We characterize sequences of Kleinian surface groups with convergent subsequences in terms of the asymptotic behavior of the ending invariants of the associated hyperbolic 3-manifolds. Asymptotic behavior of end invariants in a convergent sequence predicts the parabolic locus of the algebraic limit as well as how the a…
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48 results for “convergence behavior”
The article explores Helfrich flow with spontaneous curvature, finding singularities and convergence behaviors.
problem Understanding the long-time behavior of Helfrich flow with spontaneous curvature.
method Analyzing the gradient flow of a locally area- and volume-constrained Willmore flow, and applying it to the Helfrich flow.
result For negative spontaneous curvature, the Helfrich flow exhibits finite-time singularities; for positive spontaneous curvature, it converges globally.
The paper extends kernel ridge regression to product kernels and reveals new convergence behaviors.
problem Understanding kernel ridge regression in large dimensions with various kernels.
method Established a broad family of large dimensional kernels and derived convergence rates.
result Revealed new phenomena including minimax optimality, saturation effect, and multiple descent behavior.
Paper introduces a new topological loss for better convergence.
problem Optimizing topological losses for model's desired topological behavior.
method Introduces a new regularized topology-aware loss function.
result Guarantees efficient optimization of the new loss function.
Local Bayesian optimization shows strong performance and converges well, contrary to folklore.
problem Understanding the behavior and convergence of local Bayesian optimization methods.
method Studied the behavior of local optimization strategies and rigorously analyzed a specific algorithm.
result Local Bayesian optimization algorithms converge well and perform strongly, contrary to the folklore.
Study shows flow convergence to smooth K-Ricci outside a divisor with cusp singularity.
problem Behavior of conical Kähler-Ricci flow as cone angle approaches zero.
method Analysis of limit behavior of conical Kähler-Ricci flow as cone angle tends to zero.
result Flow converges to a unique Kähler-Ricci flow with cusp singularity along the divisor.
We construct convergent and divergent lattices in negative curvature and give a precise asymptotic description of the behavior of their counting function.
The Expectation-Maximization algorithm is perhaps the most broadly used algorithm for inference of latent variable problems. A theoretical understanding of its performance, however, largely remains lacking. Recent results established that EM enjoys global convergence for Gaussian Mixture Models. For Mixed Linear Regres…
In this paper, we prove pointwise convergence of heat kernels for mGH-convergent sequences of -spaces. We obtain as a corollary results on the short-time behavior of the heat kernel in -spaces. We use then these results to initiate the study of Weyl's law in the setting
Green functions on stationary varifolds established with inequalities and convergence results.
problem Establishing Green functions on stationary varifolds with inequalities and convergence.
method Extending Grüter and Widman's method, constructing Green functions, using local Harnack inequality.
result Green functions converge for sequences of stationary varifolds converging with multiplicity one.
Generalized belief propagation converges to optimal solutions on graphs with motifs.
problem Understanding belief propagation on loopy graphs.
method Study of generalized belief propagation on graphs with motifs.
result Generalized belief propagation converges to the global optimum of the Bethe free energy.
New method improves convergence of spatial filters in neural networks.
problem Poor convergence behavior of spatial filters in neural networks.
method Correlated initialization for spatial filters.
result Uncorrelated initialization leads to poor convergence and slow training of some parameters.
Study of tori of revolution under Willmore flow converges to Clifford Torus.
problem Long-time behavior and convergence of Willmore flow for tori of revolution.
method Gradient flow of Willmore energy for tori of revolution, analyzing energy threshold and convergence to Clifford Torus.
result Convergence of Willmore flow to Clifford Torus for initial energy below 8π.
The modified J-flow with Calabi ansatz shows convergence or blow-up behavior based on topological constants.
problem Analyzing the behavior of the modified J-flow with Calabi ansatz.
method Using the Calabi symmetry and studying the singularities of the flow.
result The modified J-flow with Calabi ansatz converges to a solution away from a variety, and blows up along the variety.
A simple function shows how neural nets can converge despite high sharpness.
problem Understanding why neural nets converge with high sharpness.
method Constructed a minimal example function and analyzed its training dynamics rigorously.
result Final converging point has sharpness close to .
We perform a rescaling analysis to analyze the future behavior of a class of -symmetric vacuum spacetimes. We show that on the universal cover, there is -convergence to a spatially homogeneous spacetime that does not satisfy the vacuum Einstein equations.
