Persistent Modulation Research Articles

Decomposition of Zero-Dimensional Persistence Modules via Rooted Subsets

AbstractWe study the decomposition of zero-dimensional persistence modules, viewed as functors valued in the category of vector spaces factorizing through sets. Instead of working directly at the level of vector spaces, we take a step back and first study the decomposition problem at the level of sets. This approach allows us to define the combinatorial notion of rooted subsets. In the case of a filtered metric space M, rooted subsets relate the clustering behavior of the points of M with the decomposition of the associated persistence module. In particular, we can identify intervals in such a decomposition quickly. In addition, rooted subsets can be understood as a generalization of the elder rule, and are also related to the notion of constant conqueror of Cai, Kim, Mémoli and Wang. As an application, we give a lower bound on the number of intervals that we can expect in the decomposition of zero-dimensional persistence modules of a density-Rips filtration in Euclidean space: in the limit, and under very general circumstances, we can expect that at least 25% of the indecomposable summands are interval modules.

Protein profile of extracellular vesicles derived from adult Parascaris spp.

BackgroundParascaris spp. represent a significant threat to equine health worldwide, particularly in foals. The long-term survival of parasites in the host necessitates persistent modulation of the host immune response. Intercellular communication achieved through the exchange of molecules via extracellular vesicles (EVs) released from the parasite could be a crucial factor in this regard. This study aimed to isolate and characterize EVs released by adult male and female Parascaris worms and conduct a proteomic analysis to identify sex-specific proteins and potential immunomodulatory factors.MethodsLive adult Parascaris worms were collected, and EVs were isolated from spent culture media using differential ultracentrifugation. Nanoparticle tracking analysis and transmission electron microscopy confirmed the size, concentration, and morphology of the isolated EVs. Proteins within the isolated EVs were analyzed using mass spectrometry-based proteomics (LC–MS/MS).ResultsProteomic analysis revealed a total of 113 proteins in Parascaris EVs, with several proteins showing homology to known helminth exosome proteins and exhibiting immunomodulatory functions. Sex-specific differences in EV protein composition were observed, with a distinct abundance of C-type lectins in female EVs, suggesting potential sex-specific roles or regulation. Gene Ontology (GO) and Kyoto Encyclopedia of Genes and Genomes (KEGG) pathway analyses revealed metabolic pathways shared between male and female Parascaris EVs, as well as differences in signal transduction, and cell growth and death pathways, indicating sex-specific variations.ConclusionsThese findings imply that Parascaris EVs and their protein cargo are complex. This data potentially opens avenues for discovering innovative approaches to managing and understanding helminth infection.Graphical

Coarse nodal count and topological persistence

Courant’s theorem implies that the number of nodal domains of a Laplace eigenfunction is controlled by the corresponding eigenvalue. Over the years, there have been various attempts to find an appropriate generalization of this statement in different directions. We propose a new take on this problem using ideas from topological data analysis. We show that if one counts the nodal domains in a coarse way, basically ignoring small oscillations, Courant’s theorem extends to linear combinations of eigenfunctions, to their products, to other operators, and to higher topological invariants of nodal sets. We also obtain a coarse version of the Bézout estimate for common zeros of linear combinations of eigenfunctions. We show that our results are essentially sharp and that the coarse count is necessary, since these extensions fail in general for the standard count. Our approach combines multiscale polynomial approximation in Sobolev spaces with new results in the theory of persistence modules and barcodes.

Bigraded Betti numbers and generalized persistence diagrams

Commutative diagrams of vector spaces and linear maps over Z2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {Z}^2$$\\end{document} are objects of interest in topological data analysis (TDA) where this type of diagrams are called 2-parameter persistence modules. Given that quiver representation theory tells us that such diagrams are of wild type, studying informative invariants of a 2-parameter persistence module M is of central importance in TDA. One of such invariants is the generalized rank invariant, recently introduced by Kim and Mémoli. Via the Möbius inversion of the generalized rank invariant of M, we obtain a collection of connected subsets I⊂Z2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$I\\subset \\mathbb {Z}^2$$\\end{document} with signed multiplicities. This collection generalizes the well known notion of persistence barcode of a persistence module over R\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {R}$$\\end{document} from TDA. In this paper we show that the bigraded Betti numbers of M, a classical algebraic invariant of M, are obtained by counting the corner points of these subsets Is. Along the way, we verify that an invariant of 2-parameter persistence modules called the interval decomposable approximation (introduced by Asashiba et al.) also encodes the bigraded Betti numbers in a similar fashion. We also show that the aforementioned results are optimal in the sense that they cannot be extended to d-parameter persistence modules for d≥3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d \\ge 3$$\\end{document}.

