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Cartograms are a technique for visually representing geographically distributed statistical data, where values of a numerical attribute are mapped to the size of geographic regions. Contiguous cartograms preserve the adjacencies of the…
Cartograms are maps in which the areas of regions (e.g., countries or provinces) are proportional to a thematic mapping variable (e.g., population or gross domestic product). A cartogram is called contiguous if it keeps geographically…
Many modern datasets don't fit neatly into $n \times p$ matrices, but most techniques for measuring statistical stability expect rectangular data. We study methods for stability assessment on non-rectangular data, using statistical learning…
Cartograms depict geographic regions with areas proportional to quantitative data. However, when created using density-equalizing map projections, cartograms may exhibit invalid topologies if boundary polygons are drawn using only a finite…
Persistent homology analysis provides means to capture the connectivity structure of data sets in various dimensions. On the mathematical level, by defining a metric between the objects that persistence attaches to data sets, we can…
Persistence diagrams are important tools in the field of topological data analysis that describe the presence and magnitude of features in a filtered topological space. However, current approaches for comparing a persistence diagram to a…
One of the main reasons for topological persistence being useful in data analysis is that it is backed up by a stability (isometry) property: persistence diagrams of $1$-parameter persistence modules are stable in the sense that the…
Computational topology has recently known an important development toward data analysis, giving birth to the field of topological data analysis. Topological persistence, or persistent homology, appears as a fundamental tool in this field.…
Delay-coordinate mapping is an effective and widely used technique for reconstructing and analyzing the dynamics of a nonlinear system based on time-series outputs. The efficacy of delay-coordinate mapping has long been supported by Takens'…
Topological data analysis offers a rich source of valuable information to study vision problems. Yet, so far we lack a theoretically sound connection to popular kernel-based learning techniques, such as kernel SVMs or kernel PCA. In this…
Distances on merge trees facilitate visual comparison of collections of scalar fields. Two desirable properties for these distances to exhibit are 1) the ability to discern between scalar fields which other, less complex topological…
Cartograms combine statistical and geographical information in thematic maps, where areas of geographical regions (e.g., countries, states) are scaled in proportion to some statistic (e.g., population, income). Cartograms make it possible…
The theory of multidimensional persistent homology was initially developed in the discrete setting, and involved the study of simplicial complexes filtered through an ordering of the simplices. Later, stability properties of…
Stability is a fundamental notion in dynamical systems and control theory that, traditionally understood, describes asymptotic behavior of solutions around an equilibrium point. This notion may be characterized abstractly as continuity of a…
The availability of devices that track moving objects has led to an explosive growth in trajectory data. When exploring the resulting large trajectory collections, visual summaries are a useful tool to identify time intervals of interest. A…
This paper addresses the ubiquity of remarkable measures on graphs, and their applications. In many queueing systems, it is necessary to take into account the compatibility constraints between users, or between supply and demands, and so…
To date, the instability of prognostic predictors in a sparse high dimensional model, which hinders their clinical adoption, has received little attention. Stable prediction is often overlooked in favour of performance. Yet, stability…
Numerical simulations of coupled map lattices (CMLs) and other complex model systems show an enormous phenomenological variety that is difficult to classify and understand. It is therefore desirable to establish analytical tools for…
Algorithmic stability is a central concept in statistics and learning theory that measures how sensitive an algorithm's output is to small changes in the training data. Stability plays a crucial role in understanding generalization,…
State space subspace algorithms for input-output systems have been widely applied but also have a reasonably well-developedasymptotic theory dealing with consistency. However, guaranteeing the stability of the estimated system matrix is a…