Proves conjecture on deformation invariance of big fundamental groups.
problem Stability of big fundamental groups under small deformations.
method Deformation regularity of equivariant pluriharmonic maps and techniques from Shafarevich conjectures.
result Deformation openness of big fundamental groups for varieties with big complex local systems.
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.
Study proves stability of big bang singularity in complex system.
problem Stability of Kasner solutions in Einstein-Maxwell-scalar field-Vlasov system.
method Detailed mathematical structures and new delicate arguments.
result Nonlinear stability with Kasner exponents in full strong sub-critical regime.
Estimates Kaehler metrics' diameter in big cohomology classes.
problem Estimating the diameter of Kaehler metrics in big cohomology classes.
method Proves uniform diameter estimates using integrability conditions and stability properties of complex Monge-Ampere equations.
result Uniform diameter estimates for Kaehler metrics in big cohomology classes.
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…
Study foundational aspects of degenerate para-CR structures and their PDE systems.
problem Understanding invariants and degeneracies of submanifolds in para-CR structures.
method Analyzing split-diffeomorphisms and Levi forms, setting up foundational material.
result Equivalence of PDE system properties under 2-nondegeneracy conditions.
Paper optimizes a big data and ML risk monitoring system for financial markets.
problem Traditional risk monitoring methods are inadequate for modern financial markets due to data complexity and volume.
method Four-layer architecture integrating big data and advanced ML algorithms (LSTM, RF, GB).
result Significantly enhances efficiency and accuracy in risk management, especially in market crash risk detection.
New algorithm learns halfspaces with adversarial noise efficiently.
problem Learning halfspaces in the presence of adversarial noise.
method Polynomial-time Perceptron-like online active learning algorithm.
result Near-optimal label and sample complexity with isotropic log-concave marginal distribution.
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.
One of the key technologies for future large-scale location-aware services covering a complex of multi-story buildings --- e.g., a big shopping mall and a university campus --- is a scalable indoor localization technique. In this paper, we report the current status of our investigation on the use of deep neural network…
As a non-parametric Bayesian model which produces informative predictive distribution, Gaussian process (GP) has been widely used in various fields, like regression, classification and optimization. The cubic complexity of standard GP however leads to poor scalability, which poses challenges in the era of big data. Hen…
The paper discusses scalable learning for wireless data-driven systems.
problem Expanding data volume and model complexity limit centralized learning solutions.
method Discusses scalable architecture and local learning strategies.
result Promising research directions in scalable data-driven wireless communications.
Localized big bang singularities found without background solutions.
problem Proving localized big bang formation without proximity to background solutions.
method Introducing a new foliation by spacelike hypersurfaces and a time function to synchronize and stabilize the singularity.
result Maximally globally hyperbolic developments have local quiescent big bang singularities with curvature blow-up.
Paper presents a data preprocessing method for PHM models.
problem Lack of consistent data preprocessing for PHM applications.
method Comprehensive pipeline for sensor data preprocessing.
result Creation of clean data sets for training machinery health state classifiers.
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 proves boundedness of envelopes in complex manifolds.
problem Regularity of envelopes in complex manifolds.
method Analyzes bounded functions and their envelopes in the context of cohomology classes and Laplacians.
result The α-psh envelope P(f) is locally bounded with locally bounded Laplacian on the ample locus of {α}. The recent trend for acquiring big data assumes that possessing quantitatively more and qualitatively finer data necessarily provides an advantage that may be critical in competitive situations. Using a model complex adaptive system where agents compete for a limited resource using information coarse-grained to differe…
Let X be a smooth projective complex variety of maximal Albanese dimension, and let L→X be a big line bundle. We prove that the moving Seshadri constants of the pull-backs of L to suitable finite abelian étale covers of X are arbitrarily large. As an application, given any integer k≥1, there exists an…
New methods prove controllability of non-linear systems, extending classical results.
problem Controllability of non-linear control systems.
method Analytic control system, graph completions, flows of vector fields, pseudogroup of local diffeomorphisms.
result Sufficient conditions for local controllability and accessibility of non-linear systems.
