Related papers: Random Geometric Graphs in Reflexive Banach Spaces

In a random graph, counts for the number of vertices with given degrees will typically be dependent. We show via a multivariate normal and a Poisson process approximation that, for graphs which have independent edges, with a possibly…

The objective of this series is to study metric geometric properties of (coarse) disjoint unions of amenable Cayley graphs. We employ the Cayley topology and observe connections between large scale structure of metric spaces and group…

Bassino et al. (arXiv:1907.08517) have shown that uniform random co-graphs (graphs without induced $P_4$) of size $n$ converge to a certain non-deterministic graphon. The edge-density of this graphon is a random variable $\Lambda \in [0,1]$…

By analogy with the Cayley graph of a group with respect to a finite generating set or the Cayley--Abels graph of a totally disconnected, locally compact group, we detail countable connected graphs associated to Polish groups that we term…

This paper introduces a new probabilistic framework for supervised learning in neural systems. It is designed to model complex, uncertain systems whose random outputs are strongly non-Gaussian given deterministic inputs. The architecture…

Johnson and Schechtman (2009) characterized superreflexivity in terms of finite diamond graphs. The present author characterized the Radon-Nikod\'ym property (RNP) for dual spaces in terms of the infinite diamond. This paper is devoted to…

Let $X$ be Banach space which is not super-reflexive. Then, for each $n\ge1$ and $\varepsilon>0$, we exhibit metric embeddings of the Laakso graph $\mathcal{L}_n$ into $X$ with distortion less than $2+\varepsilon$ and into $L_1[0,1]$ with…

We consider random geometric graphs on the plane characterized by a non-uniform density of vertices. In particular, we introduce a graph model where $n$ vertices are independently distributed in the unit disc with positions, in polar…

We study geometric properties of the Banach space $\mathcal{R}$ constructed recently by C.~Read (arXiv 1307.7958) which does not contain proximinal subspaces of finite codimension greater than or equal to two. Concretely, we show that the…

We consider the random geometric graph on $n$ vertices drawn uniformly from a $d$--dimensional sphere. We focus on the sparse regime, when the expected degree is constant independent of $d$ and $n$. We show that, when $d$ is larger than $n$…

In the anisotropic random geometric graph model, vertices correspond to points drawn from a high-dimensional Gaussian distribution and two vertices are connected if their distance is smaller than a specified threshold. We study when it is…

We study the complexities of isometry and isomorphism classes of separable Banach spaces in the Polish spaces of Banach spaces recently introduced and investigated by the authors in [14]. We obtain sharp results concerning the most…

A reflexive Banach space with an unconditional basis admits an equivalent $1$-unconditional $2R$ norm and embeds into a reflexive space with a $1$-symmetric $2R$ norm. Partial results on $1$-symmetric $2R$ renormings of spaces with a…

We discuss two types of `regularity point', points of continuity and R-points for Banach function algebras, which were introduced by the first author and Somerset in an earlier paper on non-regularity for Banach function algebras. We show…

Graph convolutional neural network (GCNN) operates on graph domain and it has achieved a superior performance to accomplish a wide range of tasks. In this paper, we introduce a Barron space of functions on a compact domain of graph signals.…

In this paper we present new proofs of the non-embeddability of countably branching trees into Banach spaces satisfying property $(\beta_p)$ and of countably branching diamonds into Banach spaces which are $p$-AMUC for $p > 1$. These proofs…

Generalizing the notion of numerical range and numerical radius of an operator on a Banach space, we introduce the notion of joint numerical range and joint numerical radius of tuple of operators on a Banach space. We study the convexity of…

Diamond graphs and binary trees are important examples in the theory of metric embeddings and also in the theory of metric characterizations of Banach spaces. Some results for these families of graphs are parallel to each other, for example…

Let $A$ be a complex Banach space with a norm $\|f\|=\|f\|_X+\|d(f)\|_Y$ for $f\in A$, where $d$ is a complex linear map from $A$ onto a Banach space $B$, and $\|\cdot\|_K$ represents the supremum norm on a compact Hausdorff space $K$. In…

Let $(x_n)$ be a sequence in a Banach space $X$ which does not converge in norm, and let $E$ be an isomorphically precisely norming set for $X$ such that \[ \sum_n |x^*(x_{n+1}-x_n)|< \infty, \; \forall x^* \in E. \qquad (*) \] Then there…