Estimates average number of common zeros of Laplacian eigenfunctions on manifolds.
problem Estimating the average number of common zeros of Laplacian eigenfunctions on compact Riemannian manifolds.
method Application of Crofton's formula for the sphere.
result Proves an estimate for the average number of common zeros of eigenfunctions, showing it does not exceed a specific expression.
Study finds average number of common zeros for eigenfunctions on a compact manifold.
problem Determining the average number of common zeros for eigenfunctions on a compact manifold.
method Computing the volume of the image of the manifold under an equivariant immersion into a sphere.
result Average number of common zeros for n eigenfunctions is computed. Average zeros of Finsler functions equals mixed symplectic volume of ellipsoids.
problem Counting isolated common zeros of Finsler functions.
method Construction of ring of normal densities and Crofton formula generalization.
result Average number of zeros equals mixed symplectic volume of Finsler ellipsoids.
This study calculates the average number of common zeros of holomorphic functions on complex manifolds.
problem Calculating the average number of common zeros of holomorphic functions.
method Defined a Hermitian mixed volume for a mix of non-negative Hermitian forms and proved the average number of common zeros equals this mixed volume.
result The average number of common zeros of holomorphic functions equals the mixed volume of the manifold.
Paper proposes a mean-field gradient descent for zero-sum games, proving convergence to Nash equilibrium.
problem Finding mixed Nash equilibria in zero-sum games with multiple players.
method Mean-field gradient descent dynamics with time-averaging, incorporating exponentially discounted gradients.
result Exponential convergence rate to mixed Nash equilibrium with respect to total variation metric.
Theorem identifies real eigenvalues of hyperbolic Laplacian uniquely.
problem Identifying real eigenvalues of the hyperbolic Laplacian.
method Average values of eigenfunctions over spheres.
result Two radii theorem for identifying eigenvalues uniquely.
Improved averaging method for noisy observations converges strongly.
problem Noisy observations from random dynamical systems require stable estimates.
method Introduced p-EMA, a modified exponential moving average with subharmonic weight decay. result Stochastic convergence guarantees for p-EMA under mild assumptions. Adam optimization algorithm can have non-zero average regret under certain conditions.
problem Non-zero average regret in Adam optimization algorithm.
method Used a three-periodic sequence of linear functions on [-1,1] with slopes c, -1, -1, and analyzed Adam variants.
result Adam optimization algorithm can have non-zero average regret under certain conditions.
Paper proves non-zero generalization boost for equivariant models.
problem Improving generalization in machine learning models.
method Analyzes simplest case of linear models, focusing on invariant/equivariant properties.
result First provably non-zero improvement in generalization for invariant/equivariant models.
Boosted CVaR Classification improves tail performance in classification tasks.
problem Maximizing tail performance in classification tasks.
method Proposed Boosted CVaR Classification framework using randomized classifiers and LPBoost algorithm.
result Minimizing CVaR loss over randomized classifiers leads to better tail performance.
Automates zero-shot classification by scoring and weighting prompts.
problem Improving zero-shot accuracy through prompt ensembling.
method Automatic prompt scoring and weighting method.
result Method outperforms existing techniques on various benchmarks.
Post-averaging improves neural network robustness against adversarial attacks.
problem Adversarial attacks on neural networks.
method Post-averaging technique to smooth high frequency components.
result Post-averaging method successfully defends over 95% of adversarial samples without significant performance degradation.
We derive a closed-form solution for the price of an average price as well as an average strike geometric Asian option, by making use of the path integral formulation. Our results are compared to a numerical Monte Carlo simulation. We also develop a pricing formula for an Asian option with a barrier on a control proces…
We present an iterative technique for finding zeroes of vector fields on Riemannian manifolds. As a special case we obtain a ``nonlinear averaging algorithm'' that computes the centroid of a mass distribution supported in a set of small enough diameter D in a Riemannian manifold M. We estimate the convergence rate of o…
The paper derives formulas for linear connections with totally anti-symmetric torsion in 3D generalized Berwald manifolds.
problem Understanding linear connections with specific torsion in generalized Berwald manifolds.
method Averaging of Finslerian quantities over indicatrix surfaces to express the torsion tensor.
result Explicit formulas for linear connections with totally anti-symmetric torsion in 3D generalized Berwald manifolds.
