New dynamical system framework explains Nesterov acceleration.
problem Understanding Nesterov's accelerated gradient method.
method Dynamical system derivation without vanishing step size.
result Acceleration arises from discretizing an ODE with semi-implicit Euler.
A new method uses higher-order Langevin dynamics with critical damping for better generative modeling.
problem Improving generative models using Langevin dynamics with auxiliary variables.
method Introducing higher-order Langevin dynamics with critical damping, providing closed-form solutions.
result Improved generative models with better performance as measured by FID metric.
New method for constructing contact Lie systems on various spaces.
problem Constructing contact Lie systems on Riemannian and Lorentzian spaces.
method Adaptation of scaling symmetries to Lie-Hamilton systems, leading to contact Lie systems.
result Curvature-dependent reductions of contact Lie systems on Cayley-Klein spaces.
TOLD++ improves convergence of diffusion models by critically damping the forward transition matrix.
problem Improving the convergence of Denoising Diffusion Probabilistic Models.
method Critically damping the Third-Order Langevin Dynamics (TOLD) forward transition matrix using eigen-analysis.
result TOLD++ converges faster than TOLD, verified on toy and real datasets.
Motivated by the classical Euler elastic curves, David A. Singer posed in 1999 the problem of determining a plane curve whose curvature is given in terms of its position. We propound the same question in Lorentz-Minkowski plane, focusing on spacelike and timelike curves. In this article, we study those curves in $\math…
New damping technique improves deep learning models by reducing noise in flat directions.
problem Improving generalization in deep learning models by reducing estimation noise in flat directions.
method Developed a novel random matrix theory based damping learner to reduce the shrinkage coefficient and improve generalization.
result Significant generalization improvements in logistic regression and deep neural networks experiments.
We prove a Weyl-type fractal upper bound for the spectrum of the damped wave equation, on a negatively curved compact manifold. It is known that most of the eigenvalues have an imaginary part close to the average of the damping function. We count the number of eigenvalues in a given horizontal strip deviating from this…
Autoencoder estimates parameters of noisy, multi-component damped signals.
problem Parameter estimation of damped sinusoidal signals under rapid decay and noise.
method Autoencoder-based approach using latent space for frequency, phase, decay, and amplitude estimation.
result High accuracy in parameter estimation, robustness to subdominant components and phase differences.
Optimization rates improved for manifolds with bounded geometry.
problem Optimizing functions on manifolds with bounded geometry.
method Riemannian gradient descent and dynamic trivialization algorithm.
result Curvature-dependent convergence rates computed explicitly for common manifolds.
Unified method for calculating financial option prices from characteristic functions.
problem Calculating financial option prices from characteristic functions in high dimensions.
method Damped Fourier-cosine expansion (COS) method.
result The method converges exponentially if the characteristic function decays exponentially.
Study shows how neural networks learn eigenfunctions of the NTK in underparameterized settings.
problem Understanding the dynamics of MSE optimization in underparameterized neural networks.
method Analysis of gradient flow dynamics, focusing on eigenfunctions of the NTK.
result Eigenfunctions of the NTK determine the learning dynamics in underparameterized networks.
DistillKac generates images quickly using damped wave equations.
problem Generating high-quality images efficiently.
method Uses damped wave equations and Kac dynamics for finite speed transport.
result Fast image generation with high quality and numerical stability.
This paper is the second part of a study of the quantum free particle on spherical and hyperbolic spaces by making use of a curvature-dependent formalism. Here we study the analogues, on the three-dimensional spherical and hyperbolic spaces, $S_\k^3$ (κ>0) and $H_\k^3$ (κ<0), to the standard {\itshape spherical wav…
Spatial statisticians and quantitative investors use the same mathematical object: a Schur complement, damped by one parameter.
problem The Schur complement is used in both spatial modeling and portfolio allocation, but the parameters are different.
method The Schur complement is interpreted as reliability shrinkage of a conditional Gaussian.
result The Schur complement is the same in both applications.
Study spherical curves with curvature dependent on distance to a great circle.
problem Understanding spherical curves with curvature dependent on distance to a great circle.
method Introducing spherical angular momentum, characterizing known curves, finding new families, and obtaining arc length parametrizations.
result New families of spherical curves with intrinsic equations in elementary or Jacobi elliptic functions.
