% papers.bib — the single source for the Papers page. % One entry per paper, NEWEST FIRST. Numbers (P1, P2, ... and 00, 01, ...) are computed from this order: % entries WITHOUT a journal/booktitle are preprints; entries WITH one are published. % To add a preprint: put the PDF in source/, paste arXiv's BibTeX at the top, add file = {source/.pdf} and abstract = {...}. % When it is published: add journal = {...} and doi = {...} and update year. Optionally move it to just above the newest published entry. % Optional: shortjournal = {...} is shown in the table instead of the full journal name. % If there is no PDF in source/, leave out the file field and the page links to arXiv instead. % Write accented names directly (Temčinas) rather than as {\v{c}}. Maths in abstracts: $ ... $ or $ ... $. % NOTE: author order below is alphabetical by surname; please check against the papers where order differs. @misc{landfall, title = {{Landfall Geometry}}, author = {Kirwan, Frances and Nanda, Vidit}, year = {2026}, eprint = {2610.30993}, archiveprefix = {arXiv}, file = {source/landfall.pdf}, abstract = {We study the negative gradient flow of analytic maps $f:X\to\R$ defined on compact analytic Riemannian manifolds without imposing any nondegeneracy or transversality assumptions. We construct tight isolating blocks around the connected components of the critical set $\Crit(f)$ and record only bounded-time contacts made after an orbit exits one block and before it makes landfall within the interior of another. These contacts assemble into a strict topological category $\cL$ -- called the landfall category -- whose classifying space $B\cL$ is shown to be simple homotopy equivalent to $X$ naturally over all down-sets of the flow partial order. The filtered homotopy type is independent of the isolating blocks. Passing to a finite subanalytic triangulation yields a flow-induced acyclic partial matching, and hence, an explicit finite chain complex generated by the non-exit simplices of the blocks which computes the homology of $X$. The finite model also computes the Morse--Conley spectral sequence over arbitrary coefficients; over a field, homological perturbation further produces a connection matrix whose entries are finite sums over descending contact itineraries.} } @misc{smtcell, title = {{Stratified Morse Theory for Cell Complexes}}, author = {Nanda, Vidit and Tombari, Francesca}, year = {2026}, eprint = {2601.18343}, archiveprefix = {arXiv}, file = {source/dsmt.pdf}, abstract = {We develop a version of discrete Morse theory for finite regular CW complexes equipped with an auxiliary stratification. The key construction is the halo of a cell, which contains all those faces in the boundary that enter closed sublevelsets precisely when the threshold reaches that cell’s value. The complement of this halo in the boundary, called the shadow, is always a subcomplex. A stratified discrete Morse function requires Forman’s conditions on each stratum together with the requirement that closures of paired cells admit filtered collapses onto their shadows. We establish fundamental Morse lemmas: filtered collapses across regular intervals, and controlled attachments at critical values. For functions satisfying only the stratum-wise Forman condition, we construct an upper envelope on the barycentric subdivision whose local Morse data decomposes into horizontal and vertical components. This yields a simplicial analogue of the standard tangential-normal splitting of local Morse data in the sense of Goresky and MacPherson.} } @misc{morsedrug, title = {{A Morse Transform for Drug Discovery}}, author = {Asaad, Aras and Cooper, Richard and Nanda, Vidit and Tanaka, Alex}, year = {2025}, eprint = {2503.04507}, archiveprefix = {arXiv}, file = {source/MorseDrug_1.pdf}, abstract = {We introduce a new ligand-based virtual screening (LBVS) framework that uses piecewise-linear (PL) Morse theory to predict ligand binding potential. We model ligands as simplicial complexes via a pruned Delaunay triangulation and catalogue the critical points across multiple directional height-functions. This produces a rich feature vector, consisting of crucial topological features – peaks, troughs and saddles – that characterise ligand surfaces relevant to binding interactions. Unlike contemporary LBVS methods that rely on computationally intensive deep neural networks, we require only a lightweight classifier. The Morse-theoretic approach achieves state-of-the-art performance on standard datasets while offering an interpretable feature vector and scalable method for ligand prioritization in early-stage drug discovery.