Constructs ancient solutions to mean curvature flow with prescribed singular sets.
problem Creating mean-convex ancient solutions with specific singular sets.
method Constructs solutions with a prescribed singular set Kimes{0} using mean curvature flow in a Riemannian metric. result Constructs ancient solutions with a first-time singular set exactly Kimes{0}. Constructs constant spacetime mean curvature surfaces for hyperboloidal initial data sets.
problem Creating a foliation of constant spacetime mean curvature surfaces for asymptotically hyperboloidal initial data sets.
method Long time limit of volume preserving spacetime mean curvature flow starting from a constant mean curvature foliation.
result Obtains a foliation of constant spacetime mean curvature surfaces as the long time limit.
Formula extends mean curvature to convex sets.
problem Extending mean curvature to convex sets.
method Derived a special formula for ovaloids.
result Proved integral formula for ovaloids.
A mean-convex set can be regarded as a barrier for the construction of minimal surfaces. Namely, if we are given a mean-convex set and a null-homotopic Jordan curve on its boundary, then there exists an embedded minimal disk with boundary the given curve contained in the starting mean-convex set. Does a mean-convex set…
The paper shows dense and residual sets of continuous maps with positive metric mean dimension.
problem Understanding the genericity of continuous maps with positive metric mean dimension.
method Analyzing continuous maps on compact Riemannian manifolds and Cantor sets.
result The set of continuous maps with metric mean dimension equal to a given value is dense and, for the dimension, residual in the space of continuous maps.
We compute an approximate Fréchet mean for sets of sparse graphs.
problem Characterizing the location of a set of graphs in a metric space.
method We use the pseudometric defined by the ℓ₂ norm of eigenvalues of adjacency matrices.
result We describe an algorithm to approximate the Fréchet mean of a set of graphs.
In this note we announce results on the mean curvature flow of mean convex sets in 3-dimensions. Loosely speaking, our results justify the naive picture of mean curvature flow where the only singularities are neck pinches, and components which collapse to asymptotically round spheres.
Diffusion means converge to extrinsic means for long times on spheres.
problem Understanding the long-time behavior of diffusion means on manifolds.
method Introduced diffusion means as a parameterized family of location statistics on manifolds, and analyzed their convergence to extrinsic means for long times.
result For real projective spaces and connected compact symmetric spaces, the long-time limit of diffusion means is conjectured to be the extrinsic mean in the isometric embedding.
The paper estimates the volume of singular points in evolving surfaces.
problem Estimating the volume of singular points in evolving surfaces.
method Uniform and sharp volume estimates for singular sets of mean curvature flows.
result Uniform and sharp volume estimates for singular sets of mean curvature flows.
Study shows thresholding scheme converges for mean curvature flow of convex sets.
problem Analyzing convergence of thresholding scheme for mean curvature flow.
method Time discretization using Merriman, Bence and Osher's scheme, focusing on two-phase mean convex settings.
result Time-integrated energy of approximation converges to limit's energy in minimizing movements interpretation.
The paper computes an approximation to the sample Frechet mean of graph sets using spectral information.
problem Characterizing the location of a set of graphs in a metric space.
method The Frechet mean is computed for sets of large graphs using the pseudometric defined by the norm between eigenvalues of adjacency matrices.
result An algorithm to approximate the sample Frechet mean of undirected unweighted graphs is described.
The paper assesses conditions for the uniqueness of k-means clustering.
problem Conditions for the uniqueness of k-means clustering.
method Analyzes the choice of k and provides necessary and sufficient conditions for uniqueness.
result Determines the asymptotic distribution of the within cluster sum of squares (WCSS) and provides a bootstrap test for uniqueness.
The study examines constant weighted mean curvature hypersurfaces in shrinking Ricci solitons.
problem Characterizing constant weighted mean curvature hypersurfaces in shrinking Ricci solitons.
method Analyzing properties of hypersurfaces in specific ambient spaces (shrinking Ricci solitons).
result Conditions for a constant weighted mean curvature hypersurface to be a level set of the potential function.
We show that a mean curvature flow starting from a compact, smoothly embedded hypersurface M remains unique past singularities, provided the singularities are of mean convex type, i.e., if around each singular point, the surface moves in one direction. Specifically, the level set flow of M does not fatten if all singul…
Alexandrov immersed surfaces maintain their properties under mean curvature flow.
problem Preserving Alexandrov immersivity in mean curvature flow.
method Mean curvature flow techniques adapted for Alexandrov immersed, 2D surfaces.
result Mean curvature flow properties hold for Alexandrov immersed surfaces.
