Non-asymptotic rates for SGD via martingale CLT.
problem Improving the convergence rates of SGD.
method Combining Stein's method and Lindeberg's argument for multivariate martingale CLT, then applying to SGD.
result Explicit rates for multivariate martingale CLT and SGD convergence.
Deep neural networks' infinite-width behavior approximated by Gaussian models.
problem Understanding the behavior of deep neural networks in the limit of infinite width.
method Using the Lindeberg exchange principle to approximate weights by Gaussian random variables.
result Quantitative bounds on the 2-Wasserstein distance between deep neural networks and Gaussian limits.
The Black-Scholes model (sometimes known as the Black-Scholes-Merton model) gives a theoretical estimate for the price of European options. The price evolution under this model is described by the Black-Scholes formula, one of the most well-known formulas in mathematical finance. For their discovery, Merton and Scholes…
Gaussian equivalence fails for simple polynomial embeddings in quadratic scaling RF models.
problem Failure of Gaussian equivalence in polynomial feature embeddings under quadratic scaling.
method Introduced Conditional Gaussian Equivalent (CGE) model to capture non-Gaussian behavior.
result Correct asymptotics derived for training and test errors in CGE model.
Data augmentation affects estimates' uncertainty and distribution in complex ways.
problem Understanding how data augmentation impacts the variance and limiting distribution of estimates.
method Developed an adaptation of Lindeberg's technique for block dependence.
result Data augmentation can increase rather than decrease uncertainty, and it may shift the double-descent peak of an empirical risk.
The paper analyzes boosting and minimum-ℓ1-norm classifiers in high dimensions.
problem Understanding the generalization error and optimal Bayes error in boosting.
method High-dimensional asymptotic theory, Gaussian comparison techniques, uniform deviation argument.
result Precise characterizations of boosting test error and optimal Bayes error.
The paper analyzes the generalization error of min-norm interpolators in transfer learning with limited test samples.
problem Characterizing the generalization error of min-norm interpolators in transfer learning with limited test samples.
method Characterizes the bias and variance of pooled min-ℓ2-norm interpolation under covariate shift and model shift. result Shows that adding data can hurt when SNR is low and is beneficial at higher SNR levels under certain conditions.
TACAM improves argument mining by integrating topic and external context.
problem Mining arguments from text without topic information leads to confusion.
method Proposes models that consider topic information and integrate external context.
result Performance boost for argument mining when topic and external context are considered.
Paper presents a Siamese network for identifying more convincing evidence.
problem Identifying the more convincing argument in discussions.
method Proposes a Siamese neural network architecture for labeling evidence as more convincing.
result The Siamese network outperforms baselines on a new challenging data set.
Paper explores arbitrage and CAPM in continuous time.
problem Understanding arbitrage and CAPM in continuous time.
method Analyzes instantaneous arbitrage and its relation to CAPM.
result Arbitrage and CAPM arguments differ in assumptions about the market portfolio.
Simplified argument for second order estimate in quaternionic Calabi-Yau problem.
problem Second order estimates for quaternionic Calabi-Yau problem on hyperkähler manifolds.
method Simplified argument to derive the estimate.
result Simplified derivation of second order estimate.
This is a continuation of our first paper in [WY16]. There are two purposes of this paper: One is to give a proof of the main result in [WY16] without going through the argument depending on numerical effectiveness. The other one is to provide a proof of our conjecture, mentioned in [TY], where the assumption of negati…
For J-holomorphic mappings for a strongly pseudo-convex manifold, we prove elliptic regularity by the argument of boots-strapping.
SpArX creates faithful explanations of neural networks' decision-making.
problem Challenges in explaining neural networks' decisions.
method Sparsifies MLPs while maintaining structure, then translates into QAFs for argumentative explanations.
result SpArX provides more faithful explanations than existing methods.
Killing fields on compact pseudo-Kähler manifolds are holomorphic.
problem Characterizing Killing fields on compact pseudo-Kähler manifolds.
method Detailed explanation and argumentation of why a previous claim was incomplete.
result Killing fields on compact pseudo-Kähler manifolds are holomorphic.
