Flow of planar curves with curvature and capacity potential.
problem Geometric flow of planar curves with curvature and capacity potential.
method Curvature flow with a nonlocal term (normal derivative of capacity potential).
result Long term existence and large time asymptotics established under convexity condition.
New bounds for Dirac eigenvalue involving boundary capacity.
problem Eigenvalue bounds for Dirac operator on hypersurfaces.
method Estimates for Dirac operator on boundaries of compact manifolds.
result Lower bounds for first eigenvalue involving boundary capacity.
We study a geometric flow where the motion of a set is driven by the mean curvature of its boundary and the normal derivative of its capacity potential. We establish local well-posedness and propose two possible weak formulations that exist after singularities.
Study n-superharmonic functions and their geometric applications.
problem Asymptotic behavior of n-superharmonic functions at isolated singularities. method Using Wolff potential, n-capacity estimates, and Adams-Moser-Trudinger inequality. result Strong n-capacity lower bound estimate for geometric applications. The electric capacity of a conductor in the 3-dimensional Euclidean space R3 is defined as a ratio of a given positive charge on the conductor to the value of potential on the surface. This definition of the capacity is independent of the given charge. The capacity of a set as a mathematical notion was defined firs…
The paper proves a geometric capacitary inequality for sub-static manifolds with harmonic potentials.
problem Proving a geometric capacitary inequality for sub-static manifolds with harmonic potentials.
method Introducing a one-parameter family of functions that are monotone along the level-set flow of the potential, up to the optimal threshold.
result Proves a geometric capacitary inequality where the capacity of the horizon plays the same role as the ADM mass in the celebrated Riemannian Penrose Inequality.
Improved model capacity for graph cut algorithms by relaxing submodularity constraints.
problem Improving graph cut algorithms for complex image processing tasks.
method Enforce probably approximately submodular pairwise potentials instead of guaranteed submodular ones.
result Substantial improvement in model capacity with reduced inference error.
Paper proves anisotropic Minkowski inequality and related inequalities.
problem Proving anisotropic Minkowski inequality and related inequalities.
method Utilizes a nonlinear potential theoretic approach.
result Sharp anisotropic Minkowski inequality and related inequalities proved.
Proves mass-capacity inequalities for critical area-normalized capacitors, improving Schwarzschild metric uniqueness.
problem Proving mass-capacity inequalities for critical area-normalized capacitors.
method Analyzes asymptotically flat manifolds with boundary capacity potential satisfying an overdetermined problem.
result Improves Schwarzschild metric uniqueness and results for spin asymptotically flat spacetimes.
Continuous solutions found for complex geometry equations.
problem Finding solutions to complex geometry equations on Hermitian manifolds.
method Proving existence of continuous quasi-plurisubharmonic solutions for specific measures.
result Existence of continuous quasi-plurisubharmonic solutions for measures dominated by capacity.
Improved mass-capacity bounds for specific 3D manifolds.
problem Sharp mass-capacity inequality and upper bounds for 3D asymptotically flat manifolds.
method Monotonicity formulas associated with a harmonic potential.
result Improved bounds on ADM mass and capacity in terms of boundary area.
New method addresses crowding in high-dimensional data visualization.
problem Crowding issue in visualizing high-dimensional data.
method Adjusting capacity of high-dimensional balls and estimating correlation dimension.
result Mitigates crowding in various distance metrics.
New complexity measure explains better generalization with over-parametrization in neural networks.
problem Why neural networks generalize better with over-parametrization.
method Developed a novel complexity measure based on unit-wise capacities.
result Established a tighter generalization bound for two layer ReLU networks.
Introduces LDM to estimate machine learning algorithm capacity.
problem Estimating the performance of supervised learning algorithms.
method Characterizes algorithm flexibility using the diversity of possible outputs.
result LDM provides valuable insight into algorithm prediction behavior.
Paper shows existence of solutions for inverse mean curvature flow on manifolds with Ricci lower bounds.
problem Existence of solutions for inverse mean curvature flow on manifolds with Ricci lower bounds.
method Approximation via p-Laplace equation and new gradient and decay estimates for p-harmonic capacity potentials. result Sharp estimates for the growth of solutions and mean curvature of level sets, well-behaved under Gromov-Hausdorff convergence.
Study on neural networks' storage capacity and solution space structure.
problem Understanding the storage capacity and solution space structure of neural networks.
method Replica method from statistical physics.
result Storage capacity per parameter remains finite even with infinite width and weights exhibit negative correlations.
Normalization layers control deep neural network capacity, improving stability and generalization.
problem Excessive capacity in deep neural networks leads to overfitting and poor generalization.
method Developed a theoretical framework to explain normalization's role in capacity control.
result Normalization layers reduce the Lipschitz constant exponentially, smoothing the loss landscape and enhancing generalization.
