This paper studies the waist size of cusps in hyperbolic 3-manifolds, proving unique smallest sizes for specific manifolds.
problem Determining the smallest waist sizes of cusps in hyperbolic 3-manifolds.
method Analyzing the shortest nontrivial curves generated by parabolic isometries in maximal cusp boundaries.
result The next two smallest waist sizes are realized uniquely for specific manifolds.
The study improves bounds on space waists using advanced geometry techniques.
problem Understanding the waist of different geometric spaces.
method Advanced geometric techniques, including Borsuk-Ulam-Crofton method, Hausdorff measure.
result Established new waist bounds for various spaces.
Study incompressible surfaces to find cable knots' representativity and waist.
problem Computing the representativity and waist of cable knots.
method Study incompressible surfaces in cable knots' exteriors.
result Compute representativity and waist for most cable knots.
Uniform waist inequalities proven for manifolds with Kazhdan groups in codimension two.
problem Proving uniform waist inequalities for manifolds with specific group properties.
method Using finite covers and Cheeger inequality for manifolds with Kazhdan fundamental groups.
result Finite covers of manifolds with Kazhdan groups satisfy uniform waist inequalities in codimension two.
We introduce two numerical invariants, the waist and the trunk of knots. The waist of a closed incompressible surface in the complement of a knot is defined as the minimal intersection number of all compressing disks for the surface in the 3-sphere and the knot. Then the waist of a knot is defined as the maximal waist …
The paper proves waist inequalities for convex bodies and their linear images.
problem Understanding geometric characteristics of convex bodies through waist inequalities.
method Connections between Gromov's and Milman's work, proving waist inequalities for convex bodies and their linear images.
result Any convex body has a linear image satisfying a waist inequality with a universal constant.
Constructs foliations for 3-manifolds with positive scalar curvature.
problem Finding surfaces in 3-manifolds with positive scalar curvature.
method Singular foliations of compact three-manifolds with controlled properties.
result Extends Urysohn and Gromov-Lawson waist inequalities.
The abstract applies waist inequality to dynamical systems and entropy.
problem Understanding the relationship between waist inequality and dynamical systems.
method Applying waist inequality to entropy and mean dimension of dynamical systems.
result Maps between dynamical systems have positive conditional metric mean dimension under certain conditions.
Continuous sweepouts cover manifolds with bounded curve lengths.
problem Covering closed Riemannian manifolds with bounded curve lengths.
method Continuous family of 1-cycles parametrized by a sphere, with length bounds in terms of volume and dimension.
result Polyhedral 1-waist equals filling radius up to a constant factor.
Study geodesics on neck-degenerate manifolds, focusing and winding behavior observed.
problem Geodesics behavior on neck-degenerate manifolds with cuspidal singularities.
method Detailed multiscale analysis, blow-up techniques.
result Geodesics exhibit focussing and winding behavior as the neck degenerates.
Study minimal annuli in a slab, estimating their area.
problem Estimating the area of minimal annuli in a slab.
method Organized minimal annuli based on winding number, deduced convexity of length function, compared to catenoid waist area.
result Deduced convexity of length function and estimated area of minimal annuli.
The waist inequality states that for a continuous map from S^n to R^q, not all fibers can have small (n-q)-dimensional volume. We construct maps for which most fibers have small (n-q)-dimensional volume and all fibers have bounded (n-q)-dimensional volume.
3-manifolds with positive scalar curvature have controlled foliations.
problem Understanding foliations in 3-manifolds with positive scalar curvature.
method Showed a singular foliation by surfaces with controlled area and diameter.
result 3-manifolds with positive scalar curvature admit controlled foliations.
Consider a non-planar orientable minimal surface S in a slab which is possibly with genus or with more than two boundary components. We show that there exists a catenoidal waist W in the slab whose flux has the same vertical component as S such that Area(S)>= Area(W), provided the intersections of S with horizontal pla…
Minimal submanifolds in octonionic hyperbolic spaces have large volume.
problem Characterizing minimal submanifolds in locally symmetric spaces.
method Analyzing higher expansion properties and volume constraints.
result Codimension two minimal submanifolds have at least linear volume in the ambient space.
The study quantifies topological expansion properties of complexes and their embeddings.
problem Understanding topological expansion properties of simplicial complexes.
method Quantifying topological expansion through sublinear functions and proving monotonicity under regular maps.
result Proves topological expanders contain graphical expanders and gives lower bounds for specific embeddings.
