R. Gonzalez-Diaz, M. Soriano-Trigueros, A. Torras-Casas
We study how to obtain partial matchings using the block function $\mathcal{M}_f$, induced by a morphism $f$ between persistence modules. $\mathcal{M}_f$ is defined algebraically and is linear with respect to direct sums of morphisms. We study some interesting properties of $\mathcal{M}_f$, and provide a way to obtain $\mathcal{M}_f$ using matrix operations.
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Ladder Decomposition for Morphisms of Persistence Modules
本文研究持久模态射的直和分解条件，改进已有结果，证明即使条形码存在嵌套条，只要态射相对嵌套尺度足够接近可逆，阶梯分解仍成立。
Induced Matchings and the Algebraic Stability of Persistence Barcodes
我们定义了一个简单的显式映射，将逐点有限维持久性模块的态射$f:M \rightarrow N$发送到$M$和$N$的条形码之间的匹配。我们的主要结果是，在精确的意义上，这种匹配的质量是严格控制的$\ker f$和$\mathop{\mathrm{coker}} f$的条形码中的最长间隔的长度。作为一个直接的推论，我们得到了一个新的证明的代数稳定性的持久性，一个基本的结果，在理论的持久性的同源性。与以前的证明相比，我们明确地显示了两个持久性模块之间的$δ$交织态射如何诱导两个模块的条形码之间的$δ$匹配。我们的主要结果还专门的子模块和持久模块的子模块的结构定理，并产生一个新的“单态射”的持久模块上的交织关系的特征。
We present an algorithm for computing the barcode of the image of a morphisms in persistent homology induced by an inclusion of filtered finite-dimensional chain complexes. These algorithms make use of the clearing optimization and can be applied to inclusion-induced maps in persistent absolute homology and persistent relative cohomology for filtrations of pairs of simplicial complexes. They form the basis for our implementation for Vietoris-Rips complexes in the framework of the software Ripser.
The matching distance is a pseudometric on multi-parameter persistence modules, defined in terms of the weighted bottleneck distance on the restriction of the modules to affine lines. It is known that this distance is stable in a reasonable sense, and can be efficiently approximated, which makes it a promising tool for practical applications. In this work, we show that in the 2-parameter setting, the matching distance can be computed exactly in polynomial time. Our approach subdivides the space of affine lines into regions, via a line arrangement. In each region, the matching distance restrict…
Lifespan Functors and Natural Dualities in Persistent Homology
本文引入寿命函子，根据有界性过滤条形码区间，用于分类内射与投射对象，推广了持久（上）同调的对偶性结果，并为高效计算态射像、核与余核的条形码奠定基础。
Switch Points of Bi-Persistence Matching Distance
本文研究双持续匹配距离计算中切换点的算法化求解，给出判定候选切换点错误或多余的条件并实现算法，通过实验分析切换点数量随关键值数量及分布的变化规律。
Matrix Algebra for Persistence Modules Yields a Proof of the Isometry Theorem
本文通过矩阵化简为单参数持续模的等距定理提供证明，发展了持续模的矩阵计算方法，实现了函子性持续同调条形码及特定条件下的同调条形码与态射的计算。
In topological data analysis, we want to discern topological and geometric structure of data, and to understand whether or not certain features of data are significant as opposed to simply random noise. While progress has been made on statistical techniques for single-parameter persistence, the case of two-parameter persistence, which is highly desirable for real-world applications, has been less studied. This paper provides an accessible introduction to two-parameter persistent homology and presents results about matching distance between 2-D persistence modules obtained from families of poin…
Properties and Stability of Persistence Matching Diagrams
本文提出由度量空间集合映射诱导的持续匹配图，给出几何定义并证明其包含持续态射像、核与余核信息，主要贡献是建立嵌入及集合单射情形下的稳定性定理。
Computing the Matching Distance of 2-Parameter Persistence Modules from Critical Values
本文面向2参数持续模块，提出在原平面的组合设定下高效计算匹配距离，给出切换点显式公式并避免考察水平、竖直线。
This short note establishes explicit and broadly applicable relationships between persistence-based distances computed locally and globally. In particular, we show that the bottleneck distance between two zigzag persistence modules restricted to an interval is always bounded above by the distance between the unrestricted versions. While this result is not surprising, it could have different practical implications. We give two related applications for metric graph distances, as well as an extension for the matching distance between multiparameter persistence modules.