Improves EM algorithm for better local optima in mixture models.
problem EM algorithm's sensitivity to initialization and bad local optima.
method Big Learning principle applied to upgrade EM algorithm.
result BigLearn-EM delivers optimal solution with high probability.
We prove a C1,1 estimate for solutions of complex Monge-Ampère equations on compact Kähler manifolds with possibly nonempty boundary, in a degenerate cohomology class. This strengthens previous estimates of Phong-Sturm. As applications we deduce the local C1,1 regularity of geodesic rays in the space of Kähle…
The goal of this work is to prove the regularity of certain quasi-plurisubharmonic upper envelopes. Such envelopes appear in a natural way in the construction of hermitian metrics with minimal singularities on a big line bundle over a compact complex manifold. We prove that the complex Hessian forms of these envelopes …
FS&P uses birth-death process to ensure global convergence of stochastic conic particle gradient descent.
problem Global optimization of non-convex objective functions over measure space.
method Introduces Fast Spawn\&Prune (FS\&P) combining CPGD with birth-death process.
result First theoretical guarantee of global convergence for discrete-time stochastic algorithms.
We study the local equivalence problem for real-analytic (Cω) hypersurfaces M5⊂C3 which, in coordinates (z1,z2,w)∈C3 with w=u+iv, are rigid: \[ u \,=\, F\big(z_1,z_2,\overline{z}_1,\overline{z}_2\big), \] with F independent of v. Specifically, we study th…
The study finds conditions for a third rank Killing tensor field on a 2D Riemannian torus.
problem Conditions for the existence of a third rank Killing tensor field on a 2D Riemannian torus.
method Analyzes the metric of the torus and uses Fourier coefficients to derive conditions for the function λ.
result Equations relating Fourier coefficients of the function λ determine the existence of a third rank Killing tensor field.
This work is devoted to a systematic study of symplectic convexity for integrable Hamiltonian systems with elliptic and focus-focus singularities. A distinctive feature of these systems is that their base spaces are still smooth manifolds (with boundary and corners), similarly to the toric case, but their associated in…
The rise of Big Data has led to new demands for Machine Learning (ML) systems to learn complex models with millions to billions of parameters, that promise adequate capacity to digest massive datasets and offer powerful predictive analytics thereupon. In order to run ML algorithms at such scales, on a distributed clust…
GT-SARAH optimizes decentralized non-convex problems with recursive variance reduction.
problem Decentralized non-convex optimization of N functions over a network. method Stochastic first-order gradient method with SARAH variance reduction and gradient tracking.
result Achieves ε-accurate first-order stationary point with improved gradient complexity. Interpretability has always been a major concern for fuzzy rule-based classifiers. The usage of human-readable models allows them to explain the reasoning behind their predictions and decisions. However, when it comes to Big Data classification problems, fuzzy rule-based classifiers have not been able to maintain the g…
This paper uses information theory to improve risk modeling in big data.
problem Insufficient application of information theory in actuarial science.
method Explores information theory to uncover performance limits of insurance big data systems.
result Guidance for risk modeling and actuarial pricing systems.
Proves left-orderability of mapping class groups of infinite-type surfaces.
problem Left-orderability of mapping class groups of infinite-type surfaces.
method Inductive construction of a stable Alexander system and ideal arc systems.
result Proves left-orderability using carefully chosen exhaustion by finite-type subsurfaces.
A method to select important experts for Gaussian processes to balance computational efficiency and uncertainty quantification.
problem Balancing computational efficiency and uncertainty quantification in Gaussian processes for big data.
method Using graphical models to select important experts and aggregate their predictions while ensuring uncertainty quantification.
result Substantially reduces computational cost of aggregating dependent experts while ensuring calibrated uncertainty quantification.