The maximum a posteriori (MAP) configuration of binary variable models with submodular graph-structured energy functions can be found efficiently and exactly by graph cuts. Max-product belief propagation (MP) has been shown to be suboptimal on this class of energy functions by a canonical counterexample where MP conver…
The paper develops methods to infer modes of linear systems from noisy data.
problem Detecting oscillations in power flow in AC electrical networks.
method Develops methods to infer modes (real or complex) from observations of linear systems forced by Gaussian noise.
result Inference of damping rates, frequencies, and mode shapes for real and complex modes.
A new method optimizes Fourier pricing for multi-asset options using adaptive quadrature.
problem Efficiently pricing multi-asset options in Lévy models.
method Optimized damping parameters and hierarchical adaptive quadrature.
result Significant speed-up in computational time for up to six dimensions.
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…
In this paper, we present an algorithm for the sparse signal recovery problem that incorporates damped Gaussian generalized approximate message passing (GGAMP) into Expectation-Maximization (EM)-based sparse Bayesian learning (SBL). In particular, GGAMP is used to implement the E-step in SBL in place of matrix inversio…
The paper investigates surfaces with boundary and mean curvature depending on the Gauss map.
problem Investigating surfaces with boundary and mean curvature depending on the Gauss map.
method Investigates H-surfaces in R3 with boundary, proving non-existence and existence of estimates. result Conditions on H ensure compact H-surfaces for certain shapes. Develops geometric framework for dissipative field equations.
problem Dissipative field equations and their geometric analysis.
method Canonical k-contact manifolds, k-contactifications, splitting results, regularity conditions, criteria for PDEs. result Explicit Hamiltonian descriptions for various nonlinear PDEs.
Using Weitzenböck techniques on any compact Riemannian spin manifold we derive inequalities that involve a real parameter and join the eigenvalues of the Dirac operator with curvature terms. The discussion of these inequalities yields vanishing theorems for the kernel of the Dirac operator D and lower bounds for the …
Since the pioneering work of Canham and Helfrich, variational formulations involving curvature-dependent functionals, like the classical Willmore functional, have proven useful for shape analysis of biomembranes. We address minimizers of the Canham-Helfrich functional defined over closed surfaces enclosing a fixed volu…
Study submanifolds in spheres with Ricci curvature bounds.
problem Topology of submanifolds in spheres with Ricci curvature constraints.
method Investigates submanifolds in spheres with Ricci curvature lower bounds.
result Strong additional information on submanifold geometry.
The energy in a square membrane Ω subject to constant viscous damping on a subset ω⊂Ω decays exponentially in time as soon as ω satisfies a geometrical condition known as the "Bardos-Lebeau-Rauch" condition. The rate τ(ω) of this decay satisfies τ(ω)=2min(−μ(ω),g(ω)) (see Lebeau [Math. Phys. Stud. …
Unified ODE model explains residual and non-residual networks.
problem Unclear relationship between residual and non-residual networks.
method Introducing a damping term in an ODE model to interpolate between ResNet and CNN architectures.
result Unified framework for understanding residual and non-residual networks.
Newton's method solves variational problems on manifolds.
problem Solving variational equations on manifolds.
method Newton's method with affine covariant damping strategy.
result Numerical results for variational problems demonstrated.
We analyse four consecutive cycles observed in the USA for employment and inflation. They are driven by three oil price shocks and an intended interest rate shock. Non-linear coupling between the rate equations for consumer products as prey and consumers as predators provides the required instability, but its natural d…
Improved sampling in generative models using CLDs with a hyperparameter.
problem Improving sampling performance in generative models.
method Extending Critically-damped Langevin Diffusions with a hyperparameter to control noise.
result Derivation of a novel upper bound on Wasserstein sampling error.
Develops a new geometric framework for field theories with dissipation.
problem Dealing with Hamiltonian field theories that include dissipation.
method Defines k-contact structure and k-contact Hamiltonian system, introduces symmetries and dissipation laws. result Analyzes two examples: damped vibrating string and Burgers' equation.
This paper proposes an alternative to the classical price-adjustment mechanism (called "tâtonnement" after Walras) that is second-order in time. The proposed mechanism, an analogue to the damped harmonic oscillator, provides a dynamic equilibration process that depends only on local information. We show how such a proc…
Improved generative models using critically-damped Langevin diffusion.
problem Current score-based generative models (SGMs) use overly simplistic diffusion processes, leading to complex denoising tasks and suboptimal performance.
method Proposed a novel critically-damped Langevin diffusion (CLD) and derived a score matching objective and sampling scheme.
result CLD-based SGMs achieve superior performance in synthesis quality compared to previous methods.
