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.
Sharp lower bound found for integral varifolds' mean curvature.
problem Finding a sharp lower bound for the mean curvature integral of integral varifolds.
method Developed a new approach using integral varifolds and mean curvature.
result A sharp lower bound on the mean curvature integral with critical power for integral varifolds.
Sharp stability in Almgren problem solved in any dimension.
problem Quantitative stability in the radial isotropic Almgren problem.
method Developed a theory for estimating the sharp modulus under minimal assumptions.
result Sharp ε2 in any dimension, solving the critical mass problem. We prove sharp pointwise decay estimates for critical Dirac equations on Rn with n≥2. They appear for instance in the study of critical Dirac equations on compact spin manifolds, describing blow-up profiles, and as effective equations in honeycomb structures. For the latter case, we find excited state…
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. 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. We find empirically a characteristic sharp peak-flat trough pattern in a large set of commodity prices. We argue that the sharp peak structure reflects an endogenous inter-market organization, and that peaks may be seen as local ``singularities'' resulting from imitation and herding. These findings impose a novel strin…
Sharp fractional Sobolev inequalities on closed manifolds identified.
problem Critical fractional Sobolev embedding on closed Riemannian manifolds.
method Intrinsic heat-kernel based framework, determining optimal coefficients, proving sharp inequalities.
result Sharp p-power inequality and almost sharp inequality established. In this article, we investigate the geometry of critical metrics of the volume functional on an n-dimensional compact manifold with (possibly disconnected) boundary. We establish sharp estimates to the mean curvature and area of the boundary components of critical metrics of the volume functional on a compact manifol…
We prove a sharp Onofri-type inequality and non-existence of extremals for a Moser-Tudinger functional on the sphere in the presence of potentials having positive order singularities. We also investigate the existence of critical points and give some sufficient conditions under symmetry or nondegeneracy assumptions.
Sharp changes in time series representing market dynamics are studied by means of the self--similar analysis suggested earlier by the authors. These sharp changes are market booms and crashes. Such crises phenomena in markets are analogous to critical phenomena in physics. A simple classification of the market crisis p…
Sharp decay found for solutions of a specific equation in Lie groups.
problem Asymptotic decay of solutions to a Yamabe type equation.
method Analysis of a specific pseudodifferential operator in a homogeneous Lie group.
result Established sharp asymptotic decay of positive solutions.
Study critical metrics on manifolds with boundary using integral and boundary estimates.
problem Investigate geometry of critical metrics on compact manifolds with boundary.
method Use generalized Reilly's formula to derive integral and boundary estimates.
result Establish new boundary estimates for critical metrics of the volume functional.
Solves Dirichlet problem for Lagrangian phase equation with critical and supercritical phase.
problem Solving Dirichlet problem for Lagrangian phase equation with critical and supercritical phase.
method Uses interior C2 estimate. result Result is sharp, showing existence of singular solutions in subcritical phase.
Let (W,M,M'), dim W > 5, be a non-trivial h-cobordism (i.e., the Whitehead torsion of (W,V) is non-zero). We prove that every smooth function f: W --> [0,1], f(M)=0, f(M')=1 has at least 2 critical points. This estimate is sharp: W possesses a function as above with precisely two critical points.
Sharpness minimization algorithms don't solely improve generalization.
problem Why do overparameterized neural networks generalize?
method Theoretical and empirical investigation of two-layer ReLU networks.
result Sharpness minimization algorithms do not always lead to better generalization.
Overparameterization enhances SAM's effectiveness in minimizing sharpness.
problem Improving generalization in deep neural networks.
method Analysis of Sharpness-Aware Minimization (SAM) under varying degrees of overparameterization.
result Overparameterization significantly improves SAM's performance, particularly in noisy and sparse settings.
