Topological Data Structure Research Articles

Simulation of topological data using a stochastic model applied to protein engineering

Topological data analysis (TDA) is a powerful tool used to extract meaningful properties from the topological structure of data, enabling the discovery of complex relationships and patterns within datasets. It provides a unique perspective by focusing on the shape and structure of data rather than traditional statistical methods. In our research, we aim to explore the impact of stochastic processes on TDA. Specifically, we investigate how introducing stochastic calculations can enhance the analysis of topological features, offering a more nuanced understanding of the data. Stochastic processes account for randomness and variability, which are inherent in many real-world datasets. By integrating these processes into TDA, we can uncover more precise relationships between data points and obtain deeper insights into the underlying topological structures. Our work highlights the advantages of this approach, particularly in improving the robustness and accuracy of TDA in the presence of noise and variability. The results of this research have the potential to broaden the applicability of TDA in fields such as biology, neuroscience, and machine learning, where complex data structures are common, and stochastic behavior often plays a critical role in shaping the data’s characteristics.

Building a 4D Cell Complex Topology Through the Extrusion Process

Abstract. This paper is a step forward towards a construction of 4D cell-complex models implemented using a topological data structure, dual half-edge. This allows for explicit representation of spatial relations among objects, such as adjacency relationship, coincidence or order of construction elements, e.g. vertices and edges. A method based on extrusion is proposed. An original 2D model is extruded to 3D, where adjacent cells sharing the same face are directly linked. Then extrusion to 4D is performed. In this case individual 4D cells sharing the same 3D cell are linked together. A model is kept valid at any stage of the construction process and links among cells are automatically provided by the construction operators. Implementation is based on the 4D data structure, which is used at each stage of construction, even in intermediate models of lower dimensionality.

A vertex-centric representation for adaptive diamond-kite meshes

We describe a concise representation for adaptive diamond-kite meshes based solely on the vertices and their stars. The representation is exact because it uses only integers, is much smaller than standard topological data structures, and is highly compressible. All topological elements are reconstructed in expected constant time per element.

Generative Quantum Machine Learning via Denoising Diffusion Probabilistic Models.

Deep generative models are key-enabling technology to computer vision, text generation, and large language models. Denoising diffusion probabilistic models (DDPMs) have recently gained much attention due to their ability to generate diverse and high-quality samples in many computer vision tasks, as well as to incorporate flexible model architectures and a relatively simple training scheme. Quantum generative models, empowered by entanglement and superposition, have brought new insight to learning classical and quantum data. Inspired by the classical counterpart, we propose the quantum denoising diffusion probabilistic model (QuDDPM) to enable efficiently trainable generative learning of quantum data. QuDDPM adopts sufficient layers of circuits to guarantee expressivity, while it introduces multiple intermediate training tasks as interpolation between the target distribution and noise to avoid barren plateau and guarantee efficient training. We provide bounds on the learning error and demonstrate QuDDPM's capability in learning correlated quantum noise model, quantum many-body phases, and topological structure of quantum data. The results provide a paradigm for versatile and efficient quantum generative learning.

Construction And Performance Evaluation of Big Data Prediction Model Based on Fuzzy Clustering Algorithm in Cloud Computing Environment

In the evolving landscape of biomedical biometrics, where multimodal approaches are increasingly crucial for reliable user authentication, this research presents a comprehensive study. The primary focus is on the construction and performance evaluation of a robust big data prediction model within a cloud computing environment. The advent of big data and cloud computing has revolutionized the field of biomedical biometrics, offering immense potential for advanced data analysis and prediction. This research presents the development and evaluation of a robust prediction model for multimodal biometric data in biomedical applications. The proposed model incorporation of Reliable Discrete Variable Topology (RDVT) into the prediction model. RDVT introduces a topological data structure that enhances data reliability and ensures the integrity of multimodal biometric information. The construction and training of the prediction model are meticulously detailed, encompassing data preprocessing, feature extraction, clustering, classification, and model evaluation. Additionally, the integration of a fuzzy clustering algorithm enhances the model's ability to handle uncertainty and imprecision in biometric data. The advancement of multimodal biometrics in the biomedical field by introducing the Reliable Discrete Variable Topology (RDVT) and a big data prediction model based on a fuzzy clustering algorithm in a cloud computing environment. The model's performance is rigorously assessed through extensive experimentation, including accuracy, precision, recall, and F1-score measurements.

