We introduce the concept of hereditarily non uniformly perfect sets, compact sets for which no compact subset is uniformly perfect, and compare them with the following: Hausdorff dimension zero sets, logarithmic capacity zero sets, Lebesgue 2-dimensional measure zero sets, and porous sets. In particular, we give an exa…
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48 results for “capacity zero”
The paper proves removable singularity for nonlocal minimal graphs.
problem Proving removable singularities for nonlocal minimal graphs.
method Analyzing -capacity zero compact sets to ensure graphs are minimal in the entire domain.
result Nonlocal minimal graphs are removable in the entire domain if they are minimal in a set of -capacity zero.
Exchanges acquire excess processing capacity to accommodate trading activity surges associated with zero-sum high-frequency trader (HFT) "duels." The idle capacity's opportunity cost is an externality of low-latency trading. We build a model of decentralized exchanges (DEX) with flexible capacity. On DEX, HFTs acquire …
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 relates symplectic homology capacity to periodic orbits in Liouville domains.
problem Relating symplectic homology capacity to periodic orbits in Liouville domains.
method Uses positive symplectic homology and Hofer-Zehnder capacity to establish bounds and existence of periodic points.
result Non-zero positive symplectic homology implies finite upper bound for Hofer-Zehnder capacity relative to skeleton and Hamiltonian diffeomorphisms.
Model for cross-border markets with limited transmission capacities.
problem Limited transmission capacities between two countries' markets.
method Developed a regime-switching process model with high-frequency approximation.
result Analytic tractability allows computation of key market quantities.
First we restate the definition of a Zero Area Singularity, recently introduced by H. Bray. We then consider several definitions of mass for these singularities. We use the Inverse Mean Curvature Flow to prove some new results about the mass of a singularity, the ADM mass of the manifold, and the capacity of the singul…
Defines a distance function on a manifold using symplectic embeddings and recovers the metric.
problem Recovering a Riemannian metric from symplectic embeddings in cotangent bundles.
method Defines a distance-like function using symplectic embeddings and recovers the metric when is the unit disc-cotangent bundle.
result The distance function recovers the Riemannian metric when is the unit disc-cotangent bundle.
Study on Privileged ERM showing limitations and providing capacity analysis.
problem Improving classification accuracy with privileged information.
method Theoretical analysis of Privileged ERM using VC dimension and generalization bounds.
result Worst-case guarantees for Privileged ERM cannot improve over standard ERM unless privileged information capacity is similar or smaller.
The paper connects mass, harmonic functions, and capacity in asymptotically flat 3-manifolds.
problem Connections among ADM mass, harmonic functions, and capacity in asymptotically flat 3-manifolds.
method New formulae for ADM mass via harmonic functions, monotone quantities, and geometric inequalities.
result The mass-to-capacity ratio is bounded below by 1 - sqrt(normalized Willmore functional of the boundary).
The paper extends IPC framework to stationary physical systems and validates it with a photonic system.
problem Characterizing the computational capabilities of stationary physical systems in a principled, data-efficient way.
method Extended IPC framework, established fundamental results, derived asymptotic bias, introduced data-efficient estimation methods.
result IPC strongly correlates with machine-learning performance and provides a reliable estimate of system dimensionality.
Study on CNNs' learning rates and approximation capacities.
problem Learning rates and approximation capacities of CNNs.
method New approximation bound and covering number analysis for CNNs.
result Derives minimax optimal convergence rates for CNNs in various learning problems.
New method lowers spherical perceptron capacity using fully lifted random duality theory.
problem Tackles the negative spherical perceptron capacity, a long-standing open problem.
method Develops fully lifted random duality theory (fl RDT) to characterize capacity.
result Shows remarkable closed-form analytical relations for practical capacity values.
Working in high-dimensional latent spaces, the internal encoding of data in Variational Autoencoders becomes naturally sparse. We discuss this known but controversial phenomenon sometimes refereed to as overpruning, to emphasize the under-use of the model capacity. In fact, it is an important form of self-regularizatio…
Overparameterized deep networks have the capacity to memorize training data with zero \emph{training error}. Even after memorization, the \emph{training loss} continues to approach zero, making the model overconfident and the test performance degraded. Since existing regularizers do not directly aim to avoid zero train…
Problems of segmentation, denoising, registration and 3D reconstruction are often addressed with the graph cut algorithm. However, solving an unconstrained graph cut problem is NP-hard. For tractable optimization, pairwise potentials have to fulfill the submodularity inequality. In our learning paradigm, pairwise poten…
LLMs predict SGD convergence without training on specific data.
problem Predicting SGD convergence without labeled data.
method Link between SGD and Markov chains, leveraging LLMs' understanding of dynamical systems.
result Zero-shot prediction of SGD convergence for new starting points.
Study on existence and properties of continuous solutions to complex Hessian equations.
problem Existence and properties of continuous solutions to complex Hessian equations.
method Established new capacity estimates and weak stability estimates for the -Hessian measure.
result Existence of continuous solutions to the complex Hessian equation under certain conditions.
