Superlevel Set Research Articles

On the integration of $$L^0$$-Banach $$L^0$$-modules and its applications to vector calculus on $$\textsf{RCD}$$ spaces

AbstractA finite-dimensional $$\textsf{RCD}$$ RCD space can be foliated into sufficiently regular leaves, where a differential calculus can be performed. Two important examples are given by the measure-theoretic boundary of the superlevel set of a function of bounded variation and the needle decomposition associated to a Lipschitz function. The aim of this paper is to connect the vector calculus on the lower dimensional leaves with the one on the base space. In order to achieve this goal, we develop a general theory of integration of $$L^0$$ L 0 -Banach $$L^0$$ L 0 -modules of independent interest. Roughly speaking, we study how to ‘patch together’ vector fields defined on the leaves that are measurable with respect to the foliation parameter.

Global minimizers of a large class of anisotropic attractive‐repulsive interaction energies in 2D

AbstractWe study a large family of Riesz‐type singular interaction potentials with anisotropy in two dimensions. Their associated global energy minimizers are given by explicit formulas whose supports are determined by ellipses under certain assumptions. More precisely, by parameterizing the strength of the anisotropic part we characterize the sharp range in which these explicit ellipse‐supported configurations are the global minimizers based on linear convexity arguments. Moreover, for certain anisotropic parts, we prove that for large values of the parameter the global minimizer is only given by vertically concentrated measures corresponding to one dimensional minimizers. We also show that these ellipse‐supported configurations generically do not collapse to a vertically concentrated measure at the critical value for convexity, leading to an interesting gap of the parameters in between. In this intermediate range, we conclude by infinitesimal concavity that any superlevel set of any local minimizer in a suitable sense does not have interior points. Furthermore, for certain anisotropic parts, their support cannot contain any vertical segment for a restricted range of parameters, and moreover the global minimizers are expected to exhibit a zigzag behavior. All these results hold for the limiting case of the logarithmic repulsive potential, extending and generalizing previous results in the literature. Various examples of anisotropic parts leading to even more complex behavior are numerically explored.

Stokes, Gibbs, and Volume Computation of Semi-Algebraic Sets

We consider the problem of computing the Lebesgue volume of compact basic semi-algebraic sets. In full generality, it can be approximated as closely as desired by a converging hierarchy of upper bounds obtained by applying the Moment-SOS (sums of squares) methodology to a certain infinite-dimensional linear program (LP). At each step one solves a semidefinite relaxation of the LP which involves pseudo-moments up to a certain degree. Its dual computes a polynomial of same degree which approximates from above the discontinuous indicator function of the set, hence with a typical Gibbs phenomenon which results in a slow convergence of the associated numerical scheme. Drastic improvements have been observed by introducing in the initial LP additional linear moment constraints obtained from a certain application of Stokes’ theorem for integration on the set. However and so far there was no rationale to explain this behavior. We provide a refined version of this extended LP formulation. When the set is the smooth super-level set of a single polynomial, we show that the dual of this refined LP has an optimal solution which is a continuous function. Therefore in this dual one now approximates a continuous function by a polynomial, hence with no Gibbs phenomenon, which explains and improves the already observed drastic acceleration of the convergence of the hierarchy. Interestingly, the technique of proof involves recent results on Poisson’s partial differential equation (PDE).

From radial symmetry to fractal behavior of aggregation equilibria for repulsive–attractive potentials

For the interaction energy with repulsive–attractive potentials, we give generic conditions which guarantee the radial symmetry of the local minimizers in the infinite Wasserstein distance. As a consequence, we obtain the uniqueness of local minimizers in this topology for a class of interaction potentials. We introduce a novel notion of concavity of the interaction potential allowing us to show certain fractal-like behavior of the local minimizers. We provide a family of interaction potentials such that the support of the associated local minimizers has no isolated points and any superlevel set has no interior points.

Polynomial superlevel set representation of the multistationarity region of chemical reaction networks

In this paper we introduce a new representation for the multistationarity region of a reaction network, using polynomial superlevel sets. The advantages of using this polynomial superlevel set representation over the already existing representations (cylindrical algebraic decompositions, numeric sampling, rectangular divisions) is discussed, and algorithms to compute this new representation are provided. The results are given for the general mathematical formalism of a parametric system of equations and so may be applied to other application domains.