Two potential bottlenecks on the expressiveness of recurrent neural networks (RNNs) are their ability to store information about the task in their parameters, and to store information about the input history in their units. We show experimentally that all common RNN architectures achieve nearly the same per-task and pe…
DNPUs improve neural network performance with high-capacity nanoelectronic nodes.
problem Limited performance of single DNPUs in solving complex classification problems.
method Developed DNPUs as high-capacity neurons and implemented multi-DNPU networks.
result Feed-forward DNPU networks improve single DNPU performance from 77% to 94% test accuracy.
Theory developed for complex Hessian measures on Hermitian manifolds.
problem Defining and analyzing complex Hessian measures on Hermitian manifolds.
method Potential theory for m-subharmonic functions with respect to a Hermitian metric.
result Equivalence between polar sets and negligible sets for m-subharmonic functions.
New complexity measure helps in agnostic reinforcement learning with or without access to MDP dynamics.
problem Understanding the number of rounds needed to learn an ε-suboptimal policy in unknown MDPs.
method Introducing spanning capacity as a new complexity measure and developing POPLER algorithm.
result There is a separation between generative and online access models for agnostic learnability.
Introduce a thermodynamically informed, temperature-transferable MLCG framework for proteins.
problem Temperature transferability of MLCG models for proteins.
method Explicit decomposition of CG potential into energetic and entropic components.
result Reproduces temperature-dependent quantities like heat capacity.
Study potential theory to detect completeness of Finsler manifolds.
problem Detecting completeness of Finsler manifolds via potential theory.
method Potential theoretic aspects of eikonal and infinity Laplace operator, Liouville properties, maximum principles at infinity, viscosity solutions.
result Forward completeness of Finsler manifolds can be detected using Liouville properties and maximum principles at infinity.
New analysis explains pathology of deep Gaussian processes.
problem Pathology of deep Gaussian processes reduces learning capacities with increased layers.
method Study nonlinear dynamic systems corresponding to DGPs, derive recurrence relations.
result Provide tighter bounds and rate of convergence for dynamic systems.
Deep networks don't improve on shallow ones for finding minima.
problem Improving representation of multidimensional mappings with deep neural networks.
method Numerical training methods to find minima in deep and shallow networks.
result Minima found with deep networks are worse than those found with shallow networks.
Study builds dataset and benchmarks ML models for accurate solar and wind power forecasting in France.
problem Accurate prediction of non-dispatchable renewable energy sources for grid stability and price prediction.
method Comprehensive methodology using machine learning models trained with spatially explicit weather data and production site capacity.
result Neural networks outperform traditional models in forecasting solar and wind power production in France.
The paper examines how insurers manage risks and liquidity in a dynamic market.
problem Model uncertainty in insurance pricing and competitive equilibrium.
method Analyzes insurers' robustness preferences and optimization strategies for underwriting and liquidity management.
result Robust insurance pricing leads to higher premiums and equity valuations compared to a benchmark.
This work addresses causal inference challenges in networked interference and proposes GNN-based estimators for individual treatment effects.
problem Estimating individual treatment effects in randomized experiments with networked interference.
method Uses Graph Neural Networks (GNNs) to capture network dependencies and derive causal effect estimators.
result Provides policy regret bounds and heuristic error bounds for GNN-based causal estimators under network interference and treatment capacity constraints.
Sparse codes improve optimal control tasks with correlated inputs.
problem Optimal control tasks with correlated feature inputs.
method Used a sparse code to represent natural images in an optimal control task solved with neuro-dynamic programming.
result An over-complete sparse code increases memory capacity and learning speed beyond a complete code.
Many state-of-the-art results obtained with deep networks are achieved with the largest models that could be trained, and if more computation power was available, we might be able to exploit much larger datasets in order to improve generalization ability. Whereas in learning algorithms such as decision trees the ratio …
Wide neural networks can degrade performance, contrary to conventional wisdom.
problem Understanding the limitations of increasing network width in neural networks.
method Using Deep Gaussian Processes to decouple capacity and width, analyzing their effects on representational power and non-Gaussianity.
result Wide neural networks can become less adaptable and more Gaussian, leading to performance degradation.
Study on removing sets and uniqueness of diffusion operators on various spaces.
problem Determining the effect of removing small sets on the self-adjointness and uniqueness of diffusion operators.
method Analyzes symmetric diffusion operators on metric measure spaces, proving a truncation result for potentials.
result Characterizes the critical size of removed sets and their effect on operator properties.
