Structural features of persistent homology and their algorithmic transformations

This chapter studies, in the setting of vector spaces over a field, the basics concerning multilinear functions, tensor products, spaces of linear functions, and algebras related to tensor products. Sections 1–5 concern special properties of bilinear forms, all vector spaces being assumed to be finite-dimensional. Section 1 associates a matrix to each bilinear form in the presence of an ordered basis, and the section shows the effect on the matrix of changing the ordered basis. It then addresses the extent to which the notion of “orthogonal complement” in the theory of inner-product spaces applies to nondegenerate bilinear forms. Sections 2–3 treat symmetric and alternating bilinear forms, producing bases for which the matrix of such a form is particularly simple. Section 4 treats a related subject, Hermitian forms when the field is the complex numbers. Section 5 discusses the groups that leave some particular bilinear and Hermitian forms invariant. Section 6 introduces the tensor product of two vector spaces, working with it in a way that does not depend on a choice of basis. The tensor product has a universal mapping property—that bilinear functions on the product of the two vector spaces extend uniquely to linear functions on the tensor product. The tensor product turns out to be a vector space whose dual is the vector space of all bilinear forms. One particular application is that tensor products provide a basis-independent way of extending scalars for a vector space from a field to a larger field. The section includes a number of results about the vector space of linear mappings from one vector space to another that go hand in hand with results about tensor products. These have convenient formulations in the language of category theory as “natural isomorphisms.” Section 7 begins with the tensor product of three and then $n$ vector spaces, carefully considering the universal mapping property and the question of associativity. The section defines an algebra over a field as a vector space with a bilinear multiplication, not necessarily associative. If $E$ is a vector space, the tensor algebra $T(E)$ of $E$ is the direct sum over $n\geq0$ of the $n$-fold tensor product of $E$ with itself. This is an associative algebra with a universal mapping property relative to any linear mapping of $E$ into an associative algebra $A$ with identity: the linear map extends to an algebra homomorphism of $T(E)$ into $A$ carrying 1 into 1. Sections 8–9 define the symmetric and exterior algebras of a vector space $E$. The symmetric algebra $S(E)$ is a quotient of $T(E)$ with the following universal mapping property: any linear mapping of $E$ into a commutative associative algebra $A$ with identity extends to an algebra homomorphism of $S(E)$ into $A$ carrying 1 into 1. The symmetric algebra is commutative. Similarly the exterior algebra $\bigwedge(E)$ is a quotient of $T(E)$ with this universal mapping property: any linear mapping $l$ of $E$ into an associative algebra $A$ with identity such that $l(v)^2=0$ for all $v\in E$ extends to an algebra homomorphism of $\bigwedge(E)$ into $A$ carrying 1 into 1. The problems at the end of the chapter introduce some other algebras that are of importance in applications, and the problems relate some of these algebras to tensor, symmetric, and exterior algebras. Among the objects studied are Lie algebras, universal enveloping algebras, Clifford algebras, Weyl algebras, Jordan algebras, and the division algebra of octonions.

In this work, we test Wasserstein distance in conjunction with persistent homology as a tool for discriminating large-scale structures of simulated universes with different values of σ8 cosmological parameter (present root-mean-square matter fluctuation averaged over a sphere of radius 8 Mpc comoving). The Wasserstein distance (a.k.a. the pair-matching distance) was proposed to measure the difference between two networks in terms of persistent homology. The advantage of this approach consists in its non-parametric way of probing the topology of the cosmic web, in contrast to graph-theoretical approach depending on linking length. By treating the haloes of the cosmic web as points in a point cloud, we calculate persistent homologies, build persistence (birth–death) diagrams, and evaluate Wasserstein distance between them. The latter showed itself as a convenient tool to compare simulated cosmic webs. We show that one can discern two cosmic webs (simulated or real) with different σ8 parameter. It turns out that Wasserstein distance’s discrimination ability depends on redshift z, as well as on the dimensionality of considered homology features. We find that the highest discriminating power this tool obtains is at z = 2 snapshots, among the considered z = 2, 1, and 0.1 ones.

