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The generalized persistence diagram (GPD) is a natural extension of the classical persistence barcode to the setting of multi-parameter persistence and beyond. The GPD is defined as an integer-valued function whose domain is the set of intervals in the indexing poset of a persistence module, and is known to be able to capture richer topological information than its single-parameter counterpart. However, computing the GPD is computationally prohibitive due to the sheer size of the interval set. Restricting the GPD to a subset of intervals provides a way to manage this complexity, compromising discriminating power to some extent. However, identifying and computing an effective restriction of the domain that minimizes the loss of discriminating power remains an open challenge. In this work, we introduce a novel method for optimizing the domain of the GPD through gradient descent optimization. To achieve this, we introduce a loss function tailored to optimize the selection of intervals, balancing computational efficiency and discriminative accuracy. The design of the loss function is based on the known erosion stability property of the GPD. We showcase the efficiency of our sparsification method for dataset classification in supervised machine learning. Experimental results demonstrate that our sparsification method significantly reduces the time required for computing the GPDs associated to several datasets, while maintaining classification accuracies comparable to those achieved using full GPDs. Our method thus opens the way for the use of GPD-based methods to applications at an unprecedented scale., Comment: 22 pages, 5 figures, 3 tables

The Generalized Persistence Diagram (GPD) for multi-parameter persistence naturally extends the classical notion of persistence diagram for one-parameter persistence. However, unlike its classical counterpart, computing the GPD remains a significant challenge. The main hurdle is that, while the GPD is defined as the M\"obius inversion of the Generalized Rank Invariant (GRI), computing the GRI is intractable due to the formidable size of its domain, i.e., the set of all connected and convex subsets in a finite grid in $\mathbb{R}^d$ with $d \geq 2$. This computational intractability suggests seeking alternative approaches to computing the GPD. In order to study the complexity associated to computing the GPD, it is useful to consider its classical one-parameter counterpart, where for a filtration of a simplicial complex with $n$ simplices, its persistence diagram contains at most $n$ points. This observation leads to the question: 'Given a $d$-parameter simplicial filtration, could the cardinality of its GPD (specifically, the support of the GPD) also be bounded by a polynomial in the number of simplices in the filtration?' This is the case for $d=1$, where we compute the persistence diagram directly at the simplicial filtration level. If this were also the case for $d\geq2$, it might be possible to compute the GPD directly and much more efficiently without relying on the GRI. We show that the answer to the question above is negative, demonstrating the inherent difficulty of computing the GPD. More specifically, we construct a sequence of $d$-parameter simplicial filtrations where the cardinalities of their GPDs are not bounded by any polynomial in the the number of simplices. Furthermore, we show that several commonly used methods for constructing multi-parameter filtrations can give rise to such "wild" filtrations., Comment: 23 pages, 9 figures

Given an unknown $\mathbb{R}^n$-valued function $f$ on a metric space $X$, can we approximate the persistent homology of $f$ from a finite sampling of $X$ with known pairwise distances and function values? This question has been answered in the case $n=1$, assuming $f$ is Lipschitz continuous and $X$ is a sufficiently regular geodesic metric space, and using filtered geometric complexes with fixed scale parameter for the approximation. In this paper we answer the question for arbitrary $n$, under similar assumptions and using function-geometric multifiltrations. Our analysis offers a different view on these multifiltrations by focusing on their approximation properties rather than on their stability properties. We also leverage the multiparameter setting to provide insight into the influence of the scale parameter, whose choice is central to this type of approach., Comment: 18 pages

Persistent (co)homology is a central construction in topological data analysis, where it is used to quantify prominence of features in data to produce stable descriptors suitable for downstream analysis. Persistence is challenging to compute in parallel because it relies on global connectivity of the data. We propose a new algorithm to compute persistent cohomology in the distributed setting. It combines domain and range partitioning. The former is used to reduce and sparsify the coboundary matrix locally. After this initial local reduction, we redistribute the matrix across processors for the global reduction. We experimentally compare our cohomology algorithm with DIPHA, the only publicly available code for distributed computation of persistent (co)homology; our algorithm demonstrates a significant improvement in strong scaling., Comment: 11 pages, 16 figures