} } @misc{harmsel, title = {{Quiver Laplacians and Feature Selection}}, author = {Harrington, Heather and Nanda, Vidit and Sumray, Otto}, year = {2024}, eprint = {2404.06993}, archiveprefix = {arXiv}, file = {source/HarmSel.pdf}, abstract = {The challenge of selecting the most relevant features of a given dataset arises ubiquitously in data analysis and dimensionality reduction. However, features found to be of high importance for the entire dataset may not be relevant to subsets of interest, and vice versa. Given a feature selector and a fixed decomposition of the data into subsets, we describe a method for identifying selected features which are compatible with the decomposition into subsets. We achieve this by re-framing the problem of finding compatible features to one of finding sections of a suitable quiver representation. In order to approximate such sections, we then introduce a Laplacian operator for quiver representations valued in Hilbert spaces. We provide explicit bounds on how the spectrum of a quiver Laplacian changes when the representation and the underlying quiver are modified in certain natural ways. Finally, we apply this machinery to the study of peak-calling algorithms which measure chromatin accessibility in single-cell data. We demonstrate that eigenvectors of the associated quiver Laplacian yield locally and globally compatible features.} } @article{realstrat, title = {{Effective Whitney Stratification of Real Algebraic Varieties}}, author = {Helmer, Martin and Leykin, Anton and Nanda, Vidit}, year = {2025}, journal = {Mathematics of Computation}, shortjournal = {Math. Comp.}, url = {https://www.ams.org/journals/mcom/0000-000-00/S0025-5718-2025-04157-7/}, eprint = {2307.05427}, archiveprefix = {arXiv}, file = {source/realstrat_X.pdf}, abstract = {We describe an algorithm to compute Whitney stratifications of real algebraic varieties. The basic idea is to first stratify the complexified version of the given real variety using conormal techniques, and then to show that the resulting stratifications admit a description using only real polynomials. We also extend these methods to stratification problems involving the so-called full semialgebraic sets as well as real algebraic maps.} } @article{goodfit, title = {{Goodness-of-fit via Count Statistics in Dense Random Simplicial Complexes}}, author = {Nanda, Vidit and Reinert, Gesine and Temčinas, Tadas}, year = {2025}, journal = {Foundations of Data Science}, shortjournal = {Found. Data Sci.}, doi = {10.3934/fods.2025013}, eprint = {2309.14017}, archiveprefix = {arXiv}, file = {source/GoodFit_X.pdf}, abstract = {A key object of study in stochastic topology is a random simplicial complex. In this work we study a multi-parameter random simplicial complex model, where the probability of including a $k$-simplex, given the lower dimensional structure, is fixed. This leads to a conditionally independent probabilistic structure. This model includes the Erdős–Rényi random graph, the random clique complex as well as the Linial-Meshulam complex as special cases. The model is studied from both probabilistic and statistical points of view. We prove multivariate central limit theorems with bounds and known limiting covariance structure for the subcomplex counts and the number of critical simplices under a lexicographical acyclic partial matching. We use the CLTs to develop a goodness-of-fit test for this random model and evaluate \gr{its} empirical performance. In order for the test to be applicable in practice, we also prove that the MLE estimators are asymptotically unbiased, consistent, uncorrelated and normally distributed.} } @article{singdet, title = {{HADES: Fast Singularity Detection with Local Measure Comparison}}, author = {Lim, Uzu and Nanda, Vidit and Oberhauser, Harald}, year = {2025}, journal = {SIAM Journal on Mathematics of Data Science}, shortjournal = {SIMODS}, doi = {10.1137/24M1665763}, eprint = {2311.04171}, archiveprefix = {arXiv}, file = {source/SingDet_X.pdf}, abstract = {We introduce Hades , an unsupervised algorithm to detect singularities in data. This algorithm employs a kernel goodness-of-fit test, and as a consequence it is much faster and far more scaleable than the existing topology-based alternatives. Using tools from differential geometry and optimal transport theory, we prove that Hades correctly detects singularities with high probability when the data sample lives on a transverse intersection of equidimensional manifolds. In computational experiments, Hades recovers singularities in synthetically generated data, branching points in road network data, intersection rings in molecular conformation space, and anomalies in image data. Note: This paper is accompanied by Uzu's wonderful singularity detection software, written in python and available on github !} } @article{hqrep, title = {{Multilinear Hyperquiver Representations}}, author = {Muller, Tommi and Nanda, Vidit and Seigal, Anna}, year = {2025}, journal = {Foundations of Computational Mathematics}, shortjournal = {FoCM}, doi = {10.1007/s10208-025-09692-z}, eprint = {2305.05622}, archiveprefix = {arXiv}, file = {source/MultQuivRep.pdf}, abstract = {We count singular vector tuples of a system of tensors assigned to the edges of a directed hypergraph. To do so, we study the generalisation of quivers to directed hypergraphs. Assigning vector spaces to the nodes of a hypergraph and multilinear maps to its hyperedges gives a hyperquiver representation. Hyperquiver representations generalise quiver representations (where all hyperedges are edges) and tensors (where there is only one multilinear map). The singular vectors of a hyperquiver representation are a compatible assignment of vectors to the nodes. We compute the dimension and degree of the variety of singular vectors of a sufficiently generic hyperquiver representation. Our formula specialises to known results that count the singular vectors and eigenvectors of a generic tensor.