Assessing performance of spatial statistical stream network models in predicting alkalinity: A comparative study of two river basins in Türkiye

Developing and implementing water quality models that utilize sophisticated processes and plenty data in large river basins can be a challenging task. Factors such as sampling design, prediction at unsampled sites, and derivation of new spatial information in monitoring studies can restrict the methodologies applicable to water quality modeling. In recent years, spatial stream network (SSN) models, incorporating hydrologic distance and topological data structures, have emerged as a novel approach for predicting water quality in extensive study areas. These models have integrated conventional geostatistical methodologies and have demonstrated their effectiveness in analyzing water quality within large river basins. In this study, we aimed to investigate the feasibility of using SSN to predict alkalinity through a comparative analysis of two distinct river basins in Türkiye. We evaluated the prediction performance of SSN and the spatial characterization of the basins using spatial data sets as model components. The models are based on moving-average (MA) functions and constructed by evaluating alkalinity observations at 90 locations in Kucuk Menderes River Basin and 224 locations in the Coruh River Basin, using torgegram, correlogram, principal component analysis, and exhaustive search algorithm for covariate selection. In this research, we evaluated the optimal combination of covariates obtained from 20 different variables within the context of five spatial data set classes: topography, geology, climate, land use, and anthropology. The outcomes of the model indicate that impact of anthropogenic activities on autocorrelation based on hydrologic distance, impact of spatial covariates on variance, and root mean square prediction error (RMSPE) values of the predictions in Kucuk Menderes are more effective than those in the Coruh. The findings from the model predictions show that modeling water quality on stream network scale by using topological relationships and autocorrelation based on hydrologic distance can be recommended for rivers affected by complex and detailed processes.

PlayNet: real-time handball play classification with Kalman embeddings and neural networks

AbstractReal-time play recognition and classification algorithms are crucial for automating video production and live broadcasts of sporting events. However, current methods relying on human pose estimation and deep neural networks introduce high latency on commodity hardware, limiting their usability in low-cost real-time applications. We present PlayNet, a novel approach to real-time handball play classification. Our method is based on Kalman embeddings, a new low-dimensional representation for game states that enables efficient operation on commodity hardware and customized camera layouts. Firstly, we leverage Kalman filtering to detect and track the main agents in the playing field, allowing us to represent them in a single normalized coordinate space. Secondly, we utilize a neural network trained in nonlinear dimensionality reduction through fuzzy topological data structure analysis. As a result, PlayNet achieves real-time play classification with under 55 ms of latency on commodity hardware, making it a promising addition to automated live broadcasting and game analysis pipelines.

Painting with Data: Visually based open-source tool for geo-computing

In this article we present, the conceptual and technical background of a tool for mapping and geo-computing, Painting with Data (PWD). PWD was inspired by early computer mapping algorithms that focused on drawing as an intuitive interface between spatial data and architectural practice. PWD embraces the potential of such techniques, coupled with alternative computational representations, modes of interaction, and computational interfaces, to encourage public participation in planning and design. Essential to this task, is PWD’s integration of open-source, collaborative, interactive, and web-based technologies to create an online software with a visually based approach to spatial analysis and mapping that dramatically reduces the steep learning curve required for GIS software. As a high-level graphical interface, PWD allows users to iterate by intuitively creating spatial models on-the-fly based on their subjective understanding of information. In PWD, we deploy voxels, a data structure that organizes information as 3D pixels, which allows users to compute with spatial information visually, to potentially inform the ways in which users build quantitative models. In PWD, multiple layers of information can be visualized concurrently, and visual correlations can be made instantly. Users build spatial models by directly manipulating the map itself instead of manipulating a database that then produces a map. PWD’s high-level interaction is made possible by custom data structures that leverage GPU processing, which makes them significantly faster than traditional topological data structures. Computationally, user interactions generate visualization specifications and declarative queries that are compiled and executed by the platform. Lastly, PWD introduces a visual programming language, which enables intuitive geospatial modeling and visualization. Practical work has shown the value of PWD’s approach to design students, planning agencies and community non-profits. Throughout the process, PWD’s open-source spatial models generated a user community with more than 3000 users, including designers, students, and educators.