This thesis addresses the theory of topological spaces and the foundations of persistence theory. We will discuss chain complexes and the associated simplicial homology groups, as well as their relationship with singular homology theory. Moreover, we present the fundamental concepts of algebraic topology, including exact and short exact sequences and relative homology groups derived from quotienting with subspaces of a topological space. These tools are used to prove the Excision Theorem in algebraic topology. Subsequently, the theorem is applied to demonstrate the equivalence of simplicial and singular homology for triangulable topological spaces, i.e. those topological spaces which admit a simplicial structure. This enables a more general theory of homology to be adopted in the study of filtrations of point clouds. The chapter on homological persistence makes use of these tools throughout. We develop the theory of persistent homology, the homology of filtrations of topological spaces, and the corresponding dual concept of persistent cohomology. This work aims to provide mathematicians with a robust foundation for productive engagement with the aforementioned theories. The majority of the proofs have been rewritten to clarify the relationships between the techniques discussed. The novel aspect of this contribution is the canonical presentation of persistence theory and the associated ideas through a rigorous mathematical treatment for triangulable topological spaces and closing some gaps in the existing literature., Comment: Bachelor's thesis

We propose an improvement to the Mapper algorithm that removes the assumption of a single resolution scale across semantic space, and improves the robustness of the results under change of parameters. This eases parameter selection, especially for datasets with highly variable local density in the Morse function $f$ used for Mapper. This is achieved by incorporating this density into the choice of cover for Mapper. Furthermore, we prove that for covers with some natural hypotheses, the graph output by Mapper still converges in bottleneck distance to the Reeb graph of the Rips complex of the data, but captures more topological features than when using the usual Mapper cover. Finally, we discuss implementation details, and include the results of computational experiments. We also provide an accompanying reference implementation., Comment: 29 pages, 8 figures

When working with (multi-parameter) persistence modules, one usually makes some type of tameness assumption in order to obtain better control over their algebraic behavior. One such notion is Ezra Millers notion of finite encodability, which roughly states that a persistence module can be obtained by pulling back a finite dimensional persistence module over a finite poset. From the perspective of homological algebra, finitely encodable persistence have an inconvenient property: They do not form an abelian category. Here, we prove that if one restricts to such persistence modules which can be constructed in terms of topologically closed and sufficiently constructible (piecewise linear, semi-algebraic, etc.) upsets then abelianity can be restored.

Gradient regularization (GR), which aims to penalize the gradient norm atop the loss function, has shown promising results in training modern over-parameterized deep neural networks. However, can we trust this powerful technique? This paper reveals that GR can cause performance degeneration in adaptive optimization scenarios, particularly with learning rate warmup. Our empirical and theoretical analyses suggest this is due to GR inducing instability and divergence in gradient statistics of adaptive optimizers at the initial training stage. Inspired by the warmup heuristic, we propose three GR warmup strategies, each relaxing the regularization effect to a certain extent during the warmup course to ensure the accurate and stable accumulation of gradients. With experiments on Vision Transformer family, we confirm the three GR warmup strategies can effectively circumvent these issues, thereby largely improving the model performance. Meanwhile, we note that scalable models tend to rely more on the GR warmup, where the performance can be improved by up to 3\% on Cifar10 compared to baseline GR. Code is available at \href{https://github.com/zhaoyang-0204/gnp}{https://github.com/zhaoyang-0204/gnp}., Comment: ICML 2024 paper

In computer vision, keypoint detection is a fundamental task, with applications spanning from robotics to image retrieval; however, existing learning-based methods suffer from scale dependency and lack flexibility. This paper introduces a novel approach that leverages Morse theory and persistent homology, powerful tools rooted in algebraic topology. We propose a novel loss function based on the recent introduction of a notion of subgradient in persistent homology, paving the way toward topological learning. Our detector, MorseDet, is the first topology-based learning model for feature detection, which achieves competitive performance in keypoint repeatability and introduces a principled and theoretically robust approach to the problem., Comment: Accepted to ICML 2024