Paper analyzes EM algorithm's trajectory in 2MLR, revealing cycloid behavior.
problem Understanding the convergence and trajectory of EM algorithm in 2MLR.
method Explicit closed-form expressions for EM updates, recurrence relation derivation at population level.
result EM iterations lie on a cycloid trajectory, leading to theoretical estimate of convergence exponent.
Linear Q-learning converges to a bounded set without divergence.
problem Proving linear Q-learning does not diverge and converges to a bounded set.
method No modifications to the original linear Q-learning algorithm, no Bellman completeness or near-optimality assumptions, only an ε-softmax behavior policy with adaptive temperature.
result First convergence rate of linear Q-learning iterates to a bounded set.
Boosted additive models reveal new insights and potential pathologies.
problem Theoretical understanding of boosted additive models (BAMs) and their convergence behavior.
method Study of solution paths of BAMs and derivation of convergence results.
result Uncovering pathologies of boosting for certain additive model classes.
This work studies the implicit bias of mini-batch SGD in classification.
problem Understanding the implicit bias of mini-batch SGD in multi-class classification.
method Characterizes how batch size, momentum, and variance reduction affect convergence and max-margin behavior under different norms.
result Momentum enables small-batch convergence to an approximate max-margin solution, while variance reduction recovers the exact full-batch bias.
The paper studies harmonic map heat flow to flat tori, proving ergodic behavior and convergence to hyperbolic measure.
problem Analyzing the behavior of harmonic map heat flow to moduli space of flat tori.
method Investigates stability and ergodic behavior of harmonic map heat flow using hyperbolic structure and relative entropy.
result The flow converges weak-- to the normalized hyperbolic measure on the moduli space.
Study of Moncrief lines' behavior in curved space-times.
problem Understanding the asymptotic behavior of Moncrief lines in curved space-times.
method Analysis of geodesic laminations and convergence to Thurston boundary.
result Moncrief lines converge to a unique point in the Thurston boundary.
In this paper we study the convergence behavior of grafting rays to the Thurston boundary of Teichmuller space. When the grafting is done along a weighted system of simple closed curves or along a maximal uniquely ergodic lamination this behavior is the same as for Teichmuller geodesics and lines of minima. We also sho…
The paper studies the free elastic flow of closed curves and finds their asymptotic shape converges to a circle.
problem Challenges in studying the asymptotic behavior of the free elastic flow for closed curves.
method Analysis of the free elastic flow as an -gradient flow for Euler's elastic energy.
result An appropriate rescaling of initial curves geometrically close to circles converges to a unique round circle.
Method uses DNNs to approximate functions with specific asymptotic behavior.
problem Approximating functions with given asymptotic behavior.
method Specifically constructed terms combined with unconstrained DNN.
result Enforcing asymptotic behavior leads to better approximation and faster convergence.
The study examines the dynamic behavior of RMSprop and Adam algorithms.
problem Understanding the training loss curve patterns of adaptive gradient algorithms.
method Careful numerical experiments and theoretical explanations using the signGD flow.
result Adam converges smoother and faster when momentum factors are close to each other.
In manifold learning, algorithms based on graph Laplacians constructed from data have received considerable attention both in practical applications and theoretical analysis. In particular, the convergence of graph Laplacians obtained from sampled data to certain continuous operators has become an active research topic…
We determine the long-time asymptotic behavior of a relativistic diffusion taking values in the unitary tangent bundle of a Robertson-Walker space-time. We prove in particular that when approaching the explosion time of the diffusion, its projection on the base manifold almost surely converges to a random point of the …
Using elements from the theory of ergodic backward stochastic differential equations (BSDE), we study the behavior of forward entropic risk measures. We provide their general representation results (via both BSDE and convex duality) and examine their behavior for risk positions of long maturities. We show that forward …
Study explores learning behavior of GFlowNets, revealing key mechanisms.
problem Lack of theoretical understanding of GFlowNets' learning dynamics.
method Rigorous theoretical investigation of four dimensions: convergence, sample complexity, implicit regularization, and robustness.
result Elucidates mechanisms underlying GFlowNet's learning dynamics, providing insights into performance factors.
We study the asymptotic behavior of convex Cauchy hypersurfaces on maximal globally hyperbolic spatially compact space-times of constant curvature. We generalise the result of [11] to the (2+1) de Sitter and anti de Sitter cases. We prove that in these cases the level sets of quasi-concave times converge in the Gromov …
In this paper, we study the backward Ricci flow on locally homogeneous 3-manifolds. We describe the long time behavior and show that, typically and after a proper re-scaling, there is convergence to a sub-Riemannian geometry. A similar behavior was observed by the authors in the case of the cross curvature flow.