Persistent homology is used for computing topological features of a space at different spatial resolutions. It is one of the main tools from computational topology that is applied to the problems of data analysis. Despite several attempts to reduce its complexity, persistent homology remains expensive in both time and space. These limits are such that the largest data sets to which the method can be applied have the number of points of the order of thousands in ℝ3. This paper explores a technique intended to reduce the number of data points while preserving the salient topological features of the data. The proposed technique enables the computation of persistent homology on a reduced version of the original input data without affecting significant components of the output. Since the run time of persistent homology is exponential in the number of data points, the proposed data reduction method facilitates the computation in a fraction of the time required for the original data. Moreover, the data reduction method can be combined with any existing technique that simplifies the computation of persistent homology. The data reduction is performed by creating small groups of similar data points, called nano-clusters, and then replacing the points within each nano-cluster with its cluster center. The persistence homology of the reduced data differs from that of the original data by an amount bounded by the radius of the nano-clusters. The theoretical analysis is backed by experimental results showing that persistent homology is preserved by the proposed data reduction technique.

Persistent homology is a recently developed theory in the field of algebraic topology to study shapes of datasets. It is an effective data analysis tool that is robust to noise and has been widely applied. We demonstrate a general pipeline to apply persistent homology to study time series, particularly the instantaneous heart rate time series for the heart rate variability (HRV) analysis. The first step is capturing the shapes of time series from two different aspects—the persistent homologies and hence persistence diagrams of its sub-level set and Taken's lag map. Second, we propose a systematic and computationally efficient approach to summarize persistence diagrams, which we coined persistence statistics. To demonstrate our proposed method, we apply these tools to the HRV analysis and the sleep-wake, REM-NREM (rapid eyeball movement and non rapid eyeball movement) and sleep-REM-NREM classification problems. The proposed algorithm is evaluated on three different datasets via the cross-database validation scheme. The performance of our approach is better than the state-of-the-art algorithms, and the result is consistent throughout different datasets.

Topological Data Analysis (TDA) is a data mining technique to characterize the topological features of data. Persistent Homology (PH) is an important tool of TDA that has been applied to a wide range of applications. However its time and space complexities motivates a need for new methods to compute the PH of high-dimensional data. An important, and memory intensive, element in the computation of PH is the complex constructed from the input data. In general, PH tools use and focus on optimizing simplicial complexes; less frequently cubical complexes are also studied. This paper develops a method to construct polytopal complexes (or complexes constructed of any mix of convex polytopes) in any dimension <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\mathbb{R}^{n}$</tex> In general, polytopal complexes are significantly smaller than simplicial or cubical complexes. This paper includes an experimental assessment of the impact that polytopal complexes have on memory complexity and output results of a PH computation.

Persistent homology (PH) is a robust method to compute multi-dimensional geometric and topological features of a dataset. Because these features are often stable under certain perturbations of the underlying data, are often discriminating, and can be used for visualization of structure in high-dimensional data and in statistical and machine learning modeling, PH has attracted the interest of researchers across scientific disciplines and in many industry applications. However, computational costs may present challenges to effectively using PH in certain data contexts, and theoretical stability results may not hold in practice. In this paper, we define, implement, and investigate a simplicial complex construction for computing persistent homology of Euclidean point cloud data, which we call the Delaunay-Rips complex (DR). By only considering simplices that appear in the Delaunay triangulation of the point cloud and assigning the Vietoris-Rips weights to simplices, DR avoids potentially costly computations in the persistence calculations. We document and compare a Python implementation of DR with other simplicial complex constructions for generating persistence diagrams. By imposing sufficient conditions on point cloud data, we are able to theoretically justify the stability of the persistence diagrams produced using DR. When the Delaunay triangulation of the point cloud changes under perturbations of the points, we prove that DR-produced persistence diagrams exhibit instability. Since we cannot guarantee that real-world data will satisfy our stability conditions, we demonstrate the practical robustness of DR for persistent homology in comparison with other simplicial complexes in machine learning applications. We find in our experiments that using DR in an ML-TDA pipeline performs comparatively well as using other simplicial complex constructions.

