Multivariate Data Analysis Using Persistence-Based Filtering and Topological Signatures

A Survey of Designs for Combined 2D+3D Visual Representations

We examine visual representations of data that make use of combinations of both 2D and 3D data mappings. Combining 2D and 3D representations is a common technique that allows viewers to understand multiple facets of the data with which they are interacting. While 3D representations focus on the spatial character of the data or the dedicated 3D data mapping, 2D representations often show abstract data properties and take advantage of the unique benefits of mapping to a plane. Many systems have used unique combinations of both types of data mappings effectively. Yet there are no systematic reviews of the methods in linking 2D and 3D representations. We systematically survey the relationships between 2D and 3D visual representations in major visualization publications-IEEE VIS, IEEE TVCG, and EuroVis-from 2012 to 2022. We closely examined 105 articles where 2D and 3D representations are connected visually, interactively, or through animation. These approaches are designed based on their visual environment, the relationships between their visual representations, and their possible layouts. Through our analysis, we introduce a design space as well as provide design guidelines for effectively linking 2D and 3D visual representations.

AffectiveTDA: Using Topological Data Analysis to Improve Analysis and Explainability in Affective Computing

We present an approach utilizing Topological Data Analysis to study the structure of face poses used in affective computing, i.e., the process of recognizing human emotion. The approach uses a conditional comparison of different emotions, both respective and irrespective of time, with multiple topological distance metrics, dimension reduction techniques, and face subsections (e.g., eyes, nose, mouth, etc.). The results confirm that our topology-based approach captures known patterns, distinctions between emotions, and distinctions between individuals, which is an important step towards more robust and explainable emotion recognition by machines.

Attribute-based Explanation of Non-Linear Embeddings of High-Dimensional Data

Embeddings of high-dimensional data are widely used to explore data, to verify analysis results, and to communicate information. Their explanation, in particular with respect to the input attributes, is often difficult. With linear projects like PCA the axes can still be annotated meaningfully. With non-linear projections this is no longer possible and alternative strategies such as attribute-based color coding are required. In this paper, we review existing augmentation techniques and discuss their limitations. We present the Non-Linear Embeddings Surveyor (NoLiES) that combines a novel augmentation strategy for projected data (rangesets) with interactive analysis in a small multiples setting. Rangesets use a set-based visualization approach for binned attribute values that enable the user to quickly observe structure and detect outliers. We detail the link between algebraic topology and rangesets and demonstrate the utility of NoLiES in case studies with various challenges (complex attribute value distribution, many attributes, many data points) and a real-world application to understand latent features of matrix completion in thermodynamics.

Understanding Changes in the Topology and Geometry of Financial Market Correlations during a Market Crash

In econophysics, the achievements of information filtering methods over the past 20 years, such as the minimal spanning tree (MST) by Mantegna and the planar maximally filtered graph (PMFG) by Tumminello et al., should be celebrated. Here, we show how one can systematically improve upon this paradigm along two separate directions. First, we used topological data analysis (TDA) to extend the notions of nodes and links in networks to faces, tetrahedrons, or k-simplices in simplicial complexes. Second, we used the Ollivier-Ricci curvature (ORC) to acquire geometric information that cannot be provided by simple information filtering. In this sense, MSTs and PMFGs are but first steps to revealing the topological backbones of financial networks. This is something that TDA can elucidate more fully, following which the ORC can help us flesh out the geometry of financial networks. We applied these two approaches to a recent stock market crash in Taiwan and found that, beyond fusions and fissions, other non-fusion/fission processes such as cavitation, annihilation, rupture, healing, and puncture might also be important. We also successfully identified neck regions that emerged during the crash, based on their negative ORCs, and performed a case study on one such neck region.

Weighted persistent homology for osmolyte molecular aggregation and hydrogen-bonding network analysis

