The paper studies symplectic forms on projective limits of Banach bundles and their Darboux Theorem.
problem Conditions for weak symplectic forms on projective limits of Banach bundles.
method Analyzing projective sequences of Banach bundles and applying Darboux Theorem.
result Necessary and sufficient conditions for the Darboux Theorem on projective limits of Banach manifolds.
New research shows label refinement and weak training have limitations for aligning LLMs.
problem Limitations of refinement methods for aligning large language models.
method Analyzed probabilistic assumptions and alternative approaches to label refinement and weak training.
result Label refinement and weak training suffer from irreducible error, leaving a performance gap.
In this article, we study the relationship between the weak limit of a sequence of integral currents in a metric space and the possible Hausdorff limit of the sequence of supports. Due to cancellation, the weak limit is in general supported in a strict subset of the Hausdorff limit. We exhibit sufficient conditions in …
The study explores the strengths and weaknesses of models that generalize from weak to strong supervision.
problem Understanding the limitations and capabilities of models that generalize from weak to strong supervision.
method Theoretical analysis and experimental validation in both classification and regression settings.
result Theoretical bounds reveal the importance of strong generalization and calibration of the weak model and a careful balance in the training process.
New framework transfers latent knowledge from weak to strong models.
problem Aligning superhuman LLMs with human feedback.
method Transfer learning framework using refinement approach.
result Proves weak-to-strong generalization is possible.
The study examines Fisher-Riemann geodesics for nonparametric probability densities.
problem Understanding nonparametric probability densities using Fisher-Riemann geometry.
method Obtaining Fisher-Riemann geodesics as a limit of parametric cases with increasing parameters.
result The weak limit approach for nonparametric probability densities.
Stability of Yang-Mills connections' Morse indices and nullity in 4D.
problem Stability of Yang-Mills connections' Morse indices and nullity in 4D under weak convergence.
method Proves stability results of the Morse index plus nullity of Yang-Mills connections in dimension 4 under weak convergence.
result Stability of the sum of Morse indices and nullity of a sequence of Yang-Mills connections.
The problem of prescribing conformally the scalar curvature of a closed Riemannian manifold as a given Morse function reduces to solving an elliptic partial differential equation with critical Sobolev exponent. Two ways of attacking this problem consist in subcritical approximations or negative pseudo gradient flows. W…
Suppose M be the projective limit of weak symplectic Banach manifolds \{(M_i,φ_{ij})\}_{i,j\in\mathbb N}, where M_i are modeled over reflexive Banach space and σis compatible with the inverse system(defined in the article). We associate to each point x\in M, a Fréchet space H_x(defined in section 3). We prove that if H…
Preserves scalar curvature bounds under weak convergence of 3-manifolds.
problem Preserving scalar curvature bounds under weak convergence of 3-manifolds.
method Comparison between μ-bubbles in M_k and M.
result Scalar curvature lower bounds are preserved under weak convergence.
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.
In this paper we study utility maximization with proportional transaction costs. Assuming extended weak convergence of the underlying processes we prove the convergence of the corresponding utility maximization problems. Moreover, we establish a limit theorem for the optimal trading strategies. The proofs are based on …
Paper reconciles different Ricci flow approaches and proves weak solutions.
problem Proving weak solutions for Ricci flows with singularities.
method Introducing a novel hitting estimate for Brownian motion, compensating for lack of lower heat kernel bounds.
result Every noncollapsed limit of Ricci flows and singular Ricci flows are weak solutions.
Active WeaSuL uses active learning to improve weak supervision for better model performance.
problem Limited labelled data in machine learning.
method Combines active learning with weak supervision to improve probabilistic labels.
result Active WeaSuL outperforms weak supervision and active learning with limited labelled data.
Framework detects fake news using weak social signals from multiple sources.
problem Lack of annotated data for early fake news detection.
method Jointly uses weak social signals and clean data to train deep neural networks in a meta-learning framework.
result Framework outperforms state-of-the-art baselines for early fake news detection.
Estimates ends of Ricci shrinkers, focusing on smooth and singular cases.
problem Understanding the structure of ends in Ricci shrinkers, especially singular ones.
method Analyzes general and asymptotically conical ends, applies to weak convergence.
result No new conical end can form in the limit of sequences of Ricci shrinkers.