Tensorized topological graph learning for generalized incomplete multi-view clustering

The success of the current multi-view clustering lies in the default assumption of completeness on each view, while it is hardly satisfied for real-world applications. Incomplete multi-view clustering (IMC) studies clustering multi-view instances that are partially observed due to data corruption or sensor failure. Although making some progress, the current research suffers from the following obstacles: (1) Existing works are always built on pairwise similarity measurement, which cannot fully capture the topological structure of incomplete multi-view samples for precise clustering; (2) Due to the absence of partially-observed samples across multiple views, how could we recover and enhance the high-order similarities across different views becomes one of the main bottlenecks of IMC; (3) Current research mainly aims to handle specific dual-view IMC, which greatly limits the deployment in real-world applications. In this paper, we propose a novel Tensorized Topological Graph Learning (TTGL) for the generalized incomplete multi-view clustering, which jointly considers the topological graph construction, missing feature completion, and high-order uncertain correlation enhancement. Specifically, instead of using pairwise similarity measurement, we formulate a consensus topological graph construction module, which can coalesce affinity matrices to analyze the topological structure of incomplete multi-view data. Moreover, we build an iterative missing similarity imputation scheme for feature completion, which ensures the intrinsic similarity preservation on observed instances as well as an imputation on unobserved ones. Furthermore, a low-rank tensor structure is imposed to capture the high-order similarities via both feature and similarity enhancement and completion across different views. Our method can cluster incomplete data with an arbitrary number of views and any missing statuses. Extensive experiments on several benchmark datasets validate the effectiveness of our method when compared with a series of state-of-the-art IMC clustering algorithms. The source code of our TTGL is available at: https://github.com/DarrenZZhang/TTGL.

Shape reconstruction for undetectable regions of abdominal organs based on a graph convolutional network

Although computed tomography (CT) and magnetic resonance imaging (MRI) produce high-resolution images, during surgery or radiotherapy, only low-resolution cone-beam CT and low-dimensional X-ray images are usually obtained. Furthermore, because the duodenum and stomach are filled with air, it may be hard to accurately segment their contours, even on high-resolution images. Although denoising and image enhancement can be used to assist in the segmentation and reconstruction of 3D images, some regions of organs may remain difficult to detect. To approach this issue, we propose a graph convolutional network (GCN) to reconstruct organs that are hard to detect on medical images. The method uses a framework to transform an initial template into an estimated shape of the patient’s organs based on previous information attainted from different detectable organs from different patients. To overcome the inaccurate estimation results of triangular surface mesh data with the GCN, we propose a method to add internal vertices and edges, which has the effect of improving the calculation accuracy by enhancing the topological structure of the mesh data. In this study, the organ data of 124 patients who underwent pancreatic tumor treatment were used to verify the predictive effect of the proposed method. The 3D organ contours were defined by board-certified radiation oncologists and used as the ground truth surface meshes. For 100 randomly selected detectable feature vertices, the average Euclidean distance error of the liver was 3.72 mm. The topological enhancement proposed in this study reduced the prediction error of the liver by an average of 0.84 mm.

Surface network and drainage network: towards a common data structure

The surface network is an application of the Morse-Smale complex to digital terrain models connecting ridges and thalwegs of the terrain in a planar, undirected graph. Although it provides a topological structure embedding critical elements of the terrain, its application to morphological analysis and hydrology remains limited mainly because the drainage network is the most relevant structure for analysis and it cannot be derived from the surface network. The drainage network is a directed, hierarchical graph formed by streams. Ridges of the surface network are not equivalent to drainage divides, which are not contained in the drainage network, and there is no direct association between thalwegs and streams. Therefore, this paper proposes to extend the surface network into a new structure that also embeds the drainage network. This is done by (1) revising the definition of ridges so that they include drainage divides and (2) assigning a flow direction to each thalweg, taking into account spurious depressions to avoid flow interruption. We show that this extended surface network can be used to compute the flow accumulation and different hydrographic features such as drainage basins and the Strahler order. The drainage network extracted from the extended surface network is compared to drainage networks computed with the traditional D8 approach in three case studies. Differences remain minor and are mainly due to the elevation inaccuracy in flat or slightly convex areas. Hence, the extended surface network provides a richer data structure allowing the use of a common topological data structure in both terrain analysis and hydrology.

Hessian-based semi-supervised feature selection using generalized uncorrelated constraint

Feature selection (FS) aims to eliminate redundant features and choose the informative ones. Since labeled data are not always easily available and abundant unlabeled data are often accessible, the importance of semi-supervised FS (SSFS), which selects the informative features utilizing the labeled and unlabeled data, becomes apparent. Semi-supervised sparse FS based on graph Laplacian (GL) regularization has become a popular technique. The GL regularization biases the solution towards a constant geodesic function, has a poor extrapolating ability, and cannot preserve well the topological structure of data. Traditional ridge regression is widely utilized by the GL-based semi-supervised sparse FS, but the results cannot gain the closed-form solution and carry out the manifold structure. To tackle these problems, we propose a Hessian-based SSFS framework using the generalized uncorrelated constraint (HSFSGU) and the mixed ℓ2,p-norm (0<p≤1) regularization. We also propose a Hessian–Laplacian-based SSFS framework using the generalized uncorrelated constraint (HLSFSGU) which utilizes the combination of GL and Hessian matrices. Our frameworks use the Hessian regularization for maintaining the topological structure of data well, the ℓ2,p-norm regularization for making the projection matrix appropriate for FS, and the generalized uncorrelated constraint for preventing exceeding row sparsity of the projection matrix and providing the elegant closed-form solution for our frameworks. We also present a unified algorithm to solve the frameworks for convex and non-convex cases, and prove its convergence. The results of the experiments depict the ability of the Hessian-based frameworks under generalized uncorrelated constraint in selecting the informative features.