Hyperbolic spaces have many cells in their fibers.
problem Finding lower bounds on the topological complexity of fibered spaces
method Using a freedom theorem for ideals in group rings of hyperbolic groups
result For large injectivity radii, there are many cells of dimension k in the fiber p^{-1}(z)
Tune simplifies hyperparameter search for distributed computing.
problem Adapting hyperparameter search to distributed environments.
method Unified framework for model selection and training.
result Simplifies implementation of state-of-the-art search algorithms.
New maxfaces with catenoid or planar ends constructed using node-opening technique.
problem Lack of examples of maxfaces with catenoid or planar ends.
method Adapted node-opening technique to construct maxfaces of high genus.
result Singularities on constructed maxfaces form curves around the waists of the necks, with most singularities being cuspidal edges and the rest swallowtails.
The study finds infinitely many periodic orbits just above a critical value on a 2-sphere.
problem Finding periodic orbits just above a critical value on a 2-sphere.
method Introduced a new critical value c∞(L) and showed its strict inequality to the Mañé critical value c(L), proving the existence of infinitely many periodic orbits on energy levels e∈(c(L),c∞(L)). result Infinitely many periodic orbits exist on energy levels just above the Mañé critical value.
Machine learning predicts obesity causes using genetic and imaging data.
problem Predicting causes of obesity in children and adults.
method Use ML techniques like decision trees, SVM, RF, GBM, LASSO, BN, and ANN on genetic and imaging data.
result ML models accurately predict obesity causes and chronic diseases.
Study of minimal surfaces in 4D with specific ends.
problem Characterize minimal surfaces in R4 with specific ends. method Modification of Costa and Hoffman-Meeks method, generalized Weierstrass representation.
result Minimal surfaces with specific ends are J-holomorphic under certain conditions. Method predicts NAFLD risk with high accuracy and distribution-free coverage guarantees.
problem Insufficient population-level screening tools for NAFLD.
method Gradient-boosted decision trees with conformal prediction.
result Method achieves AUROC of 0.912 internally and 0.891 externally, superior to other models.
Paper compares ML models for a wall-following robot, achieving high accuracy.
problem Improving prediction accuracy of a wall-following robot's direction.
method Trained various machine learning models on a dataset of ultrasound sensor readings.
result Presented machine learning models with higher accuracy than previous work.
Automates size normalization for fashion items.
problem Reduce merchandise returns in e-commerce.
method Uses sales data to automate size mapping.
result Automated size mappings comparable to human-generated ones.
Active learning performance degrades with larger batch sizes, but can be mitigated with smaller window sizes.
problem Impact of batch size on stopping active learning for text classification.
method Analyzed the impact of batch size on a stopping method for active learning in text classification, finding that larger batch sizes degrade performance and that using smaller window sizes mitigates this effect.
result Mitigating batch size degradation in active learning for text classification can be achieved by adjusting the window size parameter.
AdaBatch dynamically adjusts batch size during training for deep learning models.
problem Choosing optimal batch size for deep neural networks.
method Adaptive batch size adjustment during training.
result Adaptive batch sizes improve performance by up to 6.25x on 4 GPUs with minimal accuracy loss.
Mixed-size training improves CNN accuracy and speed.
problem Training CNNs on fixed image sizes limits their adaptability to various image sizes.
method Mixed-size training: training on multiple image sizes at once.
result Models trained with mixed-size images achieve higher accuracy and faster inference.
A new scaling law predicts optimal batch size for training models.
problem Finding the optimal batch size for training models efficiently.
method Proposed a three-term scaling law that considers model size, training data, training steps, and batch size.
result The three-term law accurately recovers the optimal batch size and can be robustly fit with fewer training runs.
Implicit Q-learning and SARSA adjust step-sizes automatically, improving stability and performance.
problem Numerical instability and slow progress in Q-learning and SARSA due to step-size calibration.
method Reformulate iterative updates as fixed-point equations, scaling step-sizes inversely with feature norms.
result Implicit methods maintain stability over broader step-size ranges and achieve comparable convergence rates.
Study on size and depth of neural networks for approximating benign functions, showing barriers and explicit results.
problem Understanding how size and depth of neural networks affect their ability to approximate benign functions.
method Analyzing ReLU networks for benign functions, proving barriers and explicit results.
result Explicit benign functions that cannot be approximated by networks of certain sizes or depths, showing barriers to size and depth separation.
Riemannian stochastic gradient descent converges faster with increasing batch size.
problem Improving convergence rate of Riemannian stochastic gradient descent.
method Theoretical analysis and numerical investigation of increasing batch size effects.
result Riemannian stochastic gradient descent converges faster with increasing batch size.
