New problems on NNSC fill-ins for Bartnik data in high dimensions.
problem Conditions for (n−1)-dimensional Bartnik data to be NNSC-cobordant. method Formulating three problems related to nonnegative scalar curvature fill-ins.
result Conditions for (n−1)-dimensional Bartnik data to be NNSC-cobordant. New algorithm finds approximate stationary points faster under differential privacy constraints.
problem Finding approximate stationary points of smooth and Lipschitz functions under differential privacy constraints.
method Developed an efficient algorithm that improves convergence rates to stationary points.
result Achieved faster rates of convergence to stationary points in both finite-sum and stochastic settings.
New algorithm for private non-convex optimization with optimal rates.
problem Private optimization of non-convex functions under KL condition.
method Variance-reduced gradient descent and proximal point method.
result Achieves nearly optimal rates for excess empirical risk.
In Part III of this study, we apply the price dynamical model with big buyers and big sellers developed in Part I of this paper to the daily closing prices of the top 20 banking and real estate stocks listed in the Hong Kong Stock Exchange. The basic idea is to estimate the strength parameters of the big buyers and the…
Big data sets must be carefully partitioned into statistically similar data subsets that can be used as representative samples for big data analysis tasks. In this paper, we propose the random sample partition (RSP) data model to represent a big data set as a set of non-overlapping data subsets, called RSP data blocks,…
Establishes Kobayashi-Hitchin correspondence for nef and big classes.
problem Analyzing stability and positivity in algebraic geometry.
method Introducing adapted currents and metrics to establish correspondence.
result Equality cases of Bogomolov-Gieseker and Miyaoka-Yau inequalities.
This article provides the role of big idea statisticians in future of Big Data Science. We describe the `United Statistical Algorithms' framework for comprehensive unification of traditional and novel statistical methods for modeling Small Data and Big Data, especially mixed data (discrete, continuous).
Geometric quantization extended to big line bundles.
problem Quantization of line bundles with large curvature.
method Proving asymptotic isometry and submultiplicative norms equivalence, showing Mabuchi geodesic rays.
result Bounded submultiplicative filtrations on big line bundles lead to Mabuchi geodesic rays.
Metric study on Kähler manifolds with prescribed singularities.
problem Defining a metric space for Kähler potentials with prescribed singularities.
method Introducing a distance d and dA on the relative finite energy class and showing convergence. result The space XA is complete and converges in Gromov-Hausdorff sense. New DP algorithms achieve near-optimal regret bounds for online learning problems.
problem Online learning problems with zero-loss solutions and differential privacy constraints.
method Developed new Differentially Private algorithms with near-optimal regret bounds.
result Achieved near-optimal regret bounds for various online prediction and convex optimization problems.
The paper calculates large genus limits for two types of Siegel-Veech constants.
problem Large genus asymptotics for Siegel-Veech constants in Abelian differentials.
method Combining combinatorial analysis and large genus asymptotics of Masur-Veech volumes.
result The saddle connection and area Siegel-Veech constants converge to specific values as genus grows large.
DPASF stream preprocesses Big Data streams efficiently.
problem Efficient preprocessing of streaming Big Data.
method Implemented six preprocessing algorithms in Apache Flink.
result Preprocessing improves data accuracy in streaming Big Data.
New insights into spectral statistics of sample covariance matrix for stable linear systems.
problem Estimating high-dimensional stable state transition matrices from noisy data.
method Combining spectral theorem for non-Hermitian operators, concentration of measure, and perturbation theory.
result The spectral radius of the sample covariance matrix exhibits phase transitions in high dimensions.
Refines spinorial Sobolev inequality on sphere, proving stability and new properties of Killing spinors.
problem Stability of spinorial Sobolev inequalities on the unit sphere.
method Establishes a refined spinorial Sobolev inequality and proves stability.
result Stability inequality and new properties of Killing spinors.
Study assesses 'big data' in materials science, highlighting challenges.
problem Understanding what constitutes 'big data' in materials science.
method Selected examples of machine learning models, data quality, and infrastructure requirements.
result Big data presents unique challenges in materials science.
Online learning improves big data accuracy quickly.
problem Heterogeneity in big data analysis.
method Online machine learning for big data.
result Online learning converges quickly to batch accuracy.
Uniform volume estimate for Kähler metrics in big cohomology classes.
problem Estimating volume for singular Kähler metrics in big cohomology classes.
method Generalized mixed energy estimate for functions in complex Sobolev space to big cohomology classes.
result Uniform non-collapsing volume estimate for local Kähler metrics.
