Proposes a new SGD momentum optimizer using online parabola modeling.
problem Optimizing step size in stochastic gradient descent for neural networks.
method Online parabola modeling in a single direction using normalized momentum vector.
result Improves step size estimation for faster convergence and saddle repulsion.
Archimedes knew that the area between a parabola and any chord AB on the parabola is four thirds of the area of triangle ΔABP where P is the point on the parabola at which the tangent is parallel to AB. We consider whether this property (and similar ones) characterizes parabolas. We present five conditions which …
Archimedes showed that the area between a parabola and any chord AB on the parabola is four thirds of the area of triangle ΔABP, where P is the point on the parabola at which the tangent is parallel to the chord AB. Recently, this property of parabolas was proved to be a characteristic property of parabolas. With…
Study of parabolas in Funk metric on unit disk.
problem Understanding parabolas in Funk metric on a disk.
method Analyzing four types of parabolas due to Funk metric's non-reversibility.
result Two known conics and two irreducible quartics found.
Archimedes determined the center of gravity of a parabolic section as follows. For a parabolic section between a parabola and any chord AB on the parabola, let us denote by P the point on the parabola where the tangent is parallel to AB and by V the point where the line through P parallel to the axis of the p…
A flat complete causal Lorentzian manifold is called {\it strictly causal} if the past and the future of each its point are closed near this point. We consider strictly causal manifolds with unipotent holonomy groups and assign to a manifold of this type four nonnegative integers (a signature) and a parabola in the con…
It is well known that the area U of the triangle formed by three tangents to a parabola X is half of the area T of the triangle formed by joining their points of contact. In this article, we consider whether this property and similar ones characterizes parabolas. As a result, we present three conditions which are…
It is well known that the area U of the triangle formed by three tangents to a parabola X is half of the area T of the triangle formed by joining their points of contact. In this article, we study some properties of U and T for strictly convex plane curves. As a result, we establish a characterization for par…
At each point in an immersed surface in R4 there is a curvature ellipse in the normal plane which codifies all the local second order geometry of the surface. More recently, at the singular point of a corank 1 singular surface in R3, a curvature parabola in the normal plane which codifies all the …
New curvature defined for corank 1 singular surfaces in 3D.
problem Defining curvature for singular surfaces.
method Introducing axial vector and curvature parabola to define axial curvature.
result Relates axial curvature to Gaussian curvature of a blow-up for certain singularities.
We establish a link between Archimedes' method of integration for calculating areas, volumes and centers of mass of segments of parabolas and quadrics of revolution by factorization via the moments of a balance and an integration technique for a particular integrable system, namely Bianchi's Bäcklund transformation for…
A method is proposed to construct spiral curves by inversion of a spiral arc of parabola. The resulting curve is rational of 4-th order. Proper selection of the parabolic arc and parameters of inversion allows to match a wide range of boundary conditions, namely, tangents and curvatures at the endpoints, including thos…
We consider smooth 1-parameter families of plane curves tangent to a semicubic parabola, when the curvature radius of their curves at the tangency point vanishes at the cusp point. We find the $\A$-normal form of these families, their envelopes and local patterns near the cusp. We obtain a new codimension 2 singularity…
The paper classifies surfaces with isotropic circles through each point.
problem Classifying surfaces with isotropic circles through each point.
method Using isotropic circles as Euclidean circles and applying Skopenkov and Krasauskas' methods.
result Surfaces containing two isotropic circles through each point have a specific parametrization.
The theory of slice regular functions of a quaternion variable is applied to the study of orthogonal complex structures on domains Ω of R^4. When Ω is a symmetric slice domain, the twistor transform of such a function is a holomorphic curve in the Klein quadric. The case in which Ω is the complement of a parabola is st…
Alternative finance models from physics for non-equilibrium systems.
problem Inequities of classical finance models in physics-based perspective.
method Physics-based insights for non-equilibrium finance models.
result Alternative models for non-equilibrium finance systems.