We propose a general technique for extracting a larger set of stable information from persistent homology computations than is currently done. The persistent homology algorithm is usually viewed as a procedure which starts with a filtered complex and ends with a persistence diagram. This procedure is stable (at least to certain types of perturbations of the input). This justifies the use of the diagram as a signature of the input, and the use of features derived from it in statistics and machine learning. However, these computations also produce other information of great interest to practitioners that is unfortunately unstable. For example, each point in the diagram corresponds to a simplex whose addition in the filtration results in the birth of the corresponding persistent homology class, but this correspondence is unstable. In addition, the persistence diagram is not stable with respect to other procedures that are employed in practice, such as thresholding a point cloud by density. We recast these problems as real-valued functions which are discontinuous but measurable, and then observe that convolving such a function with a suitable function produces a Lipschitz function. The resulting stable function can be estimated by perturbing the input and averaging the output. We illustrate this approach with a number of examples, including a stable localization of a persistent homology generator from brain imaging data.

Topology-controllable Implicit Surface Reconstruction Based on Persistent Homology

It is difficult to discriminate the amorphous state using a transmission electron microscope (TEM). We discriminated different amorphous states on TEM images using persistent homology, which is a mathematical analysis technique that employs the homology concept and focuses on ‘holes’. The structural models of the different amorphous states, that is, amorphous and liquid states, were created using classical molecular dynamic simulation. TEM images in several defocus conditions were simulated by the multi-slice method using the created amorphous and liquid states, and their persistent diagrams were calculated. Finally, logistic regression and support vector classification machine learning algorithms were applied for discrimination. Consequently, we found that the amorphous and liquid phases can be discriminated by more than 85%. Because the contrast of TEM images depends on sample thickness, focus, lens aberration, etc., radial distribution function cannot be classified; however, the persistent homology can discriminate different amorphous states in a wide focus range.

Topological Data Analysis is a set of mathematical methods for capturing the structure of data forms. TDA has been used to cope with noisy signals and time series. TDA is used to discover the key characteristics of data, which are shapes. One of the basic techniques for TDA is persistent homology which is used to do a study of time series and signal processing. Human activity analysis is a well-researched issue in the field of vision, with large research on the topic. The produced time- series data is typically high-dimensional, intrinsic, non-linear, and non-stationary, leading to pattern changes. The data acquired by a person’s numerous sensors, as well as their placement and orientation, often produces noisy and complicated data, resulting in poor model performance. The implementation method must be data-driven, independent of sensor location and orientation, and employ a coordinate and parameter-free strategy to identify the variable time series data. We need technology that can represent a low-dimensional manifold compactly, identify emergent behaviors, and globalize the system. The persistent homology having property being co- ordinate free. Sensor placement has little effect on time series factorization using persistent homology. This paper gives an overview of topological approaches as well as the problems of human activity analysis, as well as recommendations for the future.

