Paper analyzes origami slope gaps and their distribution, finding a unique pattern.
problem Analyzing slope gaps in origami surfaces.
method Derived slope gap distribution of a specific origami by considering return times under the horocycle flow.
result Found a unique distribution of origami slope gaps, not a sum of scaled Hall distributions.
Researchers compute gap distributions for saddle connection directions on specific translation surfaces.
problem Computing gap distributions for saddle connection directions on translation surfaces.
method Translation to dynamical question of return times to a transversal under the horocycle flow.
result Gap distributions have support at 0 and quadratic tail decay.
The paper calculates gap distributions for translation surfaces, focusing on the double heptagon.
problem Calculating gap distributions for translation surfaces.
method Describes a procedure to find winning holonomy vectors and applies it to the double heptagon.
result Explicitly computed gap distribution for the regular double heptagon translation surface.
Federated learning studies separate client data and distribution gaps.
problem Understanding performance differences in federated learning across different datasets.
method Proposed a framework to disentangle out-of-sample and participation gaps.
result Dataset synthesis strategy is crucial for realistic simulations of federated learning generalization.
We analyze the slope gap distribution of Veech surfaces, finding finite non-analytic points and quadratic tail decay.
problem Understanding the slope gap distribution of Veech surfaces.
method Explicit parameterization of a Poincaré section to the horocycle flow, finiteness result for the first return map.
result The limiting gap distribution of slopes of saddle connections on Veech surfaces is piecewise real-analytic with finitely many points of non-analyticity and has quadratic tail decay.
Proves effective slope gaps for lattice surfaces.
problem Proving effective slope gaps for lattice surfaces.
method Proves effective slope gap distribution for square torus and general lattice surfaces.
result Effective slope gap distribution result for lattice surfaces.
Study calculates slope gaps on polygon surfaces, finding non-unimodal distributions.
problem Understanding the distribution of slope gaps on polygon surfaces.
method Explicit computation of slope gap distributions for 2n-gons, providing bounds on non-differentiability points.
result Slope gap distributions are not always unimodal, answering a question by Athreya.
New theorem improves spectral gap for sampling from mixture distributions.
problem Sampling from multimodal distributions with simulated tempering.
method Introduced a decomposition theorem for the restricted spectral gap of simulated tempering.
result Lower bound on the restricted spectral gap for mixture distributions.
New estimates show spectral gap stability in RCD spaces, close to Beta distribution.
problem Stability of spectral gap bounds in metric-measure spaces.
method Combines L1-functional inequality and Stein's method. result Sharp quantitative estimate for spectral gap stability.
We give an explicit formula for the limiting gap distribution of slopes of saddle connections on the golden L, or any translation surface in its SL(2, R)-orbit, in particular the double pentagon. This is the first explicit computation of the distribution of gaps for a flat surface that is not a torus cover.
WR-CP reduces prediction set size and coverage gap under distribution shift.
problem Guaranteed coverage under distribution shift not achievable with i.i.d. assumption.
method Wasserstein distance, probability measure pushforwards, importance weighting, regularized representation learning.
result Reduces coverage gap to 3.2% across different confidence levels.
We explicitly compute the limiting gap distribution for slopes of saddle connections on the flat surface associated to the regular octagon with opposite sides identified. This is the first such computation where the Veech group of the translation surface has multiple cusps. We also show how to parametrize a Poincaré se…
Extracurricular learning closes the accuracy gap in knowledge distillation.
problem Accuracy gap between teacher and student models after knowledge distillation.
method Modeling student and teacher output distributions, sampling from an extended data distribution, and matching over this set.
result Extracurricular learning reduces the accuracy gap by 46% to 68%.
Factorial moments are convenient tools in particle physics to characterize the multiplicity distributions when phase-space resolution (Δ) becomes small. They include all correlations within the system of particles and represent integral characteristics of any correlation between these particles. In this letter, we sh…
We survey the use of dynamics of SL(2,R)-actions to understand gap distributions for various sequences of subsets of [0,1), particularly those arising from special trajectories of various two-dimensional dynamical systems. We state and prove an abstract theorem that gives a unified explanation for some of the ex…
Motivated by the study of billiards in polygons, we prove fine results for the distribution of gaps of directions of saddle connections on translation surfaces. As an application we prove that for almost every holomorphic differential ω on a Riemann surface of genus g≥2 the smallest gap between saddle connecti…
The distribution of returns in financial time series exhibits heavy tails. In empirical studies, it has been found that gaps between the orders in the order book lead to large price shifts and thereby to these heavy tails. We set up an agent based model to study this issue and, in particular, how the gaps in the order …
New bound limits generalization gap for large models, independent of model complexity.
problem Understanding generalization gap in large-scale machine learning models.
method Established a model-independent upper bound for generalization gap using Rényi entropy.
result Generalization gap can be maintained with arbitrarily large models if data entropy is sufficient.