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.
The paper analyzes the performance of delay-based reservoir computing using eigenvalue analysis.
problem Quantifying the performance of delay-based reservoir computing.
method Eigenvalue analysis of the dynamical system to predict reservoir computing performance.
result The performance of a reservoir computing system can be predicted by analyzing the small signal response and eigenvalue spectrum.
Estimates the generalization error of deep neural networks without relying on capacity measures.
problem Understanding how generalization error scales with training data for deep neural networks.
method Derives estimates of generalization error for deep networks based on two assumptions: zero training error and error probability proportional to distance to nearest training point.
result Estimates the generalization error of DNNs as O(1/(δN^{1/d})), matching experimental behavior.
Generative autoencoders offer a promising approach for controllable text generation by leveraging their latent sentence representations. However, current models struggle to maintain coherent latent spaces required to perform meaningful text manipulations via latent vector operations. Specifically, we demonstrate by exa…
Zero-shot KD for object detection without training data.
problem Challenges in using training data for knowledge distillation.
method Synthesizes pseudo-targets and samples using pretrained network.
result Achieves respectable mAP on object detection benchmarks.
We prove a Morrey-type theorem for Hamiltonian stationary submanifolds of . Namely, if is a Lagrangian submanifold with weakly harmonic Lagrangian phase then must be smooth. In the process we also discuss a local version of the equation, which is a nonline…
We introduce Continual Learning via Neural Pruning (CLNP), a new method aimed at lifelong learning in fixed capacity models based on neuronal model sparsification. In this method, subsequent tasks are trained using the inactive neurons and filters of the sparsified network and cause zero deterioration to the performanc…
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 -affine capacity and prove some basic properties for the general -affine capacity, such as affine invariance and monotonicity. The newly proposed general -affine capacity is compared with several classical geometric quantities, e.g., the volume, the -var…
Deep Reinforcement Learning (deep RL) has made several breakthroughs in recent years in applications ranging from complex control tasks in unmanned vehicles to game playing. Despite their success, deep RL still lacks several important capacities of human intelligence, such as transfer learning, abstraction and interpre…
Dual-objective GANs reduce training instabilities with tunable α-loss parameters.
problem Training instabilities in Generative Adversarial Networks (GANs).
method Introduce -GANs with dual objectives modeled using -loss.
result Upper bounds on estimation error show improved performance under certain conditions.
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 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 -equivariant techniques
result Extremal Lagrangian torus on the boundary of ellipsoid
Knowledge distillation deals with the problem of training a smaller model (Student) from a high capacity source model (Teacher) so as to retain most of its performance. Existing approaches use either the training data or meta-data extracted from it in order to train the Student. However, accessing the dataset on which …
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.
Develops a theory for mth order p-affine capacity for convex bodies containing the origin.
problem Defines and studies the mth order p-affine capacity for convex bodies containing the origin.
method Provides equivalent definitions, proves properties, and establishes inequalities.
result Establishes inequalities comparing to other geometric measures.
Derives an empirical capacity model for self-attention neural networks.
problem Theoretical capacity of large transformer models is not fully utilized by current optimization algorithms.
method Analyzes memory capacity of transformers using synthetic training data and common training algorithms.
result Derives an empirical capacity model (ECM) for a generic transformer.
Improves online learning algorithms for functional models with capacity assumptions.
problem Convergence rates of online stochastic gradient descent algorithms for functional linear models.
method Characterizations of slope function regularity, kernel space capacity, and sampling process covariance operator.
result Capacity assumptions can alleviate saturation of convergence rates as function regularity increases.
We introduce the concept of pseudo symplectic capacities which is a mild generalization of that of symplectic capacities. As a generalization of the Hofer-Zehnder capacity we construct a Hofer-Zehnder type pseudo symplectic capacity and estimate it in terms of Gromov-Witten invariants. The (pseudo) symplectic capacitie…
Learning capacity measures model complexity, correlating with test loss and sample size.
problem Understanding model complexity and its relation to test performance.
method Formal correspondence between thermodynamics and inference; learning capacity as a measure of effective dimensionality.
result Learning capacity correlates with test loss and is a small fraction of model parameters.
Study compares Monge-Ampère capacities on Kähler manifolds.
problem Comparing Monge-Ampère capacities on compact Kähler manifolds.
method Proved all capacities comparable, used Xia's integration by parts formula.
result All Monge-Ampère capacities are comparable.
Generalizes memory and forecasting capacities for nonlinear recurrent networks with dependent inputs.
problem Understanding memory and forecasting capabilities in networks with dependent inputs.
method Formulated bounds for memory and forecasting capacities in terms of network size and input properties.
result Proved that memory capacity for linear recurrent networks with independent inputs is given by the rank of the controllability matrix.
The Affine BV-capacitymath.DG
This paper is devoted to a geometric-measure-theoretic study of the brand new affine BV-capacity which is essentially different from the classic BV-capacity in dimension greater than one.