A gap in the proof prevents us to show that surfaces with constant mean curvature closed to 1/2 in H2 X R and having boundary with curvature greater than one, contained in a horizontal section P of H2 X R are topological disks, provided they are contained in one of the two halfspaces determined by P. This is the analog…
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48 results for “H2 error bounds”
We study stable compact constant mean curvature surfaces in the product spaces S2 X R and H2 X R and in some other Riemannian 3-manifolds.
"Ends of hyperbolic 3-manifolds should support canonical Wick Rotations, so they realize effective interactions of their ending globally hyperbolic spacetimes of constant curvature." We develop a consistent sector of WR-rescaling theory in 3D gravity, that, in particular, concretizes the above guess for many geometrica…
Enhances stability ranges for Torelli and congruence subgroup homologies.
problem Improving stability ranges for specific subgroup homologies.
method Analyzes H2(Torelli subgroup of Aut(Fn)'s), H2(Torelli subgroup of mapping class groups), and Hk(congruence subgroups of GL_n(R)'s).
result Improved central stability ranges for various subgroup homologies.
New PINNs method improves accuracy in computing Mean Escape Time from bounded domains.
problem Computing Mean Escape Time from bounded domains with high accuracy.
method Boundary-adapted Physics-Informed Neural Networks (PINNs) with exact Dirichlet boundary enforcement.
result Derivation of a priori error bounds for PINNs with normalized distance approximations.
Let M be an orientable hyperbolic surface without boundary and let be a closed geodesic in M. We prove that any side of any triangle formed by distinct lifts of in H2 is shorter than .
We investigate contact magnetic curves in the real special linear group of degree 2. They are geodesics of the Hopf tubes over the projection curve. We prove that periodic contact magnetic curves in SL(2,R) can be quantized in the set of rational numbers. Finally, we study contact homogeneous magnetic trajectories in S…
New geometric invariant from disc intersections captures all coloured Jones polynomials.
problem Constructing a universal knot invariant from configuration spaces.
method Defining a new local system and Lagrangian submanifolds in the disc.
result The new invariant recovers Habiro's universal invariant and more.
In this paper, we define Mannheim partner curves in a three dimensional Lie group G with a bi-invariant metric. And then the main result in this paper is given as (Theorem 3.3): A curve α with the Frenet apparatus {T,N,B,κ,τ} in G is a Mannheim partner curve if and only if λκ(1+H2)=1, where λ is constant and H is the h…
We classify all closed non-orientable P2-irreducible 3-manifolds having complexity up to 6 and we describe some having complexity 7. We show in particular that there is no such manifold with complexity less than 6, and that those having complexity 6 are precisely the 4 flat non-orientable ones and the filling of the Gi…
Enhances scenario approach for certifying design properties post-design.
problem Certifying additional useful properties in designs not considered during the design phase.
method Two-level framework of appropriateness: baseline and post-design. Distribution-free upper bounds on risk derived.
result Distribution-free upper bounds on the risk of failing to meet post-design appropriateness.
Given a compact Kaehler manifold, we consider the complement U of a divisor with normal crossings and a unitary local system V on it. We consider a differential graded Lie algebra (DGLA) of forms with holomorphic logarithmic singularities and vanishing residues. We construct a spectral sequence corresponding to the ant…
This is a survey on the global theory of constant mean curvature surfaces in Riemannian homogeneous 3-manifolds. These ambient 3-manifolds include the eight canonical Thurston 3-dimensional geometries, i.e. R3, H3, S3, H2 \times R, S2 \times R, the Heisenberg space Nil3, the universal cover of PSL2(R) and the Lie group…
This paper concerns model reduction of dynamical systems using the nuclear norm of the Hankel matrix to make a trade-off between model fit and model complexity. This results in a convex optimization problem where this trade-off is determined by one crucial design parameter. The main contribution is a methodology to app…
Study analyzes smart contract adoption under bounded risk, showing stable adoption but fragile financial outcomes.
problem Understanding smart contract adoption in derivative markets under risk constraints.
method Structural theory linked with simulation and real-world validation.
result Adoption intensity is stable but profitability and service outcomes are sensitive to volatility.
We study constant mean curvature 1/2 surfaces in H2xR that admit a compactification of the mean curvature operator. We show that a particular family of complete entire graphs over H2 admits a structure of infinite dimensional manifold with local control on the behaviors at infinity. These graphs also appear to have a h…
Using results by Donaldson and Auroux on pseudo-holomorphic curves as well as Duval's rational convexity construction, the paper investigates the existence of smooth Lagrangian surfaces representing 2-dimensional homology classes in complex projective surfaces. We prove that if the projective surface X is minimal, of g…
Model financial dynamics using 2-manifold geometries, revealing the torus as best for cyclical data.
problem Financial forecasting using complex market data.
method Embedding market data onto 2-manifolds (S2, R2, H2, T) guided by uniformization theorem, inferring latent curvature.
result The torus geometry best predicts cyclical financial data, aligning with IS-LM theory.
