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Register a new userOn the complexity of zero-dimensional multiparameter persistence
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Multiparameter persistence is a natural extension of the well-known persistent homology, which has attracted a lot of interest. However, there are major theoretical obstacles preventing the full development of this promising theory. In this paper we consider the interesting special case of multiparameter persistence in zero dimensions which can be regarded as a form of multiparameter clustering. In particular, we consider the multiparameter persistence modules of the zero-dimensional homology of filtered topological spaces when they are finitely generated. Under certain assumptions, we characterize such modules and study their decompositions. In particular we identify a natural class of representations that decompose and can be extended back to form zero-dimensional multiparameter persistence modules. Our study of this set of representations concludes that despite the restrictions, there are still infinitely many classes of indecomposables in this set.
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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 theorem characterizes finitely encoded modules as admitting appropriately finite presentations and resolutions. The geometric and algebraic theory focuses on modules over real polyhedral groups (real vector spaces with polyhedral positive cones) and a parallel theory over discrete polyhedral groups (abelian groups with finitely generated positive cones). Existence of primary decomposition is proved over arbitrary polyhedral partially ordered abelian groups, but the real and discrete cases carry enough geometry and, crucially in the real case, topology to induce complete theories of minimal primary and secondary decomposition, associated and attached faces, minimal generators and cogenerators, socles and tops, minimal upset covers and downset hulls, Matlis duality, and minimal fringe presentation. Real semialgebraic properties of data are preserved by functorial constructions. Tops and socles become functorial birth and death spaces for multiparameter persistence modules. They yield functorial QR codes and elder morphisms for modules over real and discrete polyhedral groups that generalize and categorify the bar code and elder rule for persistent homology in one parameter. The disparate ways that QR codes and elder morphisms model bar codes coalesce, in one parameter, to functorial bar codes.
We study the multiparty communication complexity of high dimensional permutations, in the Number On the Forehead (NOF) model. This model is due to Chandra, Furst and Lipton (CFL) who also gave a nontrivial protocol for the Exactly-n problem where three players receive integer inputs and need to decide if their inputs sum to a given integer $n$. There is a considerable body of literature dealing with the same problem, where $(mathbb{N},+)$ is replaced by some other abelian group. Our work can be viewed as a far-reaching extension of this line of work. We show that the known lower bounds for that group-theoretic problem apply to all high dimensional permutations. We introduce new proof techniques that appeal to recent advances in Additive Combinatorics and Ramsey theory. We reveal new and unexpected connections between the NOF communication complexity of high dimensional permutations and a variety of well known and thoroughly studied problems in combinatorics. Previous protocols for Exactly-n all rely on the construction of large sets of integers without a 3-term arithmetic progression. No direct algorithmic protocol was previously known for the problem, and we provide the first such algorithm. This suggests new ways to significantly improve the CFL protocol. Many new open questions are presented throughout.
The aim of applied topology is to use and develop topological methods for applied mathematics, science and engineering. One of the main tools is persistent homology, an adaptation of classical homology, which assigns a barcode, i.e. a collection of intervals, to a finite metric space. Because of the nature of the invariant, barcodes are not well-adapted for use by practitioners in machine learning tasks. We can circumvent this problem by assigning numerical quantities to barcodes and these outputs can then be used as input to standard algorithms. It is the purpose of this paper to identify tropical coordinates on the space of barcodes and prove that they are stable with respect to the bottleneck distance and Wasserstein distances.
Biological and physical systems often exhibit distinct structures at different spatial/temporal scales. Persistent homology is an algebraic tool that provides a mathematical framework for analyzing the multi-scale structures frequently observed in nature. In this paper a theoretical framework for the algorithmic computation of an arbitrarily good approximation of the persistent homology is developed. We study the filtrations generated by sub-level sets of a function $f : X to mathbb{R}$, where $X$ is a CW-complex. In the special case $X = [0,1]^N$, $N in mathbb{N}$ we discuss implementation of the proposed algorithms. We also investigate a priori and a posteriori bounds of the approximation error introduced by our method.
Let $mathrm{TC}_r(X)$ denote the $r$-th topological complexity of a space $X$. In many cases, the generating function $sum_{rge 1}mathrm{TC}_{r+1}(X)x^r$ is a rational function $frac{P(x)}{(1-x)^2}$ where $P(x)$ is a polynomial with $P(1)=mathrm{cat}(X)$, that is, the asymptotic growth of $mathrm{TC}_r(X)$ with respect to $r$ is $mathrm{cat}(X)$. In this paper, we introduce a lower bound $mathrm{MTC}_r(X)$ of $mathrm{TC}_r(X)$ for a rational space $X$, and estimate the growth of $mathrm{MTC}_r(X)$.
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