Study on fourth order Lamm-Riviere system for biharmonic mappings in 4D.
problem Higher order regularity and sharp Holder continuity of weak solutions.
method Optimal higher order regularity and sharp Holder continuity through analysis of the Lamm-Riviere system.
result Derive weak compactness for sequences of weak solutions with uniformly bounded energy.
In this paper we discuss the possibility of using multilevel Monte Carlo (MLMC) methods for weak approximation schemes. It turns out that by means of a simple coupling between consecutive time discretisation levels, one can achieve the same complexity gain as under the presence of a strong convergence. We exemplify thi…
Study shows comparison principle for weak solutions in narrow domains.
problem Weak solutions to degenerate quasilinear elliptic equations in unbounded tubular domains.
method Showed weak comparison principle holds under narrow conditions.
result Weak comparison principle holds in narrow tubular domains.
Proposes a new regression framework for mixed strong and weak guidance.
problem Current regression frameworks cannot use both strong and weak guidance.
method Introduces a probabilistic formulation for weak guidance based on relative orderings, bounds, and similarity relations.
result Optimization problems with weak guidance are convex.
Let (W,S) be a finite rank Coxeter system with W infinite. We prove that the limit weak order on the blocks of infinite reduced words of W is encoded by the topology of the Tits boundary of the Davis complex X of W. We consider many special cases, including W word hyperbolic, and X with isolated flats. We establish tha…
Paper investigates conditions for independence of weak gradients on metric spaces.
problem Dependence of weak gradients on p in arbitrary metric measure spaces. method Investigates the Bounded Interpolation Property to ensure independence of weak gradients.
result Bounded Interpolation Property guarantees independence of weak gradients.
Study improves weak error estimates for rough volatility models.
problem Efficient numerical schemes for non-Markovian stochastic processes with rough volatility.
method Analyzes weak rates for a class of stochastic processes with rough stochastic volatility.
result Weak rate is of order min{3H+0.5, 1} for a large class of test functions.
This paper studies the Glosten Milgrom model whose risky asset value admits an arbitrary discrete distribution. Contrast to existing results on insider's models, the insider's optimal strategy in this model, if exists, is not of feedback type. Therefore a weak formulation of equilibrium is proposed. In this weak formul…
A new method for averaging probability distributions based on optimal weak mass transport.
problem Averaging probability distributions in a geometric way.
method Weak barycenters based on optimal weak mass transport.
result Extracts common geometric information shared by all input distributions.
Introduces weak (p,k)-Dirac structures in geometric settings.
problem Defining and analyzing new geometric structures.
method Introducing and studying weak (p,k)-Dirac structures in TM⊕ΛpT∗M. result Weak (p,k)-Dirac structures contain more information than (p,k)-Lagrangian structures. Paper explores weak solutions' regularity in critical dimensions without conservation law.
problem Regularity of weak solutions to higher order elliptic systems in critical dimensions.
method Elementary and unified treatment, without conservation law.
result Interior Hölder continuity for solutions in critical dimensions.
Establishes a microstructural foundation for a rough log-normal volatility model.
problem Developing a robust model for financial volatility under microstructural effects.
method Introduced a sequence of order-driven financial market models with Poisson process arrivals and analyzed their convergence to a log-normal rough volatility model.
result Weak convergence of price-volatility process to a log-normal rough volatility model with established weak error rates.
Removes singularity order for Willmore immersions, reducing bubbling scenarios.
problem Understanding the singularity order of weak limits of Willmore immersions.
method Obtains removability result on singularity order, reducing bubbling scenarios.
result Only three out of twelve non-planar minimal surfaces may occur as bubbles of Willmore immersions.
Snorkel DryBell uses weak supervision to speed up machine learning model development.
problem Costly label data in machine learning applications.
method Flexible ingestion of organizational knowledge, cross-feature production serving, scalable execution.
result Comparable quality to hand-labeled models, 52% performance improvement on average.
Study on error rates for approximating rough volatility models.
problem Simulation of rough volatility models with fractional Brownian motion.
method Analysis of weak error rates for numerical schemes, focusing on fBm and cubic test functions.
result Convergence rates for approximations are (3H+21)∧1 for exact left-point discretization and H+21 for hybrid schemes. This paper studies a limit order book (LOB) model, in which the order dynamics depend on both, the current best available prices and the current volume density functions. For the joint dynamics of the best bid price, the best ask price, and the standing volume densities on both sides of the LOB we derive a weak law of …
Using a method developped in [1] and [2], we prove the existence of weak non trivial solutions to fourth order elliptic equations with singularities and with critical Sobolev growth.
