New method for better initial centers in clustering with improved accuracy and privacy.
problem Improving the quality of clustering centers in metric spaces.
method HST initialization based on metric embedding tree structure, combined with efficient search algorithm and DP extension.
result HST initialization produces better initial centers than k-median++ with comparable efficiency and improved privacy. Refines geometric center of mass analysis for Einstein field equations.
problem Analyzing the geometric center of mass of Willmore surfaces in initial data for Einstein field equations.
method Refined Lyapunov-Schmidt analysis to study geometric center of mass of area-constrained Willmore surfaces.
result The geometric center of mass agrees with the Hamiltonian center of mass under specific conditions.
New distributed clustering algorithms show resilience to initialization issues.
problem Resilience of distributed gradient-based clustering algorithms to center initialization.
method Distributed gradient-based clustering algorithms with novel center initialization.
result The algorithms are more resilient to initialization compared to baseline methods.
This article briefly introduced Arthur and Vassilvitshii's work on \textbf{k-means++} algorithm and further generalized the center initialization process. It is found that choosing the most distant sample point from the nearest center as new center can mostly have the same effect as the center initialization process in…
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.
Proposes a new ridge estimator for smooth covariates with adaptive centering.
problem Estimating coefficients and center function for smooth covariates in linear models.
method SACR framework with convex formulation, roughness penalty, and adaptive centering.
result Improves prediction and variable selection for smooth covariates.
In the first half of this article, we survey the new quasi-local and total angular momentum and center of mass defined in [9] and summarize the important properties of these definitions. To compute these conserved quantities involves solving a nonlinear PDE system (the optimal isometric embedding equation), which is ra…
We propose a new foliation of asymptotically Euclidean initial data sets by 2-spheres of constant spacetime mean curvature (STCMC). The leaves of the foliation have the STCMC-property regardless of the initial data set in which the foliation is constructed which asserts that there is a plethora of STCMC 2-spheres in a …
The paper studies constant harmonic mean curvature surfaces in Schwarzschild spaces, proving they foliate the space.
problem Investigating constant harmonic mean curvature surfaces in Schwarzschild spaces.
method Volume-preserving harmonic mean curvature flow in asymptotically Schwarzschild spaces.
result These surfaces form a foliation of the space outside a large ball.
We define the (total) center of mass for suitably asymptotically hyperbolic time-slices of asymptotically anti-de Sitter spacetimes in general relativity. We do so in analogy to the picture that has been consolidated for the (total) center of mass of suitably asymptotically Euclidean time-slices of asymptotically Minko…
The paper studies a flow of surfaces in spacetime with a focus on curvature evolution.
problem Constructing spacetime foliations and center of mass in General Relativity.
method Volume preserving curvature flow with mean curvature speed.
result The flow converges to a constant curvature limit, providing a new foliation method.
We prove the existence and uniqueness of constant mean curvature foliations for initial data sets which are asymptotically flat satisfying the Regge-Teitelboim condition near infinity. It is known that the (Hamiltonian) center of mass is well-defined for manifolds satisfying this condition. We also show that the foliat…
In this article, we consider the limit of quasi-local conserved quantities [31,9] at the infinity of an asymptotically hyperbolic initial data set in general relativity. These give notions of total energy-momentum, angular momentum, and center of mass. Our assumption on the asymptotics is less stringent than any previo…
The study extends conserved quantities theory to non-compact boundary initial data sets.
problem Extending conserved quantities theory to initial data sets with non-compact boundaries.
method Analysis of scalar curvature and mean curvature in the interior and boundary.
result Rigidity/flexibility phenomena in positive mass theorems and Penrose inequalities.
Recently, a general method for analyzing the statistical accuracy of the EM algorithm has been developed and applied to some simple latent variable models [Balakrishnan et al. 2016]. In that method, the basin of attraction for valid initialization is required to be a ball around the truth. Using Stein's Lemma, we exten…
We define quasi-local conserved quantities in general relativity by using the optimal isometric embedding in [26] to transplant Killing fields in the Minkowski spacetime back to the 2-surface of interest in a physical spacetime. To each optimal isometric embedding, a dual element of the Lie algebra of the Lorentz group…
The center of mass in General Relativity is hard to define due to coordinate freedom.
problem Defining the center of mass in General Relativity rigorously and consistently.
method Analyzing the challenges in Newtonian Gravity and using Bartnik's asymptotic harmonic coordinates.
result Examples of initial data sets in General Relativity that do not satisfy center of mass definitions.