} } @article{highsig1, title = {{A Topological Approach to Mapping Space Signatures}}, author = {Giusti, Chad and Lee, Darrick and Nanda, Vidit and Oberhauser, Harald}, year = {2025}, journal = {Advances in Applied Mathematics}, shortjournal = {Adv. Appl. Math.}, url = {https://www.sciencedirect.com/science/article/abs/pii/S0196885824001192}, eprint = {2202.00491}, archiveprefix = {arXiv}, file = {source/MSSig1.pdf}, abstract = {A common approach for describing classes of functions and probability measures on a topological space $\mathcal{X}$ is to construct a suitable map $\Phi$ from $\mathcal{X}$ into a vector space, where linear methods can be applied to address both problems. The case where $\mathcal{X}$ is a space of paths $[0,1] \to \mathbb{R}^n$ and $\Phi$ is the path signature map has received much attention in stochastic analysis and related fields. In this article we develop a generalized $\Phi$ for the case where $\mathcal{X}$ is a space of maps $[0,1]^d \to \mathbb{R}^n$ for any $d \in \mathbb{N}$, and show that the map $\Phi$ generalizes many of the desirable algebraic and analytic properties of the path signature to $d \ge 2$. The key ingredient to our approach is topological; in particular, our starting point is a generalisation of K-T Chen's path space cochain construction to the setting of cubical mapping spaces.} } @article{hnpmod, title = {{Harder-Narasimhan Filtrations of Persistence Modules}}, author = {Fersztand, Marc and Jacquard, Emile and Nanda, Vidit and Tillmann, Ulrike}, year = {2024}, journal = {Transactions of the London Mathematical Society}, shortjournal = {Trans. LMS}, doi = {10.1112/tlm3.70003}, eprint = {2303.16075}, archiveprefix = {arXiv}, file = {source/HNPers.pdf}, abstract = {The Harder-Narasimhan type of a quiver representation is a discrete invariant parameterised by a real-valued function (called a central charge) defined on the vertices of the quiver. In this paper, we investigate the strength and limitations of Harder-Narasimhan types for several families of quiver representations which arise in the study of persistence modules. We introduce the skyscraper invariant, which amalgamates the HN types along central charges supported at single vertices, and generalise the rank invariant from multiparameter persistence modules to arbitrary quiver representations. Our four main results are as follows: (1) we show that the skyscraper invariant is strictly finer than the rank invariant in full generality, (2) we characterise the set of complete central charges for zigzag (and hence, ordinary) persistence modules, (3) we extend the preceding characterisation to rectangle-decomposable multiparameter persistence modules of arbitrary dimension; and finally, (4) we show that although no single central charge is complete for nestfree ladder persistence modules, a finite set of central charges is complete.} } @article{dimdet, title = {{Tangent Space and Dimension Estimation with the Wasserstein Distance}}, author = {Lim, Uzu and Nanda, Vidit and Oberhauser, Harald}, year = {2024}, journal = {SIAM Journal on Applied Algebra and Geometry}, shortjournal = {SIAGA}, doi = {10.1137/22M1522711}, eprint = {2110.06357}, archiveprefix = {arXiv}, file = {source/DimDet.pdf}, abstract = {We provide explicit bounds on the number of sample points required to estimate tangent spaces and intrinsic dimensions of (smooth, compact) Euclidean submanifolds via local principal component analysis. Our approach directly estimates local covariance matrices, which in turn allows us to simultaneously estimate both the tangent spaces and the intrinsic dimension of a manifold. The key arguments involve a matrix concentration inequality, a Wasserstein bound for flattening a manifold, and a Lipschitz relation for the covariance matrix with respect to the Wasserstein distance.} } @article{mcog, title = {{Morse Theory for Complexes of Groups}}, author = {Nanda, Vidit and Yerolemou, Naya}, year = {2024}, journal = {Journal of Pure and Applied Algebra}, shortjournal = {JPAA}, url = {https://www.sciencedirect.com/science/article/pii/S0022404924000033}, eprint = {2203.00539}, archiveprefix = {arXiv}, file = {source/MCoG_X.pdf}, abstract = {We construct an equivariant version of discrete Morse theory for simplicial complexes endowed with group actions. The key ingredient is a 2-categorical criterion for making acyclic partial matchings on the quotient space compatible with an overlaid complex of groups. We use the discrete flow category of any such compatible matching to build the corresponding Morse complex of groups. Our main result establishes that the development of the Morse complex of groups recovers the original simplicial complex up to equivariant homotopy equivalence.