Minimal graph level sets are concave if boundary is concave.
problem Understanding curvature of minimal graph level sets.
method Proved an inequality and showed geometric properties.
result Level sets of minimal graphs are concave if boundary is concave.
Characterizes level-set families of harmonic functions without critical points.
problem Understanding level-set families of harmonic functions without critical points.
method Characterization via local differential-geometric condition and construction from geometric data.
result Evolution of gradient of harmonic functions determined by mean curvature of level sets.
New scoring rules for multivariate distributions and level sets.
problem Evaluating forecast accuracy for multivariate distributions and level sets.
method Theoretical framework for scoring rules, decomposition of multivariate scoring functions, numerical algorithm for computation.
result New scoring functions for multivariate distributions and level sets, including density and cumulative distribution level sets.
Minimal graph level sets are strictly convex in curved spaces.
problem Regularity and convexity of minimal graph level sets in curved spaces.
method Continuity method to prove strict convexity.
result Minimal graph level sets are strictly convex.
The article uses surgery on mean curvature flow to study level set flow's regularity and stability.
problem Analyzing the regularity and stability of level set flow.
method Using mean curvature flow with surgery to derive estimates.
result Demonstrates stability of the plane under level set flow.
Novel equations for nonsmooth level sets on Heisenberg group.
problem Parametrizing level sets of irregular maps on the Heisenberg group.
method Rough path theory equations for sub-Riemannian geometry.
result Well-posedness and calculus on nonsmooth level sets.
The paper characterizes potential functions whose level sets are orbits in mechanical systems.
problem Characterizing smooth potential energy functions on the plane with specific level set properties.
method Analyzing inverse curvature flow and properties of level sets.
result Analytic or functions with totally path-disconnected critical sets must be radial, while every compact convex set is a critical set of a Levi potential.
Continuity of second derivative in level set flow determined.
problem Regularity of level set flow solutions.
method Analysis of singular times and singular sets.
result Second derivative is continuous if and only if flow has a single singular time.
Study on transnormal functions and their level sets on Finsler manifolds.
problem Understanding transnormal functions and their geometric properties on Finsler manifolds.
method Proving smoothness of focal varieties and regular level sets of transnormal functions.
result Focal varieties of a C2 transnormal function are smooth submanifolds and regular level sets are tubes over these varieties.
We propose and analyze a constrained level-set method for semi-automatic image segmentation. Our level-set model with constraints on the level-set function enables us to specify which parts of the image lie inside respectively outside the segmented objects. Such a-priori information can be expressed in terms of upper a…
Paper presents a robust transfer learning method for active level set estimation.
problem Efficiently identifying regions of a black-box function with limited function evaluations.
method Incorporates prior knowledge from a related function while locally adapting it.
result The method achieves better convergence of level sets compared to standard transfer learning.
Find conditions for starshapedness of level sets in Heisenberg group.
problem Ensure starshapedness of level sets of p-capacitary potentials. method Examine horizontally p-harmonic functions in the Heisenberg group. result Sharp conditions for strictly starshaped level sets.
Smooth flow of a complex curve proven.
problem Proving smoothness of level-set flow for complex curves.
method Analyzing the topologist's sine curve and its evolution under level-set flow.
result First example of a non-locally-connected set evolving smoothly.
Proposes a method to improve hierarchical clustering using set-level structural priors.
problem Lack of supervision for non-leaf structure in hierarchical clustering.
method Introduces set-level structural priors for semi-supervised hyperbolic hierarchical clustering.
result Improves label consistency and similarity-based tree quality over baselines.
Study approximates unknown function levels with queries.
problem Approximating unknown function levels through sequential queries.
method Introduce Bisect and Approximate algorithms to reduce to local function approximation.
result Rate-optimal sample complexity guarantees for H{ö}lder functions.
A method to control neural level sets for improved generalization and robustness.
problem Improving the properties of neural networks, particularly their decision boundaries and robustness.
method Sampling neural level sets and relating them to network parameters through a sample network.
result High fidelity surface reconstruction from raw 3D point clouds and comparable robust accuracy to state-of-the-art methods.
Proves convexity of level sets of general inverse σ_k equations.
problem Convexity of level sets of general inverse σ_k equations.
method Analyzes level sets of degree n general inverse σ_k equations and uses numerical conditions to verify convexity.
result Proves convexity of level sets of general inverse σ_k equations.