On the bottleneck stability of rank decompositions of multi-parameter persistence modules

A significant part of modern topological data analysis is concerned with the design and study of algebraic invariants of poset representations—often referred to as persistence modules. One such invariant is the minimal rank decomposition, which encodes the ranks of all the structure morphisms of the persistence module by a single ordered pair of rectangle-decomposable modules, interpreted as a signed barcode. This signed barcode generalizes the concept of persistence barcode from one-parameter persistence to any number of parameters, raising the question of its bottleneck stability. We show in this paper that the minimal rank decomposition is not stable under the natural notion of signed bottleneck matching between signed barcodes. We remedy this by turning our focus to the rank exact decomposition, a related signed barcode induced by the minimal projective resolution of the module relative to the so-called rank exact structure, which we prove to be bottleneck stable under signed matchings. As part of our proof, we obtain two intermediate results of independent interest: we compute the global dimension of the rank exact structure on the category of finitely presentable multi-parameter persistence modules, and we prove a bottleneck stability result for hook-decomposable modules. We also give a bound for the size of the rank exact decomposition that is polynomial in the size of the usual minimal projective resolution, we prove a universality result for the dissimilarity function induced by the notion of signed matching, and we compute, in the two-parameter case, the global dimension of a different exact structure related to the upsets of the indexing poset. This set of results combines concepts from topological data analysis and from the representation theory of posets, and we believe is relevant to both areas.

Repopulated microglia after pharmacological depletion decrease dendritic spine density in adult mouse brain.

Microglia are innate immune cells in the brain and show exceptional heterogeneity. They are key players in brain physiological development regulating synaptic plasticity and shaping neuronal networks. In pathological disease states, microglia-induced synaptic pruning mediates synaptic loss and targeting microglia was proposed as a promising therapeutic strategy. However, the effect of microglia depletion and subsequent repopulation on dendritic spine density and neuronal function in the adult brain is largely unknown. In this study, we investigated whether pharmacological microglia depletion affects dendritic spine density after long-term permanent microglia depletion and after short-term microglia depletion with subsequent repopulation. Long-term microglia depletion using colony-stimulating-factor-1 receptor (CSF1-R) inhibitor PLX5622 resulted in increased overall spine density, especially of mushroom spines, and increased excitatory postsynaptic current amplitudes. Short-term PLX5622 treatment with subsequent repopulation of microglia had an opposite effect resulting in activated microglia with increased synaptic phagocytosis and consequently decreased spine density and reduced excitatory neurotransmission, while Barnes maze and elevated plus maze testing was unaffected. Moreover, RNA sequencing data of isolated repopulated microglia showed an activated and proinflammatory phenotype. Long-term microglia depletion might be a promising therapeutic strategy in neurological diseases with pathological microglial activation, synaptic pruning, and synapse loss. However, repopulation after depletion induces activated microglia and results in a decrease of dendritic spines possibly limiting the therapeutic application of microglia depletion. Instead, persistent modulation of pathological microglia activity might be beneficial in controlling synaptic damage.