} } @article{hnzz, title = {{Harder-Narasimhan Filtrations and Zigzag Persistence}}, author = {Fersztand, Marc and Nanda, Vidit and Tillmann, Ulrike}, year = {2023}, journal = {Advances in Applied Mathematics}, shortjournal = {Adv. Appl. Math.}, url = {https://www.sciencedirect.com/science/article/pii/S0196885823001525}, eprint = {2211.07553}, archiveprefix = {arXiv}, file = {source/HNZZ_X.pdf}, abstract = {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.} } @article{multclt, title = {{Multivariate Central Limit Theorems for Random Clique Complexes}}, author = {Nanda, Vidit and Reinert, Gesine and Temčinas, Tadas}, year = {2023}, journal = {Journal of Applied and Computational Topology}, shortjournal = {J. Appl. Comput. Topol.}, doi = {10.1007/s41468-023-00146-5}, eprint = {2112.08922}, archiveprefix = {arXiv}, file = {source/MultCLT_X.pdf}, abstract = {Motivated by open problems in applied and computational algebraic topology, we establish multivariate normal approximation theorems for three random vectors which arise organically in the study of random clique complexes. These are: 1. the vector of critical simplex counts attained by a lexicographical Morse matching, 2. the vector of simplex counts in the link of a fixed simplex, and 3. the vector of total simplex counts. The first of these random vectors forms a cornerstone of modern homology algorithms, while the second one provides a natural generalisation for the notion of vertex degree, and the third one may be viewed from the perspective of $U$-statistics. To obtain distributional approximations for these random vectors, we extend the notion of dissociated sums to a multivariate setting and prove a new central limit theorem for such sums using Stein's method.} } @article{coninf, title = {{Topological Inference of the Conley Index}}, author = {Nanda, Vidit and Yim, Ka Man}, year = {2023}, journal = {Journal of Dynamics and Differential Equations}, shortjournal = {J. Dyn. Diff. Equ.}, doi = {10.1007/s10884-023-10310-1}, eprint = {2206.10198}, archiveprefix = {arXiv}, file = {source/ConleyInf_X.pdf}, abstract = {The Conley index of an isolated invariant set is a fundamental object in the study of dynamical systems. Here we consider smooth functions on closed submanifolds of Euclidean space and describe a framework for inferring the Conley index of any compact, connected isolated critical set of such a function with high confidence from a sufficiently large finite point sample. The main construction of this paper is a specific index pair which is local to the critical set in question. We establish that these index pairs have positive reach and hence admit a sampling theory for robust homology inference. This allows us to estimate the Conley index, and as a direct consequence, we are also able to estimate the Morse index of any critical point of a Morse function using finitely many local evaluations.} } @article{vecsur, title = {{A Survey of Vectorization Methods in Topological Data Analysis}}, author = {Ali, Dashti and Asaad, Aras and Jimenez, Maria-Jose and Nanda, Vidit and Paluzo-Hidalgo, Eduardo and Soriano-Trigueros, Manuel}, year = {2023}, journal = {IEEE Transactions on Pattern Analysis and Machine Intelligence}, shortjournal = {IEEE TPAMI}, url = {https://ieeexplore.ieee.org/document/10235748}, eprint = {2212.09703}, archiveprefix = {arXiv}, file = {source/persfeat_X.pdf}, abstract = {Attempts to incorporate topological information in supervised learning tasks have resulted in the creation of several techniques for vectorizing persistent homology barcodes. In this paper, we study thirteen such methods. Besides describing an organizational framework for these methods, we comprehensively benchmark them against three well-known classification tasks. Surprisingly, we discover that the best-performing method is a simple vectorization, which consists only of a few elementary summary statistics. Finally, we provide a convenient web application which has been designed to facilitate exploration and experimentation with various vectorization methods.} } @article{clam, title = {{Complex Links and Hilbert-Samuel Multiplicities}}, author = {Helmer, Martin and Nanda, Vidit}, year = {2023}, journal = {SIAM Journal on Applied Algebra and Geometry}, shortjournal = {SIAGA}, doi = {10.1137/22M1475533}, eprint = {2006.10452}, archiveprefix = {arXiv}, file = {source/CLHSM_X.pdf}, abstract = {We describe a framework for estimating Hilbert-Samuel multiplicities $e_XY$ for pairs of projective varieties $X < Y$ from finite point samples rather than defining equations. The first step involves proving that this multiplicity remains invariant under certain hyperplane sections which reduce $X$ to a point $p$ and $Y$ to a curve $C$. Next, we establish that $e_pC$ equals the Euler characteristic (and hence, the cardinality) of the complex link of $p$ in $C$. Finally, we provide explicit bounds on the number of uniform point samples needed (in an annular neighborhood of $p$ in $C$) to determine this Euler characteristic with high confidence.