In this paper, we study pointwise finite-dimensional (p.f.d.) $2$-parameter persistence modules where each module admits a finite convex isotopy subdivision. We show that a p.f.d. $2$-parameter persistence module $M$ (with a finite convex isotopy subdivision) is isomorphic to a $2$-parameter persistence module $N$ where the restriction of $N$ to each chamber of the parameter space $(\mathbb{R},\leq)^2$ is a constant functor. Moreover, we show that the convex isotopy subdivision of $M$ induces a finite encoding of $M$. Finally, we prove that every indecomposable thin $2$-parameter persistence module is isomorphic to a polytope module., Comment: 19 Pages

In this position paper, we argue that when hypergraphs are used to capture multi-way local relations of data, their resulting topological features describe global behaviour. Consequently, these features capture complex correlations that can then serve as high fidelity inputs to autoencoder-driven anomaly detection pipelines. We propose two such potential pipelines for cybersecurity data, one that uses an autoencoder directly to determine network intrusions, and one that de-noises input data for a persistent homology system, PHANTOM. We provide heuristic justification for the use of the methods described therein for an intrusion detection pipeline for cyber data. We conclude by showing a small example over synthetic cyber attack data.

Cell biologists study in parallel the morphology of cells with the regulation mechanisms that modify this morphology. Such studies are complicated by the inherent heterogeneity present in the cell population. It remains difficult to define the morphology of a cell with parameters that can quantify this heterogeneity, leaving the cell biologist to rely on manual inspection of cell images. We propose an alternative to this manual inspection that is based on topological data analysis. We characterise the shape of a cell by its contour and nucleus. We build a filtering of the edges defining the contour using a radial distance function initiated from the nucleus. This filtering is then used to construct a persistence diagram that serves as a signature of the cell shape. Two cells can then be compared by computing the Wasserstein distance between their persistence diagrams. Given a cell population, we then compute a distance matrix that includes all pairwise distances between its members. We analyse this distance matrix using hierarchical clustering with different linkage schemes and define a purity score that quantifies consistency between those different schemes, which can then be used to assess homogeneity within the cell population. We illustrate and validate our approach to identify sub-populations in human mesenchymal stem cell populations., Comment: 21 pages, 7 Figures

We use tools from applied topology for feature selection on vector-valued time series data. We employ persistent homology and sliding window embeddings to quantify the coordinated dynamics of time series. We describe an algorithm for gradient descent to assign scores, or weights, to the variables of the time series based on their contribution to the dynamics as quantified by persistent homology; the result is a convex combination of a subset of the variables. In this setting, we prove persistence vineyards are piecewise linear and we give a simple formula for the derivatives of the vines. We demonstrate our method of topological feature selection with synthetic data and C. elegans neuronal data., Comment: 15 pages

We apply persistent homology, the main method in topological data analysis, to the study of demographic data. Persistence diagrams efficiently summarize information about clusters or peaks in a region's demographic data. To illustrate how persistence diagrams can be used for exploratory analysis, we undertake a study of the 100 largest U.S.~cities and their Black and Hispanic populations. We use our method to find clusters in individual cities, determine which cities are outliers and why, measure and describe change in demographic patterns over time, and roughly categorize cities into distinct groups based on the topology of their demographics. Along the way, we highlight the advantages and disadvantages of persistence diagrams as a tool for analyzing geospatial data.