Cytometry experiments yield high-dimensional point cloud data that is difficult to interpret manually. Boolean gating techniques coupled with comparisons of relative abundances of cellular subsets is the current standard for cytometry data analysis. However, this approach is unable to capture more subtle topological features hidden in data, especially if those features are further masked by data transforms or significant batch effects or donor-to-donor variations in clinical data. We present that persistent homology, a mathematical structure that summarizes the topological features, can distinguish different sources of data, such as from groups of healthy donors or patients, effectively. Analysis of publicly available cytometry data describing non-naïve CD8+ T cells in COVID-19 patients and healthy controls shows that systematic structural differences exist between single cell protein expressions in COVID-19 patients and healthy controls. We identify proteins of interest by a decision-tree based classifier, sample points randomly and compute persistence diagrams from these sampled points. The resulting persistence diagrams identify regions in cytometry datasets of varying density and identify protruded structures such as ‘elbows’. We compute Wasserstein distances between these persistence diagrams for random pairs of healthy controls and COVID-19 patients and find that systematic structural differences exist between COVID-19 patients and healthy controls in the expression data for T-bet, Eomes, and Ki-67. Further analysis shows that expression of T-bet and Eomes are significantly downregulated in COVID-19 patient non-naïve CD8+ T cells compared to healthy controls. This counter-intuitive finding may indicate that canonical effector CD8+ T cells are less prevalent in COVID-19 patients than healthy controls. This method is applicable to any cytometry dataset for discovering novel insights through topological data analysis which may be difficult to ascertain otherwise with a standard gating strategy or existing bioinformatic tools.

The microstructure of porous electrodes of solid oxide cells (SOCs) significantly impacts their electrochemical performance. Therefore, various attempts have been made to analyze their complex porous microstructure in three dimensions. The obtained microstructures have been quantitatively evaluated using structural metrics, such as volume fraction, surface area density, and tortuosity factor of the constituent phases, as well as double- and triple-phase boundaries. These structural characteristics are useful to correlate the electrode performance with their microstructure, thereby determining optimal electrode design. However, these intuitive metrics often fails to explain the electrode performance. In fact, structural changes during long term operation of SOCs are subtle so that the performance degradation of the electrodes cannot be fully detected by these intuitive characteristics. Therefore, it is necessary to find a way to extract hidden structural metrics that characterize the electrode microstructure and to clarify their correlation with electrode performance.One of the promising candidates for such structural metrics is topological information. Topology is a structural property that is invariant to successive deformation operations, such as connected structures and the number of holes in the structures. Topology-based structural analysis has many applications in material science, where macroscopic properties of materials are correlated with topological information about their internal structures (e.g., crystal structures). For example, Wang et al. [1] performed a topological analysis on the crystal structure of double-phase steel to characterize its structure and to predict its macroscopic properties, such as stress-strain curves. Application of topological data analysis is also found in the analysis of SOFC electrodes; Pawlowski et al. [2] attempted to capture the structural changes in SOFC anodes during long-term operation from a topological perspective.In addition, topological analysis is regarded as a tool for dimensional reduction of complex structural information. Therefore, the information extracted from the topological analysis can be used to evaluate conventional metrics of porous microstructures, such as surface area and triple-phase boundary. If these metrics can be accurately evaluated from a limited number of information from the topological analysis, the numerical cost is expected to be significantly reduced.Therefore, this study investigates the applicability of the topological analysis to the electrode microstructure analysis. Fig. 1 shows the schematic diagram of the structural analysis in this study. First, persistent homology analysis [3] is employed to extract the topological information of the electrodes, where the birth and death of topological features are detected during the filtration process within the structure datasets to produce the persistent diagram (PD). The persistent diagram is then discretized and concatenated to obtain the persistent image (PI). Subsequently, principal component analysis (PCA) is conducted to further reduce the dimensionality of the structural information extracted in the persistent image. The obtained values of the principal components are correlated with the conventional structural metrics of the porous electrodes, such as volume fraction, surface area density, and triple-phase boundary density.Moreover, neural networks (NNs) consisting of fully-connected layers are constructed to quantify the structural metrics from the principal components. The constructed neural networks are trained using the real electrode microstructure datasets obtained using the focused ion beam and scanning electron microstructure [4]. To validate the developed neural networks, they are compared with the convolutional neural network (CNN), which directly quantifies the microstructural metrics from the three-dimensional structures, in terms of quantification accuracy, the number of training datasets, and required training time.The values obtained from the topological analysis followed by the principal component analysis are found to contain essential structural information in the porous electrodes. This is implied by the fact that the principal component values obtained from the real structure datasets are significantly different from those from the artificial sphere-packing structures, even though the conventional structural metrics, i.e., volume fraction and surface density, are identical between the real and artificial structures. In addition, the principal component values have sensitivity to the conventional statistical metrics, because the structures with different solid compositions form clusters in different locations in a principal component space. These suggest that the topological information will be useful not only to detect subtle structural changes in the porous electrodes undetectable in conventional structural metrics, but also to reduce the dimensionality of the complex porous electrodes without losing the information quality.Since the information extracted using the persistent homology and principal component analysis possess essential information about the electrode structures, conventional microstructural metrics are accurately quantified by the constructed neural network. It should be emphasized that the number of parameters in the neural network that need to be adjusted during the training process is significantly reduced compared with the convolutional neural network. As a result, the number of required datasets for the training of the network is significantly reduced. These clearly indicate the effectiveness of the dimensional reduction in the quantification of the electrode microstructures.