We classify local minimizers of ∫σ2+∮H2 among all conformally flat metrics in the Euclidean (n+1)-ball, 4≤n≤5, for which the boundary has unit volume, subject to an ellipticity assumption. We also classify local minimizers of the analogous functional in the critical dimension n+1=4. If minimiz…
Study critical quasilinear equations on Riemannian manifolds with curvature constraints.
problem Investigate critical quasilinear elliptic equations on Riemannian manifolds with nonnegative Ricci curvature.
method Utilize a new nonlinear Kato inequality and Cheng-Yau type gradient estimates for positive solutions.
result Classify positive solutions to the critical p-Laplace equation and show rigidity concerning the ambient manifold. When trading incurs proportional costs, leverage can scale an asset's return only up to a maximum multiple, which is sensitive to its volatility and liquidity. In a model with one safe and one risky asset, with constant investment opportunities and proportional costs, we find strategies that maximize long term returns …
This thesis covers different aspects of the p-Laplace operators on Riemannian manifolds. Chapter 2. Potential theoretic aspects: the Khasmkinskii condition. Chapter 3: sharp eigenvalue estimates with Ricci curvature lower bounds. Chapter 4: Critical sets of (2-)harmonic functions.
We investigate fourth order Paneitz equations of critical growth in the case of n-dimensional closed conformally flat manifolds, n≥5. Such equations arise from conformal geometry and are modelized on the Einstein case of the geometric equation describing the effects of conformal changes of metrics on the Q-cu…
In this paper, based on the local comparison principle in [12], we study the local behavior of the difference of two spacelike graphs in a neighborhood of a second contact point. Then we apply it to the constant mean curvature equation in 3-dimensional Lorentz-Minkowski space L3 and get the uniqueness of cr…
The paper proves inequalities for hypersurfaces in weighted manifolds.
problem Willmore-type inequalities for closed hypersurfaces in weighted manifolds.
method Analyzes weighted manifolds with nonnegative Bakry-Émery Ricci curvature, proving sharp inequalities and characterizing equality cases.
result Derives sharp Willmore-type and Willmore-like inequalities in steady and shrinking gradient Ricci solitons.
Modeling financial markets with sandpile model to understand price volatility and arbitrage constraints.
problem Understanding price volatility and arbitrage constraints in financial markets.
method Uses a sandpile model to represent information and price changes, linking size of price volatility to the scaling law of avalanches.
result Identifies a structural tension between non-arbitrage condition and price adjustments consistent with a constant Sharpe ratio.
Continuous phase transitions identified in Doi-Onsager, noisy transformer, and Hegselmann-Krause models.
problem Phase transitions in multimodal models and their properties.
method Sharp coercivity estimate and constrained Lebedev--Milin inequality.
result Continuous phase transitions at critical coupling strengths for Doi-Onsager, noisy transformer, and Hegselmann-Krause models.
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.
Sharp constants in curl-Sobolev inequalities on spheres determined.
problem Determining sharp constants in curl-Sobolev inequalities on spheres.
method Analyzing conformally invariant Sobolev quotients and using local stability estimates.
result Strict upper bound for the sharp constant of the J2 inequality. Willmore flow converges globally for surfaces with rotational symmetry below a specific energy threshold.
problem Global existence and convergence of Willmore flow with Dirichlet boundary conditions.
method Considered surfaces with rotational symmetry, proved global existence and convergence for initial data below a sharp energy threshold.
result Sharp threshold for global existence and convergence of Willmore flow depends on boundary conditions.
We derive a selection of energy estimates for a generalisation of a critical equation on the unit disc in R2 introduced by Rivière. Applications include sharp regularity results and compactness theorems which generalise a large amount of previous geometric PDE theory, including some of the theory of harmoni…
Investment strategy using fractional Kelly portfolios for better growth expectations.
problem Understanding optimal growth strategies for investors with varying risk appetites.
method Developed a mathematical framework for fractional-Kelly portfolios, analyzing Sharpe ratios and log-returns.
result Fractional Kelly portfolios provide a simple distributional relationship between Sharpe ratio, fractional coefficient, and log-returns.
Tyler's M-estimator's phase transition at DS-SNR = 1 is resolved.
problem Robust Subspace Recovery
method Tyler's M-estimator
result TME converges exactly to the true subspace for DS-SNR >= 1 under a new stability condition.
In complex systems like financial market, risk tolerance of individuals is crucial for system resilience.The single-security price limit, designed as risk tolerance to protect investors by avoiding sharp price fluctuation, is blamed for feeding market panic in times of crash.The relationship between the critical market…
We prove sharp inequalities for determinants of Toeplitz operators and twisted Laplace operators on the two-sphere, generalizing the Moser-Trudinger-Onofri inequality. In particular a sharp version of conjectures of Gillet-Soule and Fang motivated by Arakelov geometry is obtained; applications to SU(2)-invariant determ…
Study on 4D Riemannian manifolds solves curvature problem.
problem Resonant prescribed T-curvature problem on compact manifolds.
method Variational theory, energy and gradient estimates, Morse lemma, Liouville technique.
result New existence results for critical points at infinity.