The extended persistent homology transform of manifolds with boundary

A shortcoming of persistent homology is that when two domains have different numbers of components or holes the persistence diagrams of any filtration will have an infinite distance between them. We address this issue by revisiting the theory of extended persistence, initially developed by Cohen-Steiner, Edelsbrunner and Harer in 2009 to quantify the support of the essential homology classes for Morse functions on manifolds. We simplify the mathematical treatment of extended persistence by formulating it as a persistence module derived from a sequence of relative homology groups for pairs of spaces. Then, for n-manifolds with boundary embedded in Rn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {R}}^n$$\\end{document}, we use Morse theory to show that the extended persistent homology of a height function over M can be deduced from the extended persistent homology of the same height function restricted to ∂M\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\partial M$$\\end{document}. As an application, we describe the extended persistent homology transform (XPHT); a topological transform which takes as input a shape embedded in Euclidean space, and to each unit vector assigns the extended persistence module of the height function over that shape with respect to that direction. We define a distance between two shapes by integrating over the sphere the distance between the respective extended persistence modules. By using extended persistence we get finite distances between shapes even when they have different numbers of essential classes. We study the application of the XPHT to binary images; outlining an algorithm for efficient calculation of the XPHT exploiting relationships between the PHT of the boundary curves to the extended persistence of the foreground.

On tau-tilting subcategories

Abstract The main theme of this paper is to study $\tau $ -tilting subcategories in an abelian category $\mathscr {A}$ with enough projective objects. We introduce the notion of $\tau $ -cotorsion torsion triples and investigate a bijection between the collection of $\tau $ -cotorsion torsion triples in $\mathscr {A}$ and the collection of support $\tau $ -tilting subcategories of $\mathscr {A}$ , generalizing the bijection by Bauer, Botnan, Oppermann, and Steen between the collection of cotorsion torsion triples and the collection of tilting subcategories of $\mathscr {A}$ . General definitions and results are exemplified using persistent modules. If $\mathscr {A}=\mathrm{Mod}\mbox {-}R$ , where R is a unitary associative ring, we characterize all support $\tau $ -tilting (resp. all support $\tau ^-$ -tilting) subcategories of $\mathrm{Mod}\mbox {-}R$ in terms of finendo quasitilting (resp. quasicotilting) modules. As a result, it will be shown that every silting module (resp. every cosilting module) induces a support $\tau $ -tilting (resp. support $\tau ^{-}$ -tilting) subcategory of $\mathrm{Mod}\mbox {-}R$ . We also study the theory in $\mathrm {Rep}(Q, \mathscr {A})$ , where Q is a finite and acyclic quiver. In particular, we give an algorithm to construct support $\tau $ -tilting subcategories in $\mathrm {Rep}(Q, \mathscr {A})$ from certain support $\tau $ -tilting subcategories of $\mathscr {A}$ .

𝐾-theory of multiparameter persistence modules: Additivity

Persistence modules stratify their underlying parameter space, a quality that makes persistence modules amenable to study via invariants of stratified spaces. In this article, we extend a result previously known only for one-parameter persistence modules to grid multiparameter persistence modules. Namely, we show the K K -theory of grid multiparameter persistence modules is additive over strata. This is true for both standard monotone multi-parameter persistence as well as multiparameter notions of zig-zag persistence. We compare our calculations for the specific group K 0 K_0 with the recent work of Botnan, Oppermann, and Oudot, highlighting and explaining the differences between our results through an explicit projection map between computed groups.

Persistent transcendental Bézout theorems

Abstract An example of Cornalba and Shiffman from 1972 disproves in dimension two or higher a classical prediction that the count of zeros of holomorphic self-mappings of the complex linear space should be controlled by the maximum modulus function. We prove that such a bound holds for a modified coarse count inspired by the theory of persistence modules originating in topological data analysis.

Signal Processing with Computational Topology

The objective of this research is to analyze the mathematical methods of signal processing on abstract geometric spaces of the given sensor network, develop an algorithm on a tangible software development environment, and visualize the usefulness of those algorithms. Specifically, the scope of this research covers the notions of persistence in computational topology and how this notion can be observed in certain mathematical features of the given network. Moreover, we will be analyzing sheaves constructed over a network of sensors in the topological space, from the point of persistence modules and persistent sheaf cohomology, visualizing the persistent sheaf cohomology over barcode diagrams. The research objective is analyzed in detail in the introduction. In the first part of the main body, we have analyzed the mathematical background and previous research works in this domain to derive insight into developing an algorithm. In the second section of the main body, discussed and developed the algorithms, based on the computational topology theoretic approaches. In this paper, we will develop the algorithms to construct simplicial complexes, obtain filtration, construct sheaves, and compute persistent sheaf cohomology. The resulting barcode diagrams are analyzed in terms of certain notions, e.g. robustness, stability, and the Betti numbers. In the end, the primary findings will be succinctly expressed in several points.