It has long been observed that trimethylamine N-oxide (TMAO) and urea demonstrate dramatically different properties in a protein folding process. Even with the enormous theoretical and experimental research work on these two osmolytes, various aspects of their underlying mechanisms still remain largely elusive. In this paper, we propose to use the weighted persistent homology to systematically study the osmolytes molecular aggregation and their hydrogen-bonding network from a local topological perspective. We consider two weighted models, i.e., localized persistent homology (LPH) and interactive persistent homology (IPH). Boltzmann persistent entropy (BPE) is proposed to quantitatively characterize the topological features from LPH and IPH, together with persistent Betti number (PBN). More specifically, from the localized persistent homology models, we have found that TMAO and urea have very different local topology. TMAO is found to exhibit a local network structure. With the concentration increase, the circle elements in these networks show a clear increase in their total numbers and a decrease in their relative sizes. In contrast, urea shows two types of local topological patterns, i.e., local clusters around 6 Å and a few global circle elements at around 12 Å. From the interactive persistent homology models, it has been found that our persistent radial distribution function (PRDF) from the global-scale IPH has same physical properties as the traditional radial distribution function. Moreover, PRDFs from the local-scale IPH can also be generated and used to characterize the local interaction information. Other than the clear difference of the first peak value of PRDFs at filtration size 4 Å, TMAO and urea also shows very different behaviors at the second peak region from filtration size 5 Å to 10 Å. These differences are also reflected in the PBNs and BPEs of the local-scale IPH. These localized topological information has never been revealed before. Since graphs can be transferred into simplicial complexes by the clique complex, our weighted persistent homology models can be used in the analysis of various networks and graphs from any molecular structures and aggregation systems.

Gait Rhythm Dynamics for Neuro-Degenerative Disease Classification via Persistence Landscape- Based Topological Representation

Neuro-degenerative disease is a common progressive nervous system disorder that leads to serious clinical consequences. Gait rhythm dynamics analysis is essential for evaluating clinical states and improving quality of life for neuro-degenerative patients. The magnitude of stride-to-stride fluctuations and corresponding changes over time-gait dynamics-reflects the physiology of gait, in quantifying the pathologic alterations in the locomotor control system of health subjects and patients with neuro-degenerative diseases. Motivated by algebra topology theory, a topological data analysis-inspired nonlinear framework was adopted in the study of the gait dynamics. Meanwhile, the topological representation-persistence landscapes were used as input of classifiers in order to distinguish different neuro-degenerative disease type from healthy. In this work, stride-to-stride time series from healthy control (HC) subjects are compared with the gait dynamics from patients with amyotrophic lateral sclerosis (ALS), Huntington's disease (HD), and Parkinson's disease (PD). The obtained results show that the proposed methodology discriminates healthy subjects from subjects with other neuro-degenerative diseases with relatively high accuracy. In summary, our study is the first attempt to provide a topological representation-based method into the disease classification with gait rhythms measured from the stride intervals to visualize gait dynamics and classify neuro-degenerative diseases. The proposed method could be potentially used in earlier interventions and state monitoring.

Weighted persistent homology for biomolecular data analysis

In this paper, we systematically review weighted persistent homology (WPH) models and their applications in biomolecular data analysis. Essentially, the weight value, which reflects physical, chemical and biological properties, can be assigned to vertices (atom centers), edges (bonds), or higher order simplexes (cluster of atoms), depending on the biomolecular structure, function, and dynamics properties. Further, we propose the first localized weighted persistent homology (LWPH). Inspired by the great success of element specific persistent homology (ESPH), we do not treat biomolecules as an inseparable system like all previous weighted models, instead we decompose them into a series of local domains, which may be overlapped with each other. The general persistent homology or weighted persistent homology analysis is then applied on each of these local domains. In this way, functional properties, that are embedded in local structures, can be revealed. Our model has been applied to systematically study DNA structures. It has been found that our LWPH based features can be used to successfully discriminate the A-, B-, and Z-types of DNA. More importantly, our LWPH based principal component analysis (PCA) model can identify two configurational states of DNA structures in ion liquid environment, which can be revealed only by the complicated helical coordinate system. The great consistence with the helical-coordinate model demonstrates that our model captures local structure variations so well that it is comparable with geometric models. Moreover, geometric measurements are usually defined in local regions. For instance, the helical-coordinate system is limited to one or two basepairs. However, our LWPH can quantitatively characterize structure information in regions or domains with arbitrary sizes and shapes, where traditional geometrical measurements fail.

Persistent homology analysis of osmolyte molecular aggregation and their hydrogen-bonding networks

Dramatically different patterns can be observed in the topological fingerprints for hydrogen-bonding networks from two types of osmolyte systems.

