Related papers: Branched Signature Model
The signature of a path can be described as its full non-commutative exponential. Following T. Lyons we regard its expectation, the expected signature, as path space analogue of the classical moment generating function. The logarithm…
As neural networks (NNs) are increasingly introduced into safety-critical domains, there is a growing need to formally verify NNs before deployment. In this work we focus on the formal verification problem of NN equivalence which aims to…
A signed graph (SG) is a graph where edges carry sign information attached to it. The sign of a network can be positive, negative, or neutral. A signed network is ubiquitous in a real-world network like social networks, citation networks,…
The paper introduces a Signed Generalized Random Dot Product Graph (SGRDPG) model, which is a variant of the Generalized Random Dot Product Graph (GRDPG), where, in addition, edges can be positive or negative. The setting is extended to a…
Our goal is to provide a novel method of representing 2D shapes, where each shape will be assigned a unique fingerprint - a computable approximation to a conformal map of the given shape to a canonical shape in 2D or 3D space (see page 22…
The aim of this article is to develop an explicit procedure that enables one to reconstruct any $C^1$ path (at natural parametrization) from its signature. We also explicitly quantify the distance between the reconstructed path and the…
A signed directed graph is a graph with sign and direction information on the edges. Even though signed directed graphs are more informative than unsigned or undirected graphs, they are more complicated to analyze and have received less…
Arc permutations, which were originally introduced in the study of triangulations and characters, have recently been shown to have interesting combinatorial properties. The first part of this paper continues their study by providing signed…
Let $X^4$ and $Y^4$ be smooth manifolds and $f: X\to Y$ a branched cover with branching set $B$. Classically, if $B$ is smoothly embedded in $Y$, the signature $\sigma(X)$ can be computed from data about $Y$, $B$ and the local degrees of…
Ray tracing has become a standard for accurate radio propagation modeling, but suffers from exponential computational complexity, as the number of candidate paths scales with the number of objects raised to the interaction order. This…
Graphs drawn in the plane are ubiquitous, arising from data sets through a variety of methods ranging from GIS analysis to image classification to shape analysis. A fundamental problem in this type of data is comparison: given a set of such…
We present a framework to define a large class of neural networks for which, by construction, training by gradient flow provably reaches arbitrarily low loss when the number of parameters grows. Distinct from the fixed-space global…
We introduce the notions of tree-like path and tree-like equivalence between paths and prove that the latter is an equivalence relation for paths of finite length. We show that the equivalence classes form a group with some similarity to a…
Signed distance map (SDM) is a common representation of surfaces in medical image analysis and machine learning. The computational complexity of SDM for 3D parametric shapes is often a bottleneck in many applications, thus limiting their…
Diffusion models form an important class of generative models today, accounting for much of the state of the art in cutting edge AI research. While numerous extensions beyond image and video generation exist, few of such approaches address…
Real networks often grow through the sequential addition of new nodes that connect to older ones in the graph. However, many real systems evolve through the branching of fundamental units, whether those be scientific fields, countries, or…
Building on techniques from complex analysis and topology, we establish a remarkable property of branched covers and formulate a complete criterion for the existence of specific types of branched covers between 2-spheres. Our results extend…
The success of Deep Learning and its potential use in many safety-critical applications has motivated research on formal verification of Neural Network (NN) models. In this context, verification involves proving or disproving that an NN…
Graph-structured data arise ubiquitously in many application domains. A fundamental problem is to quantify their similarities. Graph kernels are often used for this purpose, which decompose graphs into substructures and compare these…
Graph neural networks (GNNs) have recently become the standard approach for learning with graph-structured data. Prior work has shed light into their potential, but also their limitations. Unfortunately, it was shown that standard GNNs are…