The paper addresses fairness in facility location by considering local population densities.
problem Ensuring fair distribution of facilities based on local population sizes.
method Formulated a fairness concept involving neighborhood radius and applied an approximation algorithm.
result The algorithm guarantees a factor of at most 2 in all metric spaces and outperforms standard clustering algorithms.
Random translation surfaces converge to a Poisson plane as genus grows.
problem Understanding the geometric behavior of high genus translation surfaces.
method Proving convergence of random translation surfaces to a Poisson plane using statistical local geometric properties.
result The radius-r neighborhood of a random point in an MSV-distributed random translation surface converges in distribution to the radius r neighborhood of the root in a Poisson translation plane. We prove a descriptive theorem on the extrinsic geometry of an embedded minimal surface of injectivity radius zero in a homogeneously regular Riemannian three-manifold, in a certain small intrinsic neighborhood of a point of almost-minimal injectivity radius. This structure theorem includes a limit object which we call…
Develops a new method for robust risk measurement by averaging nearby payoffs.
problem Measuring risk under uncertainty with a focus on robustness.
method Averaging nearby payoffs weighted by a chosen metric.
result The method leads to a convex risk measure and provides stability under large neighborhoods.
Spheres in curve complexes are almost simply connected.
problem Understanding connectivity of spheres in curve complexes.
method Defining spheres as induced subgraphs and showing almost simple connectivity.
result Spheres in high-complexity surfaces are almost simply connected.
Nonuniform tubular neighborhoods of curves in Euclidean n-space are studied by using weighted distance functions and generalizing the normal exponential map. Different notions of injectivity radii are introduced to investigate singular but injective exponential maps. A generalization of the thickness formula is obtaine…
A simple geometrical proof shows that any target function can be found in a random network's neighborhood.
problem Finding any target function in a random network's neighborhood.
method Geometrical proof using a simple model of a high-dimensional sphere projected onto a low-dimensional subspace.
result High-dimensional geometry ensures that a uniform distribution over a sphere reduces to a Gaussian distribution with negligible covariances, enabling the presence of any target function in a random network's neighborhood.
This paper sets a lower limit for the size of Weinstein's Lagrangian tubular neighborhoods.
problem Finding the minimum size of Weinstein's Lagrangian tubular neighborhoods.
method Using the curvature tensor and second fundamental form of the submanifold.
result Explicit lower bounds for the radii of tubular neighborhoods are derived.
Bayesian method improves SOM training for dynamic data.
problem Training Self Organizing Maps (SOM) for non-stationary data.
method Variational Bayesian approach with adaptive neighborhood radius.
result Method outperforms other adaptive methods in high dimensions.
Develops a robust hedging valuation adjustment measure for dynamic hedging under liquidity-demand stress.
problem Dynamic hedging under liquidity-demand stress
method Define robust HVA as the worst-case expected loss over a relative-entropy neighborhood of the loss distribution generated by simulated rebalancing and maturity-unwind trades.
result Distinguishes fixed-radius convention from fixed benchmark-stress convention and shows wider no-trade bands lower rebalancing costs but raise hedge-error risk.
The Moebius energy of a knot is an energy functional for smooth curves based on an idea of self-repelling. If a knot has a thick tubular neighborhood, we would intuitively expect the energy to be low. In this paper, we give explicit bounds for energy in terms of the ropelength of the knot, i.e. the ratio of the length …
Paper develops a robust HVA measure for dynamic hedging under liquidity stress.
problem Valuation of dynamic hedging under liquidity stress.
method Defines robust HVA as worst-case expected loss over a relative-entropy neighborhood of loss distributions for no-trade bands.
result Wider no-trade bands lower rebalancing costs but increase hedge-error risk.
This paper begins the study of relations between Riemannian geometry and contact topology in any dimension and continues this study in dimension 3. Specifically we provide a lower bound for the radius of a geodesic ball in a contact manifold that can be embedded in the standard contact structure on Euclidean space, tha…
We prove the simultaneous (k,n-k)-systolic freedom, for a pair of adjacent integers k smaller than n/2, of a simply connected n-manifold X. Our construction, related to recent results of I. Babenko, is concentrated in a neighborhood of suitable k-dimensional submanifolds of X. We employ calibration by differential form…
We prove distance bounds for graphs possessing positive Bakry-Émery curvature apart from an exceptional set, where the curvature is allowed to be non-positive. If the set of non-positively curved vertices is finite, then the graph admits an explicit upper bound for the diameter. Otherwise, the graph is a subset of the …
We study closed non-positively curved Riemannian manifolds M which admit `fat k-flats': that is, the universal cover M~ contains a positive radius neighborhood of a k-flat on which the sectional curvatures are identically zero. We investigate how the fat k-flats affect the cardinality of the collection …
Fair near neighbor search ensures equal probability of reporting points in a query's radius.
problem Ensuring equal probability of reporting points within a query's radius.
method LSH-based algorithms modified to ensure uniform probability of reporting points.
result Algorithm reports points in the query's radius with almost uniform probability.