We obtain in arbitrary codimension a removability result on the order of singularity of weak limits and bubbles of Willmore immersions measured by the second residue. This permits to reduce significantly the number of possible bubbling scenarii. As a consequence, out of the twelve families of non-planar minimal surface…
The paper studies limits of flows on Kähler surfaces, proving convergence to solutions of equations.
problem Analyzing limits of flows on Kähler surfaces and their convergence to solutions of equations.
method Using a property of limits of viscosity subsolutions.
result Proves convergence of flows to weak solutions of the Monge-Ampère equation.
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…
Study shows superdiffusive behavior in geodesic flows on curved surfaces.
problem Understanding the statistical behavior of geodesic flows on curved surfaces.
method Proved nonstandard central limit theorem with superdiffusive normalisation (tlogt)1/2 for geodesic flows on nonpositively curved surfaces. result Geodesic flows exhibit superdiffusive behavior with correlations decaying at rate t−1. Neural networks trained with actor-critic algorithms converge to ODEs under weak convergence analysis.
problem Challenges in convergence analysis due to changing data distributions in online learning.
method Geometric ergodicity of data samples, Poisson equation, weak convergence techniques.
result Actor and critic networks converge to solutions of ODEs with random initial conditions.
Study the limits of discrete DPPs to continuous DPPs as set size grows.
problem Characterize the behavior of discrete DPPs as they approach continuous DPPs.
method Non-asymptotic characterization of the limit in terms of weak coherency.
result Sufficient conditions for weak coherency are identified.
Study finds weak solutions for complex map flows with optimal lifespan.
problem Existence of weak solutions for two-phase matrix-valued harmonic map flows.
method Modified minimizing movement scheme, discretizing time and interpolating solutions.
result Existence of weak solutions with optimal lifespan for the limiting system.
Characterizes when differential forms have weak exterior derivatives based on limiting behavior of integration over simplices.
problem Characterizing differential forms with weak exterior derivatives.
method Uses integration over simplices to characterize the limiting behavior of differential forms.
result Proves a direct analogue of the Bourgain-Brezis-Mironescu characterization for differential forms.
The paper proves stability of critical points for conformally invariant Lagrangians.
problem Stability of critical points for conformally invariant Lagrangians under weak convergence.
method Upper-semi-continuity of Morse index plus nullity established for critical points.
result The sum of Morse indices and nullity is bounded from above by the sum of the Morse indices plus the nullity of the weak limit and bubbles.
Smooth Yang-Mills fields proved in supercritical dimensions.
problem Regularity of weak Yang-Mills connections in high dimensions.
method ε-regularity theorem, Coulomb gauges construction.
result Stationary Yang-Mills fields are smooth away from a small singular set.
Consider the Kahler-Ricci flow with finite time singularities over any closed Kahler manifold. We prove the existence of the flow limit in the sense of current towards the time of singularity. This answers affirmatively a problem raised by Tian on the uniqueness of the weak limit from sequential convergence constructio…
Random feature models can outperform a weak teacher with early stopping.
problem Generalization from a weak to a strong model in random feature networks.
method Random feature models, early stopping, proving weak-to-strong generalization.
result Random feature models can outperform a weak teacher with early stopping.
Study shows how weak inverse anisotropic mean curvature flow behaves at infinity.
problem Understanding the asymptotic behavior of anisotropic mean curvature flow.
method Established local gradient estimates for anisotropic p-harmonic functions and weak solutions of IAMCF. result Weak IAMCF is asymptotic to the expanding Wulff shape solution at infinity.
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 …
Modeling stochastic arbitrage bubbles in Black-Scholes framework.
problem Analyzing arbitrage bubbles in financial markets.
method Developed a generalized Black-Scholes equation with stochastic arbitrage bubbles.
result The Black-Scholes model is a low-energy limit of a stochastic model.