Study evaluates the impact of academic support center's face-to-face assistance on student performance.
problem Underestimation of Academic Support Center's true impact due to group bias.
method Applied causal inference theory and T-learner to evaluate conditional average treatment effect (CATE) of F2F personal assistance.
result Developed a new CATE function that depends on the number of F2F sessions, predicting improved CATE performance.
Constructs foliations of critical surfaces for Hawking energy in asymptotically flat initial data sets.
problem Positivity and rigidity of Hawking quasi-local energy in asymptotically flat spacetimes.
method Lyapunov-Schmidt reduction within a Willmore-foliation framework.
result Existence and uniqueness of foliations by Hawking surfaces, positivity and large-sphere limit of Hawking energy.
Deep Boltzmann machines are in principle powerful models for extracting the hierarchical structure of data. Unfortunately, attempts to train layers jointly (without greedy layer-wise pretraining) have been largely unsuccessful. We propose a modification of the learning algorithm that initially recenters the output of t…
In 2002, Isenberg-Mazzeo-Pollack (IMP) constructed a series of vacuum initial data sets via a gluing construction. In this paper, we investigate some local geometry of these initial data sets as well as implications regarding their spacetime developments. In particular, we state conditions for the existence of outer tr…
In this paper, we study the problem of learning a mixture of Gaussians with streaming data: given a stream of N points in d dimensions generated by an unknown mixture of k spherical Gaussians, the goal is to estimate the model parameters using a single pass over the data stream. We analyze a streaming version of …
We give examples of asymptotically flat three-manifolds (M,g) which admit arbitrarily large constant mean curvature spheres that are far away from the center of the manifold. This resolves a question raised by G. Huisken and S.-T. Yau in 1996. On the other hand, we show that such surfaces cannot exist when (M,g) ha…
The paper studies the stability of volume and area preserving mean curvature flows in Schwarzschild and asymptotic Schwarzschild spaces.
problem Investigating the stability of mean curvature flows in specific spacetime geometries.
method Combining center manifold analysis with global existence results for flows near isoperimetric hypersurfaces.
result Global existence and convergence to constant mean curvature (CMC) hypersurfaces for flows in asymptotic Schwarzschild space.
We study the isoperimetric structure of asymptotically flat Riemannian 3-manifolds (M,g) that are C^0-asymptotic to Schwarzschild of mass m>0. Refining an argument due to H. Bray we obtain an effective volume comparison theorem in Schwarzschild. We use it to show that isoperimetric regions exist in (M, g) for all suffi…
We show that it is possible to perturb arbitrary vacuum asymptotically flat spacetimes to new ones having exactly the same energy and linear momentum, but with center of mass and angular momentum equal to any preassigned values measured with respect to a fixed affine frame at infinity. This is in contrast to the axisym…
In this paper we present a new method for motion tracking of tumors in liver ultrasound image sequences. Our algorithm has two main steps. In the first step, we apply mean shift algorithm with multiple features to estimate the center of the target in each frame. Target in the first frame is defined using an ellipse. Ed…
The paper studies stability of discretized Anosov flows.
problem Global stability of discretized Anosov flows.
method Defined and proved equivalence with previous definitions, showed properties through C1 openness and closedness, and established integrability and uniqueness of invariant foliations. result Discretized Anosov flows are globally stable.
Determinantal consensus clustering improves clustering robustness.
problem Robustness of clustering algorithms.
method Use of determinantal point processes (DPP) for random restart of clustering algorithms.
result Determinantal consensus clustering outperforms classical algorithms.
We present an iterative technique for finding zeroes of vector fields on Riemannian manifolds. As a special case we obtain a ``nonlinear averaging algorithm'' that computes the centroid of a mass distribution supported in a set of small enough diameter D in a Riemannian manifold M. We estimate the convergence rate of o…
Gradient-based clustering method for various cost functions.
problem Distance-based clustering for various cost functions.
method Iterative alternating update procedure for cluster assignments and centers.
result Converges to fixed points under mild assumptions.
Traditionally, practitioners initialize the {\tt k-means} algorithm with centers chosen uniformly at random. Randomized initialization with uneven weights ({\tt k-means++}) has recently been used to improve the performance over this strategy in cost and run-time. We consider the k-means problem with semi-supervised inf…
This paper proves a canonical foliation on null infinity for Kerr-like black holes.
problem Establishing well-defined physical quantities on null infinity for Kerr-like black holes.
method Existence and uniqueness results for GCM spheres by Klainerman-Szeftel.
result Existence of a canonical foliation on future null infinity with well-defined physical quantities.