Persistent homology analysis of ion aggregations and hydrogen-bonding networks

Despite the great advancement of experimental tools and theoretical models, a quantitative characterization of the microscopic structures of ion aggregates and their associated water hydrogen-bonding networks still remains a challenging problem. In this paper, a newly-invented mathematical method called persistent homology is introduced, for the first time, to quantitatively analyze the intrinsic topological properties of ion aggregation systems and hydrogen-bonding networks. The two most distinguishable properties of persistent homology analysis of assembly systems are as follows. First, it does not require a predefined bond length to construct the ion or hydrogen-bonding network. Persistent homology results are determined by the morphological structure of the data only. Second, it can directly measure the size of circles or holes in ion aggregates and hydrogen-bonding networks. To validate our model, we consider two well-studied systems, i.e., NaCl and KSCN solutions, generated from molecular dynamics simulations. They are believed to represent two morphological types of aggregation, i.e., local clusters and extended ion networks. It has been found that the two aggregation types have distinguishable topological features and can be characterized by our topological model very well. Further, we construct two types of networks, i.e., O-networks and H2O-networks, for analyzing the topological properties of hydrogen-bonding networks. It is found that for both models, KSCN systems demonstrate much more dramatic variations in their local circle structures with a concentration increase. A consistent increase of large-sized local circle structures is observed and the sizes of these circles become more and more diverse. In contrast, NaCl systems show no obvious increase of large-sized circles. Instead a consistent decline of the average size of the circle structures is observed and the sizes of these circles become more and more uniform with a concentration increase. As far as we know, these unique intrinsic topological features in ion aggregation systems have never been pointed out before. More importantly, our models can be directly used to quantitatively analyze the intrinsic topological invariants, including circles, loops, holes, and cavities, of any network-like structures, such as nanomaterials, colloidal systems, biomolecular assemblies, among others. These topological invariants cannot be described by traditional graph and network models.

Representability of algebraic topology for biomolecules in machine learning based scoring and virtual screening

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.

Multiscale Persistent Functions for Biomolecular Structure Characterization

In this paper, we introduce multiscale persistent functions for biomolecular structure characterization. The essential idea is to combine our multiscale rigidity functions (MRFs) with persistent homology analysis, so as to construct a series of multiscale persistent functions, particularly multiscale persistent entropies, for structure characterization. To clarify the fundamental idea of our method, the multiscale persistent entropy (MPE) model is discussed in great detail. Mathematically, unlike the previous persistent entropy (Chintakunta et al. in Pattern Recognit 48(2):391-401, 2015; Merelli et al. in Entropy 17(10):6872-6892, 2015; Rucco et al. in: Proceedings of ECCS 2014, Springer, pp 117-128, 2016), a special resolution parameter is incorporated into our model. Various scales can be achieved by tuning its value. Physically, our MPE can be used in conformational entropy evaluation. More specifically, it is found that our method incorporates in it a natural classification scheme. This is achieved through a density filtration of an MRF built from angular distributions. To further validate our model, a systematical comparison with the traditional entropy evaluation model is done. It is found that our model is able to preserve the intrinsic topological features of biomolecular data much better than traditional approaches, particularly for resolutions in the intermediate range. Moreover, by comparing with traditional entropies from various grid sizes, bond angle-based methods and a persistent homology-based support vector machine method (Cang et al. in Mol Based Math Biol 3:140-162, 2015), we find that our MPE method gives the best results in terms of average true positive rate in a classic protein structure classification test. More interestingly, all-alpha and all-beta protein classes can be clearly separated from each other with zero error only in our model. Finally, a special protein structure index (PSI) is proposed, for the first time, to describe the "regularity" of protein structures. Basically, a protein structure is deemed as regular if it has a consistent and orderly configuration. Our PSI model is tested on a database of 110 proteins; we find that structures with larger portions of loops and intrinsically disorder regions are always associated with larger PSI, meaning an irregular configuration, while proteins with larger portions of secondary structures, i.e., alpha-helix or beta-sheet, have smaller PSI. Essentially, PSI can be used to describe the "regularity" information in any systems.

Visualizing High-Dimensional Data: Advances in the Past Decade

Massive simulations and arrays of sensing devices, in combination with increasing computing resources, have generated large, complex, high-dimensional datasets used to study phenomena across numerous fields of study. Visualization plays an important role in exploring such datasets. We provide a comprehensive survey of advances in high-dimensional data visualization that focuses on the past decade. We aim at providing guidance for data practitioners to navigate through a modular view of the recent advances, inspiring the creation of new visualizations along the enriched visualization pipeline, and identifying future opportunities for visualization research.