Learning disentangled representations that correspond to factors of variation in real-world data is critical to interpretable and human-controllable machine learning. Recently, concerns about the viability of learning disentangled representations in a purely unsupervised manner has spurred a shift toward the incorporat…
We study the evolution of hypersurfaces in spacetime initial data sets by their null mean curvature. A theory of weak solutions is developed using the level-set approach. Starting from an arbitrary mean convex, outer untapped hypersurface ∂Ω0, we show that there exists a weak solution to the null mean curvatu…
We establish the weak continuity of the Gauss-Coddazi-Ricci system for isometric embedding with respect to the uniform Lp-bounded solution sequence for p>2, which implies that the weak limit of the isometric embeddings of the manifold is still an isometric embedding. More generally, we establish a compensated comp…
Generative models enhance weak supervision for better image classification.
problem Lack of labeled data in supervised learning.
method Fusion of generative adversarial networks and weak supervision.
result Model improves multiclass image classification performance.
Paper analyzes weak-to-strong generalization in CNNs, identifying data-scarce and data-abundant regimes.
problem Weak-to-strong generalization in CNNs trained on weak models.
method Formal analysis of gradient descent dynamics in data-scarce and data-abundant regimes.
result Identifies two regimes and distinct mechanisms of generalization in each.
In this paper, we study the weak compactness of the set of conformal metrics in any Riemann surface without boundary whose Calabi energy and area are uniformly bounded. We prove that for any sequence of such metrics, there alwasy exists a subsequence which converges in H\sp{2,2}_\sb{loc} everywhere except a finite numb…
We will study the blowup behavior of a surface sequence immersed in R2 with bounded Willmore functional and fixed genus.
New method for geometric flows with surgery without smooth estimates.
problem Existence of geometric flows with surgery.
method Hybrid compactness theorem for weak limits.
result Existence of geometric flows with surgery in mean-convex surfaces.
Smooth flows with surgery approximate weak mean curvature flows with spherical and neck-pinch singularities.
problem Approximating weak mean curvature flows with singularities using smooth flows.
method Combining Choi-Haslhofer-Hershkovits and Choi-Haslhofer-Hershkovits-White work on canonical neighbourhoods and barriers to flows with surgery.
result Smooth flows with surgery can approximate weak mean curvature flows with spherical and neck-pinch singularities.
New method estimates model performance bounds without ground truth labels.
problem Evaluation of weakly supervised models without direct access to ground truth labels.
method Formulates model evaluation as a partial identification problem and uses Fréchet bounds for performance estimation.
result Derives accurate and computationally efficient bounds for key metrics like accuracy, precision, recall, and F1-score.
Stability results for complex Monge-Ampère equations in various classes.
problem Stability of solutions to complex Monge-Ampère equations.
method Weak stability results followed by Ck,α stability proofs. result Proves stability of solutions in relative full mass classes and on quasi-projective varieties.
We prove that the limit hypersurfaces of converging curvature flows are stable, if the initial velocity has a weak sign, and give a survey of the existence and regularity results.
We are concerned with the global weak continuity of the Cartan structural system -- or equivalently, the Gauss--Codazzi--Ricci system -- on semi-Riemannian manifolds with lower regularity. For this purpose, we first formulate and prove a geometric compensated compactness theorem on vector bundles over semi-Riemannian m…
Defines weak normals for irregular curves in high-dimensional spaces.
problem Dealing with irregular curves in high-dimensional Euclidean spaces.
method Using sequences of inscribed polygonals and Gram-Schmidt procedure, introduces a relaxed notion of weak normals.
result Weak normals for irregular curves are the strong limit of approximating polygonals and agree with relaxed energy.
We study the minimization problem for the Yang-Mills energy under fixed boundary connection in supercritical dimension n≥5. We define the natural function space A_{G} in which to formulate this problem in analogy to the space of integral currents used for the classical Plateau problem. The space A_{G} can be also…
Fix a translation surface X, and consider the measures on X coming from averaging the uniform measures on all the saddle connections of length at most R. Then as R→∞, the weak limit of these measures exists and is equal to the Lebesgue measure on X. We also show that any weak limit of a subsequence of …
ProbKT uses probabilistic logical reasoning to train object detection models with weak supervision.
problem Training object detection models requires instance-level annotations, which are often unavailable.
method ProbKT, a framework based on probabilistic logical reasoning, uses arbitrary types of weak supervision.
result ProbKT leads to significant improvement and better generalization compared to existing baselines.