Comparison theorems in centro-affine differential geometry
problem rigidity phenomena of comparison theorems
method study of centro-affine differential geometry
result examples of rigidity phenomena
We discuss stochastic modeling of volatility persistence and anti-correlations in electricity spot prices, and for this purpose we present two mean-reverting versions of the multifractal random walk (MRW). In the first model the anti-correlations are modeled in the same way as in an Ornstein-Uhlenbeck process, i.e. via…
Study shows curvature constraints force submanifolds to have specific topology or geometry.
problem Curvature constraints on submanifolds in nonnegative curvature spaces.
method Investigates submanifolds with lower bounds on sectional curvature and mean curvature.
result Curvature constraints force submanifolds to have specific topology or geometry.
Stability of black holes proven in full subextremal range with positive cosmological constant.
problem Stability of Kerr-de Sitter black holes in the full subextremal range.
method Similar to previous proof in slowly rotating case, with implementation of constraint damping and verification of subprincipal symbol condition.
result Stability of Kerr-de Sitter black holes proven in the full subextremal range.
In this paper we consider the composite self-concordant (CSC) minimization problem, which minimizes the sum of a self-concordant function f and a (possibly nonsmooth) proper closed convex function g. The CSC minimization is the cornerstone of the path-following interior point methods for solving a broad class of co…
Machine learning predicts nuclear physics parameters with high accuracy.
problem Predicting nuclear physics parameters for superheavy elements.
method Gradient boosted trees algorithm trained on nuclear data.
result Predictions have standard deviation from 0.00035 to 0.73.
Study gives bounds on filling radius for Riemannian manifolds.
problem Finding bounds on the filling radius of Riemannian manifolds.
method Curvature-dependent bounds for all closed manifolds and upper bounds for submersion and submetry cases.
result Upper and lower bounds on the filling radius for specific types of manifolds.
Improved numerical solution for BSDEs with reduced boundary errors.
problem Boundary errors in numerical solution of BSDEs.
method Modified damping and shifting schemes to transform target function into a bounded periodic function, applying Fourier transforms.
result Significant reduction in boundary errors with improved accuracy and convergence.
Logarithmic-time schedules boost large-scale language model training efficiency.
problem Improving performance and efficiency in large-scale language model training.
method Designing time-varying hyperparameters (β1,β2,λ) for AdamW, specifically logarithmic-time scheduling with damping mechanisms. result ADANA optimizer achieves up to 40% compute efficiency compared to tuned AdamW, with gains persisting as model scale increases.
New algorithm for computing Wasserstein barycenters with guarantees.
problem Computing Wasserstein barycenters with varying regularization strengths.
method Damped Sinkhorn iterations followed by exact maximization/minimization steps.
result First non-asymptotic convergence guarantees for approximating Wasserstein barycenters.
Momentum is a simple and widely used trick which allows gradient-based optimizers to pick up speed along low curvature directions. Its performance depends crucially on a damping coefficient β. Large β values can potentially deliver much larger speedups, but are prone to oscillations and instability; hence one typic…
New method predicts quasar continuum near Lyman-α with high precision and accuracy.
problem Precise measurement of quasar red damping wing for epoch of reionization.
method Fully probabilistic approach using conditional neural spline flows.
result Achieved state-of-the-art precision and accuracy in predicting quasar continua.
The main aim of this paper is to extend Bochner's technique to statistical structures. Other topics related to this technique are also introduced to the theory of statistical structures. It deals, in particular, with Hodge's theory, Bochner-Weitzenbock and Simon's type formulas. Moreover, a few global and local theorem…
Study heat kernel on quaternionic contact manifolds, finding linear dependence of coefficients on curvature.
problem Analyzing heat kernel on quaternionic contact manifolds.
method Explicit computation of heat kernel coefficients and dependence on curvature.
result Second coefficient of heat kernel's small time asymptotics depends linearly on the qc scalar curvature.
Using a modified damped harmonic oscillator model equivalent to a model of market dynamics with price expectations, we analyze the reaction of financial markets to shocks. In order to do this, we gather data from indices of a variety of financial markets for the 1987 Black Monday, the Russian crisis of 1998, the crash …