Proves a function's locally least gradient property if its level sets are minimal laminations.
problem Understanding the relationship between 1-harmonic functions and minimal laminations.
method Analyzes minimal laminations and their convergence properties, then applies to 1-harmonic functions.
result Proves a function is 1-harmonic if its level sets are minimal laminations.
New algorithm estimates level sets of black-box functions efficiently.
problem Estimating level sets of black-box functions from noisy queries.
method Hierarchical Gaussian Process with multiscale partitioning.
result Algorithm has lower computational cost and tighter information gain bounds.
Proves smoothness of conical singularities in mean curvature flow.
problem Resolving singularities in mean curvature flow.
method Analyzes smooth hypersurfaces with isolated conical singularities.
result Smoothness of level set flow through asymptotically conical singularities.
Paper studies generic dynamics of MCFs with spherical singularities.
problem Characterizing the generic behavior of mean curvature flow with spherical singularities.
method Level set formulation of mean curvature flow, analysis of arrival time function.
result Generically, the arrival time function has at most C2 regularity. Generative model learns conditional distributions on collective variable levels.
problem Modeling conditional probability distributions on collective variable levels.
method General and efficient learning approach, data enrichment strategy.
result Effective generative models on different level-sets of collective variables.
Study lower bounds for connectivity of distance function level sets in convex sets.
problem Understanding connectivity of distance function level sets in convex sets.
method Lower bound calculation using critical points of the distance function.
result Provide a lower bound for the range of connectivity.
The paper studies stability and singularities of a two-convex level set flow.
problem Stability and singularities of a two-convex level set flow.
method Assumes two-convex initial hypersurface and finitely many singular times, then shows the singular set has finitely many connected components.
result Near each connected component of the singular set, the perturbed flow has the same type of singular set.
The paper explores using set-level ratings for better user-item preference prediction in recommender systems.
problem Capturing user preferences on individual items using set-level ratings.
method Developed collaborative filtering-based methods to model user behaviors in set-level ratings.
result Collaborative filtering-based models can recover and predict user preferences on individual items using set-level ratings.
New method for analyzing elliptic and parabolic equations.
problem Analyzing elliptic and parabolic equations.
method Level set version of partial uniform ellipticity.
result Effective approach to investigate equations.
The level set tree approach of Hartigan (1975) provides a probabilistically based and highly interpretable encoding of the clustering behavior of a dataset. By representing the hierarchy of data modes as a dendrogram of the level sets of a density estimator, this approach offers many advantages for exploratory analysis…
Proves positive mass theorem for 3-manifolds with a boundary.
problem Proving the positive mass theorem for specific 3-manifolds.
method Uses harmonic level set approach.
result Validates the positive mass theorem for new class of manifolds.
Proposes methods for online conformal prediction with nested prediction sets across multiple confidence levels.
problem Need for uncertainty quantification with multiple confidence levels in diverse applications.
method Online optimization perspective to enforce nestedness of prediction sets while controlling quantile estimation error.
result Achieves stable coverage across all levels, strictly nested prediction sets, and improved efficiency.
Unified algorithm for Bayesian optimization and level-set estimation.
problem Efficiently optimizing and estimating in settings with pointwise costs and heteroscedastic noise.
method Truncated Variance Reduction (TruVaR) algorithm that greedily shrinks a sum of truncated variances.
result Unified theoretical guarantee for TruVaR covering pointwise costs and heteroscedastic noise.
Level set flow's singularities are type I under 2-convexity, leading to specific curvature blow-up rates.
problem Understanding the nature and behavior of singularities in level set flow.
method Analytical approach using Lojasiewicz inequality and curvature blow-up rates.
result The arrival time is C2 near a critical point if and only if it satisfies a Lojasiewicz inequality. Study finds nodal solutions of Yamabe equation constant along isoparametric levels.
problem Existence of nodal solutions for the Yamabe equation.
method Constant along isoparametric levels of a function.
result Existence of nodal solutions proven.
Deep learning predicts curvature of 2D interfaces in level-set method.
problem Estimating curvature in level-set method for complex interfaces.
method Deep learning using feed-forward neural networks trained on synthetic data.
result Deep learning models approximate curvature with comparable precision to traditional methods.