Visualization and Extraction of Carvings for Heritage Conservation

We present novel techniques for visualizing, illustrating, analyzing, and generating carvings in surfaces. In particular, we consider the carvings in the plaster of the cloister of the Magdeburg cathedral, which dates to the 13th century. Due to aging and weathering, the carvings have flattened. Historians and restorers are highly interested in using digitalization techniques to analyze carvings in historic artifacts and monuments and to get impressions and illustrations of their original shape and appearance. Moreover, museums and churches are interested in such illustrations for presenting them to visitors. The techniques that we propose allow for detecting, selecting, and visualizing carving structures. In addition, we introduce an example-based method for generating carvings. The resulting tool, which integrates all techniques, was evaluated by three experienced restorers to assess the usefulness and applicability. Furthermore, we compared our approach with exaggerated shading and other state-of-the-art methods.

Object-oriented Persistent Homology

Persistent homology provides a new approach for the topological simplification of big data via measuring the life time of intrinsic topological features in a filtration process and has found its success in scientific and engineering applications. However, such a success is essentially limited to qualitative data classification and analysis. Indeed, persistent homology has rarely been employed for quantitative modeling and prediction. Additionally, the present persistent homology is a passive tool, rather than a proactive technique, for classification and analysis. In this work, we outline a general protocol to construct object-oriented persistent homology methods. By means of differential geometry theory of surfaces, we construct an objective functional, namely, a surface free energy defined on the data of interest. The minimization of the objective functional leads to a Laplace-Beltrami operator which generates a multiscale representation of the initial data and offers an objective oriented filtration process. The resulting differential geometry based object-oriented persistent homology is able to preserve desirable geometric features in the evolutionary filtration and enhances the corresponding topological persistence. The cubical complex based homology algorithm is employed in the present work to be compatible with the Cartesian representation of the Laplace-Beltrami flow. The proposed Laplace-Beltrami flow based persistent homology method is extensively validated. The consistence between Laplace-Beltrami flow based filtration and Euclidean distance based filtration is confirmed on the Vietoris-Rips complex for a large amount of numerical tests. The convergence and reliability of the present Laplace-Beltrami flow based cubical complex filtration approach are analyzed over various spatial and temporal mesh sizes. The Laplace-Beltrami flow based persistent homology approach is utilized to study the intrinsic topology of proteins and fullerene molecules. Based on a quantitative model which correlates the topological persistence of fullerene central cavity with the total curvature energy of the fullerene structure, the proposed method is used for the prediction of fullerene isomer stability. The efficiency and robustness of the present method are verified by more than 500 fullerene molecules. It is shown that the proposed persistent homology based quantitative model offers good predictions of total curvature energies for ten types of fullerene isomers. The present work offers the first example to design object-oriented persistent homology to enhance or preserve desirable features in the original data during the filtration process and then automatically detect or extract the corresponding topological traits from the data.

A topological approach for protein classification

Protein function and dynamics are closely related to its sequence and structure.However, prediction of protein function and dynamics from its sequence and structure is still a fundamental challenge in molecular biology. Protein classification, which is typically done through measuring the similarity between proteins based on protein sequence or physical information, serves as a crucial step toward the understanding of protein function and dynamics. Persistent homology is a new branch of algebraic topology that has found its success in the topological data analysis in a variety of disciplines, including molecular biology. The present work explores the potential of using persistent homology as an independent tool for protein classification. To this end, we propose a molecular topological fingerprint based support vector machine (MTF-SVM) classifier. Specifically,we construct machine learning feature vectors solely fromprotein topological fingerprints,which are topological invariants generated during the filtration process. To validate the presentMTF-SVMapproach, we consider four types of problems. First, we study protein-drug binding by using the M2 channel protein of influenza A virus. We achieve 96% accuracy in discriminating drug bound and unbound M2 channels. Secondly, we examine the use of MTF-SVM for the classification of hemoglobin molecules in their relaxed and taut forms and obtain about 80% accuracy. Thirdly, the identification of all alpha, all beta, and alpha-beta protein domains is carried out using 900 proteins.We have found a 85% success in this identification. Finally, we apply the present technique to 55 classification tasks of protein superfamilies over 1357 samples and 246 tasks over 11944 samples. Average accuracies of 82% and 73% are attained. The present study establishes computational topology as an independent and effective alternative for protein classification.

Multiresolution persistent homology for excessively large biomolecular datasets

Although persistent homology has emerged as a promising tool for the topological simplification of complex data, it is computationally intractable for large datasets. We introduce multiresolution persistent homology to handle excessively large datasets. We match the resolution with the scale of interest so as to represent large scale datasets with appropriate resolution. We utilize flexibility-rigidity index to access the topological connectivity of the data set and define a rigidity density for the filtration analysis. By appropriately tuning the resolution of the rigidity density, we are able to focus the topological lens on the scale of interest. The proposed multiresolution topological analysis is validated by a hexagonal fractal image which has three distinct scales. We further demonstrate the proposed method for extracting topological fingerprints from DNA molecules. In particular, the topological persistence of a virus capsid with 273 780 atoms is successfully analyzed which would otherwise be inaccessible to the normal point cloud method and unreliable by using coarse-grained multiscale persistent homology. The proposed method has also been successfully applied to the protein domain classification, which is the first time that persistent homology is used for practical protein domain analysis, to our knowledge. The proposed multiresolution topological method has potential applications in arbitrary data sets, such as social networks, biological networks, and graphs.

