Sharp bounds on hyperbolic metrics in Ptolemaic spaces are derived.
problem Finding sharp bounds on hyperbolic metrics in Ptolemaic spaces.
method Construction of metrics on open subsets of Ptolemaic spaces.
result Sharp parameter bounds for hyperbolic and strongly hyperbolic metrics are derived.
Sharp boundaries for detecting dense subhypergraphs established.
problem Detecting dense subhypergraphs in random hypergraphs.
method Established sharp detection boundaries for known and unknown edge probabilities.
result Sharp detectable regions differ significantly from graph counterparts.
We conduct a post hoc analysis of solar flare predictions made by a Long Short Term Memory (LSTM) model employing data in the form of Space-weather HMI Active Region Patches (SHARP) parameters calculated from data in proximity to the magnetic polarity inversion line where the flares originate. We train the the LSTM mod…
Sharp bounds on ATE with unmeasured confounders, valid even when misspecified.
problem Bounding average treatment effects with unmeasured confounders.
method Distributionally robust optimization, double sharpness, double validity.
result Proposes estimators with robustness properties for valid bounds.
A method to assess sensitivity to unmeasured confounding with sharp bounds.
problem Assessing the impact of unmeasured confounding on causal effects.
method Sets two intuitive parameters to estimate sensitivity intervals.
result Bounds on true causal effects can be tighter than existing methods.
Noise in linear networks minimizes sharpness and leads to shrinkage-thresholding.
problem Minimizing sharpness in diagonal linear networks.
method Stochastic sharpness-aware minimization (SAM) with isotropic noise.
result Noise forces shrinkage-thresholding of true parameters.
Paper proposes a method to estimate scientific parameters in hybrid models without relying on model architecture.
problem Estimating unknown parameters in hybrid models combining machine learning and scientific models.
method Sharpness-aware minimization adapted for hybrid modeling, focusing on model simplicity.
result Demonstrates effectiveness of SAM-based hybrid model learning for scientific parameter estimation.
Double descent in portfolio optimization shows improved performance with complexity, then declines, due to overfitting.
problem Improving portfolio optimization performance with model complexity.
method Investigates the relationship between model complexity and out-of-sample performance in mean-variance portfolio optimization.
result Performance of low-dimensional models initially improves with complexity but declines due to overfitting. High-dimensional models show double ascent Sharpe ratio curve.
ASAM improves deep neural network generalization by adapting sharpness to scale.
problem Fixed-radius sharpness measure is sensitive to parameter scaling, weakening its connection to generalization.
method Introduces adaptive sharpness, a scale-invariant measure, and proposes ASAM for deep learning.
result ASAM significantly improves model generalization performance across various datasets.
We obtain new sharp isoperimetric inequalities on a Riemannian manifold equipped with a probability measure, whose generalized Ricci curvature is bounded from below (possibly negatively), and generalized dimension and diameter of the convex support are bounded from above (possibly infinitely). Our inequalities are shar…
When the in-sample Sharpe ratio is obtained by optimizing over a k-dimensional parameter space, it is a biased estimator for what can be expected on unseen data (out-of-sample). We derive (1) an unbiased estimator adjusting for both sources of bias: noise fit and estimation error. We then show (2) how to use the adjust…
Monge SAM improves deep learning by making sharpness-aware minimization invariant to reparametrizations.
problem Non-invariance of sharpness-aware minimization (SAM) to reparametrizations.
method Introduces Monge SAM, a reparametrization-invariant version of SAM using a Riemannian metric.
result Monge SAM enhances robustness and generalization compared to previous methods.
Develops a nonlocal PINN framework using PDDO for better solution of PDEs with sharp gradients.
problem Dealing with sharp gradients in solutions of PDEs using traditional PINN approaches.
method Integrates long-range interactions (nonlocality) into PINN using Peridynamic Differential Operator (PDDO).
result Nonlocal PINN approach improves solution accuracy and parameter inference for problems with sharp gradients.
Averaged SGD optimizes a smoothed objective, leading to better generalization.
problem Improving generalization performance in machine learning models.
method Analyzed the smoothed objective function of SGD and proved that averaged SGD can optimize this smoothed function efficiently.
result Averaged SGD can efficiently optimize a smoothed objective, leading to better generalization.
This paper optimizes portfolio selection by penalizing tracking error, improving Sharpe ratio.
problem Optimizing portfolio allocation with a penalty for deviation from a reference portfolio.
method Formulated as a McKean-Vlasov control problem, provides explicit solutions and asymptotic expansions.
result The penalized portfolio strategy outperforms standard mean-variance and reference portfolios in most cases.
Estimates true Sharpe ratio of selected assets with various methods.
problem Estimating the true Sharpe ratio of a selected asset with high in-sample ratio.
method Polyhedral lemma, James Stein shrinkage, debiasing, thresholding, empirical Bayes.
result James Stein estimator performs best across various parameter values.