We study the geometry of surfaces in R4 with corank 1 singularities. For such surfaces the singularities are isolated and at each point we define the curvature parabola in the normal space. This curve codifies all the second order information of the surface. Also, using this curve we define asymptotic a…
The cone projection fR(z)=z/(1+∣z∣/R) maps lines to conic arcs with specific properties.
problem Mapping lines to conic arcs with specific properties.
method Using a reciprocal lens identity and radial homeomorphism.
result The Self-Directrix Theorem and Confocal--Codirectrix Theorem.
New method constructs translationally equivariant hyperbolic affine spheres.
problem Constructing translationally equivariant hyperbolic affine spheres.
method Noncompact Iwasawa factorization via DPW method and Weierstrass elliptic functions.
result Every translationally equivariant hyperbolic affine sphere is equiaffinely equivalent to one with a circle, hyperbola, or parabola slice curve.
Study the geometry of a surface formed by extending a Whitney umbrella.
problem Investigate the geometric properties of a specific surface formed by extending a Whitney umbrella.
method Analyze the intersection with the normal plane, geodesic and normal curvatures, Gaussian and mean curvatures.
result Determine the zeros of curvature functions and deduce geometric relationships.
The study examines spectral properties of the Laplacian on forms for open Riemannian manifolds.
problem Investigating spectral properties of the Laplacian on forms for open Riemannian manifolds.
method Finding sufficient conditions for the Weyl criterion to hold for the Lp-spectrum of the Laplacian on k-forms, proving the decomposition of the Lp-spectrum, and analyzing the resolvent set of the Laplacian. result The Lp-spectrum of the Laplacian on k-forms over hyperbolic space is described in detail. Unified description of aesthetic curves through self-affinities.
problem Characterizing log-aesthetic curves and their properties.
method Reformulating and proving self-affinities of planar curves, integrating equiaffine geometry.
result Unified characterization of constant curvature curves in similarity and equiaffine geometries.
This paper studies the continuous time mean-variance portfolio selection problem with one kind of non-linear wealth dynamics. To deal the expectation constraint, an auxiliary stochastic control problem is firstly solved by two new generalized stochastic Riccati equations from which a candidate portfolio in feedback for…
The Strong Cosmic Censorship conjecture states that for generic initial data to Einstein's field equations, the maximal globally hyperbolic development is inextendible. We prove this conjecture in the class of orthogonal Bianchi class B perfect fluids and vacuum spacetimes, by showing that unboundedness of certain curv…
Researchers create new types of surfaces in hyperbolic space.
problem Constructing biconservative surfaces in hyperbolic space.
method Intrinsic and extrinsic approaches using parametric equations and gluing.
result Found three families of simply connected, complete, non-CMC biconservative surfaces. Solitons are special polygon midpoints under affine transformations.
problem Characterizing polygons whose midpoints under affine transformations form a new polygon.
method Analyzing midpoints polygons and their relationship to affine transformations and differential equations.
result A large class of polygons are on an orbit of a one-parameter subgroup of the affine group, and these curves are solutions to a specific differential equation.
Using the complex parabolic rotations of holomorphic null curves in C4, we transform minimal surfaces in Euclidean space R3⊂R4 to a family of degenerate minimal surfaces in Euclidean space R4. Applying our deformation to holomorphic null curves in ${…
Paper classifies conic submanifolds in control systems.
problem Characterizing and classifying conic submanifolds in control systems.
method Feedback equivalence of control-affine and fully nonlinear systems.
result Complete description of non-degenerate conic submanifolds.
Introduces a new geometry based on difference angles, showing unique properties.
problem Defining angles independently of circles or rotations.
method Axiomatic system for difference angles, defining new geometric constructs.
result Explicit confirmation of the concurrency of the parabolic Miquel configuration.
The paper generalizes a mean value theorem for solutions of the ultrahyperbolic equation.
problem Establishing a mean value property for solutions of the ultrahyperbolic equation.
method Using conformal maps of the pseudo-Euclidean space of signature 2+2, the paper extends Asgeirsson's theorem to a more general class of pairs of curves.
result The mean value property is proven for non-degenerate conjugate conics, including conjugate circles, hyperbolae, parabolae, and line-empty pairs.
Online boosting method improves weak to strong learner.
problem Online learning of weak to strong learner.
method Extends batch GentleAdaBoost to online approach with line search.
result Online boosting performs better than other methods.