In this article I describe my recent geometric localization argument dealing with actions of NONcompact groups which provides a geometric bridge between two entirely different character formulas for reductive Lie groups and answers the question posed in [Sch]. A corresponding problem in the compact group setting was so…
Study extends Elkalla's work on subnormal subgroups to PD3-groups, but L2-Betti numbers need verification.
problem Verifying L2-Betti numbers for PD3-groups and group pairs. method Algebraic arguments extending Elkalla's work, but reliant on unproven L2-Betti number hypothesis. result Need further research on L2-Betti numbers for general PD3-groups. Study improves curvature estimate for stable marginally outer trapped hypersurfaces with a free boundary.
problem Curvature estimate for stable marginally outer trapped hypersurfaces with a free boundary.
method Iteration argument based on uniform area bound.
result Improved curvature estimate for stable marginally outer trapped hypersurfaces.
New method for hedging path-dependent options with price impact using probabilistic arguments.
problem Hedging of path-dependent options with price impact.
method Dual formulation using probabilistic arguments, proving existence of perfect hedging portfolios.
result Existence of a perfect hedging portfolio for path-dependent options with price impact.
Simplified Obata-Vétois argument for Einstein manifolds with nonnegative scalar curvature.
problem Identifying conditions for closed conformally Einstein manifolds to be Einstein.
method Simplified Obata-Vétois argument, identifying a closed interval containing zero.
result Closed conformally Einstein manifolds with nonnegative scalar curvature are Einstein if they satisfy certain conditions.
New argument suggests torsion cannot be part of gravity models.
problem The presence of torsion in gravity models is debated.
method Used spectral geometry and pseudo-differential calculus.
result No well-defined functional for torsion in spectral formulation.
The paper simplifies arguments for stationary varifolds results.
problem Height bound and Lipschitz approximation for stationary varifolds.
method Simpler arguments to obtain height bound and Lipschitz approximation.
result Excess decay as a consequence of height bound and Lipschitz approximation.
The paper develops axioms for uniquely decomposing functions with real arguments.
problem Decomposing functions with real arguments while preserving their overall structure.
method Developing axioms to uniquely decompose Borel measurable functions.
result Unique decompositions for all Borel measurable functions are achieved.
Paper uses ML to predict utility in APS dialogue outcomes.
problem Predict utility for different user subpopulations in APS.
method Develops EAI and EDS ML methods to predict utility functions.
result EDS more effective at predicting utility functions.
A new method extracts events and their arguments efficiently from text.
problem Efficiently extract event information from texts with long-range dependencies and associations.
method Graph Convolutional Networks with shortest dependency paths to capture syntactic relationships.
result Significant improvement over state-of-the-art methods.
Complete Calabi-Yau metrics made on special 3D spaces.
problem Creating complete Calabi-Yau metrics on complex 3D spaces.
method Used gluing construction and perturbation argument.
result Produced complete Calabi-Yau metrics with unbounded curvature.
New argument for 3-manifold cohomology with F2 coefficients.
problem Characterization of 3-manifold cohomology rings with F2 coefficients. method New argument based on Postnikov's 1948 characterization using intersection rings.
result A new proof for the characterization of 3-manifold cohomology rings.
This note presents the handlebody argument for modifying achiral Lefschetz singularities into broken Lefschetz fibrations, yielding a handlebody proof of the existence of broken Lefschetz fibrations on arbitrary closed smooth oriented 4-manifolds based on the earlier work of Gay and Kirby. Appeared in Geometry and Topo…
We equip many non compact non simply connected surfaces with smooth Riemannian metrics whose isoperimetric profile is smooth, a highly non generic property. The computation of the profile is based on a calibration argument, a rearrangement argument, the Bol-Fiala curvature dependent inequality, together with new result…
Extends arguments to limit structure in Calabi-Yau degenerations.
problem Understanding Gromov-Hausdorff limits in degenerating Calabi-Yau manifolds.
method Reduces conjecture to partial second-order estimate.
result Extends arguments to new settings.
We provide sharp empirical estimates of expectation, variance and normal approximation for a class of statistics whose variation in any argument does not change too much when another argument is modified. Examples of such weak interactions are furnished by U- and V-statistics, Lipschitz L-statistics and various error f…
Causal discovery algorithms can help generate legal arguments.
problem Leveraging causal discovery algorithms in legal decision-making.
method Prepared a legal dataset, annotated with 17 legal concepts, applied causal discovery algorithms, and quantified degrees of belief.
result Some causal relationships help generate viable legal arguments.