Currently, the world is witnessing a mounting avalanche of data due to the increasing number of mobile network subscribers, Internet websites, and online services. This trend is continuing to develop in a quick and diverse manner in the form of big data. Big data analytics can process large amounts of raw data and extr…
Study convexity of Mabuchi functional in big cohomology classes.
problem Convexity of Mabuchi functional in big cohomology classes.
method Defined an invariant related to transcendental Fujita approximations and established convexity under vanishing of this invariant.
result Established almost convexity along weak geodesics in big cohomology classes.
Mobile big data contains vast statistical features in various dimensions, including spatial, temporal, and the underlying social domain. Understanding and exploiting the features of mobile data from a social network perspective will be extremely beneficial to wireless networks, from planning, operation, and maintenance…
Constructing flat metrics with prescribed coframings on 3-manifolds.
problem Constructing a flat metric on a 3-manifold given a set of closed 2-forms.
method Using a triple of closed 2-forms to construct a coframing and a flat metric.
result The problem is solvable on a neighborhood of a point in the nonsingular case, with the solution depending on three arbitrary functions of two variables.
Big data transforms accounting and auditing, enhancing insights but posing challenges.
problem Challenges in data privacy and security with increased data sources.
method Utilizing AI and machine learning for efficient data analysis and anomaly detection.
result Enhanced analytics tools and continuous learning are key to overcoming challenges.
The paper explores machine learning in mobile big data analysis.
problem Challenges in mobile big data analysis.
method Discussion and review of existing methods.
result Identification of main challenges and future directions.
Big Data classifiers perform similarly to Small Data classifiers, suggesting scalability tradeoffs.
problem Comparing Big Data classifiers to Small Data classifiers for performance and scalability.
method Empirical study comparing Big Data classifiers to Small Data classifiers.
result Big Data classifiers are slightly inferior but catching up with Small Data classifiers.
Unified view on big bang singularities from initial data.
problem Understanding quiescent big bang singularities from initial data.
method Unified geometric perspective on various results.
result Solutions of Oude Groeniger et al. induce data on the singularity.
We consider model-free reinforcement learning for infinite-horizon discounted Markov Decision Processes (MDPs) with a continuous state space and unknown transition kernel, when only a single sample path under an arbitrary policy of the system is available. We consider the Nearest Neighbor Q-Learning (NNQL) algorithm to…
The paper examines percentiles of non-identical random variables and provides non-asymptotic bounds.
problem Investigating percentiles of independent but non-identical random variables.
method Analyzing the 100(1−p)%-th percentile X(pn) for a wide class of distributions. result Discovering a connection between the median and the harmonic mean of standard deviations for certain distributions.
Big Data bring new opportunities to modern society and challenges to data scientists. On one hand, Big Data hold great promises for discovering subtle population patterns and heterogeneities that are not possible with small-scale data. On the other hand, the massive sample size and high dimensionality of Big Data intro…
The paper tackles machine unlearning by designing efficient algorithms for adaptive query classes.
problem Designing efficient unlearning algorithms for machine learning models.
method Formalizes the problem and gives efficient unlearning algorithms for linear and prefix-sum query classes.
result Improved guarantees for stochastic convex optimization with reduced unlearning query complexity.
In this paper, we consider the problem of sequentially optimizing a black-box function f based on noisy samples and bandit feedback. We assume that f is smooth in the sense of having a bounded norm in some reproducing kernel Hilbert space (RKHS), yielding a commonly-considered non-Bayesian form of Gaussian process …
Explosive growth in data and availability of cheap computing resources have sparked increasing interest in Big learning, an emerging subfield that studies scalable machine learning algorithms, systems, and applications with Big Data. Bayesian methods represent one important class of statistic methods for machine learni…
Characterizes and analyzes the large scale geometry of big mapping class groups of surfaces.
problem Analyzing the large scale geometry of big mapping class groups of surfaces with a unique maximal end.
method Building on previous work, the paper characterizes and analyzes the large scale geometry of big mapping class groups of surfaces with a unique maximal end.
result Proves that any locally CB big mapping class group is CB generated and gives an explicit criterion for determining which big mapping class groups are CB generated.
Proves existence of Kähler-Einstein metrics in big cohomology classes.
problem Existence of Kähler-Einstein metrics in big cohomology classes.
method Using a divisorial stability condition and Fujita-Odaka type delta invariants, building up from scratch the theory of pluripotential theory.
result Uniform Yau-Tian-Donaldson existence theorem for Kähler-Einstein metrics in the big cohomology class setting.