Multi-parameter Module Approximation: an efficient and interpretable invariant for multi-parameter persistence modules with guarantees
针对多参数持久化模的拓扑描述问题，提出带精度参数δ的MMA算法，该算法基于相容匹配函数生成可控近似分解，有界复杂度且鲁棒性可证，在多个数据集上达到SOTA性能。
Additive Partial Matchings Induced by Persistence Maps
提出持续映射的新不变量，由定义域和值域持续图之间的部分匹配构成，该不变量对直和分解可加、比像不变量更具区分力，且可在三次时间内计算。
Asymptotic Improvements on the Exact Matching Distance for 2-parameter Persistence
针对2参数持续模的精确匹配距离计算，本文利用对偶线排列判定与三维平面排列的随机增量搜索，将复杂度改进至期望 $O(n^5\log^3 n)$ 时间和 $O(n^2)$ 空间。
The Convex Matching Distance in Multiparameter Persistence
本文提出了一种新的多参数持续性度量方法，通过凸组合优化计算效率，验证其在不同数据集上的有效性。
We first introduce the notion of meta-rank for a 2-parameter persistence module, an invariant that captures the information behind images of morphisms between 1D slices of the module. We then define the meta-diagram of a 2-parameter persistence module to be the Möbius inversion of the meta-rank, resulting in a function that takes values from signed 1-parameter persistence modules. We show that the meta-rank and meta-diagram contain information equivalent to the rank invariant and the signed barcode. This equivalence leads to computational benefits, as we introduce an algorithm for computing th…
Computing Flat-Injective Presentations of Multiparameter Persistence Modules
针对多参数持久模块，提出其平坦-内射表示的显式构造方法，基于有界链复形的边界态射得到拟同构，可从平坦分解矩阵计算表示矩阵，为不变量计算提供起点并实现Julia包。
We develop a unifying framework for the treatment of various persistent homology architectures using the notion of correspondence modules. In this formulation, morphisms between vector spaces are given by partial linear relations, as opposed to linear mappings. In the one-dimensional case, among other things, this allows us to: (i) treat persistence modules and zigzag modules as algebraic objects of the same type; (ii) give a categorical formulation of zigzag structures over a continuous parameter; and (iii) construct barcodes associated with spaces and mappings that are richer in geometric in…
Efficient Algorithms for Complexes of Persistence Modules with Applications
本文扩展持久性算法至非自由模复形，通过表示与高效算法计算同调，应用于单纯形塔及持久层，实现高效持久（上）同调计算。
In order to better understand and to compare interleavings between persistence modules, we elaborate on the algebraic structure of interleavings in general settings. In particular, we provide a representation-theoretic framework for interleavings, showing that the category of interleavings under a fixed translation is isomorphic to the representation category of what we call a shoelace. Using our framework, we show that any two interleavings of the same pair of persistence modules are themselves interleaved. Furthermore, in the special case of persistence modules over $\mathbb{Z}$, we show tha…
One of the main reasons for topological persistence being useful in data analysis is that it is backed up by a stability (isometry) property: persistence diagrams of $1$-parameter persistence modules are stable in the sense that the bottleneck distance between two diagrams equals the interleaving distance between their generating modules. However, in multi-parameter setting this property breaks down in general. A simple special case of persistence modules called rectangle decomposable modules is known to admit a weaker stability property. Using this fact, we derive a stability-like property fo…
Triangulation, Persistence, and Fukaya categories
本文提出三角化持久性范畴（TPC）这一新代数概念，证明导出Fukaya范畴在适当条件下可被TPC精化，并由此在Liouville流形的紧致正合Lagrangian子流形空间上构造出具有内蕴辛性质的度量，得到整体刚性结果。
Topological Quality of Subsets via Persistence Matching Diagrams
本文提出持久匹配图这一拓扑不变量，结合嵌入与持久同调，通过最小生成树算法衡量子集对数据集的代表质量，并用于估计 Hausdorff 距离界和解释监督学习性能不佳的原因。
We present a generalization of the induced matching theorem and use it to prove a generalization of the algebraic stability theorem for $\mathbb{R}$-indexed pointwise finite-dimensional persistence modules. Via numerous examples, we show how the generalized algebraic stability theorem enables the computation of rigorous error bounds in the space of persistence diagrams that go beyond the typical formulation in terms of bottleneck (or log bottleneck) distance.