Let G/H be a contractible homogeneous Sasaki manifold. A compact locally homogeneous aspherical Sasaki manifold Γ\G/H is by definition a quotient of G/H by a discrete uniform subgroup Γ≤G. We show that a compact locally homogeneous aspherical Sasaki manifold is always quasi-regular, that is, $…
Improved algorithm reduces stochastic gradient complexity for large-scale learning problems.
problem High stochastic gradient complexity for large-scale learning problems.
method Hybrid Stochastic-Deterministic Minibatch Proximal Gradient (HSDMPG) algorithm.
result Achieves nearly optimal generalization in less than a single pass over data.
We establish various stability results for solutions of complex Monge-Ampère equations in big cohomology classes, generalizing results that were known to hold in the context of Kähler classes.
In this survey, we remind some fibrations structure theorems (also called Milnor's fibrations) recently proved in the real and complex case, in the local and global settings. We give several Poincaré-Hopf type formulae which relates the Euler-Poincaré characteristic of these fibers (also called Milnor's fibers) and ind…
Sharp constants in curl-Sobolev inequalities on spheres determined.
problem Determining sharp constants in curl-Sobolev inequalities on spheres.
method Analyzing conformally invariant Sobolev quotients and using local stability estimates.
result Strict upper bound for the sharp constant of the J2 inequality. New method improves likelihood-free parameter estimation in complex models.
problem Estimating parameters in simulation-based models with unknown likelihood.
method Nested multi-time-scale stochastic approximation (NMTS) method.
result Eliminates bias and accelerates convergence in likelihood-free inference.
Efficiently learns sparse halfspaces with noisy labels.
problem Learning sparse halfspaces with arbitrary bounded noise.
method Polynomial time algorithm for active learning with improved label complexity.
result First efficient algorithm with label complexity polynomial in 1−2η1. 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.
The paper studies positivity properties of cotangent bundles in complex hyperbolic manifolds with cusps.
problem Positivity properties of cotangent bundles in complex hyperbolic manifolds with cusps.
method Analyzes intrinsic positivity properties of cotangent bundles using toroidal compactifications and ample line bundles.
result The cotangent bundle is ample modulo the boundary divisor for sufficiently small rational numbers.
Proves properties of complex algebraic varieties and local systems.
problem Properties of complex algebraic varieties and local systems.
method Analytic Zariski open subsets and algebraic maps.
result Trivializing covering spaces of complex algebraic varieties.
Agent-based modeling is a powerful simulation technique to understand the collective behavior and microscopic interaction in complex financial systems. Recently, the concept for determining the key parameters of the agent-based models from empirical data instead of setting them artificially was suggested. We first revi…
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.
Continuity of complex Monge-Ampère potentials on Kähler manifolds.
problem Continuity of solutions to complex Monge-Ampère equations on compact Kähler manifolds.
method Extending DiNezza-Lu's approach to big cohomology classes, proving continuity on Zariski open sets.
result Singular Kähler-Einstein metrics have continuous potentials on the ample locus outside of the non-klt part.
The world is witnessing an unprecedented growth of cyber-physical systems (CPS), which are foreseen to revolutionize our world {via} creating new services and applications in a variety of sectors such as environmental monitoring, mobile-health systems, intelligent transportation systems and so on. The {information and …
The following problem is addressed: A 3-manifold M is endowed with a triple Ω=(Ω1,Ω2,Ω3) of closed 2-forms. One wants to construct a coframing ω=(ω1,ω2,ω3) of M such that, first, dωi=Ωi for i=1,2,3, and, second, the Riemannian metric $g=\big(ω^1\big)^2+\big(ω^2\big)^2+\…
This paper studies the problem of distributed stochastic optimization in an adversarial setting where, out of the m machines which allegedly compute stochastic gradients every iteration, an α-fraction are Byzantine, and can behave arbitrarily and adversarially. Our main result is a variant of stochastic gradient de…