arXiv research

Study sharp decay of capacity for subharmonic functions on compact Hermitian manifolds.

Sharp upper bounds derived for capacities in hyperbolic and Euclidean spaces.

In this paper, we investigate the common scenario where every candidate item for recommendation is characterized by a maximum capacity, i.e., number of seats in a Point-of-Interest (POI) or size of an item's inventory. Despite the prevalence of the task of recommending items under capacity constraints in a variety of s…

Study semicontinuity of capacity in non-smooth spaces using intrinsic flat convergence.

The electric capacity of a conductor in the 3-dimensional Euclidean space $R^3$ 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…

This note develops certain sharp inequalities relating the fractional Sobolev capacity of a set to its standard volume and fractional perimeter.

Study capacity constraints in continual learning with a simple model.

New algorithm bounds regret in mediator feedback bandit problems.

Here, the concept of electric capacity on Finsler spaces is introduced and the fundamental conformal invariant property is proved, i.e. the capacity of a compact set on a connected non-compact Finsler manifold is conformal invariant. This work enables mathematicians and theoretical physicists to become more familiar wi…

Study online learning with delays and capacity constraints, achieving optimal regret bounds.

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…

Develops a theory for mth order p-affine capacity for convex bodies containing the origin.

Study derives error decay rates for kernel classification under source and capacity conditions.

Sharp bounds for anisotropic p-capacity of Euclidean compact sets derived using flow methods.

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.

Memory capacity of DAM scales exponentially with feature separation, unaffected by correlations.

A variety of large-scale machine learning problems can be cast as instances of constrained submodular maximization. Existing approaches for distributed submodular maximization have a critical drawback: The capacity - number of instances that can fit in memory - must grow with the data set size. In practice, while one c…

Study characterizes hulls and capacities on Riemannian manifolds, proving isoperimetric inequalities.

Scattering networks maximize separation on low-dimensional data.

Optimal hidden-target learning for online inventory optimization on general convex sets.

Study proposes local effective dimension to measure model capacity and generalization error.

Designing neural network architectures is a task that lies somewhere between science and art. For a given task, some architectures are eventually preferred over others, based on a mix of intuition, experience, experimentation and luck. For many tasks, the final word is attributed to the loss function, while for some ot…

The paper tackles imbalanced classification under operational constraints, proposing a framework to maximize sensitivity.

The paper proves removable singularity for nonlocal minimal graphs.

The paper defines capacities for minimal graphs over manifolds and proves the half-space property.

New analysis tightens memory capacity of Hopfield models using spherical codes.

Sharp estimates for p-capacity on manifolds with Ricci curvature bounds.

New algorithm for shareable arms with load-dependent rewards in stochastic bandits.

Improved mass-capacity bounds for specific 3D manifolds.

Study Poincaré inequality in metric spaces via separating sets.

Paper investigates Lambda Value-at-Risk under ambiguity and risk sharing.

A long standing open problem in the theory of neural networks is the development of quantitative methods to estimate and compare the capabilities of different architectures. Here we define the capacity of an architecture by the binary logarithm of the number of functions it can compute, as the synaptic weights are vari…

We prove capacity inequalities involving the total mean curvature of hypersurfaces with boundary in convex cones and the mass of asymptotically flat manifolds with non-compact boundary. We then give the analogous of Pölia-Szegö, Alexandrov-Fenchel and Penrose type inequalities in this setting. Among the techniques used…

New analysis shows capacity of treelike neural networks with various activations.

New method reduces overfitting in deep neural networks by measuring and regulating hidden unit diversity.

We provide a rigorous mathematical treatment to the crowding issue in data visualization when high dimensional data sets are projected down to low dimensions for visualization. By properly adjusting the capacity of high dimensional balls, our method makes right enough room to prepare for the embedding. A key component …

This paper focuses on the discrimination capacity of aggregation functions: these are the permutation invariant functions used by graph neural networks to combine the features of nodes. Realizing that the most powerful aggregation functions suffer from a dimensionality curse, we consider a restricted setting. In partic…

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.

Study optimizes pricing under uncertainty and capacity constraints.

Solves a discrete logarithmic Minkowski problem for electrostatic p-capacity.

CapOptix uses options theory to price capacity in electricity markets.

The study calculates the injectivity capacity of ReLU networks using a novel mathematical approach.

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…

Measures neural network complexity using tangent space diversity.

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…

Researchers analyze backdoor data poisoning attacks and identify a memorization capacity parameter.

Sharp limits found for storing and retrieving input-output associations in linear associative memories.

Study excess capacity in neural networks using Rademacher complexity.