Persistent topology for cryo-EM data analysis

In this work, we introduce persistent homology for the analysis of cryo-electron microscopy (cryo-EM) density maps. We identify the topological fingerprint or topological signature of noise, which is widespread in cryo-EM data. For low signal-to-noise ratio (SNR) volumetric data, intrinsic topological features of biomolecular structures are indistinguishable from noise. To remove noise, we employ geometric flows that are found to preserve the intrinsic topological fingerprints of cryo-EM structures and diminish the topological signature of noise. In particular, persistent homology enables us to visualize the gradual separation of the topological fingerprints of cryo-EM structures from those of noise during the denoising process, which gives rise to a practical procedure for prescribing a noise threshold to extract cryo-EM structure information from noise contaminated data after certain iterations of the geometric flow equation. To further demonstrate the utility of persistent homology for cryo-EM data analysis, we consider a microtubule intermediate structure Electron Microscopy Data (EMD 1129). Three helix models, an alpha-tubulin monomer model, an alpha-tubulin and beta-tubulin model, and an alpha-tubulin and beta-tubulin dimer model, are constructed to fit the cryo-EM data. The least square fitting leads to similarly high correlation coefficients, which indicates that structure determination via optimization is an ill-posed inverse problem. However, these models have dramatically different topological fingerprints. Especially, linkages or connectivities that discriminate one model from another, play little role in the traditional density fitting or optimization but are very sensitive and crucial to topological fingerprints. The intrinsic topological features of the microtubule data are identified after topological denoising. By a comparison of the topological fingerprints of the original data and those of three models, we found that the third model is topologically favored. The present work offers persistent homology based new strategies for topological denoising and for resolving ill-posed inverse problems.

Multidimensional persistence in biomolecular data

Persistent homology has emerged as a popular technique for the topological simplification of big data, including biomolecular data. Multidimensional persistence bears considerable promise to bridge the gap between geometry and topology. However, its practical and robust construction has been a challenge. We introduce two families of multidimensional persistence, namely pseudomultidimensional persistence and multiscale multidimensional persistence. The former is generated via the repeated applications of persistent homology filtration to high-dimensional data, such as results from molecular dynamics or partial differential equations. The latter is constructed via isotropic and anisotropic scales that create new simiplicial complexes and associated topological spaces. The utility, robustness, and efficiency of the proposed topological methods are demonstrated via protein folding, protein flexibility analysis, the topological denoising of cryoelectron microscopy data, and the scale dependence of nanoparticles. Topological transition between partial folded and unfolded proteins has been observed in multidimensional persistence. The separation between noise topological signatures and molecular topological fingerprints is achieved by the Laplace-Beltrami flow. The multiscale multidimensional persistent homology reveals relative local features in Betti-0 invariants and the relatively global characteristics of Betti-1 and Betti-2 invariants.

Persistent homology analysis of protein structure, flexibility, and folding

Proteins are the most important biomolecules for living organisms. The understanding of protein structure, function, dynamics, and transport is one of the most challenging tasks in biological science. In the present work, persistent homology is, for the first time, introduced for extracting molecular topological fingerprints (MTFs) based on the persistence of molecular topological invariants. MTFs are utilized for protein characterization, identification, and classification. The method of slicing is proposed to track the geometric origin of protein topological invariants. Both all-atom and coarse-grained representations of MTFs are constructed. A new cutoff-like filtration is proposed to shed light on the optimal cutoff distance in elastic network models. On the basis of the correlation between protein compactness, rigidity, and connectivity, we propose an accumulated bar length generated from persistent topological invariants for the quantitative modeling of protein flexibility. To this end, a correlation matrix-based filtration is developed. This approach gives rise to an accurate prediction of the optimal characteristic distance used in protein B-factor analysis. Finally, MTFs are employed to characterize protein topological evolution during protein folding and quantitatively predict the protein folding stability. An excellent consistence between our persistent homology prediction and molecular dynamics simulation is found. This work reveals the topology-function relationship of proteins.