181,436 research outputs found
Symmetries of Nonlinear PDEs on Metric Graphs and Branched Networks
This Special Issue focuses on recent progress in a new area of mathematical physics and applied analysis, namely, on nonlinear partial differential equations on metric graphs and branched networks. Graphs represent a system of edges connected at one or more branching points (vertices). The connection rule determines the graph topology. When the edges can be assigned a length and the wave functions on the edges are defined in metric spaces, the graph is called a metric graph. Evolution equations on metric graphs have attracted much attention as effective tools for the modeling of particle and wave dynamics in branched structures and networks. Since branched structures and networks appear in different areas of contemporary physics with many applications in electronics, biology, material science, and nanotechnology, the development of effective modeling tools is important for the many practical problems arising in these areas. The list of important problems includes searches for standing waves, exploring of their properties (e.g., stability and asymptotic behavior), and scattering dynamics. This Special Issue is a representative sample of the works devoted to the solutions of these and other problems
On Graph Stream Clustering with Side Information
Graph clustering becomes an important problem due to emerging applications
involving the web, social networks and bio-informatics. Recently, many such
applications generate data in the form of streams. Clustering massive, dynamic
graph streams is significantly challenging because of the complex structures of
graphs and computational difficulties of continuous data. Meanwhile, a large
volume of side information is associated with graphs, which can be of various
types. The examples include the properties of users in social network
activities, the meta attributes associated with web click graph streams and the
location information in mobile communication networks. Such attributes contain
extremely useful information and has the potential to improve the clustering
process, but are neglected by most recent graph stream mining techniques. In
this paper, we define a unified distance measure on both link structures and
side attributes for clustering. In addition, we propose a novel optimization
framework DMO, which can dynamically optimize the distance metric and make it
adapt to the newly received stream data. We further introduce a carefully
designed statistics SGS(C) which consume constant storage spaces with the
progression of streams. We demonstrate that the statistics maintained are
sufficient for the clustering process as well as the distance optimization and
can be scalable to massive graphs with side attributes. We will present
experiment results to show the advantages of the approach in graph stream
clustering with both links and side information over the baselines.Comment: Full version of SIAM SDM 2013 pape
A singular Riemannian geometry approach to Deep Neural Networks I. Theoretical foundations
Deep Neural Networks are widely used for solving complex problems in several
scientific areas, such as speech recognition, machine translation, image
analysis. The strategies employed to investigate their theoretical properties
mainly rely on Euclidean geometry, but in the last years new approaches based
on Riemannian geometry have been developed. Motivated by some open problems, we
study a particular sequence of maps between manifolds, with the last manifold
of the sequence equipped with a Riemannian metric. We investigate the
structures induced trough pullbacks on the other manifolds of the sequence and
on some related quotients. In particular, we show that the pullbacks of the
final Riemannian metric to any manifolds of the sequence is a degenerate
Riemannian metric inducing a structure of pseudometric space, we show that the
Kolmogorov quotient of this pseudometric space yields a smooth manifold, which
is the base space of a particular vertical bundle. We investigate the
theoretical properties of the maps of such sequence, eventually we focus on the
case of maps between manifolds implementing neural networks of practical
interest and we present some applications of the geometric framework we
introduced in the first part of the paper
Representability of algebraic topology for biomolecules in machine learning based scoring and virtual screening
This work introduces a number of algebraic topology approaches, such as
multicomponent persistent homology, multi-level persistent homology and
electrostatic persistence for the representation, characterization, and
description of small molecules and biomolecular complexes. Multicomponent
persistent homology retains critical chemical and biological information during
the topological simplification of biomolecular geometric complexity.
Multi-level persistent homology enables a tailored topological description of
inter- and/or intra-molecular interactions of interest. Electrostatic
persistence incorporates partial charge information into topological
invariants. These topological methods are paired with Wasserstein distance to
characterize similarities between molecules and are further integrated with a
variety of machine learning algorithms, including k-nearest neighbors, ensemble
of trees, and deep convolutional neural networks, to manifest their descriptive
and predictive powers for chemical and biological problems. Extensive numerical
experiments involving more than 4,000 protein-ligand complexes from the PDBBind
database and near 100,000 ligands and decoys in the DUD database are performed
to test respectively the scoring power and the virtual screening power of the
proposed topological approaches. It is demonstrated that the present approaches
outperform the modern machine learning based methods in protein-ligand binding
affinity predictions and ligand-decoy discrimination
- …