The paper identifies a new geometric and spectral phenomenon in the critical hyperbolic catenoid family.
problem The study investigates the critical hyperbolic catenoid family and its geometric and spectral properties.
method The approach involves analyzing the critical hyperbolic catenoid family, identifying parameter-criticality, and studying the Robin spectrum.
result The paper proves that at a parameter-critical value a♯, the Robin nullity of Σa♯ is at least 3, with an additional kernel element in mode k=0. We study the profitability of optimal mean reversion trading strategies in the US equity market. Different from regular pair trading practice, we apply maximum likelihood method to construct the optimal static pairs trading portfolio that best fits the Ornstein-Uhlenbeck process, and rigorously estimate the parameters.…
Normalization methods play an important role in enhancing the performance of deep learning while their theoretical understandings have been limited. To theoretically elucidate the effectiveness of normalization, we quantify the geometry of the parameter space determined by the Fisher information matrix (FIM), which als…
New insights into network generalization show learning rate affects both norm and sharpness.
problem Understanding the generalization of overparameterized networks.
method Empirical analysis and theoretical proof of the trade-off between norm and sharpness.
result Learning rate influences both norm and sharpness, neither alone minimizes generalization error.
In an incomplete market, including liquidly-traded European options in an investment portfolio could potentially improve the expected terminal utility for a risk-averse investor. However, unlike the Sharpe ratio, which provides a concise measure of the relative investment attractiveness of different underlying risky as…
Study on curve diffusion flows with scale-critical curvature term.
problem Analyzing stability of curve diffusion flows with scale-critical curvature.
method Introduced and studied a one-parameter family of curve diffusion flows with a scale-critical cubic curvature term. Analyzed dynamical stability of homothetic circles using variational methods.
result Established that any small perturbation of an ω-fold circle monotonically approaches the unit ω-circle after rescaling, translation, and reparametrisation. This work connects SAM to variational inference and evaluates its performance.
problem Improving generalization of gradient-based learning by finding flat minima.
method Establishes connections between SAM and Mean-Field Variational Inference (MFVI), and evaluates variational algorithms combining or interpolating between SAM and MFVI.
result SAM-like updates can be used as a drop-in replacement for the reparametrisation trick.
New optimization method improves generalization across various tasks.
problem Improving zeroth-order optimization for better generalization.
method Exponential tilting objective to connect zeroth-order optimization with sharpness-aware minimization.
result Achieves better generalization compared to vanilla zeroth-order baselines.
We study sharp asymptotics of the first eigenvalue on Riemannian surfaces obtained from a fixed Riemannian surface by attaching a collapsing flat handle or cross cap to it. Through a careful choice of parameters this construction can be used to strictly increase the first eigenvalue normalized by area if the initial su…
We establish a sharp geometric constant for the upper bound on the resonance counting function for surfaces with hyperbolic ends. An arbitrary metric is allowed within some compact core, and the ends may be of hyperbolic planar, funnel, or cusp type. The constant in the upper bound depends only on the volume of the cor…
New estimates show spectral gap stability in RCD spaces, close to Beta distribution.
problem Stability of spectral gap bounds in metric-measure spaces.
method Combines L1-functional inequality and Stein's method. result Sharp quantitative estimate for spectral gap stability.
High-performing equity factor with Sharpe ratio above 13 out-of-sample.
problem Hidden cross-sectional predictability in stock returns.
method Regime-conditional signal activation combining value and short-term reversal signals.
result Annualized returns of 158.6% with 12.0% volatility, strong performance out-of-sample.
Deep ReLU networks need Ω(N) parameters to interpolate at irregularly spaced points.
problem Interpolating at irregularly spaced data points with deep ReLU networks.
method Analyzing the number of parameters required for interpolation.
result Ω(N) parameters are necessary for interpolation when δ is exponentially small in N.
DualAdam improves generalization of Adam by integrating its update mechanisms.
problem Adam's tendency to converge to sharp minima leading to suboptimal generalization.
method DualAdam combines Adam and inverse Adam's update mechanisms to enhance generalization.
result DualAdam outperforms Adam and state-of-the-art variants in generalization performance.
We introduce a scalable measure of curvature for analyzing training dynamics of large language models.
problem Analyzing the training dynamics of large language models due to high computational cost of measuring Hessian sharpness.
method We introduce critical sharpness and relative critical sharpness as computationally efficient measures capturing Hessian sharpness phenomena.
result We provide the first demonstration of sharpness phenomena at scale up to 7B parameters.