New collaborative algorithm improves personalized mean estimation in online settings.
problem Online estimation of means from multiple, possibly overlapping distributions.
method Novel collaborative strategy for active querying and mean estimation.
result Algorithm improves mean estimates through communication among agents.
We prove two new estimates for the level set flow of mean convex domains in Riemannian manifolds. Our estimates give control - exponential in time - for the infimum of the mean curvature, and the ratio between the norm of the second fundamental form and the mean curvature. In particular, the estimates remove a stumblin…
Generic level sets in mean curvature flow are BV solutions.
problem Understanding the behavior of level sets in mean curvature flow.
method Using the framework of sets of finite perimeter and distributional solutions, the paper extends Evans and Spruck's work.
result Generic level sets are distributional solutions with optimal energy dissipation rate.
Robustly estimates mean with quantized data and corruption.
problem Mean estimation under quantization and adversarial corruption.
method Constructs multivariate robust estimators in two settings.
result Optimal estimators up to logarithmic factors.
We give background which shows the connection between the mean value theorem and the obstacle problem, and then we prove that a set is a mean value set for an elliptic operator of the form Lu:=∂i(aij(x)∂ju(x)) if and only if it arises as the noncontact set of an obstacle problem involving the …
Sharp upper bounds derived for capacities in hyperbolic and Euclidean spaces.
problem Finding upper limits for the capacity of compact sets in hyperbolic and Euclidean spaces.
method Inverse mean curvature flow, unit-speed normal flow, weak inverse mean curvature flow, inverse anisotropic mean curvature flow.
result Various sharp upper bounds for the p-capacity of compact sets in hyperbolic and Euclidean spaces are derived. A possible evolution of a compact hypersurface in R^n by mean curvature past singularities is defined via the level set flow. In the case that the initial hypersurface has positive mean curvature, we show that the Brakke flow associated to the level set flow is actually a Brakke flow with equality. We obtain as a conse…
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.
Study constructs solutions for evolving hypersurfaces using inverse spacetime mean curvature.
problem Evolution of hypersurfaces in spacetime.
method Weak solutions for hypersurfaces evolving along inverse spacetime mean curvature in asymptotically flat maximal initial data sets.
result Weak solution detects both future- and past-trapped apparent horizons.
We study the evolution of hypersurfaces in spacetime initial data sets by their null mean curvature. A theory of weak solutions is developed using the level-set approach. Starting from an arbitrary mean convex, outer untapped hypersurface ∂Ω0, we show that there exists a weak solution to the null mean curvatu…
Improved multi-task averaging reduces mean squared error in high-dimensional data.
problem Joint estimation of multiple distributions using independent data sets.
method Exploits similarities between tasks by shrinking naive estimators towards local averages.
result The method provides a significant reduction in mean squared error, especially in high-dimensional spaces.
The paper identifies the minimum mean-variance spanning set and its importance in asset evaluation.
problem Estimating the minimum subset of assets that span the efficient frontier.
method Established identification conditions and developed a novel procedure for MSS estimation and inference.
result The MSS estimator accurately covers the true MSS and converges to it at any desired confidence level.
The classical k-means algorithm for partitioning n points in Rd into k clusters is one of the most popular and widely spread clustering methods. The need to respect prescribed lower bounds on the cluster sizes has been observed in many scientific and business applications. In this paper, we present an…
Study of mean curvature flow with obstacles using singular perturbation.
problem Obstacle problem associated to mean curvature flow.
method Geometric vanishing-viscosity approximation with singular perturbation.
result Generic level sets are distributional solutions of the obstacle problem.
Study on estimating Gaussian mean from coarse data, resolving identifiability and computational efficiency questions.
problem Estimating the mean of a Gaussian distribution from coarse data (sets containing true samples rather than exact values).
method Analyzes the conditions for mean identifiability and computable estimation under convex partitions.
result Resolves the identifiability and computational efficiency questions for Gaussian mean estimation from coarse data.
Generalizes mean-value inequality to orbifold setting.
problem Mean-value inequality for orbifold setting.
method Generalizes fundamental results in Kähler geometry to orbifolds.
result Shows mean-value inequality is insensitive to quotient singularities.