Study restricts line arrangements with odd points using topological arguments.
problem Restrictions on line arrangements with singular points of odd multiplicity.
method Topological arguments on locally-flat spheres in 4-manifolds.
result No line arrangement with 13 lines and only triple points exists.
We consider the classical "Serrin symmetry result" for the overdetermined boundary value problem related to the equation Δu=−1 in a model manifold of non-negative Ricci curvature. Using an extension of the Weinberger classical argument we prove a Euclidean symmetry result under a suitable "compatibility" assumption b…
We prove the smoothness of the L^2-analytic torsion form on some fiber bundles with non-compact fibers of positive Novikov-Shubin invariant. We do so by generalizing the arguments of Azzali-Goette-Schick to an appropriate Sobolev space, and proving that the Novikov-Shubin invariant remains positive in the Sobolev setti…
Derives Lagrangian for minimal surfaces, proving tangential variations vanish.
problem Variational calculus for minimal surfaces.
method Lagrangian formulation, pullback covariant derivative, geometric argument.
result Tangential variations vanish for minimal surfaces.
Embeds Teichmüller space into geodesic currents, proving independence.
problem Embedding Teichmüller space into geodesic currents.
method Algebraic method for Teichmüller space, ergodic argument for negatively curved surfaces.
result Embedding is totally linearly independent.
In the present paper, we discuss contra-arguments concerning the use of Pareto-Levý distributions for modeling in Finance. It appears that such probability laws do not provide sufficient number of outliers observed in real data. Connection with the classical limit theorem for heavy-tailed distributions with such type o…
The study examines the long-term behavior of mean curvature flows in closed 3-manifolds.
problem Understanding the long-term behavior of mean curvature flows in closed 3-manifolds.
method The approach involves constructing piecewise almost regular flows and applying perturbative arguments.
result The study constructs minimal surfaces in 3-manifolds via parabolic methods.
Direct proof shows adaptive gradient descent converges near-linearly for convex functions.
problem Proving near-linear convergence of adaptive gradient descent for convex functions.
method Direct Lyapunov-based argument for convex functions with unique minimizer.
result Direct proof of near-linear convergence for convex functions.
It is argued that arguments for strict prohibition of interests must be based on the use of arguments from authority. This is carried out by first making a survey of so-called dialectical roots for interest prohibition and then demonstrating that for at least one important positive interest bearing financial product, t…
It has been pointed out to the author by David Glickenstein that the proof of the (closely related) Lemmas 1.2 and 3.2 in the title paper is incorrect. The statements of both Lemmas are correct, and the purpose of this note is to give a correct argument. The argument is of some interest in its own right.
Study solves complex equation on specific types of manifolds.
problem Solving complex Monge-Ampère equation on Kähler manifolds.
method Flow-based arguments to establish existence of smooth solutions.
result Existence of smooth solutions under decreasing right-hand side.
Proves quantitative Alexandrov theorem for capillary surfaces.
problem Proving a quantitative version of the Alexandrov theorem for capillary hypersurfaces.
method Quantitative analysis of Montiel-Ros-type argument.
result Generalizes Julin-Niinikoski's result to capillary case.
Explain Arnold's proof of the Morse index theorem using Maslov index.
problem Proving the Morse index theorem in Riemannian geometry.
method Using symplectic arguments and the Maslov index.
result Self-contained exposition of Arnold's proof.
Paper provides a formula for translating solitons and singular minimal surfaces.
problem Representing translating solitons and singular minimal surfaces in 3D space.
method Develops a Weierstrass representation formula.
result Solves a general Cauchy problem for the class of surfaces.
A novel fact-checking method using debate dynamics on knowledge graphs.
problem Fact-checking on knowledge graphs with user comprehension and interactive reasoning.
method Reinforcement learning agents debate on paths in the graph to classify facts as true or false.
result Interactive reasoning and user understanding of AI decisions on knowledge graphs.
A method for reasoning on knowledge graphs using debate dynamics.
problem Automatic reasoning on knowledge graphs with interpretability.
method Reinforcement learning agents debate over facts, judge decides truth.
result Method outperforms baselines on triple classification and link prediction tasks.