We study the mean curvature flow of hypersurfaces in Rn+1, with initial surfaces sufficiently close to the standard n-dimensional sphere. The closeness is in the Sobolev norm with the index greater than 2n+1 and therefore it does not impose restrictions of the mean curvature of the initial surface. W…
We establish the inequality for Henneaux-Teitelboim's total energy-momentum for asymptotically anti-de Sitter initial data sets which are asymptotic to arbitrary t-slice in anti-de Sitter spacetime. In particular, when t=0, it generalizes Chruściel-Maerten-Tod's inequality in the center of AdS mass coordinates. We …
For regular particle filter algorithm or Sequential Monte Carlo (SMC) methods, the initial weights are traditionally dependent on the proposed distribution, the posterior distribution at the current timestamp in the sampled sequence, and the target is the posterior distribution of the previous timestamp. This is techni…
Center manifold analysis can be used in order to investigate the stability of the stationary solutions of various PDEs. This can be done by considering the PDE as an ODE between certain Banach spaces and linearising about the stationary solution. Here we investigate the volume preserving mean curvature flow using such …
We describe explicitly the large volume isoperimetric regions of a natural class of asymptotically flat manifolds, in any dimension. These isoperimetric regions detect the mass and the center of mass of such manifolds when viewed as initial data sets for the Einstein equations in general relativity. Using the positivit…
A network-based approach identifies financial factors from asset interactions, explaining market dynamics.
problem Characterizing joint financial asset behavior through underlying drivers.
method Modeling market as coupled iterated maps, where asset returns depend on past returns and interactions.
result Stable patterns of co-movement (financial factors) emerge from asset interactions, explaining asset variance.
Researchers geometrically define asymptotic coordinates in General Relativity.
problem Understanding the asymptotic behavior of relativistic initial data sets.
method Geometrization of asymptotic flatness and analysis of geometric invariants.
result Geometrically defined asymptotic coordinates for mass, energy, momentum, and angular momentum.
Recently, several studies have explored methods for using KG embedding to answer logical queries. These approaches either treat embedding learning and query answering as two separated learning tasks, or fail to deal with the variability of contributions from different query paths. We proposed to leverage a graph attent…
We study the problem of training deep neural networks with Rectified Linear Unit (ReLU) activation function using gradient descent and stochastic gradient descent. In particular, we study the binary classification problem and show that for a broad family of loss functions, with proper random weight initialization, both…
DEMVC improves multi-view clustering with collaborative training and deep autoencoders.
problem Existing multi-view clustering methods have high computation and space complexities or lack representation capability.
method DEMVC learns embedded representations of multiple views individually using deep autoencoders and collaboratively trains all views.
result DEMVC achieves significant improvements over state-of-the-art methods on multi-view datasets.
Randomly initialized wide neural networks with zero-mean activations are nearly independent, potentially solving AI interpretability limits.
problem Measuring the limits of AI interpretability.
method Randomly initialized neural networks with large width and zero-mean activation functions.
result Neural networks with zero-mean activations are nearly independent, solving the computational no-coincidence conjecture.
The paper solves circle packings on surfaces with boundaries.
problem Circle packing on surfaces with boundaries and finite genus.
method Using Thurston's algorithm and discrete Schwarz-Pick lemma.
result A unique solution to the boundary value problem exists.
In this paper, we address the issue of linear stability of Schwarzschild space- time subject to certain axisymmetric perturbations. In particular, we prove that associ- ated solutions to the linearized vacuum Einstein equations centered at a Schwarzschild metric, with suitably regular initial data, decay to a linearize…
The entropy of a hypersurface is a geometric invariant that measures complexity and is invariant under rigid motions and dilations. It is given by the supremum over all Gaussian integrals with varying centers and scales. It is monotone under mean curvature flow, thus giving a Lyapunov functional. Therefore, the entropy…
Bayesian Federated Inference improves survival model analysis without merging data.
problem Accurately estimating survival model parameters requires sufficient data and events, which is often lacking in practice.
method Bayesian Federated Inference (BFI) for survival models, where local centers perform analyses and combine results.
result Results from BFI are similar to those from merged data analyses, demonstrating excellent performance.