Paper tackles online optimization with memory and competitive control.
problem Minimizing hitting and switching costs in online optimization problems.
method Optimistic Regularized Online Balanced Descent algorithm.
result Achieves a constant, dimension-free competitive ratio.
Study on computable online learning with new conditions and complexities.
problem Characterizing optimal online learning under varying optimality requirements.
method Introduced anytime optimal (a-optimal) online learning and explored computational separations.
result Found a computational separation between a-optimal and optimal online learning.
We study the task of online boosting--combining online weak learners into an online strong learner. While batch boosting has a sound theoretical foundation, online boosting deserves more study from the theoretical perspective. In this paper, we carefully compare the differences between online and batch boosting, and pr…
Proposes an online method for high-dimensional streaming data.
problem Increasing variable dimensions with sample size in online kernel sliced inverse regression.
method Introduces approximate linear dependence condition and dictionary variable sets to address the problem. Transforms into online generalized eigen-decomposition problem and uses stochastic optimization for updates.
result Achieves close performance to batch processing kernel sliced inverse regression.
Boosts weak online learners to strong ones with sublinear regret.
problem Online learning agnostic setting without strong guarantees.
method Reduction to online convex optimization, boosting via marginally-better-than-trivial regret guarantees.
result First agnostic online boosting algorithm with sublinear regret.
Improved online classification with accurate predictions.
problem Online classification challenges with limited data.
method Designing an online learner that uses predictions to reduce regret.
result Expected regret is better than worst-case analysis, especially with accurate predictions.
Continuous-time algorithms improve online learning performance.
problem Online learning with sequential data and minimizing overall regret.
method Extending discrete-time algorithms to continuous-time models for online linear optimization, adversarial bandit, and adversarial linear bandit.
result Optimal regret bounds are proven for continuous-time settings.
Extends boosting to multiclass online agnostic classification.
problem Online multiclass classification with weak learners.
method Reduces multiclass online agnostic boosting to online convex optimization.
result First boosting algorithm for online agnostic multiclass classification.
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.
New private algorithms for online learning improve regret in high privacy regimes.
problem Private online learning from experts and convex optimization.
method Transformed lazy algorithms for differential privacy.
result Improved regret bounds for DP-OPE and DP-OCO.
Online-iForest detects anomalies in streaming data efficiently.
problem Offline anomaly detection methods are impractical for streaming contexts.
method Online-iForest tracks evolving data processes in real-time without periodic retraining.
result Online-iForest outperforms all competitors in efficiency.
Study online learning with set-valued feedback, showing differences between deterministic and randomized approaches.
problem Online learning with set-valued feedback, where labels are sets rather than single labels.
method Introduced new combinatorial dimensions (Set Littlestone and Measure Shattering) to characterize learnability.
result Characterized deterministic and randomized online learnability, and established bounds for various learning settings.
Transforms offline algorithms to online with low regret in random order model.
problem Developing online algorithms with low approximate regret from offline approximation algorithms.
method General reduction theorem and coreset construction method.
result Achieves polylogarithmic ε-approximate regret for various online problems.
New setup for continuous online learning improves understanding of imitation learning.
problem Challenges in capturing regularity in online problems.
method Continuous Online Learning (COL) setup, focusing on continuous gradient changes.
result Fundamental equivalence between sublinear dynamic regret and solving certain EPs.
Paper proposes an online transfer learning framework using online bagging.
problem Difficulty in obtaining sufficient labeled data in the target domain.
method Ensemble approach with online bagging for anytime transfer learning.
result Effectiveness of the proposed algorithms demonstrated on real data sets.
Private learning can be used to efficiently solve online learning problems.
problem The relationship between differentially private learning and online learning efficiency.
method Derive an efficient black-box reduction from differentially private learning to online learning from expert advice.
result An efficient differentially private learner implies an efficient online learner.
Predicts student performance in interactive online question pools using GNNs.
problem Predicting student performance in interactive online question pools with evolving knowledge.
method Proposes R^2GCN, a GNN model for heterogeneous networks to predict student performance.
result Achieves higher accuracy in student performance prediction than traditional methods.