In this paper we give Hamilton's Laplacian estimates for the heat equation on complete noncompact manifolds with nonnegative Ricci curvature. As an application, combining Li-Yau's lower and upper bounds of the heat kernel, we give an estimate on Laplacian form of the heat kernel on complete manifolds with nonnegative R…
Weight decay stabilizes training dynamics by slowing progressive sharpening.
problem Understanding how weight decay affects training stability in deep learning models.
method Analyzing weight decay effects at the Edge of Stability, developing a mathematical framework.
result Weight decay dampens oscillations and stabilizes sharpness in CNNs, causing a phase transition in MLPs.
The study predicts solar flare productivity using magnetic data from SDO/HMI.
problem Forecasting solar flares, especially M- and X-class, to mitigate space weather effects.
method Statistical and machine learning methods applied to 563 ARs' magnetic data.
result Improved accuracy in predicting AR's Flare Index, especially for large values.
Sharp estimates for p-capacity on manifolds with Ricci curvature bounds.
problem Estimating p-capacity on manifolds with Ricci curvature constraints.
method Sharp comparison inequalities, warped-product model ends, and scale-invariant quantities.
result Characterization of equality cases and optimal ranges for normalization parameters.
The study bounds the stability of Gaussian mixtures under small perturbations.
problem Stability of Gaussian mixtures under small changes in distribution.
method Deriving an explicit bound on parameter stability of spherical Gaussian Mixture Models (sGMM) in a pre-defined model class.
result Upper bound on parameter distance of close sGMMs to the original sGMM, dependent only on the original model.
The study examines MCMC methods for arbitrary objectives and finds likelihood sharpness impacts performance and regularization.
problem Limitations of MCMC methods for arbitrary objective functions.
method Two-block MCMC framework with Metropolis-Hastings and Gibbs sampling, exploring likelihood curvature and sharpness.
result Likelihood sharpness governs in-sample performance and regularization inferred by training data.
Forward regression is a statistical model selection and estimation procedure which inductively selects covariates that add predictive power into a working statistical regression model. Once a model is selected, unknown regression parameters are estimated by least squares. This paper analyzes forward regression in high-…
The paper establishes bounds on the smoothness parameter in Gaussian process interpolation.
problem Estimating the smoothness parameter in Gaussian process models.
method Approximation theory in Sobolev spaces and general theorems on parameter estimation.
result Maximum likelihood estimation recovers the true smoothness for certain classes of functions.
In Deep Learning, Stochastic Gradient Descent (SGD) is usually selected as a training method because of its efficiency; however, recently, a problem in SGD gains research interest: sharp minima in Deep Neural Networks (DNNs) have poor generalization; especially, large-batch SGD tends to converge to sharp minima. It bec…
Develops new methods to evaluate data influence in SAM for improved model training.
problem Challenges in mislabeled noisy data and privacy concerns in SAM.
method Two innovative data valuation methods based on influence functions (IF) for SAM.
result Demonstrates effectiveness in identifying mislabeled data and enhancing interpretability.
Optimizes PnL using linear signals in quantitative finance.
problem Maximizing profit and loss in financial trading.
method Unsupervised machine learning approach that maximizes Sharpe Ratio through linear relationships and parameter optimization.
result Empirical validation and effectiveness of the model on U.S. Treasury ETF.
Study on convergence rates for optimal transport with regularization.
problem Convergence analysis of divergence-regularized optimal transport.
method Novel methodology using quantization and martingale couplings.
result Sharp rates for various divergences and transport costs.
Improves model generalization by minimizing loss sharpness.
problem Overparameterized models often fail to generalize well despite low training loss.
method Sharpness-Aware Minimization (SAM) minimizes both loss value and sharpness.
result SAM improves model generalization across various datasets and models.
SAM minimizes loss sharpness, improving adversarial transferability.
problem Improving adversarial transferability of deep neural networks.
method Evaluating surrogate models trained with seven minimizers, focusing on loss sharpness and flat neighborhoods.
result SAM minimizes loss sharpness, leading to better adversarial transferability.
Sharp bounds derived for test error of finite-rank kernel ridge regression.
problem Loose bounds on test error for finite-rank kernels in machine learning.
method Sharp non-asymptotic upper and lower bounds for KRR test error.
result Tighter bounds on finite-rank KRR test error, valid for any regularization parameters.
New algorithm avoids spurious sharpness minimization for NLP models.
problem SAM fails in NLP, leading to performance degradation.
method Developed Functional-SAM, which modifies logit statistics instead of function geometry.
result Functional-SAM and combined methods outperform AdamW and SAM in NLP tasks.
Large learning rates cause parameter instability, leading to better generalization.
problem Understanding why deep neural networks perform well despite operating outside the traditional stability regime.
method Analyzing the effect of large learning rates on the orientation of Hessian eigenvectors and parameter exploration.
result Large learning rates induce parameter instability, leading to better generalization through exploration of flatter regions of the loss landscape.