Data consisting of a graph with a function mapping into $\mathbb{R}^d$ arise in many data applications, encompassing structures such as Reeb graphs, geometric graphs, and knot embeddings. As such, the ability to compare and cluster such objects is required in a data analysis pipeline, leading to a need for distances between them. In this work, we study the interleaving distance on discretization of these objects, called mapper graphs when $d=1$, where functor representations of the data can be compared by finding pairs of natural transformations between them. However, in many cases, computation of the interleaving distance is NP-hard. For this reason, we take inspiration from recent work by Robinson to find quality measures for families of maps that do not rise to the level of a natural transformation, called assignments. We then endow the functor images with the extra structure of a metric space and define a loss function which measures how far an assignment is from making the required diagrams of an interleaving commute. Finally we show that the computation of the loss function is polynomial with a given assignment. We believe this idea is both powerful and translatable, with the potential to provide approximations and bounds on interleavings in a broad array of contexts., Comment: Major updates and cleanup for resubmission

For safety and robustness of AI systems, we introduce topological parallax as a theoretical and computational tool that compares a trained model to a reference dataset to determine whether they have similar multiscale geometric structure. Our proofs and examples show that this geometric similarity between dataset and model is essential to trustworthy interpolation and perturbation, and we conjecture that this new concept will add value to the current debate regarding the unclear relationship between overfitting and generalization in applications of deep-learning. In typical DNN applications, an explicit geometric description of the model is impossible, but parallax can estimate topological features (components, cycles, voids, etc.) in the model by examining the effect on the Rips complex of geodesic distortions using the reference dataset. Thus, parallax indicates whether the model shares similar multiscale geometric features with the dataset. Parallax presents theoretically via topological data analysis [TDA] as a bi-filtered persistence module, and the key properties of this module are stable under perturbation of the reference dataset., Comment: 18 pages, 6 figures. Preprint submitted to NeurIPS 2023

While decomposition of one-parameter persistence modules behaves nicely, as demonstrated by the algebraic stability theorem, decomposition of multiparameter modules is well known to be unstable in a certain precise sense. Until now, it has not been clear that there is any way to get around this and build a meaningful stability theory for multiparameter module decomposition, as naive attempts to do this tend to fail. We introduce tools and definitions, in particular $\epsilon$-refinements and the $\epsilon$-erosion neighborhood of a module, to make sense of the question of how to build such a theory. Having built this foundation, we define the $\epsilon$-pruning of a module, which is a new invariant acting like a ``refined barcode'' that shows great promise to extract features from a module by approximately decomposing it. The main theorem of the paper can be interpreted as a generalization of the algebraic stability theorem to multiparameter modules up to a factor of $2r$, where $r$ is the maximal pointwise dimension of one of the modules. Another interpretation is that the pruning of a module $M$ detects all summands appearing in modules in a neighborhood of $M$. Furthermore, we show that the factor $2r$ appearing in the theorem is close to optimal. We also define a distance that we conjecture to be a stable bottleneck-type distance, which has long been a missing piece in multipersistence theory. Finally, we discuss whether the main theorem can be improved for modules that decompose into pointwise low-dimensional summands, and pose a conjecture phrased purely in terms of basic linear algebra and graph theory that seems to capture the difficulty of doing this. We also show that this conjecture is relevant for other areas of multipersistence, like the complexity of approximating the interleaving distance, and recent applications of homological algebra to multipersistence., Comment: 40 pages, 7 figures

We consider persistent homology obtained by applying homology to the open Rips filtration of a compact metric space $(X,d)$. We show that each decrease in zero-dimensional persistence and each increase in one-dimensional persistence is induced by local minima of the distance function $d$. When $d$ attains local minimum at only finitely many pairs of points, we prove that each above mentioned change in persistence is induced by a specific critical edge in Rips complexes, which represents a local minimum of $d$. We use this fact to develop a theory (including interpretation) of critical edges of persistence. The obtained results include upper bounds for the rank of one-dimensional persistence and a corresponding reconstruction result. Of potential computational interest is a simple geometric criterion recognizing local minima of $d$ that induce a change in persistence. We conclude with a proof that each locally isolated minimum of $d$ can be detected through persistent homology with selective Rips complexes. The results of this paper offer the first interpretation of critical scales of persistent homology (obtained via Rips complexes) for general compact metric spaces., Comment: 20 pages, 5 figures