} } @article{barbase, title = {{The Space of Barcode Bases for Persistence Modules}}, author = {Jacquard, Emile and Nanda, Vidit and Tillmann, Ulrike}, year = {2022}, journal = {Journal of Applied and Computational Topology}, shortjournal = {J. Appl. Comput. Topol.}, doi = {10.1007/s41468-022-00094-6}, eprint = {2111.03700}, archiveprefix = {arXiv}, file = {source/Barbase_X.pdf}, abstract = {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 modules can be represented via a partial matching between bars provided that neither source nor target admits nested bars in its barcode. We also generalise the algorithm and results described above to work for zizag modules.} } @article{constr, title = {{Conormal Spaces and Whitney Stratifications}}, author = {Helmer, Martin and Nanda, Vidit}, year = {2022}, journal = {Foundations of Computational Mathematics}, shortjournal = {FoCM}, doi = {10.1007/s10208-022-09574-8}, eprint = {2106.14555}, archiveprefix = {arXiv}, file = {source/Constrat_XI.pdf}, abstract = {We describe a new algorithm for computing Whitney stratifications of complex projective varieties. The main ingredients are (a) an algebraic criterion, due to Lê and Teissier, which reformulates Whitney regularity in terms of conormal spaces and maps, and (b) a new interpretation of this conormal criterion via primary decomposition, which can be practically implemented on a computer. We show that this algorithm improves upon the existing state of the art by several orders of magnitude, even for relatively small input varieties. En route, we introduce related algorithms for efficiently stratifying affine varieties, flags on a given variety, and algebraic maps.} } @article{qpc, title = {{Principal Components along Quiver Representations}}, author = {Harrington, Heather and Nanda, Vidit and Seigal, Anna}, year = {2022}, journal = {Foundations of Computational Mathematics}, shortjournal = {FoCM}, doi = {10.1007/s10208-022-09563-x}, eprint = {2104.10666}, archiveprefix = {arXiv}, file = {source/QPC_X.pdf}, abstract = {Quiver representations arise naturally in many areas across mathematics. Here we describe an algorithm for calculating the vector space of sections, or compatible assignments of vectors to vertices, of any finite-dimensional representation of a finite quiver. Consequently, we are able to define and compute principal components with respect to quiver representations. These principal components are solutions to constrained optimisation problems defined over the space of sections, and are eigenvectors of an associated matrix pencil.} } @inproceedings{sfgnn, title = {{Dist2Cycle: A Simplicial Neural Network for Homology Localization}}, author = {Keros, Alex and Nanda, Vidit and Subr, Kartic}, year = {2022}, booktitle = {Proceedings of the AAAI Conference on Artificial Intelligence}, shortjournal = {AAAI}, url = {https://ojs.aaai.org/index.php/AAAI/article/view/20673}, eprint = {2110.15182}, archiveprefix = {arXiv}, file = {source/SFGNN_X.pdf}, abstract = {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 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.} } @article{esr, title = {{Equivariant Simplicial Reconstruction}}, author = {Carbone, Lisa and Nanda, Vidit and Naqvi, Yusra}, year = {2020}, journal = {SIAM Journal on Applied Algebra and Geometry}, shortjournal = {SIAGA}, doi = {10.1137/20M1327483}, eprint = {1807.09396}, archiveprefix = {arXiv}, file = {source/EqSimRecX.pdf}, abstract = {We introduce and analyze parallelizable algorithms to compress and accurately reconstruct finite simplicial complexes that have non-trivial automorphisms. The compressed data – called a complex of groups – amounts to a functor from (the poset of simplices in) the orbit space to the 2-category of groups, whose higher structure is prescribed by isomorphisms arising from conjugation. Using this functor, we show how to algorithmically recover the original complex up to equivariant simplicial isomorphism. Our algorithms are derived from generalizations (by Bridson-Haefliger, Carbone-Rips and Corson, among others) of the classical Bass-Serre theory for reconstructing group actions on trees.} } @article{gadata, title = {{Geometric Anomaly Detection in Data}}, author = {Harrington, Heather and Nanda, Vidit and Stolz, Bernadette and Tanner, Jared}, year = {2020}, journal = {Proceedings of the National Academy of Sciences}, shortjournal = {PNAS}, url = {https://www.pnas.org/content/early/2020/07/31/2001741117}, eprint = {1812.05593}, archiveprefix = {arXiv}, file = {source/gadata_X.pdf}, abstract = {This paper describes the systematic application of local topological methods for detecting interfaces and related anomalies in complicated high-dimensional data. By examining the topology of small regions around each point, one can optimally stratify a given dataset into clusters, each of which is in turn well-approximable by a suitable submanifold of the ambient space. Since these approximating submanifolds might have different dimensions, we are able to detect non-manifold like singular regions in data even when none of the data points have been sampled from those singularities. We showcase this method by identifying the intersection of two surfaces in the 24-dimensional space of cyclo-octane conformations, and by locating all the self-intersections of a Henneberg minimal surface immersed in 3-dimensional space. Due to the local nature of the required topological computations, the algorithmic burden of performing such data stratification is readily distributable across several processors.