Following an approach of the second author for conformally invariant variational problems in two dimensions, we show in four dimensions the existence of a conservation law for fourth order systems, which includes both intrinsic and extrinsic biharmonic maps. With the help of this conservation law we prove the continuit…
New deep learning architecture learns martingales efficiently.
problem Efficiently learning martingales in financial derivatives pricing.
method High-order weak approximation algorithms of Runge-Kutta type.
result Deep neural networks based on this architecture learn martingales effectively.
Using the method of Nehari manifold, we prove the existence of at least two distinct weak solutions to elliptic equation of four order with singulatities and with critical Sobolev growth.
This paper investigates higher order generalizations of well known results for Lie algebroids and bialgebroids. It is proved that n-Lie algebroid structures correspond to n-ary generalization of Gerstenhaber algebras and are implied by n-ary generalization of linear Poisson structures on the dual bundle. A Nambu-…
Study rough volatility models using path-dependent PDEs and fractional Brownian motions.
problem Modeling and analyzing rough volatility in financial markets.
method Showed conditional expectations are unique classical solutions to path-dependent PDEs derived from functional Itô formula. Leverage these to study weak rates of convergence for discretized stochastic integrals.
result Obtained optimal weak error rates for approximating log-stock prices in rough volatility models.
New high-order approximations for CIR process using random grids.
problem Approximating the Cox-Ingersoll-Ross process with high order.
method Combining discretization schemes on different random grids.
result Weak approximations of order 2k for all k∈N∗. Study introduces weak elastic energy for curves on Riemannian surfaces.
problem Detecting curvature of curves on Riemannian surfaces.
method Relaxation starting from inscribed geodesic polygonals, defined in normalized isothermal coordinates.
result Relaxed energy detects intrinsic second-order Sobolev regularity and agrees with geodesic curvature.
Tensor completion requires fewer samples with weak side information.
problem Tensor completion with limited samples and side information.
method Algorithm utilizing weak side information to reduce sample complexity.
result Consistent estimator with O(n1+κ) samples for any small constant κ>0. Improved accuracy in quantization methods for financial derivatives.
problem Efficient numerical methods for evaluating functionals of stochastic differential equations.
method Recursive Marginal Quantization of higher-order schemes (Euler, Milstein, simplified weak order 2.0).
result Higher-order schemes provide improved weak order convergence and accurate marginal distributions.
The paper defines weak lower scalar curvature bounds for C0 metrics and shows their stability under Ricci flow.
problem Defining and proving stability of weak lower scalar curvature bounds for metrics with low regularity.
method Proposes local definitions of weak lower scalar curvature bounds for C0 metrics, shows stability under perturbation, and defines a Ricci flow for C0 initial data. result Weak lower scalar curvature bounds are preserved under Ricci flow from C0 initial data. New algorithm for duelling bandits with weak regret in adversarial settings.
problem Improving performance in duelling bandits with weak regret.
method Developed an algorithm for duelling bandits in adversarial environments, considering the Borda winner.
result Algorithm provides theoretical guarantees in both utility-based and unrestricted settings.
Develops Poisson structures on weak Sobolev loop spaces for integrable systems.
problem Analyzing integrable systems on low regularity loop spaces.
method Extending Mokhov's constructions to weak Sobolev spaces, constructing presymplectic and Poisson structures.
result Valid Poisson structures and deformations for weak Sobolev loops, extending Hamiltonian formalisms.
Weak correlations explain linear dynamics in deep learning models.
problem Understanding the linear structure in gradient-based learning algorithms.
method Characterization of weak correlations between derivatives and parameters.
result Weak correlations are the underlying principle for linearization in deep learning models.
Study on Kähler manifolds proves weak decompositions and relates harmonic forms.
problem Analyzing harmonic forms on Kähler manifolds.
method Proves weak W1,2 Bott-Chern and Dolbeault decompositions. result Strict relation between W1,2 Bott-Chern harmonic forms and the W1,2 Bott-Chern decomposition. Faster weak supervision framework using triplet methods.
problem Computational inefficiency in weak supervision models.
method Closed-form solution for latent variable models, avoiding iterative methods.
result Orders of magnitude faster than previous approaches.