Big data analytics improves healthcare through early detection and quality life.
problem Limited access to healthcare data hinders evidence-based decision-making.
method Analysis of healthcare data using various tools and techniques.
result Big data analytics enhances healthcare quality and patient outcomes.
Study local equivalence of rigid 5D hypersurfaces in C^3, finding necessary and sufficient conditions for rigid biholomorphism.
problem Local equivalence problem for real-analytic rigid hypersurfaces in C^3.
method Cartan-type reduction to an appropriate {e}-structure, finding primary invariants.
result Identify necessary and sufficient condition for rigid biholomorphism using invariants.
Big mapping class groups of infinite type surfaces have infinite asymptotic dimension.
problem Understanding asymptotic dimension of big mapping class groups of infinite type surfaces.
method Analyzing big mapping class groups with coarsely bounded generating sets and essential shifts.
result Big mapping class groups of infinite type surfaces have infinite asymptotic dimension.
Paper defines new stability and metrics for complex spaces.
problem Stability and metrics for complex spaces with big cohomology classes.
method Introduces slope stability and Hermitian-Einstein metrics for big cohomology classes.
result Kobayashi Hitchin correspondence and Bogomolov Gieseker inequality proved.
New obstruction prevents certain spacetimes with both big bang and big crunch.
problem Preventing spacetimes with both big bang and big crunch.
method Analyzing initial data sets subject to dominant energy condition and enlargeability obstruction.
result Pairs of spacetimes with both big bang and big crunch are not connected in certain cases.
Uniform Ding stability implies existence of Kähler-Einstein metric on big anticanonical manifolds.
problem Existence of Kähler-Einstein metrics on manifolds with big anticanonical class.
method Developed a theory of Deligne functionals and slope formulas for singular metrics, proving a slope formula for the Ding functional in the big setting.
result Existence of a unique Kähler-Einstein metric implies uniform Ding stability.
Improved bounds for algebraic degeneracy and hyperbolicity of hypersurfaces.
problem Improving bounds for algebraic degeneracy and hyperbolicity of hypersurfaces in complex projective space.
method Combining techniques from Diverio-Merker-Rousseau, Bérczi, and Darondeau with computer explorations.
result New degree bounds for algebraic degeneracy and hyperbolicity, improving previous results.
The paper proposes a new model using financial big data to improve portfolio risk analysis.
problem Addressing potential information loss in portfolio risk measurement.
method Uses financial big data to incorporate out-of-target-portfolio information and overcomes the curse of dimensionality.
result The use of financial big data improves small portfolio risk analysis.
Extends K-stability theory to projective klt pairs with a big anticanonical class.
problem Behavioral pathologies in K-stability for projective klt pairs with a big anticanonical class.
method Extends K-stability theory to projective klt pairs with a big anticanonical class, observing that K-semistability forces a klt anticanonical model with the same stability property.
result K-semistability forces projective klt pairs with a big anticanonical class to have a klt anticanonical model with the same stability property.
To any g-manifold M are associated two dglas tot(Λ∙g∨⊗kTpoly∙) and tot(Λ∙g∨⊗kDpoly∙), whose cohomologies $H_{\operatorn…
New invariants prove existence of Kahler-Einstein metrics on big classes.
problem Existence of Kahler-Einstein metrics on varieties with klt singularities.
method Introducing new invariants and proving a generalization of the Tian-Odaka-Sano Theorem.
result Proves existence of twisted Kahler-Einstein metrics on big classes.
We consider spacetimes N satisfying some structural conditions, which are still fairly general, and prove convergence results for the leaves of an inverse mean curvature flow. Moreover, we define a new spacetime N^ by switching the light cone and using reflection to define a new time function, such that the two…
Proves inequalities for hypersurfaces in the sphere, solving a long-standing problem.
problem Proving inequalities for hypersurfaces in the sphere.
method Using mixed volumes and quermassintegrals, the authors prove inequalities equivalent to a sharp relation among three adjacent quermassintegrals.
result Proves inequalities for hypersurfaces in the sphere, equivalent to a sharp relation among three adjacent quermassintegrals.
The paper improves smoothed analysis for online problems with adaptive adversaries.
problem Online prediction, discrepancy minimization, and online optimization with adaptive adversaries.
method General technique to prove smoothed guarantees against adaptive adversaries, reducing to simpler oblivious adversaries.
result Strong smoothed guarantees for three online problems, matching or improving previous results.
This study designs a financial risk control platform using big data and machine learning.
problem Traditional risk management models are inadequate for modern financial complexities.
method Big data mining, real-time streaming data processing, statistical analysis, and precise customer behavior mining.
result The platform effectively identifies and responds to potential risks in real-time.