Persistence diagrams are common descriptors of the topological structure of data appearing in various classification and regression tasks. They can be generalized to Radon measures supported on the birth-death plane and endowed with an optimal transport distance. Examples of such measures are expectations of probability distributions on the space of persistence diagrams. In this paper, we develop methods for approximating continuous functions on the space of Radon measures supported on the birth-death plane, as well as their utilization in supervised learning tasks. Indeed, we show that any continuous function defined on a compact subset of the space of such measures (e.g., a classifier or regressor) can be approximated arbitrarily well by polynomial combinations of features computed using a continuous compactly supported function on the birth-death plane (a template). We provide insights into the structure of relatively compact subsets of the space of Radon measures, and test our approximation methodology on various data sets and supervised learning tasks., Comment: 16 pages, 4 figures

The Extended Persistent Homology Transform (XPHT) is 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 can define a distance between two shapes by integrating over the sphere the distance between their respective extended persistence modules. By using extended persistence we get finite distances between shapes even when they have different Betti numbers. We use Morse theory to show that the extended persistence of a height function over a manifold with boundary can be deduced from the extended persistence for that height function restricted to the boundary, alongside labels on the critical points as positive or negative critical. 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., Comment: 42 pages, 9 figures

Advances in AI based computer vision has led to a significant growth in synthetic image generation and artificial image tampering with serious implications for unethical exploitations that undermine person identification and could make render AI predictions less explainable.Morphing, Deepfake and other artificial generation of face photographs undermine the reliability of face biometrics authentication using different electronic ID documents.Morphed face photographs on e-passports can fool automated border control systems and human guards.This paper extends our previous work on using the persistent homology (PH) of texture landmarks to detect morphing attacks.We demonstrate that artificial image tampering distorts the spatial distribution of texture landmarks (i.e. their PH) as well as that of a set of image quality characteristics.We shall demonstrate that the tamper caused distortion of these two slim feature vectors provide significant potentials for building explainable (Handcrafted) tamper detectors with low error rates and suitable for implementation on constrained devices., Comment: 6 pages, 7 figures, 3 tables

Motivated by persistent homology and topological data analysis, we consider formal sums on a metric space with a distinguished subset. These formal sums, which we call persistence diagrams, have a canonical 1-parameter family of metrics called Wasserstein distances. We study the topological and metric properties of these spaces. Some of our results are new even in the case of persistence diagrams on the half-plane. Under mild conditions, no persistence diagram has a compact neighborhood. If the underlying metric space is $\sigma$-compact then so is the space of persistence diagrams. However, under mild conditions, the space of persistence diagrams is not hemicompact and the space of functions from this space to a topological space is not metrizable. Spaces of persistence diagrams inherit completeness and separability from the underlying metric space. Some spaces of persistence diagrams inherit being path connected, being a length space, and being a geodesic space, but others do not. We give criteria for a set of persistence diagrams to be totally bounded and relatively compact. We also study the curvature and dimension of spaces of persistence diagrams and their embeddability into a Hilbert space. As an important technical step, which is of independent interest, we give necessary and sufficient conditions for the existence of optimal matchings of persistence diagrams., Comment: 40 pages, v3: improvements based on referees' comments

Given a filtration function on a finite simplicial complex, stability theorem of persistent homology states that the corresponding barcode is continuous with respect to changes in the filtration function. However, due to the discrete setting of simplicial complexes, the critical simplices terminating matched bars cannot change continuously for arbitrary perturbations of filtration functions. In this paper we provide a sufficient condition for rigidity of a terminal simplex, i.e., a condition on $\epsilon > 0$ implying that the terminal simplex of a homology class or a bar in persistent homology remains constant through $\epsilon$-perturbations of filtration function. The condition for a homology class or a bar in dimension n depends on the barcodes in dimensions n and n+1., Comment: 21 pages, 6 figures