• A novel foundation for topological pattern recognition of point cloud data, through persistent homology of graph approximations. • An extension of persistent homology to point cloud data of metric trees, including theoretical and experimental verification. • The first charting of cell trajectory data sets that explains some of the difficulties this field is confronted with. The rising field of Topological Data Analysis (TDA) provides a new approach to learning from data through persistence diagrams , which are topological signatures that quantify topological properties of data in a comparable manner. For point clouds, these diagrams are often derived from the Vietoris-Rips filtration—based on the metric equipped on the data—which allows one to deduce topological patterns such as components and cycles of the underlying space. In metric trees these diagrams often fail to capture other crucial topological properties, such as the present leaves and multifurcations. Prior methods and results for persistent homology attempting to overcome this issue mainly target Rips graphs, which are often unfavorable in case of non-uniform density across our point cloud. We therefore introduce a new theoretical foundation for learning a wider variety of topological patterns through any given graph . Given particular powerful functions defining persistence diagrams to summarize topological patterns, including the normalized centrality or eccentricity , we prove a new stability result, explicitly bounding the bottleneck distance between the true and empirical diagrams for metric trees. This bound is tight if the metric distortion obtained through the graph and its maximal edge-weight are small. Through a case study of gene expression data, we demonstrate that our newly introduced diagrams provide novel quality measures and insights into cell trajectory inference.

We propose an algorithm for persistence homology computation of orientable 2-dimensional (2D) manifolds with or without boundary (meshes) represented by 2D combinatorial maps. Having as an input a real function h on the vertices of the mesh, we first compute persistent homology of filtrations obtained by adding cells incident to each vertex of the mesh, The cells to add are controlled by both the function h and a parameter \(\delta \). The parameter \(\delta \) is used to control the number of cells added to each level of the filtration. Bigger \(\delta \) produces less levels in the filtration and consequently more cells in each level. We then simplify each level (cluster) by merging faces of the same cluster. Our experiments demonstrate that our method allows fast computation of persistent homology of big meshes and it is persistent-homology aware in the sense that persistent homology does not change in the simplification process when fixing \(\delta \).

Topological Data Analysis is a machine learning method that summarizes the topological features of a space. Persistent Homology (PH) can identify these topological features as they persist within a point cloud; persisting in respect to the connectedness of the point cloud at increasing distances. The utility of PH is apparent in several fields including bioinformatics, network security, and object classification. However, the memory complexity of PH limits the application to relatively small point clouds for low-dimensional topological feature identification. For this reason, numerous approaches to optimize and approximate the PH have been introduced for providing results over large point clouds. One solution, Partitioned Persistent Homology (PPH), has shown favorable approximation on a single node with significant performance improvement. However, the single-node approach is limited by the available system memory, leading to the need for a distributed approach for additional (especially memory) resources. This paper studies a distributed version of PPH for use with large point clouds over a high-performance compute cluster. Experimental results of the distributed algorithm against previous studies is presented along with scalability of the distributed library.