In this paper, we prove short time existence and uniqueness of smooth evolution by mean curvature in Rn+1 starting from any n-dimensional (ε,R)-Reifenberg flat set with ε sufficiently small. More precisely, we show that the level set flow in such a situation is non-fattening and …
We study the provenance of singularity formation under mean curvature flow and volume preserving mean curvature flow in an axially symmetric setting. We prove that if the mean curvature is uniformly bounded on any finite time interval, then no singularities can develop during that time under both mean curvature flow an…
Proves higher regularity for anisotropic inverse mean curvature flow.
problem Higher regularity of solutions to anisotropic inverse mean curvature flow.
method Proves Harnack estimate and constructs smooth solutions from C1 initial sets. result Smooth solutions become smooth outside a compact set.
We show compactness in the locally smooth topology for certain natural families of asymptotically conical self-expanding solutions of mean curvature flow. Specifically, we show such compactness for the set of all two-dimensional self-expanders of a fixed topological type and, in all dimensions, for the set of self-expa…
Model selection for time series forecasting can be biased by the distribution of scores.
problem Model selection for probabilistic forecasting on time series data.
method Using proper scoring rules to aggregate scores across multiple time series.
result The mean score is immune to the skewness of the score distribution.
We consider the problem of estimating the mean of a symmetric log-concave distribution under the constraint that only a single bit per sample from this distribution is available to the estimator. We study the mean squared error as a function of the sample size (and hence the number of bits). We consider three settings:…
Two algorithms improve K-means clustering speed without sacrificing quality.
problem Improving clustering quality of K-means while speeding up the process.
method Divisive K-means and Parallel Two-Phase K-means.
result Achieved empirically global optimum clustering results with lower complexity.
We consider K-means clustering in networked environments (e.g., internet of things (IoT) and sensor networks) where data is inherently distributed across nodes and processing power at each node may be limited. We consider a clustering algorithm referred to as networked K-means, or NK-means, which relies only on l…
In this paper, we generalize White's regularity and structure theory for mean-convex mean curvature flow to the setting with free boundary. A major new challenge in the free boundary setting is to derive an a priori bound for the ratio between the norm of the second fundamental form and the mean curvature. We establish…
IVF k-means algorithm improves performance on large sparse data sets.
problem Efficiently clustering large-scale sparse data sets with numerous classes.
method Sparse data representation and inverted-file structure for high-speed and low-memory clustering.
result IVF achieves better performance than other algorithms on real document data sets.
Surface diffusion and mean curvature flows converge to stable critical sets in flat tori.
problem Stability of surface diffusion and mean curvature flows in flat tori.
method Existence and convergence of flows starting close to stable critical sets, proven for all times.
result Flows converge exponentially fast to stable critical sets in flat tori.
Study of nonlocal mean curvature and symmetry of surfaces.
problem Symmetry of surfaces with ordered nonlocal mean curvature.
method Generalization of Alexandrov's moving plane method.
result Similar result to classical setting proved in nonlocal setting.
New algorithm solves mean-field control problems using actor-critic learning with moment neural networks.
problem Solving mean-field control problems in continuous time reinforcement learning.
method Gradient-based policy and value function learning with moment neural networks on the Wasserstein space.
result Effective solution for diverse mean-field control problems, including multi-dimensional and nonlinear settings.
Instead of controlling "symmetric" risks measured by central moments of investment return or terminal wealth, more and more portfolio models have shifted their focus to manage "asymmetric" downside risks that the investment return is below certain threshold. Among the existing downside risk measures, the lower-partial …
Flow preserves volume on flat torus, converging to stable set.
problem Volume preservation in discrete mean curvature flow on flat torus.
method Discrete mean curvature flow, quantitative Alexandrov estimate, characterization in 2D.
result Flow converges exponentially fast to stable set.
We provide a self-contained treatment of set-theoretic subsolutions to flow by mean curvature, or, more generally, to flow by mean curvature plus an ambient vector field. The ambient space can be any smooth Riemannian manifold. Most importantly, we show that if two such set-theoretic subsolutions are initially disjoint…
Compact solutions persist even with linear perturbations of the mean curvature term.
problem Compactness of solutions to the Yamabe problem on manifolds with boundary.
method Linear perturbation of the mean curvature term, proving compactness of solutions.
result Set of solutions remains compact even with negative perturbations.