New bounds on machine learning model generalization error moments.
problem Understanding the performance of machine learning models.
method Information-theoretic bounds on the moments of the generalization error of learning algorithms.
result Proposed bounds on generalization error moments and their high-probability bounds.
This work bounds classification error in machine learning for low Bayes error conditions.
problem Understanding the error mismatch between Bayes error and model-based classification error.
method Applying classification error bounds to study the relationship with Kullback-Leibler divergence and proposing a linear approximation for low Bayes error conditions.
result A linear approximation of the classification error bound for low Bayes error conditions is proposed.
PAC-Bayesian bounds for stochastic LTI systems derived.
problem Error bounds for stochastic LTI systems.
method PAC-Bayesian theory applied to autonomous stochastic LTI models.
result Error bounds for stochastic LTI systems derived.
Upper bounds for CV errors apply to lasso and other models.
problem Bounding CV errors for lasso and similar models.
method Rademacher complexity and Orlicz- norm.
result Upper bounds are tight and stable for lasso.
This paper examines error bounds for deep learning classifiers with noisy labels.
problem Understanding the performance of classifiers trained on noisy data.
method Derives error bounds for excess risk, decomposing it into statistical and approximation errors. Uses independent block construction for statistical dependencies and vector-valued setting for approximation error.
result Established theoretical results for error bounds in deep learning with noisy labels, mitigating the impact of high-dimensional input spaces.
We introduce the speculate-correct method to derive error bounds for local classifiers. Using it, we show that k nearest neighbor classifiers, in spite of their famously fractured decision boundaries, have exponential error bounds with O(sqrt((k + ln n) / n)) error bound range for n in-sample examples.
The paper bounds the mean absolute error in DNN vector-to-vector regression.
problem Bounding the mean absolute error in deep neural network based vector-to-vector regression.
method Error decomposition techniques in statistical learning theory and non-convex optimization theory were used to derive upper bounds for approximation, estimation, and optimization errors.
result Theoretical upper bounds for mean absolute error in DNN vector-to-vector regression were derived and validated experimentally.
PAC-Bayesian-Like Error Bound for a Class of Linear Time-Invariant Stochastic State-Space Modelsstat.ML
Paper derives an error bound for stochastic LTI systems.
problem Stochastic LTI systems with inputs in control engineering and econometrics.
method PAC-Bayesian-Like error bound derivation.
result Derived an error bound for stochastic LTI systems.
A new error bound improves safety in Bayesian optimization.
problem Ensuring safety in Bayesian optimization with probabilistic models.
method Introducing a novel error bound using Wiener kernel regression for Gaussian processes and noise.
result The new error bound provides larger safety regions than previous methods.
The paper provides tighter error bounds for GPR under bounded support noise.
problem Rigorous error quantification for safety-critical applications with bounded noise.
method Using concentration inequalities and low complexity assumptions in RKHS, the paper derives probabilistic and deterministic error bounds for GPR.
result The derived error bounds are substantially tighter than existing state-of-the-art bounds and are particularly well-suited for GPR with neural network kernels.
This paper tackles worst-class error rate in classification tasks.
problem Minimizing worst-class error rate in classification tasks, especially in medical image classification.
method Designing a boosting approach to bound the worst-class error rate using Deep Neural Networks (DNNs).
result The proposed boosting approach lowers worst-class test error rates while avoiding overfitting.
Derives error bounds for stochastic iterative algorithms using Stein's method.
problem Bounding errors in stochastic iterative algorithms like SGD and SGLD.
method Uses infinite-dimensional Stein's method of exchangeable pairs to derive functional approximation error bounds.
result Establishes non-asymptotic error bounds for algorithm sample paths and variance of iterate averages.
In this paper, we improve the PAC-Bayesian error bound for linear regression derived in Germain et al. [10]. The improvements are twofold. First, the proposed error bound is tighter, and converges to the generalization loss with a well-chosen temperature parameter. Second, the error bound also holds for training data t…
New PAC-Bayes bound controls multiple error types simultaneously.
problem Current PAC-Bayes bounds are limited to scalar metrics.
method Bounding KL divergence between empirical and true probabilities of multiple error types.
result First PAC-Bayes bound for rich information-rich certificates.