The theory of persistence modules on the commutative ladders $CL_n(τ)$ provides an extension of persistent homology. However, an efficient algorithm to compute the generalized persistence diagrams is still lacking. In this work, we view a persistence module $M$ on $CL_n(τ)$ as a morphism between zigzag modules, which can be expressed in a block matrix form. For the representation finite case ($n\leq 4)$, we provide an algorithm that uses certain permissible row and column operations to compute a normal form of the block matrix. In this form an indecomposable decomposition of $M$, and thus its …
Projective Constant Decompositions of Persistence Modules over Noetherian Rings
本文研究了诺特环上持续模块的投射常数分解，提出了投射常数类型模块的概念，并建立了分解存在的代数条件及唯一性理论。
Complex Matching Distance and Stability for Minimal Projective Resolutions, with Applications to Persistence
本文研究了有限度量偏序集P的最小投射解析的稳定性理论，引入了新的复杂匹配距离，并应用于持续性图的稳定性分析。
On the bottleneck stability of rank decompositions of multi-parameter persistence modules
本文研究了多参数持续模秩分解的瓶颈稳定性，证明了最小秩分解在带符号瓶颈匹配下不稳定，转而证明秩正合分解具有瓶颈稳定性，并得出多个独立理论结果。
Motivated both by theoretical and practical considerations in topological data analysis, we generalize the $p$-Wasserstein distance on barcodes to multiparameter persistence modules. For each $p\in [1,\infty]$, we in fact introduce two such generalizations $d_{\mathcal I}^p$ and $d_{\mathcal M}^p$, such that $d_{\mathcal I}^\infty$ equals the interleaving distance and $d_{\mathcal M}^\infty$ equals the matching distance. We show that on 1- or 2-parameter persistence modules over prime fields, $d_{\mathcal I}^p$ is the universal (i.e., largest) metric satisfying a natural stability property; th…
We propose and implement a method to analyze time series with a neural network using a matrix of area-normalized persistence landscapes obtained through topological data analysis. We include a gating layer in the network's architecture that is able to identify the most relevant landscape levels for the classification task, thus working as an importance attribution system. Next, we perform a matching between the selected landscape functions and the corresponding critical points of the original time series. From this matching we are able to reconstruct an approximate shape of the time series tha…
A theory of modules over posets is developed to define computationally feasible, topologically interpretable data structures, in terms of birth and death of homology classes, for persistent homology with multiple real parameters. To replace the noetherian hypothesis in the general setting of modules over posets, a "finitely encoded" condition is defined combinatorially and developed algebraically. It captures topological tameness of persistent homology. Poset-modules satisfying it can be specified by "fringe presentations" that reflect birth-and-death descriptions of persistence. A syzygy theo…
Exact structures for persistence modules
本文研究多参数持久模的精确结构与相对同调代数，提出一个新伴随将有限偏序集上的经典论证提升到无限偏序集，并给出若干精确结构下相对投射模间不可约态射的显式描述。
Matching distance via the extended Pareto grid
本文利用扩展 Pareto 网格分析双参数持久化匹配距离计算中真正起作用的过滤直线，并刻画其有限特殊类型。
An ideal invariant for multiparameter persistence would be discriminative, computable and stable. In this work we analyse the discriminative power of a stable, computable invariant of multiparameter persistence modules: the fibered bar code. The fibered bar code is equivalent to the rank invariant and encodes the bar codes of the 1-parameter submodules of a multiparameter module. This invariant is well known to be globally incomplete. However in this work we show that the fibered bar code is locally complete for finitely presented modules by showing a local equivalence of metrics between the i…
A Theory of Sub-Barcodes