Study of 3D steady gradient Ricci solitons using level set flow.
problem Characterizing the behavior of level sets in 3D steady gradient Ricci solitons.
method Analysis of scalar curvature and umbilical ratio using level set flow.
result The umbilical ratio of level sets is bounded by specific functions of the scalar curvature.
Study connects spectral clustering to maximum margin and level set estimation.
problem Connecting spectral clustering to maximum margin and level set estimation.
method Obtained bounds on eigenvectors of graph Laplacian matrices in terms of cluster separation and connectivity. Showed sensitivity mitigation by removing outliers and estimating level sets.
result Spectral clustering converges to maximum margin clustering as scaling parameter approaches zero.
Study shows zero level sets of solutions to Allen-Cahn equation are minimal surfaces with zero mean curvature.
problem Understanding phase transitions through entire solutions of the Allen-Cahn equation.
method Proving minimality of the zero level set with respect to a perimeter functional with density and showing zero mean curvature.
result The zero level set of entire solutions of the Allen-Cahn equation has zero mean curvature and is minimal.
Adaptive coverage policies improve conformal prediction accuracy.
problem Fixed coverage levels in traditional conformal prediction lead to uninformative predictions.
method Optimizes adaptive coverage policy using a neural network trained on leave-one-out calibration.
result Adaptive coverage policies produce more informative and flexible prediction sets.
Study on harmonic functions on nonnegative curvature 3D manifolds.
problem Analyzing harmonic functions on specific 3D manifolds.
method Inspired by Miao, developed a monotonic quantity for level sets of harmonic functions on (R3∖{0},g) with nonnegative scalar curvature. result Established a rigidity result for the derived monotonic quantity.
The clusters of a distribution are often defined by the connected components of a density level set. However, this definition depends on the user-specified level. We address this issue by proposing a simple, generic algorithm, which uses an almost arbitrary level set estimator to estimate the smallest level at which th…
Preserves metric space properties under certain function constraints.
problem Understanding functions that preserve specific geometric properties in metric spaces.
method Formulating and proving conjectures about isometries and level sets in complete Riemannian manifolds.
result Functions preserving at least one level set of a metric space are isometries under certain conditions.
Study evaluates GP metamodels and sequential designs for noisy level set estimation.
problem Efficiently reconstructing the level set of a noisy function.
method Investigates Gaussian process (GP) and Student-t process (TP) metamodels, along with various acquisition functions.
result GPs with Student-t observations and TPs perform better than classification GPs in noisy conditions.
Estimating the level set of a signal from measurements is a task that arises in a variety of fields, including medical imaging, astronomy, and digital elevation mapping. Motivated by scenarios where accurate and complete measurements of the signal may not available, we examine here a simple procedure for estimating the…
BDMBC clusters data with varying densities using a new PLLS measure.
problem Finding clusters with varying densities in data.
method Bagged k-distance with PLLS for mode estimation. result BDMBC achieves optimal convergence rates for mode and level set estimation.
Bayesian Neural Networks improve high-dimensional level set estimation.
problem Scalability issue in existing LSE methods for high-dimensional inputs.
method Bayesian Neural Networks with information-based acquisition functions.
result Proposed method achieves better results than state-of-the-art approaches.
The paper studies motion of level sets by general curvature, overcoming well-definedness issues.
problem Motion of level sets by general curvature with well-definedness issues.
method Introduced a new approximation function and used an elliptic approach to extend the existence of a weak solution.
result Extended existence of a weak solution to outside the admissible cone.
DBSCAN estimates density level sets on manifolds with i.i.d. samples.
problem Estimating connected components of density level sets on manifolds.
method DBSCAN algorithm applied to i.i.d. samples.
result Rates of estimation error for different data settings.
This paper introduces a more efficient method for estimating level sets with a stopping criterion.
problem Efficiently estimating regions where a function exceeds a threshold without exhaustive evaluations.
method Acquisition strategy with a stopping criterion for ε-accurate level set estimation. result The method satisfies ε-accuracy with a confidence level of 1−δ and guarantees on lower bounds of performance metrics. The paper examines how the topology of level sets changes with critical points in Morse theory.
problem Understanding how the topology of level sets changes with critical points in Morse theory.
method Study of sublevel sets and level sets of Morse functions, analysis of critical points and their indices.
result For a general class of functions, the topology of a regular level set changes when passing a single critical point, unless the index is half the dimension of the manifold.