Develops an online Gaussian process method that maintains convergence guarantees without sample complexity issues.
problem The computational intractability of Gaussian processes with streaming data.
method Parsimonious Online Gaussian Processes (POG) that maintains asymptotic consistency with bounded memory.
result POG preserves convergence guarantees to the population posterior with finite memory, even for constant error radius.
Let (M,g) be a compact Riemannian manifold of dimension N≥2. We prove the existence of a family (Ωε)ε∈(0,ε0) of self-Cheeger sets in (M,g) . The domains Ωε⊂M are perturbations of geodesic balls of radius ε c…
Let (M,g) be a compact Riemannian manifold of dimension N≥2. We prove the existence of a family (Ωε)ε∈(0,ε0) of self-Cheeger sets in (M,g) . The domains Ωε⊂M are perturbations of geodesic balls of radius ε cen…
New findings on domains without parabolic minimal submanifolds and weakly hyperbolic domains.
problem Characterizing domains without parabolic minimal submanifolds and weakly hyperbolic domains.
method Analyzing properties of domains and their boundaries, using tubular neighborhoods and conformal harmonic maps.
result Domains without parabolic minimal submanifolds and weakly hyperbolic domains have specific geometric properties.
New adaptive stepsize method for stochastic approximation converges to target point.
problem Finding optimal step sizes for stochastic approximation algorithms.
method Adaptive block-coordinate stepsizes using online estimates of second moment.
result New method converges almost surely to a small neighborhood of the target point.
We establish a new fundamental relationship between total curvature of knots and crossing number. If K is a smooth knot in 3-space, R the cross-section radius of a uniform tube neighborhood of K, L the arclength of K, and k the total curvature of K, then (up to a coefficient independent of K), crossing number of K < (k…
This paper studies mean curvature flows near cylindrical singularities.
problem Understanding the behavior of mean curvature flows near cylindrical singularities.
method Proved the rescaled flow converges to a graph over a cylinder, defined nondegeneracy, and showed properties of nondegenerate singularities.
result Nondegenerate cylindrical singularities are isolated, have a mean convex neighborhood, and are type-I.
We prove a general theorem providing smoothed analysis estimates for conic condition numbers of problems of numerical analysis. Our probability estimates depend only on geometric invariants of the corresponding sets of ill-posed inputs. Several applications to linear and polynomial equation solving show that the estima…
Information mapping is a popular application of Multivoxel Pattern Analysis (MVPA) to fMRI. Information maps are constructed using the so called searchlight method, where the spherical multivoxel neighborhood of every voxel (i.e., a searchlight) in the brain is evaluated for the presence of task-relevant response patte…
The paper proves properties for random graphs based on geometric submanifolds.
problem Establishing measure-metric properties of random geometric graphs.
method Analyzing ε-neighborhood graphs with specific conditions on submanifold and distribution. result Volume doubling and local Poincaré inequalities hold for random geometric graphs with high probability.
Estimates causal effects in networks with varying interference.
problem Estimating causal effects in settings with network interference.
method Proposes neighborhood adaptive estimators for average direct treatment effect on the treated.
result Establishes rates of convergence and distributional results for proposed estimators.
GraLSP improves graph neural networks by incorporating local structural patterns.
problem GNNs struggle with identifying common structural patterns in graphs.
method GraLSP uses random anonymous walks to capture local graph structures and incorporates these into feature aggregation mechanisms.
result GraLSP outperforms other models in various prediction tasks on multiple datasets.
We will prove that \emph{there are no stable complete hypersurfaces of R4 with zero scalar curvature, polynomial volume growth and such that H3(−K)≥c>0 everywhere, for some constant c>0}, where K denotes the Gauss-Kronecker curvature and H denotes the mean curvature of the immersion. …
This work links SOMs and GMMs, providing a mathematical basis for their use.
problem Understanding the relationship between SOMs and GMMs.
method Mathematical treatment showing SOMs as gradient descent on a GMM log-likelihood.
result SOMs can be interpreted as probabilistic models, justifying their use in various applications.