} } @inproceedings{bistrat, title = {{Canonical Stratifications along Bisheaves}}, author = {Nanda, Vidit and Patel, Amit}, year = {2020}, booktitle = {Topological Data Analysis: The Abel Symposium 2018}, shortjournal = {Abel Symposia}, doi = {10.1007/978-3-030-43408-3_15}, eprint = {1812.05593}, archiveprefix = {arXiv}, file = {source/bistrat_XI.pdf}, abstract = {A theory of bisheaves has been recently introduced to measure the homological stability of fibers of maps to manifolds. A bisheaf over a topological space is a triple consisting of a sheaf, a cosheaf, and compatible maps from the stalks of the sheaf to the stalks of the cosheaf. In this note we describe how, given a bisheaf constructible (i.e., locally constant) with respect to a triangulation of its underlying space, one can explicitly determine the coarsest stratification of that space for which the bisheaf remains constructible.} } @article{strat, title = {{Local Cohomology and Stratification}}, author = {Nanda, Vidit}, year = {2019}, journal = {Foundations of Computational Mathematics}, shortjournal = {FoCM}, doi = {10.1007/s10208-019-09424-0}, eprint = {1707.00354}, archiveprefix = {arXiv}, file = {source/Strat_X.pdf}, abstract = {We outline an algorithm to recover the canonical (or, coarsest) stratification of a given finite-dimensional regular CW complex into cohomology manifolds, each of which is a union of cells. The construction proceeds by iteratively localizing the poset of cells about a family of subposets; these subposets are in turn determined by a collection of cosheaves which capture variations in cohomology of cellular neighborhoods across the underlying complex. The result is a nested sequence of categories, each containing all the cells as its set of objects, with the property that two cells are isomorphic in the last category if and only if they lie in the same canonical stratum. The entire process is amenable to efficient distributed computation.} } @article{flowcat, title = {{Discrete Morse Theory and Localization}}, author = {Nanda, Vidit}, year = {2019}, journal = {Journal of Pure and Applied Algebra}, shortjournal = {JPAA}, doi = {10.1016/j.jpaa.2018.04.001}, eprint = {1510.01907}, archiveprefix = {arXiv}, file = {source/Flowcat_press.pdf}, abstract = {Incidence relations among the cells of a regular CW complex produce a 2-category of entrance paths whose classifying space is homotopy-equivalent to that complex. We show here that each acyclic partial matching on (the cells of) such a complex corresponds precisely to a homotopy-preserving localization of the associated entrance path category. Restricting attention further to the full localized subcategory spanned by critical cells, we obtain the discrete flow category whose classifying space is also shown to lie in the homotopy class of the original CW complex. This flow category forms a combinatorial and computable counterpart to the one described here by Cohen , Jones and Segal in the context of smooth Morse theory.} } @article{sigs, title = {{Persistence Paths and Signature Features in Topological Data Analysis}}, author = {Chevyrev, Ilya and Nanda, Vidit and Oberhauser, Harald}, year = {2018}, journal = {IEEE Transactions on Pattern Analysis and Machine Intelligence}, shortjournal = {IEEE TPAMI}, url = {https://ieeexplore.ieee.org/document/8567922}, eprint = {1806.00381}, archiveprefix = {arXiv}, file = {source/paths.pdf}, abstract = {We introduce a new feature map for barcodes that arise in persistent homology computation. The main idea is to first realize each barcode as a path in a convenient vector space, and to then compute its path signature which takes values in the tensor algebra of that vector space. The composition of these two operations — barcode to path, path to tensor series — results in a feature map that has several desirable properties for statistical learning, such as universality and characteristicness, and achieves state-of-the-art results on common classification benchmarks.} } @article{discjs, title = {{Discrete Morse Theory and Classifying Spaces}}, author = {Nanda, Vidit and Tamaki, Dai and Tanaka, Kohei}, year = {2018}, journal = {Advances in Mathematics}, shortjournal = {Adv. Math.