arXiv research

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…

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

Employing the affine normal flow, we prove a stability version of the $p$-affine isoperimetric inequality for $p\geq1$ in $\mathbb{R}^2$ in the class of origin-symmetric convex bodies. That is, if $K$ is an origin-symmetric convex body in $\mathbb{R}^2$ such that it has area $π$ and its $p$-affine perimeter is close en…

New $L_p$-Steiner quermassintegrals defined from Steiner formula.

Study on $L_p$ affine surface areas and their inequalities for convex bodies.

We prove an analogue of the classical Steiner formula for the $L_p$ affine surface area of a Minkowski outer parallel body for any real parameters $p$. We show that the classical Steiner formula and the Steiner formula of Lutwak's dual Brunn Minkowski theory are special cases of this new Steiner formula. This new Stein…

Study on curvature measures in non-Euclidean spaces linked to Euclidean geometry.

In this paper, we introduce several mixed $L_p$ geominimal surface areas for multiple convex bodies for all $p\neq -n$. Our definitions are motivated from an equivalent formula for the mixed $p$-affine surface area. Some properties, such as the affine invariance, for these mixed $L_p$ geominimal surface areas are prove…

Study spherical convex bodies using $L_p$-floating areas and curvature entropy.

The paper develops inequalities for log-concave functions and related surface areas.

In this paper, we introduce the $L_p$ geominimal surface area for all $-n\neq p<1$, which extends the classical geominimal surface area ($p=1$) by Petty and the $L_p$ geominimal surface area by Lutwak ($p>1$). Our extension of the $L_p$ geominimal surface area is motivated by recent work on the extension of the $L_p$ a…

Given $L$ a convex body, the $L_p$-Busemann Random Simplex Inequality is closely related to the centroid body $Γ_p L$ for $p=1$ and $2$, and only in these cases it can be proved using the $L_p$-Busemann-Petty centroid inequality. We define a convex body $N_p L$ and prove an isoperimetric inequality for $(N_p L)^\circ$ …

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.

CapOptix uses options theory to price capacity in electricity markets.

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.

Study rigidity by logarithmic capacity and related functions.

Study binary perceptrons' capacity using random duality theory.

Study capacity constraints in continual learning with a simple model.

New complete panel dataset for LMICs helps analyze innovation and development.

Upper bounds for Lagrangian capacities of Liouville domains

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

Proves local maximizers for higher Ekeland-Hofer capacities in 4D star-shaped domains.

Derives an empirical capacity model for self-attention neural networks.

Improves online learning algorithms for functional models with capacity assumptions.

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…

Study relates symplectic homology capacity to periodic orbits in Liouville domains.

Learning capacity measures model complexity, correlating with test loss and sample size.

Study compares Monge-Ampère capacities on Kähler manifolds.

Generalizes memory and forecasting capacities for nonlinear recurrent networks with dependent inputs.

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.

For any Lie group $G$, we construct a $G$-equivariant analogue of symplectic capacities and give examples when $G = \mathbb{T}^k\times\mathbb{R}^{d-k}$, in which case the capacity is an invariant of integrable systems. Then we study the continuity of these capacities, using the natural topologies on the symplectic $G$-…

Study proves inequalities for mass-capacity on curved spaces.

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

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…

Introduces Rashomon Capacity to measure predictive multiplicity in probabilistic classifiers.

Inspired by the work of G. Lu on pseudo symplectic capacities we obtain several results on the Gromov width and the Hofer--Zehnder capacity of Hermitian symmetric spaces of compact type. Our results and proofs extend those obtained by Lu for complex Grassmannians to Hermitian symmetric spaces of compact type. We also c…

BestChanID identifies the channel with maximal capacity using training sequences.

Capacity analysis has been recently introduced as a way to analyze how linear models distribute their modelling capacity across the input space. In this paper, we extend the notion of capacity allocation to the case of neural networks with non-linear layers. We show that under some hypotheses the problem is equivalent …

Estimates for $p$-capacities on symmetric manifolds.

The study evaluates memory and capacity of graph embedding methods.

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

In this paper we address the following question, given a face representation, how many identities can it resolve? In other words, what is the capacity of the face representation? A scientific basis for estimating the capacity of a given face representation will not only benefit the evaluation and comparison of differen…

Recurrent neural networks are powerful models for processing sequential data, but they are generally plagued by vanishing and exploding gradient problems. Unitary recurrent neural networks (uRNNs), which use unitary recurrence matrices, have recently been proposed as a means to avoid these issues. However, in previous …

Study shows how activation functions impact the storage capacity of treelike neural networks.

Enhances CNN feature extractors' separation capacity analysis.

Existence and uniqueness of the solution to the discrete Lp Minkowski problem for $\mathfrak{p}$-capacity are proved when $p \geq 1$ and $1<\mathfrak{p}<n$. For general Lp Minkowski problem for $\mathfrak{p}$-capacity, existence and uniqueness of the solution are given when $p \geq 1$ and $1<\mathfrak{p}\le 2$. These r…