The degree-Rips bifiltration is the most computable of the parameter-free, density-sensitive bifiltrations in topological data analysis. It is known that this construction is stable to small perturbations of the input data, but its robustness to outliers is not well understood. In recent work, Blumberg-Lesnick prove a result in this direction using the Prokhorov distance and homotopy interleavings. Based on experimental evaluation, they argue that a more refined approach is desirable, and suggest the framework of homology inference. Motivated by these experiments, we consider a probability measure that is uniform with high density on an annulus, and uniform with low density on the disc inside the annulus. We compute the degree-Rips complexes of this probability space up to homotopy type, using the Adamaszek-Adams computation of the Vietoris-Rips complexes of the circle. These degree-Rips complexes are the limit objects for the Blumberg-Lesnick experiments. We argue that the homology inference approach has strong explanatory power in this case, and suggest studying the limit objects directly as a strategy for further work., Comment: 14 pages, 3 figures. To appear in SoCG 2022. Comments welcome

An Important tool in the field topological data analysis is known as persistent Homology (PH) which is used to encode abstract representation of the homology of data at different resolutions in the form of persistence diagram (PD). In this work we build more than one PD representation of a single image based on a landmark selection method, known as local binary patterns, that encode different types of local textures from images. We employed different PD vectorizations using persistence landscapes, persistence images, persistence binning (Betti Curve) and statistics. We tested the effectiveness of proposed landmark based PH on two publicly available breast abnormality detection datasets using mammogram scans. Sensitivity of landmark based PH obtained is over 90% in both datasets for the detection of abnormal breast scans. Finally, experimental results give new insights on using different types of PD vectorizations which help in utilising PH in conjunction with machine learning classifiers., Comment: 14 pages

The homology of an unknown subspace of Euclidean space can be determined from the intrinsic \v{C}ech complex of a sample of points in the subspace, without reference to the ambient Euclidean space. More precisely, given a subspace $X$ of Euclidean space and a sample $A$ of points in $X$, we give conditions for the homology of $X$ to be isomorphic to a certain persistent homology group of the intrinsic \v{C}ech complex., Comment: 16 pages

Simplicial complexes can be viewed as high dimensional generalizations of graphs that explicitly encode multi-way ordered relations between vertices at different resolutions, all at once. This concept is central towards detection of higher dimensional topological features of data, features to which graphs, encoding only pairwise relationships, remain oblivious. While attempts have been made to extend Graph Neural Networks (GNNs) to a simplicial complex setting, the methods do not inherently exploit, or reason about, the underlying topological structure of the network. We propose a graph convolutional model for learning functions parametrized by the $k$-homological features of simplicial complexes. By spectrally manipulating their combinatorial $k$-dimensional Hodge Laplacians, the proposed model enables learning topological features of the underlying simplicial complexes, specifically, the distance of each $k$-simplex from the nearest "optimal" $k$-th homology generator, effectively providing an alternative to homology localization., Comment: 9 pages, 5 figures

The ability to abstract, count, and use System~2 reasoning are well-known manifestations of intelligence and understanding. In this paper, we argue, using the example of the ``Look and Say" puzzle, that although deep neural networks can exhibit high `competence' (as measured by accuracy) when trained on large data sets (2 million examples in our case), they do not show any sign on the deeper understanding of the problem, or what D. Dennett calls `comprehension'. We report on two sets experiments: first, computing the next element of the sequence, and ,then, the previous element. We view both problems as building a translator from one set of tokens to another. We apply both standard LSTMs and Transformer/Attention-based neural networks, using publicly available machine translation software. We observe that despite the amazing accuracy, the performance of the trained programs on the actual L\&S sequence is bad, and shows no understanding of the principles behind the sequences. The ramifications of this finding include: (1) from the cognitive science perspective, we argue that we need better mathematical models of abstraction; (2) the universality of neural networks should be re-examined for functions acting on discrete data sets; (3) we hypothesize topology can provide a definition of without the reference to the concept of distance., Comment: 12 pages; 5 figures

This paper proposes a stable volume and a stable volume variant, referred to as a stable sub-volume, for more reliable data analysis using persistent homology. In prior research, an optimal cycle and similar ideas have been proposed to identify the homological structure corresponding to each birth-death pair in a persistence diagram. While this is helpful for data analysis using persistent homology, the results are sensitive to noise. In this paper, stable volumes and stable sub-volumes are proposed to solve this problem. For a special case, we prove that a stable volume is the robust part of an optimal volume against noise. We implemented stable volumes and sub-volumes on HomCloud, a data analysis software package based on persistent homology, and show examples of stable volumes and sub-volumes.