Many practitioners who use the EM algorithm complain that it is sometimes slow. When does this happen, and what can be done about it? In this paper, we study the general class of bound optimization algorithms - including Expectation-Maximization, Iterative Scaling and CCCP - and their relationship to direct optimizatio…
We describe the asymptotic behavior of Palais-Smale sequences associated to certain Yamabe-type equations on manifolds with boundary. We prove that each of those sequences converges to a solution of the limit equation plus a finite number of "bubbles" which are obtained by rescaling fundamental solutions of the corresp…
The purpose of this paper is to provide a sharp analysis on the asymptotic behavior of the Durbin-Watson statistic. We focus our attention on the first-order autoregressive process where the driven noise is also given by a first-order autoregressive process. We establish the almost sure convergence and the asymptotic n…
Extends tracking guarantees for time-varying variational inequalities.
problem Tracking solutions of time-varying variational inequalities.
method Extends existing results to sublinear solution paths and periodic problems.
result Discrete dynamical systems of periodic time-varying VI can exhibit chaotic behavior or converge to the solution.
New method for analyzing learning dynamics in singular models.
problem Challenges in analyzing learning of singular models with no one-to-one parameter space.
method Relative reparameterization technique to extract regular sub-models.
result Demonstrated differences in convergence behavior due to algorithmic and intrinsic aspects.
In this paper we study backward Ricci flow of locally homogeneous geometries of -manifolds which admit compact quotients. We describe the long-term behavior of each class and show that many of the classes exhibit the same behavior near the singular time. In most cases, these manifolds converge to a sub-Riemannian ge…
Current auto loans converge to super-prime credit despite remaining underwater.
problem Inefficient consumer behavior in auto loans leading to suboptimal credit risk.
method Large-sample statistical hypothesis test on transition matrix between risk bands.
result All current risk bands converge to super-prime credit, despite remaining underwater.
New method avoids spurious critical points for low-rank matrix recovery.
problem Low-rank matrix recovery problems on Riemannian manifold.
method Riemannian gradient descent with random initialization.
result Riemannian gradient descent avoids spurious critical points and converges nearly linearly.
Study on Langevin dynamics convergence rates and their application to GAN training.
problem Understanding the long-term behavior of Langevin dynamics equations.
method Analytical and numerical methods to study convergence rates of underdamped mean-field Langevin dynamics.
result Exponential convergence rate results for the Langevin dynamics under various conditions.
In this paper we consider the problem of minimizing the relative perimeter under a volume constraint in the interior of a convex body, i.e., a compact convex set in Euclidean space with interior points. We shall not impose any regularity assumption on the boundary of the convex set. Amongst other results, we shall prov…
A new algorithm for decentralized optimization over directed graphs.
problem Decentralized stochastic optimization over directed networks.
method Gradient tracking and S-ADDOPT algorithm with constant and decaying step-sizes.
result S-ADDOPT converges linearly with constant step-size and sublinearly with decaying step-size.
SAD Neural Networks: Divergent Gradient Flows and Asymptotic Optimality via o-minimal Structurescs.LG
Gradient flows of neural networks converge to optimal values or diverge, with thresholds and asymptotic behaviors.
problem Understanding the convergence and divergence of gradient flows in neural networks.
method Analysis of gradient flows on loss landscapes of neural networks using o-minimal structures.
result Gradient flows either converge to optimal values or diverge to infinity, with thresholds and asymptotic behaviors.
We present an agent behavior based microscopic model that induces jumps, spikes and high volatility phases in the price process of a traded asset. We transfer dynamics of thermally activated jumps of an unexcited/ excited two state system discussed in the context of quantum mechanics to agent socio-economic behavior an…
Study on self-consuming generative models with diverse human curation, focusing on convergence and stability.
problem Analyzing self-consuming generative models with heterogeneous human curation.
method Investigates the asymptotic behavior of retraining dynamics using nonlinear Perron--Frobenius theory and Banach contraction mapping.
result Improves convergence results and provides stability and non-stability analyses for the model.
We study the convergence behavior of the general inverse -flow on Kähler manifolds with initial metrics satisfying the Calabi Ansatz. The limiting metrics can be either smooth or singular. In the latter case, interesting conic singularities along negatively self-intersected sub-varieties are formed as a result of …
Paper analyzes faster convergence rates for reinforcement learning from offline data.
problem Analyzing faster convergence rates for reinforcement learning from offline data.
method Fine analysis of reinforcement learning from offline data, providing fast rates for regret convergence.
result The paper provides fast rates for the regret convergence, showing that the level of exponentiation depends on the noise in the decision-making problem.