We consider strategies of investments into options and diffusion market model. It is shown that there exists a correct proportion between "put" and "call" in the portfolio such that the average gain is almost always positive for a generic Black and Scholes model. This gain is zero if and only if the market price of ris…
We present a geometric analysis of the incompressible averaged Euler equations for an ideal inviscid fluid. We show that solutions of these equations are geodesics on the volume-preserving diffeomorphism group of a new weak right invariant pseudo metric. We prove that for precompact open subsets of Rn, thi…
Increasing iterate averaging improves convergence rates for saddle-point problems.
problem Solving saddle-point problems efficiently.
method Increasing iterate averaging schemes applied to various first-order methods.
result Increasing iterate averaging preserves the O(1/T) convergence rate with no additional assumptions or overhead. This study optimizes model averaging for personalized collaborative learning.
problem Differences in data or objectives between nodes in federated learning.
method Weighted averaging between local and global models for scalar mean estimation.
result There is always some positive model averaging that reduces expected squared error.
Derives pricing formulae for power binary and normal distribution standard options.
problem Developing pricing models for binary and standard options.
method Incorporates Buchen's formulae into power binary options and derives a formula for normal distribution standard options.
result Derives pricing formulae for power binary and normal distribution standard options.
An investor is estimating net present value of a firm project and performs risk analysis. Usually it is created portfolio hierarchies and make comparison of variants of project based on these hierarchies. Then one finds that portfolio which corresponds to the particular needs of individual groups within the firm. We ha…
We estimate generic statistical properties of a structural credit risk model by considering an ensemble of correlation matrices. This ensemble is set up by Random Matrix Theory. We demonstrate analytically that the presence of correlations severely limits the effect of diversification in a credit portfolio if the corre…
We define a C^1 distance between submanifolds of a riemannian manifold M and show that, if a compact submanifold N is not moved too much under the isometric action of a compact group G, there is a G-invariant submanifold C^1-close to N. The proof involves a procedure of averaging nearby submanifolds of riemannian manif…
Two-Tailed Averaging improves generalization by optimizing the number of leading iterates to ignore.
problem Improving generalization in stochastic optimization with limited resources and hyperparameters.
method An anytime adaptive algorithm that balances the number of leading iterates to ignore for better generalization.
result Approximates the optimal tail at all optimization steps, improving generalization without hyperparameters.
Zero-shot audio classification using class label embeddings.
problem Classifying audio without labeled data.
method Bilinear model with audio feature embeddings and class label embeddings.
result Achieved accuracy up to 39.7% for natural audio categories.
Group averaging boosts model accuracy without training cost.
problem Challenging training of equivariant models in physics.
method Group averaging at test time, improving accuracy.
result Improves model accuracy by up to 37% in continuous dynamics.
ZeroSCROLLS benchmarks zero-shot natural language understanding over long texts.
problem Evaluate natural language understanding models over long texts without training data.
method Adapt six tasks from SCROLLS benchmark and add four new datasets, including novel aggregation tasks.
result Claude outperforms ChatGPT, and GPT-4 achieves highest average score.
The paper modifies asset pricing models using Taylor series expansions and market-based averages.
problem Improving asset pricing models to better reflect market dynamics.
method Derives new pricing equations using Taylor series expansions and market-based averages.
result New expressions for asset prices and volatilities derived from market data.
Mutation improves FTRL convergence in zero-sum games.
problem Lack of last-iterate convergence in FTRL variants.
method Introduced mutation to perturb action probabilities in FTRL.
result M-FTRL converges to Nash equilibria under full-information feedback.
In this paper, by combining techniques from Ricci flow and algebraic geometry, we prove the following generalization of the classical uniformization theorem of Riemann surfaces. Given a complete noncompact complex two dimensional Kähler manifold M of positive and bounded holomorphic bisectional curvature, suppose its…
The paper connects non-abelian zeta functions, Fokker-Planck equations, and projective flat connections.
problem Understanding the zeros of non-abelian zeta functions and their relation to differential equations.
method Exploring the moduli space of semi-stable lattices and averaging over these spaces.
result The zeros of non-abelian zeta functions are connected to Fokker-Planck equations and projective flat connections.
Paper proposes a new process for GZSL improving recognition performance on unseen classes.
problem Maximizing recognition performance on unseen classes in ZSL.
method Proposes a new training and evaluation process that penalizes samples from seen classes to improve performance on unseen classes.
result Significant performance boost on CUB and AwA2 datasets.
