New algorithm improves convergence rates for convex optimization problems.
problem Convex optimization problems with noisy stochastic data.
method Stochastic proximal point algorithm with weak linear regularity condition.
result Achieves $\mathcal{O}\left(\frac{1}{k}
ight)$ convergence rate for SPP.
Defines weak geodesics on specific subsets of manifolds.
problem Characterizing geodesics on prox-regular subsets of Riemannian manifolds.
method Defining weak geodesics as continuous curves with weak regularities, and characterizing them as viscosity critical points of the energy functional.
result Characterizes weak geodesics on prox-regular subsets of Riemannian manifolds.
Estimates long-term effects using past experiments as instruments with many weak instruments.
problem Estimating long-term causal effects with limited short-term outcomes and many weak instruments.
method Nonparametric instrumental variable inference with many weak instruments, using past experiments as instruments.
result Automatic debiased machine learning estimators for linear functionals of the structural function and its minimum-norm projection are efficient in the many-weak-instruments regime.
The paper proves properties of curves in Riemannian manifolds.
problem Characterizing curves in Riemannian manifolds.
method Analyzing locally minimizing and weak geodesics.
result Locally minimizing curves are weak geodesics under certain conditions.
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.
Study proves existence of weak mean curvature flow with contact angle.
problem Existence of weak mean curvature flow with prescribed contact angle.
method Compactness theorem for varifolds and Ilmanen's regularization extended to capillarity.
result Existence of weak mean curvature flow with contact angle for general θ. 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.
A new method for optimization in diffeological spaces using linearizations.
problem Optimization in spaces with low regularity.
method Generalizing linearization to diffeological spaces and constructing smooth paths.
result Achieving weak convergence to minima or critical values in diffeological spaces.
The paper proves weak continuity of Cartan structural system on semi-Riemannian manifolds with lower regularity.
problem Weak continuity of the Cartan structural system on semi-Riemannian manifolds with lower regularity.
method Formulated and proved a geometric compensated compactness theorem, deduced Lp weak continuity of the Cartan structural system. result Weak continuity of the Cartan structural system and Gauss-Codazzi-Ricci system on semi-Riemannian manifolds with lower regularity.
Proves existence and regularity of spherical minimizers for lipid membrane energy.
problem Existence and regularity of minimizers for the Canham-Helfrich energy.
method Establishes lower semicontinuity and proves existence through weak convergence of immersions.
result Proves existence and regularity of minimizers for the Canham-Helfrich energy on spheres.
New regularization method corrects over-shrinkage in small data regression.
problem Over-shrinkage in small data regression leading to underfitting.
method Negative-capable ridge family that permits negative regularization.
result Negative regularization acts as controlled anti-shrinkage, increasing effective complexity.
Sharp regularity for Pfaff system leads to isometric immersions in arbitrary dimensions.
problem Existence and regularity of isometric immersions in arbitrary dimensions.
method Proving W1,2-regularity for Pfaff system with antisymmetric L2-coefficient matrix. result Equivalence between W2,2-isometric immersions and weak solubility of Gauss--Codazzi--Ricci equations. Study optimal transport on globally hyperbolic spacetimes, focusing on weak Kantorovich potentials' regularity.
problem Investigate regularity of weak Kantorovich potentials on globally hyperbolic spacetimes.
method Apply insights from Riemannian and Lorentzian cases to study π-solutions. result Conclude existence, uniqueness, and structure of optimal transport maps.
We review recent work on the Einstein equations of general relativity when the curvature is defined in a weak sense. Weakly regular spacetimes are constructed, in which impulsive gravitational waves, as well as shock waves, propagate.
This work improves trace norm regularization for multi-task learning with limited data.
problem Learning from few samples across multiple tasks.
method Trace norm regularization for a linear shared representation model.
result First estimation error bound for trace norm regularized estimator with scarce data.
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.