Characterizes preferences for decision-making under uncertainty using a leader-follower game model.
problem Decision-making under uncertainty and ambiguity aversion.
method Characterizes niveloidal preferences through a leader-follower game model, satisfying specific axioms.
result The leader's strategy space can serve as an ambiguity aversion index.
A new machine learning method solves high-dimensional Kolmogorov PDEs efficiently.
problem Solving high-dimensional Kolmogorov PDEs and SDEs.
method Stochastic weighted minimization and stochastic gradient descent with Malliavin weights.
result Accurate approximation of high-dimensional Kolmogorov PDEs and SDEs without curse of dimensionality.
Weak harmonic Weyl metrics found on all 4D closed manifolds.
problem Finding canonical metrics on 4D closed manifolds.
method Critical points of a quadratic functional involving the divergence of the Weyl tensor.
result Every 4D closed manifold admits a unique weak harmonic Weyl metric.
The Lebesgue property (order-continuity) of a monotone convex function on a solid vector space of measurable functions is characterized in terms of (1) the weak inf-compactness of the conjugate function on the order-continuous dual space, (2) the attainment of the supremum in the dual representation by order-continuous…
AMP with Gaussian initialization shows weak-recovery threshold for phase retrieval.
problem Phase retrieval with noiseless data.
method Approximate message passing with random initialization.
result Random initialization attains weak-recovery threshold \( \delta_{ ext{weak}} = 1/2 \).
We consider spaces of smooth immersed plane curves (modulo translations and/or rotations), equipped with reparameterization invariant weak Riemannian metrics involving second derivatives. This includes the full H2-metric without zero order terms. We find isometries (called R-transforms) from some of these spaces i…
Efficient simulation scheme for rough Heston model reduces computational cost.
problem Accurate and efficient simulation of the rough Heston model for option pricing.
method Weak simulation scheme based on Markovian approximations of the rough Heston process.
result The new scheme exhibits second order weak convergence with linear computational cost.
In usual stochastic volatility models, the process driving the volatility of the asset price evolves according to an autonomous one-dimensional stochastic differential equation. We assume that the coefficients of this equation are smooth. Using Itô's formula, we get rid, in the asset price dynamics, of the stochastic i…
Study geodesics on finite-dimensional manifolds.
problem Understanding geodesics on finite-dimensional manifolds.
method Develops concepts from differential geometry on finite-dimensional smooth manifolds.
result Provides a detailed description of key concepts for geodesics.
In this paper, we are interested in the strong convergence properties of the Ninomiya-Victoir scheme which is known to exhibit weak convergence with order 2. We prove strong convergence with order 1/2. This study is aimed at analysing the use of this scheme either at each level or only at the finest level of a multil…
The geodesic distance vanishes on the group of compactly supported diffeomorphisms of a Riemannian manifold M of bounded geometry, for the right invariant weak Riemannian metric which is induced by the Sobolev metric Hs of order 0≤s<21 on the Lie algebra Xc(M) of vector fields with compact …
Self-test loss functions improve data-driven modeling of weak-form operators and gradient flows.
problem Challenges in selecting test functions for data-driven modeling involving weak-form operators and gradient flows.
method Introducing self-test loss functions that depend on unknown parameters and are quadratic.
result Self-test loss functions conserve energy for gradient flows and coincide with log-likelihood ratios for stochastic differential equations.
Managing a portfolio to a risk model can tilt the portfolio toward weaknesses of the model. As a result, the optimized portfolio acquires downside exposure to uncertainty in the model itself, what we call "second order risk." We propose a risk measure that accounts for this bias. Studies of real portfolios, in asset-by…
Based on ideas of L. Alías, D. Impera and M. Rigoli developed in "Hypersurfaces of constant higher order mean curvature in warped products", we develope a fairly general weak/Omori-Yau maximum principle for trace operators. We apply this version of maximum principle to generalize several higher order mean curvature est…
Classifies crossings in tangles on surfaces, finding no nontrivial indices.
problem Classifying crossings in tangles on surfaces.
method Divides crossings into tribes compatible with Reidemeister moves and analyzes components, orders, and homotopy types.
result No nontrivial indices on classical knot diagrams.
We present an invariant of connected and oriented closed 3-manifolds based on a coribbon Weak Hopf Algebra H with a suitable left-integral. Our invariant can be understood as the generalization to Weak Hopf Algebras of the Hennings-Kauffman-Radford evaluation of an unoriented framed link using a dual quantum-trace. Thi…