Study reveals a log-periodic structure in ETF sizes and finds large ETFs outperform small ones.
problem Understanding the size distribution and performance of ETFs.
method Detailed statistical analyses of ETF size distribution and performance metrics.
result Large ETFs outperform small ones, with a log-periodic structure in size distribution.
SGD's performance improves with critical batch size, minimizing SFO complexity.
problem Optimizing SGD's performance with batch size and learning rate.
method Analysis of SGD using constant and decaying learning rates, focusing on batch size effects.
result SGD with critical batch size minimizes SFO complexity.
A tick size is the smallest increment of a security price. It is clear that at the shortest time scale on which individual orders are placed the tick size has a major role which affects where limit orders can be placed, the bid-ask spread, etc. This is the realm of market microstructure and there is a vast literature o…
Adaptive batch size schedules improve language model training efficiency and generalization.
problem Dilemma of choosing batch sizes in large-scale model training.
method General-purpose adaptive batch size schedules compatible with data and model parallelism.
result Adaptive batch size schedules outperform constant batch sizes and heuristic warmup schedules.
A new method SEBS optimizes SGD batch size for better performance.
problem Optimizing batch size for SGD to balance training speed and generalization.
method SEBS method uses a multi-stage geometric batch size enlargement scheme.
result SEBS reduces parameter updates without increasing generalization error.
The paper develops formulas to count sizes of Markov equivalence classes of DAGs.
problem Measuring uncertainty and complexity in causal learning from DAGs.
method Introducing core graphs and deriving polynomial size formulas via symbolic computation.
result Efficient formulas for counting sizes of Markov equivalence classes of DAGs.
New convergence results for NGVI with various step sizes and sample sizes.
problem Understanding convergence of stochastic NGVI for various schedules.
method Projected stochastic NGVI for exponential family variational distributions.
result Geometric convergence and $\mathcal{O}\left(\frac{1}{T^ρ}
ight)$ rates for different schedules.
New optimal step sizes and mini-batch sizes for SAGA.
problem Finding optimal step sizes and mini-batch sizes for SAGA.
method Provided closed-form expressions for expected smoothness constant and suggested new step sizes and mini-batch sizes.
result Total complexity of SAGA decreases linearly with mini-batch size up to an optimal value.
TIDBD adapts TD step-sizes for better performance.
problem Finding optimal step-sizes for TD learning.
method Generalizes IDBD to TD learning, adapting step-sizes per feature.
result TIDBD outperforms other TD methods in various tasks.
RMGD uses bandit theory to optimize mini-batch size for faster and better performance.
problem Determining the optimal mini-batch size for gradient descent is time-consuming.
method Resilient Mini-batch Gradient Descent (RMGD) using Multi-Armed Bandit.
result RMGD achieves better performance than grid search in less time.
Large batch sizes reduce gradient variance in DP-SGD, improving privacy.
problem Understanding why large batch sizes work in DP-SGD.
method Decomposed total gradient variance into subsampling and noise-induced variances, proving batch size independence in the limit.
result Large batch sizes reduce effective total gradient variance, improving privacy in DP-SGD.
We make policy optimization algorithms batch size-invariant by decoupling proximal and behavior policies.
problem Some policy optimization algorithms do not have batch size-invariance, leading to inefficiencies.
method We decouple the proximal policy from the behavior policy to achieve batch size-invariance.
result Our approach makes policy optimization algorithms more efficient and allows them to use stale data more effectively.
Dynamic batch size adaptation improves neural network training efficiency.
problem Inconsistent batch size and learning rate tuning for optimization stability.
method Adaptive batch size estimation and dynamic adjustment based on gradient variance.
result Dynamic batch size adaptation leads to faster convergence and simplified learning rate tuning.
A simple block configures optimal kernel sizes for time series classification.
problem Choosing the right kernel size for time series classification.
method Proposes Omni-Scale block (OS-block) with kernel sizes determined by prime numbers.
result Models with OS-block achieve state-of-the-art performance on time series benchmarks.
The paper analyzes fixed step-size SA schemes on Riemannian manifolds.
problem Developing efficient algorithms for optimization on curved spaces.
method Fixed step-size stochastic approximation schemes in a Riemannian framework.
result The schemes converge to the solution as the step-size approaches zero.
Revisits granular models explaining firm growth rates and sizes.
problem Understanding the relationship between firm size and growth rate statistics.
method Developed new theoretical insights linking firm size and growth rate statistics within granular models.
result Growth volatility distribution is size-independent but fat-tailed, challenging granular models.