本文提出子条形码理论，通过持久模同态的分解构造从分解范畴到条形码偏序集的函子，在缺乏交错的情况下仅利用上下界即可对未知条形码作出强有力断言。
Filtered cospans and interlevel persistence with boundary conditions
本文提出过滤余弦概念，用于捕捉Morse函数的互层持续性，并通过持续图和等距定理建立与同调的联系。
The coherent matching distance in 2D persistent homology
多维持久Betti数之间的比较通常基于多维匹配距离。虽然通过考虑与具有正斜率的线相关联的滤波函数的合适族来定义和计算该度量相当简单，但是它具有两个主要缺点。首先，它忘记了与彼此接近的线相关联的滤子的同调性质之间的自然联系。结果，部分有趣的同源信息丢失了。第二，它的本质上不连续的定义，使它难以研究其性质。在本文中，我们介绍了一个新的匹配距离的2D持久贝蒂数，称为相干匹配距离和匹配的变化与我们考虑的过滤相干。它的定义并不是微不足道的，因为它必须面对多维持久性中的单值性，即空间中参数化上述过滤的不同路径可以在相关持久图之间引起不同的匹配。在本文中，我们证明了相干的二维匹配距离是定义良好的和稳定的。
The barcode of a persistence module serves as a complete combinatorial invariant of its isomorphism class. Barcodes are typically extracted by performing changes of basis on a persistence module until the constituent matrices have a special form. Here we describe a new algorithm for computing barcodes which also keeps track of, and outputs, such a change of basis. Our main result is an explicit characterisation of the group of transformations that sends one barcode basis to another. Armed with knowledge of the entire space of barcode bases, we are able to show that any map of persistence modul…
Algebraic Wasserstein distances and stable homological invariants of data
本文基于代数Wasserstein距离引入了持久模间的参数化伪度量，定义了1-Lipschitz稳定的Wasserstein稳定秩不变量，并提出了高效计算这些不变量及其交错距离的算法，展示了其在数据分析中的实用性。
Interleaving Distance as a Limit
持久同源性是确定数据集的拓扑性质的一种方法。众所周知，每一个持久性模块都允许一个有限全序集的表示的结构。在之前的工作中，作者证明了鲍尔和莱斯尼克的等距定理的类似物，用于表示某一类有限偏序集。该度量介于Bubenik，de Silva和Scott的交织度量和包含代数信息的瓶颈度量之间。这两个等距定理的关键步骤是证明一个匹配定理，即交错产生相同高度的匹配。在本文中，我们继续这项工作，限制到这些偏序集所产生的数据，而更一般的度量的选择。我们首先表明，虽然交织总是产生匹配，对于任意选择的权重，它不会产生一个相同的高度。然后，我们表明，虽然在这个意义上的匹配定理失败，一个获得了“移位”匹配（正确的高度）从交织扩大的类别。然后，我们证明了这个扩展范畴上的等距定理。作为一个应用，我们精确的方式表示的有限偏序集近似的持久性模块。具体而言，给定两个有限点云的数据，我们关联一个广义序列（网）的代数，在这两个数据集的持久性模块可以进行比较。我们恢复了经典的交织距离均匀地采取限制。
In this paper we study a new metric for comparing Betti numbers functions in bidimensional persistent homology, based on coherent matchings, i.e. families of matchings that vary in a continuous way. We prove some new results about this metric, including its stability. In particular, we show that the computation of this distance is strongly related to suitable filtering functions associated with lines of slope 1, so underlining the key role of these lines in the study of bidimensional persistence. In order to prove these results, we introduce and study the concepts of extended Pareto grid for a…
Multi-Parameter Persistence Modules are Generically Indecomposable
本文证明多参数持久性模的代数稳定性定理推广显著失效，任何有限可表示模可被不可分解模任意逼近，并给出近不可分解模的分解结构，为多参数持久性研究提供新方向。
In this article we propose a novel approach for comparing the persistent homology representations of two spaces (filtrations). Commonly used methods are based on numerical summaries such as persistence diagrams and persistence landscapes, along with suitable metrics (e.g. Wasserstein). These summaries are useful for computational purposes, but they are merely a marginal of the actual topological information that persistent homology can provide. Instead, our approach compares between two topological representations directly in the data space. We do so by defining a correspondence relation betwe…
Path representations in multiparameter persistent homology
针对单参数拓扑分析捕捉差异有限的问题，研究将多参数持续同调中的匹配距离直线推广为单调路径滤过，拓展了多参数拓扑分析的路径表示形式，并提供对应软件实现。
On the Hausdorff stability of barcodes over posets
本文研究了在偏序集上区间可分解模块的豪斯多夫稳定性，提出了一项Lipschitz稳定性结果，扩展了持续性理论中关于瓶径距离与交织距离关系的研究。
Interval Multiplicities of Persistence Modules
本文针对有限偏序集上的持久模，给出了区间模在不可分解分解中重数的矩阵秩公式，并应用于检测区间可分解性、计算最大区间可分解直和项等。
On the stability of multigraded Betti numbers and Hilbert functions
本文针对多参数持续同调中的多重分次Betti数，引入高效可计算的瓶颈型相异度函数，证明其关于交错距离的稳定性，并给出2参数情形下Hilbert函数的1-Wasserstein稳定性结果。
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