arXiv research

New method uses relative capacities of geodesic balls to determine scalar curvature.

Symplectic capacities of domains near balls are well-defined, but not for all $C^1$-close domains.

Previously, Cristofaro-Gardiner, Hutchings and Ramos have proved that embedded contact homology (ECH) capacities can recover the volume of a contact 3-manifod in their paper "the asymptotics of ECH capacities" . There were two main steps to proving this theorem: The first step used an estimate for the energy of min-max…

Study the relative volume function on AH manifolds and its applications.

We study the volume of extrinsic balls and the capacity of extrinsic annuli in minimal submanifolds which are properly immersed with controlled radial sectional curvatures into an ambient manifold with a pole. The key results are concerned with the comparison of those volumes and capacities with the corresponding entit…

We define a capacity which measures the size of Weinstein tubular neighbourhoods of Lagrangian submanifolds. In symplectic vector spaces this leads to bounds on the codisc radius for any closed Lagrangian submanifold in terms of Viterbo's isoperimetric inequality. Moreover, we prove a generalization of Gromov's packing…

Defines a distance function on a manifold using symplectic embeddings and recovers the metric.

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 …

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

In this paper, we define a new capacity which allows us to control the behaviour of the Dirichlet spectrum of a compact Riemannian manifold with boundary, with "small" subsets (which may intersect the boundary) removed. This result generalizes a classical result of Rauch and Taylor ("the crushed ice theorem"). In the s…

Let $X$ be any rational ruled symplectic four-manifold. Given a symplectic embedding $ι:B_{c}\into X$ of the standard ball of capacity $c$ into $X$, consider the corresponding symplectic blow-up $\tX_ι$. In this paper, we study the homotopy type of the symplectomorphism group $\Symp(\tX_ι)$, simplifying and extending t…

We use the criteria of Lalonde and McDuff to determine a new class of examples of length minimizing paths in the group $Ham(M)$. For a compact symplectic manifold $M$ of dimension two or four, we show that a path in $Ham(M)$, generated by an autonomous Hamiltonian and starting at the identity, which induces no non-cons…

Enhances neural networks in hyperbolic space for better data structure capture.

Let $(M,g)$ be a complete non-compact Riemannian manifold together with a function $e^h$, which weights the Hausdorff measures associated to the Riemannian metric. In this work we assume lower or upper radial bounds on some weighted or unweighted curvatures of $M$ to deduce comparisons for the weighted isoperimetric qu…

New learning rates derived for Tikhonov-regularized problems without kernel assumptions.

The paper proves a new inequality linking mass and volume in 3D space.

New mass definition linked to ADM mass for general metrics.

We completely solve the symplectic packing problem with equally sized balls for any rational, ruled, symplectic 4-manifolds. We give explicit formulae for the packing numbers, the generalized Gromov widths, the stability numbers, and the corresponding obstructing exceptional classes. As a corollary, we give explicit va…

We obtain upper bounds for the isoperimetric quotients of extrinsic balls of submanifolds in ambient spaces which have a lower bound on their radial sectional curvatures. The submanifolds are themselves only assumed to have lower bounds on the radial part of the mean curvature vector field and on the radial part of the…

Let $P$ be a submanifold properly immersed in a rotationally symmetric manifold having a pole and endowed with a weight $e^h$. The aim of this paper is twofold. First, by assuming certain control on the $h$-mean curvature of $P$, we establish comparisons for the $h$-capacity of extrinsic balls in $P$, from which we ded…

Deep dynamic generative models are developed to learn sequential dependencies in time-series data. The multi-layered model is designed by constructing a hierarchy of temporal sigmoid belief networks (TSBNs), defined as a sequential stack of sigmoid belief networks (SBNs). Each SBN has a contextual hidden state, inherit…

Representation learning has become an invaluable approach for learning from symbolic data such as text and graphs. However, while complex symbolic datasets often exhibit a latent hierarchical structure, state-of-the-art methods typically learn embeddings in Euclidean vector spaces, which do not account for this propert…

Positive definite kernels and their associated Reproducing Kernel Hilbert Spaces provide a mathematically compelling and practically competitive framework for learning from data. In this paper we take the approximation theory point of view to explore various aspects of smooth kernels related to their inferential proper…

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.

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…

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.

3-balls in 4-sphere become isotopic in 5-ball.

Study capacity constraints in continual learning with a simple model.

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

New geometric quantities help classify manifolds and relate to entropy.

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.

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

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 on ball widths and minimal submanifolds in space forms.

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.