Topological Data Analysis with Bregman Divergences

Given a finite set in a metric space, the topological analysis generalizes hierarchical clustering using a 1-parameter family of homology groups to quantify connectivity in all dimensions. The connectivity is compactly described by the persistence diagram. One limitation of the current framework is the reliance on metric distances, whereas in many practical applications objects are compared by non-metric dissimilarity measures. Examples are the Kullback-Leibler divergence, which is commonly used for comparing text and images, and the Itakura-Saito divergence, popular for speech and sound. These are two members of the broad family of dissimilarities called Bregman divergences. We show that the framework of topological data analysis can be extended to general Bregman divergences, widening the scope of possible applications. In particular, we prove that appropriately generalized Cech and Delaunay (alpha) complexes capture the correct homotopy type, namely that of the corresponding union of Bregman balls. Consequently, their filtrations give the correct persistence diagram, namely the one generated by the uniformly growing Bregman balls. Moreover, we show that unlike the metric setting, the filtration of Vietoris-Rips complexes may fail to approximate the persistence diagram. We propose algorithms to compute the thus generalized Cech, Vietoris-Rips and Delaunay complexes and experimentally test their efficiency. Lastly, we explain their surprisingly good performance by making a connection with discrete Morse theory

Flood risk – Prevention and Impact on Agricultural Lands

Recent extreme weather events have resulted in an ongoing discussion on the issues of land use and compensation payments within Austrian agriculture. Building on a functional evaluation system for agricultural lands as developed within the Interreg IIIB project “ILUP”, the national project “Agriculture and Flooding” has as its goal to classify the flood-protection contribution and flood sensitivity of agricultural lands. This, in turn, enables the recommendation of targeted measures for potentially improving flood situations, as well as an estimate of their implementation costs. In addition to the digital soil map, other fundamental sources used for the project are the digital flood risk map, IACS land-use data and works by the Institute for Land and Water Management Research. Reference values and marginal returns sourced from the Federal Institute of Agricultural Economics also flow into the cost estimates for the recommended combination. The results will contribute to an understanding of the multifunctionality of agricultural lands and to the setting of priorities on a regional scale regarding packaged flood-prevention and damage-minimization. However, the results at hand can only serve as one step toward regional flood protection projects, whose development will require the cooperation of all interest groups.Agriculture, Multifunctionality, Sustainability, Flood Risk, Resource /Energy Economics and Policy, Q24, Q25, Q54, Q56,

Slice, Simplify and Stitch: Topology-Preserving Simplification Scheme for Massive Voxel Data

We focus on efficient computations of topological descriptors for voxel data. This type of data includes 2D greyscale images, 3D medical scans, but also higher-dimensional scalar fields arising from physical simulations. In recent years we have seen an increase in applications of topological methods for such data. However, computational issues remain an obstacle. We therefore propose a streaming scheme which simplifies large 3-dimensional voxel data - while provably retaining its persistent homology. We combine this scheme with an efficient boundary matrix reduction implementation, obtaining an end-to-end tool for persistent homology of large data. Computational experiments show its state-of-the-art performance. In particular, we are now able to robustly handle complex datasets with several billions voxels on a regular laptop. A software implementation called Cubicle is available as open-source: https://bitbucket.org/hubwag/cubicle

Leibniz International Proceedings in Information, LIPIcs

Smallest enclosing spheres of finite point sets are central to methods in topological data analysis. Focusing on Bregman divergences to measure dissimilarity, we prove bounds on the location of the center of a smallest enclosing sphere. These bounds depend on the range of radii for which Bregman balls are convex

Topological data analysis in information space

Various kinds of data are routinely represented as discrete probability distributions. Examples include text documents summarized by histograms of word occurrences and images represented as histograms of oriented gradients. Viewing a discrete probability distribution as a point in the standard simplex of the appropriate dimension, we can understand collections of such objects in geometric and topological terms. Importantly, instead of using the standard Euclidean distance, we look into dissimilarity measures with information-theoretic justification, and we develop the theory needed for applying topological data analysis in this setting. In doing so, we emphasize constructions that enable the usage of existing computational topology software in this context