}, url = {https://www.sciencedirect.com/science/article/pii/S000187081830402X}, eprint = {1612.08429}, archiveprefix = {arXiv}, file = {source/discrete_CJSX.pdf}, abstract = {The aim of this paper is to develop a refinement of Forman’s discrete Morse theory. To an acyclic partial matching µ on a finite regular CW complex X , Forman introduced a discrete analogue of gradient flows. Although Forman’s gradient flow has been proved to be useful in practical computations of homology groups, it is not sufficient to recover the homotopy type of X . Forman also proved the existence of a CW complex which is homotopy equivalent to X and whose cells are in one-to-one correspondence with the critical cells of µ, but the construction is ad hoc and does not have a combinatorial description. By relaxing the definition of Forman’s gradient flows, we introduce the notion of flow paths, which contains enough information to reconstruct the homotopy type of X , while retaining a combinatorial description. The critical difference from Forman’s gradient flows is the existence of a partial order on the set of flow paths, from which a 2-category C (µ) is constructed. It is shown that the classifying space of C (µ) is homotopy equivalent to X by using homotopy theory of 2-categories. This result may be also regarded as a discrete analogue of the unpublished work of Cohen, Jones, and Segal on Morse theory from the early 90’s.} } @article{pers-ricci, title = {{Topological Signals of Singularities in Ricci Flow}}, author = {Alsing, Paul and Blair, Howard and Corne, Matthew and Jones, Gordon and Miller, Warner and Mischaikow, Konstantin and Nanda, Vidit}, year = {2017}, journal = {Axioms}, url = {http://www.mdpi.com/2075-1680/6/3/24}, file = {source/PersRicci_X.pdf}, abstract = {We implement methods from computational homology to obtain a topological signal of singularity formation in a selection of geometries evolved numerically by Ricci flow. Our approach, based on persistent homology, produces precise, quantitative measures describing the behavior of an entire collection of data across a discrete sample of times. We analyze the topological signals of geometric criticality obtained numerically from the application of persistent homology to models manifesting singularities under Ricci flow. The results we obtain for these numerical models suggest that the topological signals distinguish global singularity formation (collapse to a round point) from local singularity formation (neckpinch). Finally, we discuss the interpretation and implication of these results and future applications.} } @article{pgl1, title = {{Higher Interpolation and Extension for Persistence Modules}}, author = {Bubenik, Peter and de Silva, Vin and Nanda, Vidit}, year = {2017}, journal = {SIAM Journal on Applied Algebra and Geometry}, shortjournal = {SIAGA}, doi = {10.1137/16M1100472}, file = {source/PGL_X.pdf}, abstract = {The use of topological persistence in contemporary data analysis has provided considerable impetus for investigations into the geometric and functional-analytic structure of the space of persistence modules. In this paper, we isolate a coherence criterion which guarantees the extensibility of non-expansive maps into this space across embeddings of the domain to larger ambient metric spaces. Our coherence criterion is category-theoretic, allowing Kan extensions to provide the desired extensions. As a consequence of such “higher interpolation”, it becomes possible to compare Vietoris-Rips and Cech complexes built within the space of persistence modules.} } @article{morse-sheaf, title = {{Discrete Morse Theory for Computing Cellular Sheaf Cohomology}}, author = {Curry, Justin and Ghrist, Robert and Nanda, Vidit}, year = {2016}, journal = {Foundations of Computational Mathematics}, shortjournal = {FoCM}, doi = {10.1007/s10208-015-9266-8}, eprint = {1312.6454}, archiveprefix = {arXiv}, file = {source/DMSheaf_Final.pdf}, abstract = {Sheaves and sheaf cohomology are powerful tools in computational topology, greatly generalizing persistent homology. We develop an algorithm for simplifying the computation of cellular sheaf cohomology via (discrete) Morse-theoretic techniques. As a consequence, we derive efficient techniques for computation of (ordinary) cohomology of a cell complex.} } @article{pers-comp, title = {{A Topological Measurement of Protein Compressibility}}, author = {Gameiro, Marcio and Hiraoka, Yasuaki and Izumi, Shunsuke and Kramar, Miroslav and Mischaikow, Konstantin and Nanda, Vidit}, year = {2015}, journal = {Japan Journal of Industrial and Applied Mathematics}, shortjournal = {Japan J. Ind. Appl. Math.}, doi = {10.1007/s13160-014-0153-5}, file = {source/compressibility-final.pdf}, abstract = {Given X-ray crystallography data of a protein molecule from the PDB , we build a van der Waal weighted alpha shape representation of that protein molecule by constructing cells around each atom center. Thus, to each protein we assoicate a set of persistence diagrams (one for each dimension). Using elementary physical principles, we identify certain structural features of molecules that are conjectured to impact compressibility. A simple parameter search through the persistence diagrams isolates these features and provides a robust measure which exhibits remarkable linear correlation with experimentally computed protein compressibility.} } @article{map-samp, title = {{Reconstructing Functions from Random Samples}}, author = {Ferry, Steve and Mischaikow, Konstantin and Nanda, Vidit}, year = {2014}, journal = {Journal of Computational Dynamics}, shortjournal = {J. Comput. Dyn.