an object using persistent diagrams and barcodes. The persistent homology method from computational topology provides a balance between reducing the data dimension and characterizing the internal structure of an object. Combining machine learning and persistent homology is hampered by topological representations of data, distance metrics, and representation of data objects. The paper considers mathematical models and functions for representing persistent landscape objects based on the persistent homology method. The persistent landscape functions allow you to map persistent diagrams to Hilbert space. The representations of topological functions in various machine learning models are considered. An example of finding the distance between images based on the construction of persistent landscape functions is given. Based on the algebra of polynomials in the barcode space, which are used as coordinates, the distances in the barcode space are determined by comparing intervals from one barcode to another and calculating penalties. For these purposes, tropical functions are used that take into account the basic structure of the barcode space. Methods for constructing rational tropical functions are considered. An example of finding the distance between images based on the construction of tropical functions is given. To increase the variety of parameters (machine learning features), filtering of object scanning by rows from left to right and scanning by columns from bottom to top are built. This adds spatial information to topological information. The method of constructing persistent landscapes is compatible with the approach of constructing tropical rational functions when obtaining persistent homologies.

On the heuristic power of mathematical representations

I argue that mathematical representations can have heuristic power since their construction can be ampliative. To this end, I examine how a representation (a) introduces elements and properties into the represented object that it does not contain at the beginning of its construction, and (b) how it guides the manipulations of the represented object in ways that restructure its components by gradually adding new pieces of information to produce a hypothesis in order to solve a problem.In addition, I defend an ‘inferential’ approach to the heuristic power of representations by arguing that these representations draw on ampliative inferences such as analogies and inductions. In effect, in order to construct a representation, we have to ‘assimilate’ diverse things, and this requires identifying similarities between them. These similarities form the basis for ampliative inferences that gradually build hypotheses to solve a problem.To support my thesis, I analyse two examples. The first one is intra-field (intra-mathematical), that is, the construction of an algebraic representation of 3-manifolds; the second is inter-fields, that is, the construction of a topological representation of DNA supercoiling.

Large Spillover Networks of Nonstationary Systems

This article proposes a vector error correction framework for constructing large consistent spillover networks of nonstationary systems grounded in the network theory of Diebold and Y ilmaz. We aim to provide a tailored methodology for the large nonstationary (macro)economic and financial system application settings avoiding technical and often hard to verify assumptions for general statistical high-dimensional approaches where the dimension can also increase with sample size. To achieve this, we propose an elementwise Lasso-type technique for consistent and numerically efficient model selection of VECM, and relate the resulting forecast error variance decomposition to the network topology representation. We also derive the corresponding asymptotic results for model selection and network estimation under standard assumptions. Moreover, we develop a refinement strategy for efficient estimation and show implications and modifications for general dependent innovations. In a comprehensive simulation study, we show convincing finite sample performance of our technique in all cases of moderate and low dimensions. In an application to a system of FX rates, the proposed method leads to novel insights on the connectedness and spillover effects in the FX market among the OECD countries.

Computational Topology and its Applications in Geometric Design

Background: In recent geometric design, many effective toolkits for geometric modeling and optimization have been proposed and applied in practical cases, while effective and efficient designing of shapes that have desirable topological properties remains to be a challenge. The development of computational topology, especially persistent homology, permits convenient usage of topological invariants in shape analysis, geometric modeling, and shape optimization. Persistence diagram, the useful topological summary of persistent homology, provides a stable representation of multiscale homology invariants in the presence of noise in original data. Recent works show the wide use of persistent homology tools in geometric design. Objective: In this paper, we review the geometric design based on computational topological tools in three aspects: the extraction of topological features and representations, topology-aware shape modeling, and topology-based shape optimization. Methods: By tracking the development of each aspect and comparing the methods using classical topological invariants, motivations, and key approaches of important related works based on persistent homology are clarified. Results : We review geometric design through topological extraction, topological design, and shape optimization based on topology preservation. Related works show the successful applications of computational topology tools of geometric design. Conclusion: Solutions for the proposed core problems will affect the geometric design and its applications. In the future, the development of computational topology may boost computer-aided topological design.

Prediction of Drug-Related Diseases Through Integrating Pairwise Attributes and Neighbor Topological Structures.