The role of the plasma-membrane Ca2+-ATPase in ca2+ homeostasis in Sinapis alba root hairs

The regulation of cytosolic Ca2+ has been investigated in growing root-hair cells of Sinapis alba L. with special emphasis on the role of the plasmamembrane Ca2+-ATPase. For this purpose, erythrosin B was used to inhibit the Ca2+-ATPase, and the Ca2+ ionophore A23187 was applied to manipulate cytosolic free [Ca2+] which was then measured with Ca2+-selective microelectrodes. (i) At 0.01 M, A23187 had no effect on the membrane potential but enhanced the Ca2+ permeability of the plasma membrane. Higher concentrations of this ionophore strongly depolarized the cells, also in the presence of cyanide. (ii) Unexpectedly, A23187 first caused a decrease in cytosolic Ca2+ by 0.2 to 0.3 pCa units and a cytosolic acidification by about 0.5 pH units, (iii) The depletion of cytosolic free Ca2+ spontaneously reversed and became an increase, a process which strongly depended on the external Ca2+ concentration, (iv) Upon removal of A23187, the cytosolic free [Ca2+] returned to its steady-state level, a process which was inhibited by erythrosin B. We suggest that the first reaction to the intruding Ca2+ is an activation of Ca2+ transporters (e.g. ATPases at the endoplasmic reticulum and the plasma membrane) which rapidly remove Ca2+ from the cytosol. The two observations that after the addition of A23187, (i) Ca2+ gradients as steep as-600 mV could be maintained and (ii) the cytosolic pH rapidly and immediately decreased without recovery indicate that the Ca2+-exporting plasma-membrane ATPase is physiologically connected to the electrochemical pH gradient, and probably works as an nH+/Ca2+-ATPase. Based on the finding that the Ca2+-ATPase inhibitor erythrosin B had no effect on cytosolic Ca2+, but caused a strong Ca2+ increase after the addion of A23187 we conclude that these cells, at least in the short term, have enough metabolic energy to balance the loss in transport activity caused by inhibition of the primary Ca2+-pump. We further conclude that this ATPase is a major Ca2+ regulator in stress situations where the cytosolic Ca2+ has been shifted from its steady-state level, as may be the case during processes of signal transduction

Signal Transduction in Sinapis alba Root Hairs: Auxins as External Messengers

In developing root hairs of Sinapis alba the effects of externally applied indole-3-acetic acid (IAA) and other auxins have been investigated with respect to membrane potential, membrane conductance, cytosolic Ca2 + and pH. Following a delay of roughly 30s, 10- 12 to lO- IO M IAA slowly hyperpolarize, 10- 7 M IAA rapidly depolarize the root hairs, while 10- 9 M has hardly an effect. We show that these voltage responses are not the result of a change in membrane conductance or permeability, but are presumably caused by a change in H+ ATPase activity. The other tested auxins and analogues yielded comparable effects, but with much lower effectivity (IAA > 1-NAA > 2,4-D ~ 2-NAA > 2,3-D). Cytosolic Ca2+ and pH were decreased during depolarization by 0.2 and 0.4 units, respectively. No such changes were observed during hyperpolarization or about 1 h after the first encounter of the root hairs with IAA. We propose that IAA is a natural external signal for roots while competing with neighboring organisms for nutrients and salts, and suggest a signal chain with the plasma membrane H+ ATPase as a target protein. The delay in response to IAA, the time dependency, and the extremely low effective IAA concentrations point to the existence of a IAA receptor. Since the IAA-induced shifts in cytosolic pH and Ca2+ occur simultaneously with the depolarization, the question whether these ions are cellular messengers and part of an IAA-triggered signal chain is critically discussed