Safe-by-design control for Euler–Lagrange systems

Safety-critical control is characterized as ensuring constraint satisfaction for a given dynamical system. Recent developments in zeroing control barrier functions (ZCBFs) have provided a framework for ensuring safety of a superlevel set of a single constraint function. Euler–Lagrange systems represent many real-world systems including robots and vehicles, which must abide by safety-regulations, especially for use in human-occupied environments. These safety regulations include state constraints (position and velocity) and input constraints that must be respected at all times. ZCBFs are valuable for satisfying system constraints for general nonlinear systems, however their construction to satisfy state and input constraints is not straightforward. Furthermore, the existing barrier function methods do not address the multiple state constraints that are required for safety of Euler–Lagrange systems. In this paper, we propose a methodology to construct multiple, non-conflicting control barrier functions for Euler–Lagrange systems subject to input constraints to satisfy safety regulations, while concurrently taking into account robustness margins and sampling-time effects. The proposed approach consists of a sampled-data controller and an algorithm for barrier function construction to enforce safety (i.e. satisfy position and velocity constraints). The proposed method is validated in simulation on a 2-DOF planar manipulator.

Quantitative estimates for the Bakry–Ledoux isoperimetric inequality

We establish a quantitative isoperimetric inequality for weighted Riemannian manifolds with \operatorname{Ric}_{\infty} \ge 1 . Precisely, we give an upper bound of the volume of the symmetric difference between a Borel set and a sub-level (or super-level) set of the associated guiding function (arising from the needle decomposition), in terms of the deficit in Bakry–Ledoux’s Gaussian isoperimetric inequality. This is the first quantitative isoperimetric inequality on noncompact spaces besides Euclidean and Gaussian spaces. Our argument makes use of Klartag’s needle decomposition (also called localization), and is inspired by a recent work of Cavalletti, Maggi and Mondino on compact spaces. Besides the quantitative isoperimetry, a reverse Poincaré inequality for the guiding function that we have as a key step, as well as the way we use it, are of independent interest.

Predictor-based safety control for systems with multiple time-varying delays

Control barrier functions (CBFs) have been recently considered for ensuring safety of nonlinear input-affine systems by means of appropriately designed controllers rendering a desired superlevel set of the CBF function forward invariant. In this work, we consider the safety control problem for nonlinear input-affine systems with multiple time-varying input delays. In order to ensure safety, we first design a set of predictors that estimate the state of the system at different future times by utilizing the control laws designed to ensure safety of the delay-free system. Under the assumption of perfect estimation of the future states, we show that under the designed controller, the closed-loop performance of the systems with and without the input delays is the same by the time the input with the largest delay acts on the system with delays for the first time.

Signal Temporal Logic Task Decomposition via Convex Optimization

In this letter we focus on the problem of decomposing a global Signal Temporal Logic formula (STL) assigned to a multi-agent system to local STL tasks when the team of agents is a-priori decomposed to disjoint sub-teams. The predicate functions associated to the local tasks are parameterized as hypercubes depending on the states of the agents in a given sub-team. The parameters of the functions are, then, found as part of the solution of a convex program that aims implicitly at maximizing the volume of the zero superlevel set of the corresponding predicate function. Two alternative definitions of the local STL tasks are proposed and the satisfaction of the global STL formula is proven when the conjunction of the local STL tasks is satisfied.

Asynchronous and Load-Balanced Union-Find for Distributed and Parallel Scientific Data Visualization and Analysis.

We present a novel distributed union-find algorithm that features asynchronous parallelism and k-d tree based load balancing for scalable visualization and analysis of scientific data. Applications of union-find include level set extraction and critical point tracking, but distributed union-find can suffer from high synchronization costs and imbalanced workloads across parallel processes. In this study, we prove that global synchronizations in existing distributed union-find can be eliminated without changing final results, allowing overlapped communications and computations for scalable processing. We also use a k-d tree decomposition to redistribute inputs, in order to improve workload balancing. We benchmark the scalability of our algorithm with up to 1,024 processes using both synthetic and application data. We demonstrate the use of our algorithm in critical point tracking and super-level set extraction with high-speed imaging experiments and fusion plasma simulations, respectively.

Modality for scenario analysis and maximum likelihood allocation

We study the variability of a risk from the statistical viewpoint of multimodality of the conditional loss distribution given that the aggregate loss equals an exogenously provided capital. This conditional distribution serves as a building block for calculating risk allocations such as the Euler capital allocation of Value-at-Risk. A superlevel set of this conditional distribution can be interpreted as a set of severe and plausible stress scenarios the given capital is supposed to cover. We show that various distributional properties of this conditional distribution, such as modality, dependence and tail behavior, are inherited from those of the underlying joint loss distribution. Among these properties, we find that modality of the conditional distribution is an important feature in risk assessment related to the variety of risky scenarios likely to occur in a stressed situation. Under unimodality, we introduce a novel risk allocation method called maximum likelihood allocation (MLA), defined as the mode of the conditional distribution given the total capital. Under multimodality, a single vector of allocations can be less sound. To overcome this issue, we investigate the so-called multimodality adjustment to increase the soundness of risk allocations. Properties of the conditional distribution, MLA and multimodality adjustment are demonstrated in numerical experiments. In particular, we observe that negative dependence among losses typically leads to multimodality, and thus a higher multimodality adjustment can be required.