During an operation of surgery on a Riemannian manifold and along a given embedded submanifold, one needs to replace the (old) metric induced by the exponential map on a tubular neighborhood of the submanifold by the Sasakian metric. So a good understanding of the behavior of these two metrics is important, this is our…
We show that a smooth unknotted curve in R^3 satisfies an isoperimetric inequality that bounds the area of an embedded disk spanning the curve in terms of two parameters: the length L of the curve and the thickness r (maximal radius of an embedded tubular neighborhood) of the curve. For fixed length, the expression giv…
Sharp stability results for reverse isoperimetric inequalities in 2D.
problem Reverse isoperimetric inequalities in the plane.
method Stability analysis of λ-convex bodies and convex bodies with smooth boundaries. result Sharp stability results for reverse isoperimetric inequalities, including inradius and Cheeger inequalities.
This paper explores tradeoffs between standard and adversarial risks in distributionally adversarial training.
problem Understanding the impact of adversarial training on standard risk and adversarial risk.
method Study of distributionally adversarial training with different learning settings and models.
result Derives Pareto-optimal tradeoff curves between standard and adversarial risks.
In this paper we prove the following pointwise and curvature-free estimates on convexity radius, injectivity radius and local behavior of geodesics in a complete Riemannian manifold M: 1) the convexity radius of p, $\operatorname{conv}(p)\ge \min\{\frac{1}{2}\operatorname{inj}(p),\operatorname{foc}(B_{\operatorname…
Positive injectivity radius for manifolds with Lie structure at infinity.
problem Injectivity radius positivity for manifolds with specific boundary conditions.
method Lie groupoids to prove injectivity radius positivity.
result Injectivity radius is positive for manifolds with Lie structure at infinity.
The ratio of convexity radius over injectivity radius may be made arbitrarily small within the class of compact Riemannian manifolds of any fixed dimension at least two. This is proved using Gulliver's method of constructing manifolds with focal points but no conjugate points. The approach is suggested by a characteriz…
Uniform curvature bounds for regularized metrics with bounds on Ricci tensor and injectivity radius.
problem Bounding curvature of regularized metrics with constraints on Ricci tensor and injectivity radius.
method Mollification of riemannian metrics, uniform W2,p-harmonic radius bounds, Ricci tensor bounds, injectivity radius bounds. result Uniform estimate on the change of sectional curvature for regularized metrics.
Upper bound for conjugate radius in open manifolds with scalar curvature and spectrum constraints.
problem Bounding the conjugate radius of open manifolds with specific curvature and spectrum conditions.
method Established an upper bound using scalar curvature and bottom-of-spectrum constraints.
result For certain conditions, the conjugate radius is no more than π.
Proves upper bound on filling radius for manifolds with positive scalar curvature.
problem Bounding the filling radius of manifolds with positive scalar curvature.
method Quantitative operator K-theory and index theory.
result Proves a quantitative upper bound on the filling radius.
Compact theorem for minimal surfaces with lower injectivity radius.
problem Proving compactness of minimal surfaces with lower injectivity radius.
method Variant of Choi--Schoen compactness theorem, focusing on injectivity radius.
result Proved compactness theorem for minimal surfaces.
Injectivity radius on Stiefel manifold is π.
problem Determining the injectivity radius of the compact Stiefel manifold.
method Utilized the property of geodesics being space curves of constant Frenet curvatures.
result The injectivity radius on the Stiefel manifold under the Euclidean metric is π.
Lower bound on boundary injectivity radius for specific tubes.
problem Estimating the boundary injectivity radius of Margulis tubes.
method Using curvature bounds to derive a lower bound.
result A lower bound on the boundary injectivity radius is provided.
Study spectral properties on manifolds with conical singularities, proving new inequalities.
problem Analyzing spectral properties and geometric inequalities on manifolds with conical singularities.
method Develops new inequalities for manifolds with conical singularities, not covered by existing methods.
result Proves a Bakry-Émery inequality, Hardy inequality, and spectral gap estimate.
Study gives bounds on filling radius for Riemannian manifolds.
problem Finding bounds on the filling radius of Riemannian manifolds.
method Curvature-dependent bounds for all closed manifolds and upper bounds for submersion and submetry cases.
result Upper and lower bounds on the filling radius for specific types of manifolds.
The paper proves estimates and theorems for Kähler manifolds.
problem Curvature conditions on Kähler manifolds.
method Volume comparison and rigidity theorems.
result Conjugate radius estimates for Kähler manifolds.
Study finds the covering radius of RM(4,8) is 26.
problem Determining the covering radius of RM(4,8).
method Invented a lift by derivation invariant to classify B(5,6,8).
result Covering radius of RM(4,8) is 26.