Identifying new disease indications for the approved drugs can help reduce the cost and time of drug development. Most of the recent methods focus on exploiting the various information related to drugs and diseases for predicting the candidate drug-disease associations. However, the previous methods failed to deeply integrate the neighborhood topological structure and the node attributes of an interested drug-disease node pair. We propose a new prediction method, ANPred, to learn and integrate pairwise attribute information and neighbor topology information from the similarities and associations related to drugs and diseases. First, a bi-layer heterogeneous network with intra-layer and inter-layer connections is established to combine the drug similarities, the disease similarities, and the drug-disease associations. Second, the embedding of a pair of drug and disease is constructed based on integrating multiple biological premises about drugs and diseases. The learning framework based on multi-layer convolutional neural networks is designed to learn the attribute representation of the pair of drug and disease nodes from its embedding. The sequences composed of neighbor nodes are formed based on random walk on the heterogeneous network. A framework based on fully-connected autoencoder and skip-gram module is constructed to learn the neighbor topological representations of nodes. The cross-validation results indicate the performance of ANPred is superior to several state-of-the-art methods. The case studies on 5 drugs further confirm the ability of ANPred in discovering the potential drug-disease association candidates.

A Topological Machine Learning Pipeline for Classification

In this work, we develop a pipeline that associates Persistence Diagrams to digital data via the most appropriate filtration for the type of data considered. Using a grid search approach, this pipeline determines optimal representation methods and parameters. The development of such a topological pipeline for Machine Learning involves two crucial steps that strongly affect its performance: firstly, digital data must be represented as an algebraic object with a proper associated filtration in order to compute its topological summary, the Persistence Diagram. Secondly, the persistence diagram must be transformed with suitable representation methods in order to be introduced in a Machine Learning algorithm. We assess the performance of our pipeline, and in parallel, we compare the different representation methods on popular benchmark datasets. This work is a first step toward both an easy and ready-to-use pipeline for data classification using persistent homology and Machine Learning, and to understand the theoretical reasons why, given a dataset and a task to be performed, a pair (filtration, topological representation) is better than another.

ABLE: Automated Brain Lines Extraction Based on Laplacian Surface Collapse.

The archetypical folded shape of the human cortex has been a long-standing topic for neuroscientific research. Nevertheless, the accurate neuroanatomical segmentation of sulci remains a challenge. Part of the problem is the uncertainty of where a sulcus transitions into a gyrus and vice versa. This problem can be avoided by focusing on sulcal fundi and gyral crowns, which represent the topological opposites of cortical folding. We present Automated Brain Lines Extraction (ABLE), a method based on Laplacian surface collapse to reliably segment sulcal fundi and gyral crown lines. ABLE is built to work on standard FreeSurfer outputs and eludes the delineation of anastomotic sulci while maintaining sulcal fundi lines that traverse the regions with the highest depth and curvature. First, it segments the cortex into gyral and sulcal surfaces; then, each surface is spatially filtered. A Laplacian-collapse-based algorithm is applied to obtain a thinned representation of the surfaces. This surface is then used for careful detection of the endpoints of the lines. Finally, sulcal fundi and gyral crown lines are obtained by eroding the surfaces while preserving the connectivity between the endpoints. The method is validated by comparing ABLE with three other sulcal extraction methods using the Human Connectome Project (HCP) test-retest database to assess the reproducibility of the different tools. The results confirm ABLE as a reliable method for obtaining sulcal lines with an accurate representation of the sulcal topology while ignoring anastomotic branches and the overestimation of the sulcal fundi lines. ABLE is publicly available via https://github.com/HGGM-LIM/ABLE .

Elucidating dynamics and mechanism of cyclic bioreaction networks using topologically-equivalent electrical circuits

A novel methodology is established to convert a set of elementary chemical reactions into topologically-equivalent electrical circuits. The circuit analog is then simulated to elucidate the dynamics of complex cyclic bionetworks. The pyramidal reaction network catalyzed by the universal enzyme dihydrofolate reductase – which has important pharmacological and medical relevance in connection with cancer therapy – is used to illustrate the application of our model. The finite number of mechanisms, sub-cycles, or graphs constituting the cyclic network are analyzed, their contribution to the overall steady-state reaction rate determined, and the most probable mechanism is identified. The developed methodology is elegant, modular, and permits representation and visualization of network structure and topology at a glance. It can handle linear and nonlinear kinetics/rate laws, does not require simplifying assumptions such as use of the quasi-steady-state/Bodenstein approximation or the absence of nonlinear kinetic steps in the intermediates, and is suitable for coupling and integration with other flux-based reaction models.