In the first half this paper, we generalize the theory of layer points for Lesnick- (or degree-Rips-) complexes to the more general context of $\vec{v}$-hierarchical clusterings. Layer points provide a compressed description of a hierarchical clustering by recording only the points where a cluster changes. For multi-parameter hierarchical clusterings we consider both a global notion of layer points and layer points in the direction of a single parameter. An interleaving of hierarchical clusterings of the same set induces an interleaving of global layer points. In the particular, we consider cases where a hierarchical clustering of a finite metric space, $Y$, is interleaved with a hierarchical clustering of some sample $X \subseteq Y$. In the second half, we focus on the hierarchical clustering $\pi_0 L_{-,k}(Y)$ for some finite metric space $Y$. When $X \subseteq Y$ satisfies certain conditions guaranteeing $X$ is well dispersed in $Y$ and the points of $Y$ are dense around $X$, there is an interleaving of layer points for $\pi_0 L_{-,k}(Y)$ and a truncated version of $L_{-,0}(X) = V_{-}(X)$. Under stronger conditions, this interleaving defines a retract from the layer points for $\pi_0 L_{-,k}(Y)$ to the layer points for $\pi_0 L_{-,0}(X)$., Comment: 18 pages, 4 figures

We construct a filtered simplicial complex $(X_L,f_L)$ associated to a subset $X\subset \mathbb{R}^d$, a function $f:X\rightarrow \mathbb{R}$ with compactly supported sublevel sets, and a collection of landmark points $L\subset \mathbb{R}^d$. The persistence values $f_L(\Delta)$ are defined as the minimizing values of a family of constrained optimization problems, whose domains are certain higher order Voronoi cells associated to $L$. We prove that $H_k^{a,b}(X_L)\cong H^{a,b}_k(X)$ provided that $f$ is the restriction of a smooth function, the landmarks are sufficiently dense, and $a

For a fixed $N$, we analyze the space of all sequences $z=(z_1,\dots,z_N)$, approximating a continuous function on the circle, with a given persistence diagram $P$, and show that the typical components of this space are homotopy equivalent to $S^1$. We also consider the space of functions on $Y$-shaped (resp., star-shaped) trees with a 2-point persistence diagram, and show that this space is homotopy equivalent to $S^1$ (resp., to a bouquet of circles).

In the last couple of years there were a few attempts to apply topological data analysis to text, and in particular to natural language inference. A recent work by Tymochko et al. suggests the possibility of capturing `the notion of logical shape in text,' using `topological delay embeddings,' a technique derived from dynamical systems, applied to word embeddings. In this note we reconstruct their argument and show, using several old and new examples, that the problem of connecting logic, topology and text is still very much unsolved. We conclude that there is no clear answer to the question: ``Can we find a circle in a circular argument?'' We point out some possible avenues of exploration. The code used in our experiment is also shown., Comment: 12 pages, 8 figures

In Topological Data Analysis, filtered chain complexes enter the persistence pipeline between the initial filtering of data and the final persistence invariants extraction. It is known that they admit a tame class of indecomposables, called interval spheres. In this paper, we provide an algorithm to decompose filtered chain complexes into such interval spheres. This algorithm provides geometric insights into various aspects of the standard persistence algorithm and two of its run-time optimizations. Moreover, since it works for any filtered chain complexes, our algorithm can be applied in more general cases. As an application, we show how to decompose filtered kernels with it., Comment: Revised version, decomposition of filtered kernels added

Identifying the role of network units in deep neural networks (DNNs) is critical in many aspects including giving understandings on the mechanisms of DNNs and building basic connections between deep learning and neuroscience. However, there remains unclear on which roles the units in DNNs with different generalization ability could present. To this end, we give role taxonomy of units in DNNs via introducing the retrieval-of-function test, where units are categorized into four types in terms of their functional preference on separately the training set and testing set. We show that ratios of the four categories are highly associated with the generalization ability of DNNs from two distinct perspectives, based on which we give signs of DNNs with well generalization.