A Task-Parallel Approach for Localized Topological Data Structures.

Unstructured meshes are characterized by data points irregularly distributed in the Euclidian space. Due to the irregular nature of these data, computing connectivity information between the mesh elements requires much more time and memory than on uniformly distributed data. To lower storage costs, dynamic data structures have been proposed. These data structures compute connectivity information on the fly and discard them when no longer needed. However, on-the-fly computation slows down algorithms and results in a negative impact on the time performance. To address this issue, we propose a new task-parallel approach to proactively compute mesh connectivity. Unlike previous approaches implementing data-parallel models, where all threads run the same type of instructions, our task-parallel approach allows threads to run different functions. Specifically, some threads run the algorithm of choice while other threads compute connectivity information before they are actually needed. The approach was implemented in the new Accelerated Clustered TOPOlogical (ACTOPO) data structure, which can support any processing algorithm requiring mesh connectivity information. Our experiments show that ACTOPO combines the benefits of state-of-the-art memory-efficient (TTK CompactTriangulation) and time-efficient (TTK ExplicitTriangulation) topological data structures. It occupies a similar amount of memory as TTK CompactTriangulation while providing up to 5x speedup. Moreover, it achieves comparable time performance as TTK ExplicitTriangulation while using only half of the memory space.

DDSM: Design-Oriented Dual-Scale Shape-Material Model for Lattice Material Components.

This paper proposes a new CAD model for the design of lattice material components. The CAD model better captures the user's design intent and provides a dual-scale framework to represent the geometry and material distribution. Conventional CAD model formats based on B-Rep generate millions of data files, which also makes design intent and material information missing. In the present work, a new shape-material model for lattice material components is proposed. At the macroscopic scale, a compact face-based non-manifold topological data structure is proposed to express the lattice shape-material information without ambiguity. At the microscopic scale, implicit function is adopted for the representation of lattice material components. Numerical experiments verify that the proposed CAD model provides a powerful support for design intent with minor space costs. Meanwhile, the representation method supports solid modeling queries of geometric and material information on each scale.

Graph Spectral Regularized Tensor Completion for Traffic Data Imputation

In intelligent transportation systems (ITS), incomplete traffic data due to sensor malfunctions and communication faults, seriously restricts the related applications of ITS. Recovering missing data from incomplete traffic data becomes an important issue for ITS. Existing works on traffic data imputation cannot achieve satisfactory accuracy due to inefficiently exploiting the underlying topological structure of the traffic data. In this paper, we model the topology of the road network as a graph and introduce graph Fourier transform (GFT) to process the traffic data. Then we utilize an algebraic framework termed as graph-tensor singular value decompositions (GT-SVD) to extract the hidden spatial information of traffic data. Furthermore, we propose a novel graph spectral regularized tensor completion algorithm based on GT-SVD and construct temporal regularized constraints to improve the recovery accuracy. The extensive experimental results on real traffic datasets demonstrate that the proposed algorithm outperforms the state-of-the-art methods under different missing patterns.

Loop Order Analysis of Weft-Knitted Textiles

In this paper, we describe algorithms that perform loop order analysis of weft-knitted textiles, which build upon the foundational TopoKnit topological data structure and associated query functions. During knitting, loops of yarn may be overlayed on top of each other and then stitched together with another piece of yarn. Loop order analysis aims to determine the front-to-back ordering of these overlapping loops, given a stitch pattern that defines the knitted fabric. Loop order information is crucial for the simulation of electrical current, water, force, and heat flow within functional fabrics. The new algorithms are based on the assumption that stitch instructions are executed row-by-row and for each row the instructions can be executed in any temporal order. To make our algorithms knitting-machine-independent, loop order analysis utilizes precedence rules that capture the order that stitch commands are executed when a row of yarn loops are being knitted by a two-bed flat weft knitting machine. Basing the algorithms on precedence rules allows them to be modified to adapt to the analysis of fabrics manufactured on a variety of knitting machines that may execute stitch commands in different temporal orders. Additionally, we have developed visualization methods for displaying the loop order information within the context of a TopoKnit yarn topology graph.