Persistent homology for functionals

We introduce topological conditions on a broad class of functionals that ensure that the persistent homology modules of their associated sublevel set filtration admit persistence diagrams, which, in particular, implies that they satisfy generalized Morse inequalities. We illustrate the applicability of these results by recasting the original proof of the Unstable Minimal Surface Theorem given by Morse and Tompkins in a modern and rigorous framework.

Harder-Narasimhan filtrations and zigzag persistence

We introduce a sheaf-theoretic stability condition for finite acyclic quivers. Our main result establishes that for representations of affine type A˜ quivers, there is a precise relationship between the associated Harder-Narasimhan filtration and the barcode of the periodic zigzag persistence module obtained by unwinding the underlying quiver.

PMEH: A Parallel and Write-Optimized Extendible Hashing for Persistent Memory

Emerging persistent memory (PM) has the potential to substitute DRAM due to its near-DRAM performance and durability similar to disks. However, hash tables designed for DRAM cannot be directly adopted for PM. Moreover, prior studies on hash tables using Optane DC Persistent Memory Modules (DCPMM) have shown sub-optimal scalability and write performance due to expensive lock-based concurrent control and massive data movement caused by expansion. In this paper, we propose an opportunistic lock-free parallel multi-split extendible hashing scheme (PMEH). Firstly, PMEH achieves lock-free operations for evenly distributed data by partitioning the hash table into multiple zones and assigning each zone to one thread. Secondly, PMEH employs an opportunistic lock-free parallel scheme to effectively handle skewed data distribution, which maximizes the utilization of lock-free operations by enabling dynamic switching between lock-free and locking operations. Finally, PMEH uses multi-split with gradual splitting, instead of 2-split, to reduce the frequency of hash table expansion and hence reduce the data movement during expansion. The experimental results under the widely used YCSB workloads demonstrate that PMEH achieves excellent scalability regardless of data distribution. Moreover, PMEH significantly speeds up insertions by 1.44x 15.4x, and deletion by 2.04 18.07x compared to other state-of-the-art hashing schemes. In addition, PMEH reduces at least 52% of extra writes while providing instant recovery.

Persistent cup product structures and related invariants

One-dimensional persistent homology is arguably the most important and heavily used computational tool in topological data analysis. Additional information can be extracted from datasets by studying multi-dimensional persistence modules and by utilizing cohomological ideas, e.g. the cohomological cup product. In this work, given a single parameter filtration, we investigate a certain 2-dimensional persistence module structure associated with persistent cohomology, where one parameter is the cup-length ℓ≥0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell \\ge 0$$\\end{document} and the other is the filtration parameter. This new persistence structure, called the persistent cup module, is induced by the cohomological cup product and adapted to the persistence setting. Furthermore, we show that this persistence structure is stable. By fixing the cup-length parameter ℓ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell $$\\end{document}, we obtain a 1-dimensional persistence module, called the persistent ℓ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell $$\\end{document}-cup module, and again show it is stable in the interleaving distance sense, and study their associated generalized persistence diagrams. In addition, we consider a generalized notion of a persistent invariant, which extends both the rank invariant (also referred to as persistent Betti number), Puuska’s rank invariant induced by epi-mono-preserving invariants of abelian categories, and the recently-defined persistent cup-length invariant, and we establish their stability. This generalized notion of persistent invariant also enables us to lift the Lyusternik-Schnirelmann (LS) category of topological spaces to a novel stable persistent invariant of filtrations, called the persistent LS-category invariant.