Sharp condition found for Burer-Monteiro method to work for MaxCut-type SDPs.
problem MaxCut-type semidefinite programs with low-rank solutions.
method Sharp condition on Laplacian matrix conditioning for global minimizers of non-convex problem.
result Any second-order critical point is a global minimizer under the given condition.
For any compact Riemannian surface S and any point y in S, Qy−1 denotes the set of all points in S, for which y is a critical point. We proved \cite{BIVZ} together with Imre Bárány that cardQy−1≥1, and that equality for all y∈S characterizes the surfaces homeomorphic to the sphere. Here …
We prove existence and regularity of metrics on a surface with boundary which maximize sigma_1 L where sigma_1 is the first nonzero Steklov eigenvalue and L the boundary length. We show that such metrics arise as the induced metrics on free boundary minimal surfaces in the unit ball B^n for some n. In the case of the a…
Study uses reinforcement learning to optimize portfolios under recursive utility.
problem Improving portfolio allocation using risk-sensitive objectives.
method Approximated certainty equivalent via Monte Carlo, trained actor-critic algorithms (PPO, A2C).
result Recursive-utility agent outperforms discounted baseline in Sharpe ratio, max drawdown, and cumulative return.
Several rigidity results are proved for critical points of natural Riemannian functionals on the space of metrics on 3-manifolds. Two of these results are as follows. Let (N, g) be a complete Riemannian 3-manifold, satisfying one of the following variational conditions: (i) (N, g) has non-negative scalar curvature and …
The paper examines stability of the Sobolev inequality in metric spaces with curvature dimension conditions.
problem Investigating stability of the Sobolev inequality in metric spaces with curvature dimension conditions.
method Assuming almost the same optimal constant, the paper shows that the cumulative distribution of almost extremal functions is close to that of an Aubin-Talenti bubble on the round sphere.
result Quantitative stability with sharp exponent for the Sobolev inequality in various curvature and dimension assumptions.
Study sharp geometric and topological properties of pinched 4D submanifolds.
problem Pinched submanifolds in space forms.
method Four-dimensional geometry, Riemannian manifolds with nonnegative isotropic curvature, Bochner technique.
result Sharp results extend previous work without additional assumptions.
Deep RL optimizes dynamic portfolio weights in China's stock market.
problem Traditional portfolio optimization methods struggle with dynamic asset weight adjustments.
method Developed a deep reinforcement learning framework with novel reward functions and random sampling.
result Model outperforms traditional methods in portfolio optimization and risk mitigation.
Develops new strategy for Hessian estimates in Lagrangian mean curvature equation.
problem Interior Hessian estimates for solutions with prescribed Lipschitz phases.
method Allard-type regularity theorem, geometric measure theory, geometry of Lagrangian graphs, De Giorgi-Nash-Moser iteration.
result Sharp interior Hessian estimates for solutions with critical and supercritical phases.
Paper analyzes NAC with neural networks for efficient policy optimization.
problem Improving sample and iteration complexity in policy optimization.
method Entropy regularization, averaging, neural network approximation, and optimization techniques.
result Entropy regularization and averaging ensure stability and sharp sample complexity bounds.
We obtain area growth estimates for constant mean curvature graphs in E(κ,τ)-spaces with κ≤0, by finding sharp upper bounds for the volume of geodesic balls in E(κ,τ). We focus on complete graphs and graphs with zero boundary values. For instance, we prove that entire graphs in $\mathbb{E}(κ…
Study how nodal domains change on surfaces under perturbations.
problem How eigenfunction nodal domains change on surfaces under smooth perturbations.
method Sector/graph count near nodal critical points, upper semicontinuity proof, branch-free on spectral clusters, wavelength-scale analysis.
result Upper semicontinuity of nodal domain count, no new domains created at wavelength scale, stable count in noncritical cases.
Motivated by the equation satisfied by the extremals of certain Hardy-Sobolev type inequalities, we show sharp Lq regularity for finite energy solutions of p-laplace equations involving critical exponents and possible singularity on a sub-space of Rn, which imply asymptotic behavior of the solutions at i…