We study the problem of what causes prices to change. We define the mechanical impact of a trading order as the change in future prices in the absence of any future changes in decision making, and its it informational impact as the remainder of the total impact once mechanical impact is removed. We introduce a method o…
Study shows four-genus ratio of two-bridge knots decreases as knots get more complex.
problem Understanding the relationship between smooth four-genus and Seifert genus in two-bridge knots.
method Analytical proof focusing on two-bridge knots and their crossing numbers.
result The expected value of the ratio between smooth four-genus and Seifert genus tends to zero as the crossing number increases.
This paper analyzes the posterior variance of Gaussian processes and derives a new bound.
problem Lack of suitable analysis of posterior variance for finite and infinite training data.
method Derives a novel bound for posterior variance requiring only local information.
result Proves sufficient conditions for the convergence of posterior variance to zero and demonstrates improved average learning bound.
New approach to handle ranking function variation in zero-shot NAS.
problem Variation in ranking function outputs due to randomness.
method Viewing ranking function output as a random variable and constructing a stochastic ordering.
result Stochastic ordering boosts performance in neural architecture search.
In this paper we present detailed simulation results on the wealth distribution model with quenched saving propensities. Unlike other wealth distribution models where the saving propensities are either zero or constant, this model is not found to be ergodic and self-averaging. The wealth distribution statistics with a …
Consider a random smooth Gaussian field G(x):F→R, where F is a compact in Rd. We derive a formula for average area of a surface generated by the equation G(x)=0 and give some applications. As an auxiliary result we obtain an integral expression for area of a surface induced by zeros of a \e…
Recently, we proposed to transform the outputs of each hidden neuron in a multi-layer perceptron network to have zero output and zero slope on average, and use separate shortcut connections to model the linear dependencies instead. We continue the work by firstly introducing a third transformation to normalize the scal…
We shall prove that under some volume growth condition, the essential spectrum of the Laplacian contains the interval [(n−1)2K/4,∞) if an n-dimensional Riemannian manifold has an end and the average of the part of the Ricci curvature on the end which lies below a nonpositive constant (n−1)K converges to ze…
New averaging technique speeds up Newton method convergence.
problem Superlinear convergence of stochastic Newton methods with noisy Hessians.
method Hessian averaging to reduce noise and maintain superlinear convergence.
result Hessian averaging achieves superlinear convergence with a non-asymptotic rate.
This study examines how large language models learn in-context and provides insights into their performance and architecture.
problem Understanding how large language models learn in-context and their performance metrics.
method Bayesian model averaging, parameterization, and statistical analysis of transformer architecture.
result Transformer architecture enables in-context learning through attention mechanisms and fine-grained statistical analysis.
Study on convergence of Langevin dynamics for zero-sum games in probability distributions.
problem Analyzing convergence of Langevin dynamics for zero-sum games in probability distributions.
method Proved exponential and biased convergence guarantees for mean-field and finite-particle min-max Langevin dynamics.
result Explicit iteration complexity for finite-particle algorithms to approximate equilibrium distributions.
A new gradient estimator for online optimization with two function evaluations.
problem Online optimization of convex and Lipschitz functions with noisy data.
method L1-randomization approach for gradient estimation.
result Compared or better guarantees than previous methods for canceling noise.
New algorithm for solving minimax problems over distributions converges to Nash equilibrium.
problem Solving minimax problems over probability distributions.
method Symmetric Mean-field Langevin Dynamics (MFL-AG and MFL-ABR) with weighted averaging and best response dynamics.
result Converges to mixed Nash equilibrium with average-iterate and last-iterate convergence.
New algorithms converge faster to Nash equilibrium in zero-sum games with bandit feedback.
problem Learning in zero-sum games with bandit feedback without communication.
method Developed two uncoupled algorithms achieving optimal rate of Ω(T−1/4). result Achieved optimal rate of Ω(T−1/4) for convergence of policy profiles to Nash equilibrium. Introduces variance layers to improve neural network performance and robustness.
problem The reliance on expected values for predictions and the limitations of conventional stochastic neural networks.
method Introduces variance layers where weights follow a zero-mean distribution and are only parameterized by their variance.
result Variance layers can learn well, serve as an efficient exploration tool, and provide a decent defense against adversarial attacks.
Alexander polynomial degree correlates with knot defect, proving conjecture for defect zero.
problem Characterizing knot polynomials and their defects.
method Analyzing differential expansions and degree in q±2 of Alexander polynomials. result Proved Alexander polynomial degree correlates with knot defect, especially for defect zero.