Probabilistic convergence and stability of random mapper graphs

We study the probabilistic convergence between the mapper graph and the Reeb graph of a topological space {mathbb {X}} equipped with a continuous function f: {mathbb {X}}rightarrow mathbb {R}. We first give a categorification of the mapper graph and the Reeb graph by interpreting them in terms of cosheaves and stratified covers of the real line mathbb {R}. We then introduce a variant of the classic mapper graph of Singh et al. (in: Eurographics symposium on point-based graphics, 2007), referred to as the enhanced mapper graph, and demonstrate that such a construction approximates the Reeb graph of ({mathbb {X}}, f) when it is applied to points randomly sampled from a probability density function concentrated on ({mathbb {X}}, f). Our techniques are based on the interleaving distance of constructible cosheaves and topological estimation via kernel density estimates. Following Munch and Wang (In: 32nd international symposium on computational geometry, volume 51 of Leibniz international proceedings in informatics (LIPIcs), Dagstuhl, Germany, pp 53:1–53:16, 2016), we first show that the mapper graph of ({mathbb {X}}, f), a constructible mathbb {R}-space (with a fixed open cover), approximates the Reeb graph of the same space. We then construct an isomorphism between the mapper of ({mathbb {X}},f) to the mapper of a super-level set of a probability density function concentrated on ({mathbb {X}}, f). Finally, building on the approach of Bobrowski et al. (Bernoulli 23(1):288–328, 2017b), we show that, with high probability, we can recover the mapper of the super-level set given a sufficiently large sample. Our work is the first to consider the mapper construction using the theory of cosheaves in a probabilistic setting. It is part of an ongoing effort to combine sheaf theory, probability, and statistics, to support topological data analysis with random data.

Localized Topological Simplification of Scalar Data.

This paper describes a localized algorithm for the topological simplification of scalar data, an essential pre-processing step of topological data analysis (TDA). Given a scalar field f and a selection of extrema to preserve, the proposed localized topological simplification (LTS) derives a function g that is close to f and only exhibits the selected set of extrema. Specifically, sub- and superlevel set components associated with undesired extrema are first locally flattened and then correctly embedded into the global scalar field, such that these regions are guaranteed-from a combinatorial perspective-to no longer contain any undesired extrema. In contrast to previous global approaches, LTS only and independently processes regions of the domain that actually need to be simplified, which already results in a noticeable speedup. Moreover, due to the localized nature of the algorithm, LTS can utilize shared-memory parallelism to simplify regions simultaneously with a high parallel efficiency (70%). Hence, LTS significantly improves interactivity for the exploration of simplification parameters and their effect on subsequent topological analysis. For such exploration tasks, LTS brings the overall execution time of a plethora of TDA pipelines from minutes down to seconds, with an average observed speedup over state-of-the-art techniques of up to ×36. Furthermore, in the special case where preserved extrema are selected based on topological persistence, an adapted version of LTS partially computes the persistence diagram and simultaneously simplifies features below a predefined persistence threshold. The effectiveness of LTS, its parallel efficiency, and its resulting benefits for TDA are demonstrated on several simulated and acquired datasets from different application domains, including physics, chemistry, and biomedical imaging.

An Observer for Infinite Dimensional 3D Surface Reconstruction that Converges in Finite Time

This paper proposes a method of reconstructing the dense structure of scenes from visual or depth sensors that provably converges in finite time. We represent the scene as a superlevel set of a function that resides within some potentially infinite-dimensional function space. The observer state is determined by the parameters of the function that represents the scene. Preliminary experiments show that the observer exhibits convergence behaviour on a variety of different function spaces both in simulation and with real light-field camera data.

Dynamic Nested Tracking Graphs.

This work describes an approach for the interactive visual analysis of large-scale simulations, where numerous superlevel set components and their evolution are of primary interest. The approach first derives, at simulation runtime, a specialized Cinema database that consists of images of component groups, and topological abstractions. This database is processed by a novel graph operation-based nested tracking graph algorithm (GO-NTG) that dynamically computes NTGs for component groups based on size, overlap, persistence, and level thresholds. The resulting NTGs are in turn used in a feature-centered visual analytics framework to query specific database elements and update feature parameters, facilitating flexible post hoc analysis.