New cyclicity measures defined in weighted Besov spaces, with stability and geometric analysis.
problem Characterizing cyclicity in weighted Besov spaces.
method Defining cyclicity indices based on potential theory and capacity, studying stability under perturbations, and linking zero set structure to cyclicity.
result Novel invariants and conditions for cyclicity in various function spaces.
We study various capacities on compact Kähler manifolds which generalize the Bedford-Taylor Monge-Ampère capacity. We then use these capacities to study the existence and the regularity of solutions of complex Monge-Ampère equations.
Solves a discrete logarithmic Minkowski problem for electrostatic p-capacity.
problem Characterize measures generated by electrostatic p-capacity.
method Solves the discrete logarithmic Minkowski problem for 1 < p < n.
result Solves the discrete logarithmic Minkowski problem for measures in general position.
CapOptix uses options theory to price capacity in electricity markets.
problem Traditional capacity market designs fail to account for risk and price shocks.
method Interprets capacity commitments as reliability options and uses Markov Regime Switching Process.
result CapOptix provides more accurate pricing of capacity premia compared to existing mechanisms.
In this article, we propose the notion of the general p-affine capacity and prove some basic properties for the general p-affine capacity, such as affine invariance and monotonicity. The newly proposed general p-affine capacity is compared with several classical geometric quantities, e.g., the volume, the p-var…
Extends capacity analysis to neural networks, showing how capacity is distributed across layers.
problem How capacity is distributed in neural networks with non-linear layers.
method Introduces layer decoupling to quantify non-linear activation's impact, and uses a markovian rule for capacity propagation in deep networks.
result Shows that under certain conditions, capacity allocation in neural networks is equivalent to linear capacity allocation in an extended input space.
While symplectic manifolds have no local invariants, they do admit many global numerical invariants. Prominent among them are the so-called symplectic capacities. Different capacities are defined in different ways, and so relations between capacities often lead to surprising relations between different aspects of sympl…
Study excess capacity in neural networks using Rademacher complexity.
problem Understanding how much capacity deep networks have beyond what's needed for classification.
method Unified Rademacher complexity bounds for function composition and convolutional layers, considering Lipschitz constants and initialization norms.
result There is substantial excess capacity per task, and capacity can be kept similar across different tasks.
Study rigidity by logarithmic capacity and related functions.
problem Rigidity phenomena in kernel functions and capacities.
method Exploration of Bergman kernel, logarithmic capacity, Green's function, and Euclidean distance/volume.
result Established rigidity theorems by logarithmic capacity.
Study binary perceptrons' capacity using random duality theory.
problem Characterize the capacity of binary perceptrons with general thresholds.
method Utilized fully lifted random duality theory (fl RDT) to characterize the capacity.
result Characterizations match replica symmetry breaking predictions and uncover the capacity for zero-threshold scenario.
Study capacity constraints in continual learning with a simple model.
problem Understanding optimal resource allocation for agents with limited memory and compute resources.
method Analyzes a capacity-constrained linear-quadratic-Gaussian (LQG) sequential prediction problem and demonstrates optimal capacity allocation strategies.
result Derives a solution to the capacity-constrained LQG sequential prediction problem and shows how to optimally allocate capacity across sub-problems in the steady state.
New complete panel dataset for LMICs helps analyze innovation and development.
problem Lack of complete data for empirical analyses in LMICs.
method Predictive Mean Matching multiple imputation technique.
result Created a large dataset of 47 variables for 82 LMICs from 2005-2019.
Upper bounds for Lagrangian capacities of Liouville domains
problem Lagrangian capacity of Liouville domains
method Using S1-equivariant techniques result Extremal Lagrangian torus on the boundary of ellipsoid
Memory capacity of DAM scales exponentially with feature separation, unaffected by correlations.
problem Understanding how feature correlations impact DAM's capacity.
method Developed an empirical framework to analyze DAM's capacity under varying feature correlations and pattern separations.
result Memory capacity scales exponentially with feature separation, unaffected by correlations.
Proves local maximizers for higher Ekeland-Hofer capacities in 4D star-shaped domains.
problem Finding local maximizers for higher Ekeland-Hofer capacities in specific domains.
method Analogous to 4D local Viterbo conjecture, proving maximizers for rational ellipsoids.
result Local maximizers of the k-th Ekeland-Hofer capacities are symplectomorphic to rational ellipsoids.
The paper introduces capacity allocation analysis for neural networks, focusing on spatial capacity.
problem Designing neural network architectures is challenging due to the interplay of intuition, experimentation, and luck.
method Introduces capacity allocation analysis, focusing on spatial capacity allocation in linear settings.
result Quantitative comparison of classical architectures on various synthetic tasks reveals insights into model capacity allocation.