In topological data analysis persistence modules are used to distinguish the legitimate topological features of a finite data set from noise. Interleavings between persistence modules feature prominantly in the analysis. One can show that for $\epsilon$ positive, the collection of $\epsilon$-interleavings between two persistence modules $M$ and $N$ has the structure of an affine variety, Thus, the smallest value of $\epsilon$ corresponding to a nonempty variety is the interleaving distance. With this in mind, it is natural to wonder how this variety changes with the value of $\epsilon$, and what information about $M$ and $N$ can be seen from just the knowledge of their varieties. In this paper, we focus on the special case where $M$ and $N$ are interval modules. In this situation we classify all possible progressions of varieties, and determine what information about $M$ and $N$ is present in the progression.

We apply persistent homology, the dominant tool from the field of topological data analysis, to study electoral redistricting. Our method combines the geographic information from a political districting plan with election data to produce a persistence diagram. We are then able to visualize and analyze large ensembles of computer-generated districting plans of the type commonly used in modern redistricting research (and court challenges). We set out three applications: zoning a state at each scale of districting, comparing elections, and seeking signals of gerrymandering. Our case studies focus on redistricting in Pennsylvania and North Carolina, two states whose legal challenges to enacted plans have raised considerable public interest in the last few years. To address the question of robustness of the persistence diagrams to perturbations in vote data and in district boundaries, we translate the classical stability theorem of Cohen--Steiner et al. into our setting and find that it can be phrased in a manner that is easy to interpret. We accompany the theoretical bound with an empirical demonstration to illustrate diagram stability in practice.

We define the notion of an approximate triangulation for a manifold $M$ embedded in euclidean space. The basic idea is to build a nested family of simplicial complexes whose vertices lie in $M$ and use persistent homology to find a complex in the family whose homology agrees with that of $M$. Our key examples are various Grassmann manifolds $G_k({\mathbb R}^n)$., Comment: 16 pages, 13 figures

A primary hypothesis that drives scientific and engineering studies is that data has structure. The dominant paradigms for describing such structure are statistics (e.g., moments, correlation functions) and signal processing (e.g., convolutional neural nets, Fourier series). Topological Data Analysis (TDA) is a field of mathematics that analyzes data from a fundamentally different perspective. TDA represents datasets as geometric objects and provides dimensionality reduction techniques that project such objects onto low-dimensional spaces that are composed of elementary geometric objects. Key property of these elementary objects (also known as topological features) are that they persist at different scales and that they are stable under perturbations (e.g., noise, stretching, twisting, and bending). In this work, we review key mathematical concepts and methods of TDA and present different applications in chemical engineering., Comment: 42 pages, 44 figures

In this paper we propose a computationally efficient multiple hypothesis testing procedure for persistent homology. The computational efficiency of our procedure is based on the observation that one can empirically simulate a null distribution that is universal across many hypothesis testing applications involving persistence homology. Our observation suggests that one can simulate the null distribution efficiently based on a small number of summaries of the collected data and use this null in the same way that p-value tables were used in classical statistics. To illustrate the efficiency and utility of the null distribution we provide procedures for rejecting acyclicity with both control of the Family-Wise Error Rate (FWER) and the False Discovery Rate (FDR). We will argue that the empirical null we propose is very general conditional on a few summaries of the data based on simulations and limit theorems for persistent homology for point processes., Comment: 43 pages, 16 figures