Computing Generalized Rank Invariant for 2-Parameter Persistence Modules via Zigzag Persistence and Its Applications

The notion of generalized rank in the context of multiparameter persistence has become an important ingredient for defining interesting homological structures such as generalized persistence diagrams. However, its efficient computation has not yet been studied in the literature. We show that the generalized rank over a finite interval I of a textbf{Z}^2-indexed persistence module M is equal to the generalized rank of the zigzag module that is induced on a certain path in I tracing mostly its boundary. Hence, we can compute the generalized rank of M over I by computing the barcode of the zigzag module obtained by restricting to that path. If M is the homology of a bifiltration F of t simplices (while accounting for multi-criticality) and I consists of t points, this computation takes O(t^omega ) time where omega in [2,2.373) is the exponent of matrix multiplication. We apply this result to obtain an improved algorithm for the following problem. Given a bifiltration inducing a module M, determine whether M is interval decomposable and, if so, compute all intervals supporting its indecomposable summands.

Persistent homology analysis of type 2 diabetes genome-wide association studies in protein-protein interaction networks.

Genome-wide association studies (GWAS) involving increasing sample sizes have identified hundreds of genetic variants associated with complex diseases, such as type 2 diabetes (T2D); however, it is unclear how GWAS hits form unique topological structures in protein-protein interaction (PPI) networks. Using persistent homology, this study explores the evolution and persistence of the topological features of T2D GWAS hits in the PPI network with increasing p-value thresholds. We define an n-dimensional persistent disease module as a higher-order generalization of the largest connected component (LCC). The 0-dimensional persistent T2D disease module is the LCC of the T2D GWAS hits, which is significantly detected in the PPI network (196 nodes and 235 edges, P0.05). In the 1-dimensional homology group analysis, all 18 1-dimensional holes (loops) of the T2D GWAS hits persist over all p-value thresholds. The 1-dimensional persistent T2D disease module comprising these 18 persistent 1-dimensional holes is significantly larger than that expected by chance (59 nodes and 83 edges, P0.001), indicating a significant topological structure in the PPI network. Our computational topology framework potentially possesses broad applicability to other complex phenotypes in identifying topological features that play an important role in disease pathobiology.

Distributing Persistent Homology via Spectral Sequences

We set up the theory for a distributed algorithm for computing persistent homology. For this purpose we develop linear algebra of persistence modules. We present bases of persistence modules, together with an operation boxplus that leads to a method for obtaining images, kernels and cokernels of tame persistence morphisms. Our focus is on developing efficient methods for the computation of homology of chains of persistence modules. Later we give a brief, self-contained presentation of the Mayer–Vietoris spectral sequence. Then we study the Persistent Mayer–Vietoris spectral sequence and present a solution to the extension problem. This solution is given by finding coefficients that indicate gluings between bars on the same dimension. Finally, we review PerMaViss, an algorithm that computes all pages in the spectral sequence and solves the extension problem. This procedure distributes computations on subcomplexes, while focusing on merging homological information. Additionally, some computational bounds are found which confirm the distribution of the method.

Composition series of arbitrary cardinality in modular lattices and abelian categories

For a certain family of complete modular lattices, we prove a “Jordan–Hölder–Schreier-like” theorem with no assumptions on cardinality or well-orderedness. This family includes both lattices with are both join- and meet-continuous, as well as the lattices of subobjects of any object in an abelian category satisfying properties related to Grothendieck's axioms (AB5) and (AB5⁎). We then give several examples of objects in abelian categories which satisfy these axioms, including pointwise finite-dimensional persistence modules, presheaves, and certain Prüfer modules. Moreover, we show that, over an arbitrary ring, the infinite product of isomorphic simple modules both fails to satisfy our axioms and admits at least two composition series with distinct cardinalities. We conclude by giving a lattice-theoretic proof that any object which is locally finitely generated and satisfies our axioms can be expressed as a direct sum of indecomposable subobjects. We conjecture that this decomposition is unique.

Algebraic stability theorem for derived categories of zigzag persistence modules

We study distances on zigzag persistence modules from the viewpoint of derived categories and Auslander–Reiten quivers. The derived category of ordinary persistence modules is derived equivalent to that of arbitrary zigzag persistence modules, depending on a classical tilting module. Through this derived equivalence, we define and compute distances on the derived category of arbitrary zigzag persistence modules and prove an algebraic stability theorem. We also compare our distance with the distance for purely zigzag persistence modules introduced by Botnan–Lesnick and the sheaf-theoretic convolution distance due to Kashiwara–Schapira.