The paper offers error bounds for quantized dynamical models.
problem Accuracy of dynamical models from dependent data sequences.
method Developed uniform error bounds for quantized models and imperfect optimization algorithms.
result Unified bounds for slow and fast rates, scaling with model encoding bits.
This note provides an error bound for the Hartman-Watson integral's leading term.
problem Bounding the error of the leading term of the Hartman-Watson integral.
method Asymptotic expansion analysis focusing on the regime constant.
result The error term is bounded uniformly as .
Thinking LLMs struggle with stock prediction, especially as data complexity increases.
problem Evaluating the performance of 'thinking' LLMs in stock prediction, especially under varying levels of cross-sectional complexity.
method Rolling 48m/1m walk-forward evaluation, comparing direct LLMs, TLLMs, and classical learners on cross-sectional ranking loss, MSE, and backtests with transaction costs.
result TLLMs' ranking quality deteriorates as cross-sectional complexity grows, while direct LLMs remain stable.
This paper addresses error bounds and posterior variance for Gaussian process regression.
problem Deriving performance guarantees for Gaussian process regression without prior knowledge.
method Lipschitz continuity and analysis of posterior variance function.
result Uniform error bounds for Gaussian process regression are derived.
Study error bounds in evaluating distributional computational graphs.
problem Error analysis in evaluating graphs with inputs as probability distributions.
method Establish non-asymptotic error bounds using Wasserstein-1 distance.
result Non-asymptotic error bounds for discretization errors in distributional computational graphs.
Crowdsourcing is an effective tool for human-powered computation on many tasks challenging for computers. In this paper, we provide finite-sample exponential bounds on the error rate (in probability and in expectation) of hyperplane binary labeling rules under the Dawid-Skene crowdsourcing model. The bounds can be appl…
Jensen-Shannon Information Based Characterization of the Generalization Error of Learning Algorithmscs.IT
New bound on machine learning model performance using Jensen-Shannon information.
problem Understanding the performance of machine learning models.
method Proposes a new information-theoretic bound on generalization error.
result Shows that the new bound can be tighter than mutual information-based bounds under certain conditions.
Proposes a new bound on generalization error using conditional mutual information.
problem Improving the generalization error bound in machine learning.
method Combines error decomposition and conditional mutual information techniques.
result New bound is order-wise better than previous ones in a simple Gaussian setting.
Paper derives PAC-Bayesian bounds for LTI systems learning from empirical data.
problem Characterizing predictive power of LTI systems learned from data.
method PAC-Bayesian bounds for LTI stochastic dynamical systems with inputs.
result Finite-sample error bounds for learning algorithms of LTI systems.
Exactly Tight Information-Theoretic Generalization Error Bound for the Quadratic Gaussian Problemcs.IT
New bound matches exact generalization error for quadratic Gaussian problem.
problem Understanding generalization error in quadratic Gaussian problems.
method Information-theoretic approach with new ingredients.
result Exact tight bound for generalization error.
New error bounds for flow matching methods using deterministic sampling.
problem Improving the accuracy of flow matching methods for generating probability distributions.
method Derived error bounds for flow matching methods under deterministic sampling conditions.
result Presented error bounds for flow matching methods using loss and regularity conditions.
Data-driven models are subject to model errors due to limited and noisy training data. Key to the application of such models in safety-critical domains is the quantification of their model error. Gaussian processes provide such a measure and uniform error bounds have been derived, which allow safe control based on thes…
Error bounds based on worst likely assignments use permutation tests to validate classifiers. Worst likely assignments can produce effective bounds even for data sets with 100 or fewer training examples. This paper introduces a statistic for use in the permutation tests of worst likely assignments that improves error b…
Bounds on factual and counterfactual distributions under measurement error in discrete models.
problem Measurement errors in discrete data and their impact on inference.
method Expressing modeling assumptions as linear constraints and using linear programming to derive bounds.
result Sharp bounds on factual and counterfactual distributions for various models, including instrumental variable scenarios.
We consider active, semi-supervised learning in an offline transductive setting. We show that a previously proposed error bound for active learning on undirected weighted graphs can be generalized by replacing graph cut with an arbitrary symmetric submodular function. Arbitrary non-symmetric submodular functions can be…
New bounds quantify estimation error in kernel-based system identification with unknown hyperparameters.
problem Inaccurate error bounds for kernel-based system identification with unknown hyperparameters.
method Construct a high-probability set for true hyperparameters from marginal likelihood, then find worst-case posterior covariance.
result Proposed bounds contain true model with high probability and verified in simulations.