}, url = {http://www.aimsciences.org/journals/displayArticlesnew.jsp?paperID=10624}, eprint = {1310.3390}, archiveprefix = {arXiv}, file = {source/mh_paper_Final.pdf}, abstract = {From a sufficiently large point sample lying on a compact Riemannian submanifold of Euclidean space, one can construct a simplicial complex which is homotopy-equivalent to that manifold with high confidence. We describe a corresponding result for a Lipschitz-continuous function between two such manifolds. That is, we outline the construction of a simplicial map which recovers the induced maps on homotopy and homology groups with high confidence using only finite sampled data from the domain and range, as well as knowledge of the image of every point sampled from the domain. We provide explicit bounds on the size of the point samples required for such reconstruction in terms of intrinsic properties of the domain, the co-domain and the function. This reconstruction is robust to certain types of bounded sampling and evaluation noise.} } @article{morse-maps, title = {{Discrete Morse Theoretic Algorithms for Computing Homology of Complexes and Maps}}, author = {Harker, Shaun and Mischaikow, Konstantin and Mrozek, Marian and Nanda, Vidit}, year = {2014}, journal = {Foundations of Computational Mathematics}, shortjournal = {FoCM}, doi = {10.1007/s10208-013-9145-0}, file = {source/MorseHomologyX.pdf}, abstract = {We provide explicit and efficient algorithms based on discrete Morse theory to compute homology of a very general class of complexes. A set-valued map of top-dimensional cells between such complexes is a natural discrete approximation of an underlying (and possibly unknown) continuous function, especially when the evaluation of that underlying function is subject to measurement errors. We introduce a new Morse theoretic algorithm for deriving chain maps from these set-valued maps, and hence an effective scheme for computing the map induced on homology by the approximated continuous function.} } @incollection{simp-mod, title = {{Simplicial Models and Topological Inference in Biological Systems}}, author = {Nanda, Vidit and Sazdanovic, Radmila}, year = {2014}, booktitle = {Discrete and Topological Models in Molecular Biology}, publisher = {Springer}, shortjournal = {Springer (chapter)}, doi = {10.1007/978-3-642-40193-0_6}, file = {source/PersBio-Final.pdf}, abstract = {This article is a user's guide to algebraic topological methods for data analysis with a particular focus on applications to datasets arising in experimental biology. We begin with the combinatorics and geometry of simplicial complexes and outline the standard techniques for imposing filtered simplicial structures on a general class of datasets. From these structures, one computes topological statistics of the original data via the algebraic theory of (persistent) homology. These statistics are shown to be computable and robust invariants of the shape underlying a dataset. Finally, we showcase some appealing instances of topology-driven inference in biological settings -- from the detection of a new type of breast cancer to the analysis of various neural structures.} } @inproceedings{pers-geo, title = {{Geometry in the Space of Persistence Modules}}, author = {de Silva, Vin and Nanda, Vidit}, year = {2013}, booktitle = {Proceedings of the 29th Annual Symposium on Computational Geometry}, shortjournal = {SoCG}, url = {https://dl.acm.org/citation.cfm?id=2462402}, file = {source/pgeom2.pdf}, abstract = {We study the geometry of the space of persistence modules and diagrams, with special attention to Cech and Rips complexes. The metric structures are determined in terms of interleaving maps (of modules) and matchings (between diagrams). We show that the relationship between the Cech and Rips complexes is governed by the relationship between the corresponding interleavings and matchings.} } @article{morse-pers, title = {{Morse Theory for Filtrations and Efficient Computation of Persistent Homology}}, author = {Mischaikow, Konstantin and Nanda, Vidit}, year = {2013}, journal = {Discrete \& Computational Geometry}, shortjournal = {Discrete Comput. Geom.}, doi = {10.1007/s00454-013-9529-6}, file = {source/DMPers-Final.pdf}, abstract = {We introduce an efficient preprocessing algorithm to reduce the number of cells in a filtered cell complex while preserving its persistent homology groups. The technique is based on an extension of combinatorial Morse theory from complexes to filtrations of cell complexes.} } @phdthesis{thesis, title = {{Discrete Morse Theory for Filtrations}}, author = {Nanda, Vidit}, year = {2012}, school = {Rutgers University}, url = {http://hdl.rutgers.edu/1782.1/rucore10001600001.ETD.000066925}, file = {source/Thesis.pdf}, abstract = {My Ph.D. dissertation from October 2012. The main results are from the paper Morse theory for filtrations and efficient computation of persistent homology; the final chapter uses filtered Morse theory to simplify long exact sequences via the zigzag lemma.} }