The study examines how averaging data improves model performance.
problem Understanding the generalization gap in machine learning models.
method Data averaging, covariance analysis, and stochastic gradient descent (SGD) noise modeling.
result A modified generalization gap is always non-negative for a large class of model parameter distributions.
Improves Bayesian predictive performance in misspecified models.
problem Misspecification gap between inferential and predictive risks.
method Develops a multi-sample loss (PACm) to bridge the gap. result Empirical study shows improved predictive distribution.
DRL agents perform poorly at high decision frequencies, but a new algorithm improves performance.
problem DRL agents struggle at high decision frequencies, leading to poor performance.
method Proved that DRL agents' action-conditioned return distributions collapse to their policy's return distribution as decision frequency increases. Defined superiority as a probabilistic generalization of advantage for high-frequency value-based RL.
result Proper modeling of superiority distribution improves performance of controllers at high decision frequencies.
This work analyzes how multi-agent reinforcement learning can bridge the gap to reality in distributed multi-robot systems.
problem Collaborative learning in distributed multi-robot systems with varying sensors and actuators.
method Simulation-based analysis using PPO and Bullet physics engine, considering different types of perturbations.
result PPO's robustness is affected by the presence of different types of perturbations and the number of agents experiencing them.
The paper analyzes the latent geometry of generative diffusion models.
problem The manifold overfitting phenomenon in generative models.
method Statistical physics approach to analyze the spectrum of eigenvalues and singular values of the Jacobian of the score function.
result Three distinct qualitative phases during the generative process: trivial, manifold coverage, and consolidation phases.
The paper explores how simplicity leads to better out-of-distribution generalization in models.
problem Understanding the theoretical principles behind out-of-distribution (OOD) generalization in modern models.
method Examining diffusion models in image generation to analyze compositional generalization abilities and develop a theoretical framework for simplicity-based OOD generalization.
result The true, generalizable model corresponds to the simplest among consistent models, and this simplicity can be quantified and used to establish sample complexity guarantees.
As shown in recent research, deep neural networks can perfectly fit randomly labeled data, but with very poor accuracy on held out data. This phenomenon indicates that loss functions such as cross-entropy are not a reliable indicator of generalization. This leads to the crucial question of how generalization gap should…
This work closes the theory-practice gap for distributed optimization methods by introducing a new regularity condition.
problem Existing convergence conditions for distributed optimization methods are violated by nearly all kernels used in practice.
method Introduces Hessian relative uniform continuity (HRUC) to guarantee convergence under mild conditions.
result Derives convergence guarantees for mirror descent-based gradient tracking without restrictive assumptions.
The paper studies eigenvalues in gaps of the essential spectrum of a Bochner-Schrödinger operator.
problem Eigenvalue distribution in gaps of the essential spectrum of the Bochner-Schrödinger operator.
method Trace asymptotics formula and Weyl type asymptotic formula for eigenvalue counting function.
result The spectrum of Hp in the gap is discrete. New method explains ML performance gaps without causal knowledge.
problem Understanding why ML algorithms perform differently across domains.
method Nonparametric hierarchical decomposition framework.
result Detailed variable-level explanations for performance gaps.
Study on stable commutator length in RAAGs and Coxeter groups, proving spectral gaps and hardness results.
problem Understanding stable commutator length in right-angled Artin and Coxeter groups.
method Established spectral gaps, determined sizes up to constants, and related to graph properties.
result Found that stable commutator length can be arbitrarily close to zero in some groups, contrasting uniform gaps.
Paper addresses uncertainty in model generalization under regime shifts.
problem Uncertainty in model generalization under regime changes.
method Proposes a framework to quantify and separate regime mismatch and sensitivity.
result Obtains exact decomposition and minimax lower bound for regime-aware models.