A Novel 2D Clustering Algorithm Based on Recursive Topological Data Structure

In the field of data science and data mining, the problem associated with clustering features and determining its optimum number is still under research consideration. This paper presents a new 2D clustering algorithm based on a mathematical topological theory that uses a pseudometric space and takes into account the local and global topological properties of the data to be clustered. Taking into account cluster symmetry property, from a metric and mathematical-topological point of view, the analysis was carried out only in the positive region, reducing the number of calculations in the clustering process. The new clustering theory is inspired by the thermodynamics principle of energy. Thus, both topologies are recursively taken into account. The proposed model is based on the interaction of particles defined through measuring homogeneous-energy criterion. Based on the energy concept, both general and local topologies are taken into account for clustering. The effect of the integration of a new element into the cluster on homogeneous-energy criterion is analyzed. If the new element does not alter the homogeneous-energy of a group, then it is added; otherwise, a new cluster is created. The mathematical-topological theory and the results of its application on public benchmark datasets are presented.

Learning ordinal constraint binary codes for fast similarity search

Similarity search with hashing has become one of the fundamental research topics in computer vision and multimedia. The current researches on semantic-preserving hashing mainly focus on exploring the semantic similarities between pointwise or pairwise samples in the visual space to generate discriminative hash codes. However, such learning schemes fail to explore the intrinsic latent features embedded in the high-dimensional feature space and they are difficult to capture the underlying topological structure of data, yielding low-quality hash codes for image retrieval. In this paper, we propose an ordinal-preserving latent graph hashing (OLGH) method, which derives the objective hash codes from the latent space and preserves the high-order locally topological structure of data into the learned hash codes. Specifically, we conceive a triplet constrained topology-preserving loss to uncover the ordinal-inferred local features in binary representation learning. By virtue of this, the learning system can implicitly capture the high-order similarities among samples during the feature learning process. Moreover, the well-designed latent subspace learning is built to acquire the noise-free latent features based on the sparse constrained supervised learning. As such, the latent under-explored characteristics of data are fully employed in subspace construction. Furthermore, the latent ordinal graph hashing is formulated by jointly exploiting latent space construction and ordinal graph learning. An efficient optimization algorithm is developed to solve the resulting problem to achieve the optimal solution. Extensive experiments conducted on diverse datasets show the effectiveness and superiority of the proposed method when compared to some advanced learning to hash algorithms for fast image retrieval. The source codes of this paper are available at https://github.com/DarrenZZhang/OLGH .

Risk analysis of China’s stock markets based on topological data structures

We choose 100 stocks from China’s markets and use their daily returns from January 3, 2013 to August 31, 2020 to investigate the risk situation in China’s stock markets by exploring their correlations in the sample period. We build complexes and carry out topological data analysis on them. The persistence landscapes and their LP-norms show that there are three clear turbulent periods since 2013. The dates are then detected when the stocks are strongly correlated. As is well known, the financial risks easily break out and spread in such situations, so we call the dates critical dates for risks. We can also take them as the early warning signals for potential risks. We then construct planar maximal filtered graphs on the critical dates to help discover the systematically important companies. We find that they changed obviously in three different turbulent periods. It reminds us to analyze the risks’ characteristics of the risks and implement risk prevention. The method combing topological data analysis and complex networks is shown to be effective in detecting critical information from markets, and hence is worth popularizing.

A Vertex-Directed Evolutionary Algorithm for Multiobjective Endmember Estimation

Hyperspectral unmixing including endmember extration and abundance estimation has been investigated successively in recent years due to increasingly hyperspectral processing requirements. As one type of decision after solution paradigm, multiobjective endmember estimation method is able to obtain a set of Pareto optimal solutions, thus providing a wealth of information to determine the most representative endmembers. In addition, multiobjective optimization methods also have the characteristics of flexible modeling, excellent global convergence and commendable adaptability, etc. However, the evolutionary algorithms designed for this kind of method generally use little spatial-spectral information of the hyperspectral image. In this paper, we delve into the memetic strategy by exploiting the topological structure of hyperspectral data in the high-dimensional space to establish a vertex-directed multiobjective endmember estimation method, termed VD-MoEE. According to the frequently used linear mixture model, endmembers of hyperspectral images are generally distributed at the vertices of hyperspectral data manifold. Therefore, we design a vertex-directed local search operator to guide the search direction of the individuals in evolutionary algorithms. Experimental results on synthetic as well as real data sets demonstrated that the proposed VD-MOEE is able to achieve appealing performance in terms of the solution selecting and accuracy in comparison with several classic and state-of-the-art endmember estimation methods.