Recently, Petrik et al. demonstrated that L1Regularized Approximate Linear Programming (RALP) could produce value functions and policies which compared favorably to established linear value function approximation techniques like LSPI. RALP's success primarily stems from the ability to solve the feature selection and va…
Proves higher regularity for anisotropic inverse mean curvature flow.
problem Higher regularity of solutions to anisotropic inverse mean curvature flow.
method Proves Harnack estimate and constructs smooth solutions from C1 initial sets. result Smooth solutions become smooth outside a compact set.
New boundary condition for weak inverse mean curvature flow in bounded domains.
problem Addressing the well-posedness of inverse mean curvature flow in bounded domains with an outer obstacle.
method Developed a new boundary condition, combined techniques including elliptic regularization, blow-up analysis, and parabolic estimates.
result Existence and uniqueness theorem for weak solutions in smooth bounded domains, with C1,α regularity of level sets up to the obstacle. In this paper we study constant scalar curvature equation (CSCK), a nonlinear fourth order elliptic equation, and its weak solutions on Kähler manifolds. We first define a notion of weak solution of CSCK for an L∞ Kähler metric. The main result is to show that such a weak solution (with uniform L∞ bound…
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.
The paper analyzes ℓq optimization methods for high-dimensional linear regression.
problem Estimating sparse parameters from noisy observations in high-dimensional settings.
method Introduces and analyzes ℓq optimization methods for sparse estimation. result Shows stable recovery properties and bounds for ℓq minimization and regularization methods. The paper examines partial regularity of Lipschitz solutions to minimal surface system.
problem Understanding the regularity of solutions to the minimal surface system.
method Investigation of stationary, integral weak, and viscosity solutions; interior gradient estimate using maximum principle.
result Partial regularity results for Lipschitz solutions, including interior gradient estimate.
We prove the smoothness of weak solutions to an elliptic complex Monge-Ampere equation, using the smoothing property of the corresponding parabolic flow.
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.
We find the optimal Tikhonov regularizer for linear inverse problems without prior knowledge.
problem Finding the optimal regularizer for linear inverse problems in imaging.
method Characterization of the optimal regularizer and learning from data.
result The optimal regularizer is independent of the forward operator and depends only on the mean and covariance of the random variable.
Paper proposes a new model using consistency regularization for learning from label proportions.
problem Learning from label proportions with weak labels on bags of instances.
method Consistency regularization applied to semi-supervised learning.
result LLP with consistency regularization achieves superior performance.
Online algorithm identifies PDEs from noisy data snapshots.
problem Identifying PDEs from sequential solution snapshots.
method Combines weak-form discretization with online proximal gradient descent.
result Efficiently identifies and tracks systems with time-varying coefficients.
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 conditions ensure points can be uniquely represented by combinations of variety elements.
problem Ensuring points can be uniquely represented by combinations of variety elements.
method Conditions on contact locus of general linear spaces.
result Conditions ensuring non tangential weak defectiveness of projective varieties.
We are concerned with the global weak rigidity of the Gauss-Codazzi-Ricci (GCR) equations on Riemannian manifolds and the corresponding isometric immersions of Riemannian manifolds into the Euclidean spaces. We develop a unified intrinsic approach to establish the global weak rigidity of both the GCR equations and isom…
We provide adaptive inference methods, based on ℓ1 regularization, for regular (semi-parametric) and non-regular (nonparametric) linear functionals of the conditional expectation function. Examples of regular functionals include average treatment effects, policy effects, and derivatives. Examples of non-regular f…
Under weak regularity assumptions, only, we develop a fully geometric theory of vacuum Einstein spacetimes with T2 symmetry, establish the global well-posedness of the initial value problem for Einstein's field equations, and investigate the global causal structure of the constructed spacetimes. Our weak regularity ass…
We prove weak and strong maximum principles, including a Hopf lemma, for smooth subsolutions to equations defined by linear, second-order, partial differential operators whose principal symbols vanish along a portion of the domain boundary. The boundary regularity property of the smooth subsolutions along this boundary…
The paper reveals three mechanisms for weak-to-strong generalization.
problem Understanding the mechanisms behind weak-to-strong generalization in imperfect labeling scenarios.
method Theoretical analysis of simple models including ridge regression and weighted ridge regression, and a nonlinear multi-index setting.
result A student model can compensate for a teacher's under-regularization and achieve lower test error.
