Related papers: Geometric Aspects of Data-Processing of Markov Cha…
We consider the number of crossings in a graph which is embedded randomly on a convex set of points. We give an estimate to the normal distribution in Kolmogorov distance which implies a convergence rate of order $n^{-1/2}$ for various…
This paper studies the problem of embedding very large information networks into low-dimensional vector spaces, which is useful in many tasks such as visualization, node classification, and link prediction. Most existing graph embedding…
In this chapter, we identify fundamental geometric structures that underlie the problems of sampling, optimisation, inference and adaptive decision-making. Based on this identification, we derive algorithms that exploit these geometric…
Message-passing based approaches form the default backbone of most learning architectures on graph-structured data. However, the rapid progress of modern deep learning architectures in other domains, particularly sequence modeling, raises…
Image feature matching is a fundamental part of many geometric computer vision applications, and using multiple images can improve performance. In this work, we formulate multi-image matching as a graph embedding problem then use a Graph…
This paper considers the problem of embedding directed graphs in Euclidean space while retaining directional information. We model a directed graph as a finite set of observations from a diffusion on a manifold endowed with a vector field.…
It is known that statistical model selection as well as identification of dynamical equations from available data are both very challenging tasks. Physical systems behave according to their underlying dynamical equations which, in turn, can…
We present a geometric perspective on sparse filtrations used in topological data analysis. This new perspective leads to much simpler proofs, while also being more general, applying equally to Rips filtrations and Cech filtrations for any…
Linear logics have been shown to be able to embed both rewriting-based approaches and process calculi in a single, declarative framework. In this paper we are exploring the embedding of double-pushout graph transformations into quantified…
We present an exclusion process based approach for sampling densest $k$-sub-graphs from regular graphs $L$ with connected complement. By interpreting an exclusion process as a Markov chain on a corresponding Token Graph $\mathfrak{L}_k$, we…
By taking the semantic object parsing task as an exemplar application scenario, we propose the Graph Long Short-Term Memory (Graph LSTM) network, which is the generalization of LSTM from sequential data or multi-dimensional data to general…
A fundamental challenge in developing high-impact machine learning technologies is balancing the need to model rich, structured domains with the ability to scale to big data. Many important problem areas are both richly structured and large…
Representing graphs as sets of node embeddings in certain curved Riemannian manifolds has recently gained momentum in machine learning due to their desirable geometric inductive biases, e.g., hierarchical structures benefit from hyperbolic…
The availability of relational data can offer new insights into the functioning of the economy. Nevertheless, modeling the dynamics in network data with multiple types of relationships is still a challenging issue. Stochastic block models…
We survey the progress on the study of symplectic geometry past five decades. The survey focuses on the convexity properties of a moment map, the classification of symplectic actions, the symplectic embedding problems, and the theory of…
Markov logic uses weighted formulas to compactly encode a probability distribution over possible worlds. Despite the use of logical formulas, Markov logic networks (MLNs) can be difficult to interpret, due to the often counter-intuitive…
The Markov gap, defined as the difference between reflected entropy and mutual information, serves as a diagnostic for quantum recoverability and multipartite entanglement. In holographic settings, it admits a geometric interpretation as…
We give new results for problems in computational and statistical machine learning using tools from high-dimensional geometry and probability. We break up our treatment into two parts. In Part I, we focus on computational considerations in…
The stochastic growth-fragmentation model describes the temporal evolution of a structured cell population through a discrete-time and continuous-state Markov chain. The simulations of this stochastic process and its invariant measure are…
Active area of research in AI is the theory of manifold learning and finding lower-dimensional manifold representation on how we can learn geometry from data for providing better quality curated datasets. There are however various issues…