Stochastic homology of Gaussian vs. non-Gaussian random fields: graphs towards Betti numbers and persistence diagrams

The topology and geometry of random fields—in terms of the Euler characteristic and the Minkowski functionals—has received a lot of attention in the context of the Cosmic Microwave Background (CMB), as the detection of primordial non-Gaussianities would form a valuable clue on the physics of the early Universe. The virtue of both the Euler characteristic and the Minkowski functionals in general, lies in the fact that there exist closed form expressions for their expectation values for Gaussian random fields. However, the Euler characteristic and Minkowski functionals are summarizing characteristics of topology and geometry. Considerably more topological information is contained in the homology of the random field, as it completely describes the creation, merging and disappearance of topological features in superlevel set filtrations. In the present study we extend the topological analysis of the superlevel set filtrations of two-dimensional Gaussian random fields by analysing the statistical properties of the Betti numbers—counting the number of connected components and loops—and the persistence diagrams—describing the creation and mergers of homological features. Using the link between homology and the critical points of a function—as illustrated by the Morse-Smale complex—we derive a one-parameter fitting formula for the expectation value of the Betti numbers and forward this formalism to the persistence diagrams. We, moreover, numerically demonstrate the sensitivity of the Betti numbers and persistence diagrams to the presence of non-Gaussianities.

Method of hill tunneling via weighted simplex centroid for continuous piecewise linear programming

This paper works on a heuristic algorithm with determinacy for the global optimization of continuous piecewise linear (CPWL) programming. CPWL is widely applied since it can be equivalently transformed into D.C. programming, and further, concave optimization over a polyhedron. Considering that the super-level sets of concave piecewise linear functions are polyhedra, we propose the hill tunneling via simplex centroid (HTSC) algorithm, which is able to escape a local optimum to the other side of its contour surface by cutting across the super-level set. The searching path for the hill tunneling is established by using the centroid of a constructed simplex. In the numerical experiments, the proposed HTSC algorithm is compared with CPLEX and the hill detouring (HD) method, which shows its superior performance on the numerical efficiency and the global search capability.

Starshapedness of the superlevel sets of solutions to equations involving the fractional Laplacian in starshaped rings

AbstractWe study solutions of the problem urn:x-wiley:0025584X:media:mana201700226:mana201700226-math-0001where are open sets such that , , and f is a nonlinearity. Under different assumptions on f we prove that, if D0 and D1 are starshaped with respect to the same point , then the same occurs for every superlevel set of u.

Critical points of solutions to a quasilinear elliptic equation with nonhomogeneous Dirichlet boundary conditions

In this paper, we mainly investigate the critical points associated to solutions u of a quasilinear elliptic equation with nonhomogeneous Dirichlet boundary conditions in a connected domain Ω in R2. Based on the fine analysis about the distribution of connected components of a super-level set {x∈Ω:u(x)>t} for any min∂Ωu(x)<t<max∂Ωu(x), we obtain the geometric structure of interior critical points of u. Precisely, when Ω is simply connected, we develop a new method to prove Σi=1kmi+1=N, where m1,⋯,mk are the respective multiplicities of interior critical points x1,⋯,xk of u and N is the number of global maximal points of u on ∂Ω. When Ω is an annular domain with the interior boundary γI and the external boundary γE, where u|γI=H,u|γE=ψ(x) and ψ(x) has N local (global) maximal points on γE. For the case ψ(x)≥H or ψ(x)≤H or minγEψ(x)<H<maxγEψ(x), we show that Σi=1kmi≤N (either Σi=1kmi=N or Σi=1kmi+1=N).

Nested Tracking Graphs

AbstractTracking graphs are a well established tool in topological analysis to visualize the evolution of components and their properties over time, i.e., when components appear, disappear, merge, and split. However, tracking graphs are limited to a single level threshold and the graphs may vary substantially even under small changes to the threshold. To examine the evolution of features for varying levels, users have to compare multiple tracking graphs without a direct visual link between them. We propose a novel, interactive, nested graph visualization based on the fact that the tracked superlevel set components for different levels are related to each other through their nesting hierarchy. This approach allows us to set multiple tracking graphs in context to each other and enables users to effectively follow the evolution of components for different levels simultaneously. We demonstrate the effectiveness of our approach on datasets from finite pointset methods, computational fluid dynamics, and cosmology simulations.