We study the regularity problem of the nonlinear sigma model with gravitino fields in higher dimensions. After setting up the geometric model, we derive the Euler--Lagrange equations and consider the regularity of weak solutions defined in suitable Sobolev spaces. We show that any weak solution is actually smooth under…
Study of surfaces meeting a plane orthogonally or contained in a line, deriving weak forms and proving regularity.
problem Critical points of the Willmore functional with boundary constraints.
method Weak forms of free boundary conditions derived by reflection.
result Proved regularity of the surfaces.
Flat isometric immersions in 3D are developable if they are C1,2/3 regular.
problem Characterizing flat isometric immersions in 3D with Hölder continuity.
method Weak second fundamental form, Gauss-Codazzi-Mainardi equations, and degenerate Monge-Ampère equation analysis.
result Isometric immersions of local C1,α regularity with α>2/3 are developable. New method for adaptive estimation and inference in econometric models without knowing smoothness.
problem Adaptive estimation and inference in ill-posed linear inverse problems with unknown smoothness.
method Discrepancy principle-based framework for adaptive hyperparameter selection.
result Achieves optimal rates in weak and strong metrics for linear functionals.
We prove the existence of canonical tubular neighbourhoods around complex submanifolds of Kähler manifolds that are adapted to both the holomorphic and symplectic structure. This is done by solving the complex Homogeneous Monge-Ampère equation on the deformation to the normal cone of the submanifold. We use this to est…
We give a new proof of Brakke's partial regularity theorem up to C^{1,ς} for weak varifold solutions of mean curvature flow by utilizing parabolic monotonicity formula, parabolic Lipschitz approximation and blow-up technique. The new proof extends to a general flow whose velocity is the sum of the mean curvature and an…
Smooth calibration improves forecast reliability even with leaked information.
problem Improving forecast reliability with leaked information.
method Combining nearby forecasts to ensure smooth calibration, which can be guaranteed by deterministic procedures.
result Smooth calibration can be guaranteed by deterministic procedures even with leaked forecasts, and it yields uncoupled finite-memory dynamics in games.
Paper generalizes Andreev's theorem with obtuse angles.
problem Characterizing hyperbolic polyhedra with obtuse angles.
method Established discrete analog of weak solution/regularity theory.
result Generalized Andreev's Theorem to include obtuse angles.
For d≥2, the regular genus of a closed connected PL d-manifold M is the least genus (resp., half of the genus) of an orientable (resp., a non-orientable) surface into which a crystallization of M imbeds regularly. The regular genus of every orientable surface equals its genus, and the regular genus of every…
We obtain the $C^{\a}$ regularity for weak solutions of a class of non-homogeneous ultraparabolic equation, with measurable coefficients. The result generalizes our recent $C^{\a}$ regularity results of homogeneous ultraparabolic equation.
New algorithm improves convergence of gradient boosting trees.
problem Global convergence of Newton boosting in tabular machine learning.
method Introduces Gradient Regularized Newton Descent for GBDTs, proving linear convergence for smooth, strongly convex losses and O(k21) rate for general convex losses. result Achieves globally convergent second-order GBDT algorithm with rate matching first-order boosting.
In the category of metrics with conical singularities along a smooth divisor with angle in (0,2π), we show that locally defined weak solutions (C1,1−solutions) to the Kähler-Einstein equations actually possess maximum regularity, which means the metrics are actually Hölder continuous in the singular polar coord…
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.