Cloud computing is becoming increasingly popular as a platform for distributed training of deep neural networks. Synchronous stochastic gradient descent (SSGD) suffers from substantial slowdowns due to stragglers if the environment is non-dedicated, as is common in cloud computing. Asynchronous SGD (ASGD) methods are i…
Missing data is a pervasive problem in data analyses, resulting in datasets that contain censored realizations of a target distribution. Many approaches to inference on the target distribution using censored observed data, rely on missing data models represented as a factorization with respect to a directed acyclic gra…
Improved spectral gap for MwG with adaptive RWM proposals.
problem Improving mixing efficiency of MwG for log-concave distributions.
method Using adaptive RWM proposals tuned to match conditional variances of log-concave target distributions.
result Established a spectral gap lower bound of order O(1/κd) for MwG. Paper introduces a method to control early classification accuracy gaps.
problem Maintaining accuracy in early classification without full input processing.
method Statistical framework for a calibrated stopping rule.
result Reduces up to 94% of timesteps while controlling accuracy gaps.
We prove a \emph{query complexity} lower bound for approximating the top r dimensional eigenspace of a matrix. We consider an oracle model where, given a symmetric matrix M∈Rd×d, an algorithm Alg is allowed to make T exact queries of the form $\mathsf{w}^{(i)} =…
A modern aircraft may require on the order of thousands of custom shims to fill gaps between structural components in the airframe that arise due to manufacturing tolerances adding up across large structures. These shims are necessary to eliminate gaps, maintain structural performance, and minimize pull-down forces req…
GNA optimally identifies the best arm with small gaps.
problem Best arm identification in fixed-budget settings.
method Generalized Neyman Allocation (GNA) for asymptotically locally minimax optimal BAI.
result GNA's worst-case bounds match the lower and upper bounds in the small-gap regime.
In this paper several examples of gaps (lacunes) between dimensions of maximal and submaximal symmetric models are considered, which include investigation of number of independent linear and quadratic integrals of metrics and counting the symmetries of geometric structures and differential equations. A general result c…
In this note, we study the relationship between the variational gap and the variance of the (log) likelihood ratio. We show that the gap can be upper bounded by some form of dispersion measure of the likelihood ratio, which suggests the bias of variational inference can be reduced by making the distribution of the like…
Price gap, defined as the logarithmic price difference between the first two occupied price levels on the same side of a limit order book (LOB), is a key determinant of market depth, which is one of the dimensions of liquidity. However, the properties of price gaps have not been thoroughly studied due to the less avail…
New framework improves learning across multiple distributions.
problem Modeling uncertainty in sensitive machine learning applications.
method Inspired by multi-armed bandits, provides distribution-dependent guarantees.
result Enhanced dependence on suboptimality gaps and sample size.
New research shows some distributions hard to sample via diffusions.
problem Some distributions hard to sample via diffusions.
method Learning drifts of diffusions to approximate target distributions.
result Superpolynomially close drifts can yield very far approximations.
We investigate the statistics of the gap, G_n, between the two rightmost positions of a Markovian one-dimensional random walker (RW) after n time steps and of the duration, L_n, which separates the occurrence of these two extremal positions. The distribution of the jumps η_i's of the RW, f(η), is symmetric and its Four…
This paper bridges the gap between ODE and SDE in diffusion models using Fokker-Planck equations.
problem Empirical evidence shows that ODE-based samples from score-based diffusion models are inferior to SDE-based samples.
method The paper rigorously describes dynamics and approximations in training score-based diffusion models, linking them to Fokker-Planck equations.
result Adding a regularisation term based on the Fokker-Planck residual can close the gap between ODE- and SDE-induced distributions.
This paper studies the problem of adaptively sampling from K distributions (arms) in order to identify the largest gap between any two adjacent means. We call this the MaxGap-bandit problem. This problem arises naturally in approximate ranking, noisy sorting, outlier detection, and top-arm identification in bandits. Th…
Dropout, a simple and effective way to train deep neural networks, has led to a number of impressive empirical successes and spawned many recent theoretical investigations. However, the gap between dropout's training and inference phases, introduced due to tractability considerations, has largely remained under-appreci…
Study shows exponential gap in sample complexity between noisy and non-noisy recurrent neural networks.
problem Understanding the impact of noise on the sample complexity of recurrent neural networks.
method Analyzing noisy multi-layered sigmoid recurrent neural networks with independent noise and proving lower bounds.
result Exponential gap in sample complexity between noisy and non-noisy networks, even for small noise values.
The classical Three Gap Theorem asserts that for a natural number n and a real number p, there are at most three distinct distances between consecutive elements in the subset of [0,1) consisting of the reductions modulo 1 of the first n multiples of p. Regarding it as a statement about rotations of the circle, we find …