Choice-free duality for orthocomplemented lattices by means of spectral spaces

The existing topological representation of an orthocomplemented lattice via the clopen orthoregular subsets of a Stone space depends upon Alexander’s Subbase Theorem, which asserts that a topological space X is compact if every subbasic open cover of X admits of a finite subcover. This is an easy consequence of the Ultrafilter Theorem—whose proof depends upon Zorn’s Lemma, which is well known to be equivalent to the Axiom of Choice. Within this work, we give a choice-free topological representation of orthocomplemented lattices by means of a special subclass of spectral spaces; choice-free in the sense that our representation avoids use of Alexander’s Subbase Theorem, along with its associated nonconstructive choice principles. We then introduce a new subclass of spectral spaces which we call upper Vietoris orthospaces in order to characterize up to homeomorphism (and isomorphism with respect to their orthospace reducts) the spectral spaces of proper lattice filters used in our representation. It is then shown how our constructions give rise to a choice-free dual equivalence of categories between the category of orthocomplemented lattices and the dual category of upper Vietoris orthospaces. Our duality combines Bezhanishvili and Holliday’s choice-free spectral space approach to Stone duality for Boolean algebras with Goldblatt and Bimbó’s choice-dependent orthospace approach to Stone duality for orthocomplemented lattices.

Patchworking oriented matroids

In a previous work, we gave a construction of (not necessarily realizable) oriented matroids from a triangulation of a product of two simplices. In this follow-up paper, we use a variant of Viro's patchworking to derive a topological representation of the oriented matroid directly from the polyhedral structure of the triangulation, hence finding a combinatorial manifestation of patchworking besides tropical algebraic geometry. We achieve this by rephrasing the patchworking procedure as a controlled cell merging process, guided by the structure of tropical oriented matroids. A key insight is a new promising technique to show that the final cell complex is regular.

Infusing common-sensical prior knowledge into topological representations of learning robots

Infusing common-sensical prior knowledge into topological representations of learning robots

Influence of interactions on Integer Quantum Hall Effect

Conductivity of Integer Quantum Hall Effect (IQHE) may be expressed as the topological invariant composed of the two-point Green function. Such a topological invariant is known both for the case of homogeneous systems with intrinsic Anomalous Quantum Hall Effect (AQHE) and for the case of IQHE in the inhomogeneous systems. In the latter case we may speak, for example, of the AQHE in the presence of elastic deformations and of the IQHE in presence of magnetic field. The topological invariant for the general case of inhomogeneous systems is expressed through the Wigner transformed Green functions and contains Moyal product. When it is reduced to the expression for the IQHE in the homogeneous systems the Moyal product is reduced to the ordinary one while the Wigner transformed Green function (defined in phase space) is reduced to the Green function in momentum space. Originally the mentioned above topological representation has been derived for the non-interacting systems. We demonstrate that in a wide range of different cases in the presence of interactions the Hall conductivity is given by the same expression, in which the noninteracting two-point Green function is substituted by the complete two-point Green function with the interactions taken into account. Several types of interactions are considered including the contact four-fermion interactions, Yukawa and Coulomb interactions. We present the complete proof of this statement up to the two loops, and argue that the similar result remains to all orders of perturbation theory. It is based on the incorporation of Wigner–Weyl calculus to the perturbation theory. We, therefore, formulate Feynmann rules of diagram technique in terms of the Wigner transformed propagators.

A new perspective on cooperative control of multi-agent systems through different types of graph Laplacians

In the field of systems and control, many cooperative control problems of multi-agent systems have been actively studied in the past two decades. This article aims to organize extensive existing work on different cooperative control problems into three categories, based on three different types of graph Laplacian matrices involved. A Laplacian matrix is an important representation of graph topology, and depending on the field of its entries, there are three types: ordinary Laplacian (non-negative diagonal entries and non-positive off-diagonal entries), signed Laplacian (arbitrary real entries), and complex Laplacian (arbitrary complex entries). Each type of graph Laplacian is useful in modeling and solving a different set of cooperative control problems. In particular, their algebraic properties are fundamental in characterizing stability and performance of the respective solution algorithms. To our best knowledge, organizing the literature on multi-agent cooperative control through the lens of different graph Laplacians is new.