arXiv subjects

Jürgen Jost

Publications and source records attributed to Jürgen Jost.

At least 91 records · Page 5Linked to original sources

arXiv2019

Manifolds of classical probability distributions and quantum density operators in infinite dimensions

The manifold structure of subsets of classical probability distributions and quantum density operators in infinite dimensions is investigated in the context of $C^{*}$-algebras and actions of Banach-Lie groups. Specificaly, classical probability distributions and quantum density operators may be both described as states (in the functional analytic sense) on a given $C^{*}$-algebra $\mathscr{A}$ which is Abelian for Classical states, and non-Abelian for Quantum states. In this contribution, the space of states $\mathscr{S}$ of a possibly infinite-dimensional, unital $C^{*}$-algebra $\mathscr{A}$ is partitioned into the disjoint union of the orbits of an action of the group $\mathscr{G}$ of invertible elements of $\mathscr{A}$. Then, we prove that the orbits through density operators on an infinite-dimensional, separable Hilbert space $\mathcal{H}$ are smooth, homogeneous Banach manifolds of $\mathscr{G}=\mathcal{GL}(\mathcal{H})$, and, when $\mathscr{A}$ admits a faithful tracial state $τ$ like it happens in the Classical case when we consider probability distributions with full support, we prove that the orbit through $τ$ is a smooth, homogeneous Banach manifold for $\mathscr{G}$.

math-ph↗

arXiv2019

A Simple Differential Geometry for Networks and its Generalizations

Based on two classical notions of curvature for curves in general metric spaces, namely the Menger and Haantjes curvatures, we introduce new definitions of sectional, Ricci and scalar curvature for networks and their higher dimensional counterparts. These new types of curvature, that apply to weighted and unweighted, directed or undirected networks, are far more intuitive and easier to compute, than other network curvatures. In particular, the proposed curvatures based on the interpretation of Haantjes definition as geodesic curvature, and derived via a fitting discrete Gauss-Bonnet Theorem, are quite flexible. We also propose even simpler and more intuitive substitutes of the Haantjes curvature, that allow for even faster and easier computations in large-scale networks.

math.MG↗

arXiv2019

Decomposing Textual Information For Style Transfer

This paper focuses on latent representations that could effectively decompose different aspects of textual information. Using a framework of style transfer for texts, we propose several empirical methods to assess information decomposition quality. We validate these methods with several state-of-the-art textual style transfer methods. Higher quality of information decomposition corresponds to higher performance in terms of bilingual evaluation understudy (BLEU) between output and human-written reformulations.

cs.CL↗

arXiv2019

Energy quantization for a singular super-Liouville boundary value problem

In this paper, we develop the blow-up analysis and establish the energy quantization for solutions to super-Liouville type equations on Riemann surfaces with conical singularities at the boundary. In other problems in geometric analysis, the blow-up analysis usually strongly utilizes conformal invariance, which yields a Noether current from which strong estimates can be derived. Here, however, the conical singularities destroy conformal invariance. Therefore, we develop another, more general, method that uses the vanishing of the Pohozaev constant for such solutions to deduce the removability of boundary singularities.

math.DG↗

arXiv2019

Synaptic Time-Dependent Plasticity Leads to Efficient Coding of Predictions

Latency reduction of postsynaptic spikes is a well-known effect of Synaptic Time-Dependent Plasticity. We expand this notion for long postsynaptic spike trains, showing that, for a fixed input spike train, STDP reduces the number of postsynaptic spikes and concentrates the remaining ones. Then we study the consequences of this phenomena in terms of coding, finding that this mechanism improves the neural code by increasing the signal-to-noise ratio and lowering the metabolic costs of frequent stimuli. Finally, we illustrate that the reduction of postsynaptic latencies can lead to the emergence of predictions.

q-bio.NC↗

arXiv2019

Ollivier Ricci Curvature of Directed Hypergraphs

We develop a definition of Ricci curvature on directed hypergraphs and explore the consequences of that definition. The definition generalizes Ollivier's definition for graphs. It involves a carefully designed optimal transport problem between sets of vertices.

cs.DM↗

arXiv2019

Evolving neural networks to follow trajectories of arbitrary complexity

Many experiments have been performed that use evolutionary algorithms for learning the topology and connection weights of a neural network that controls a robot or virtual agent. These experiments are not only performed to better understand basic biological principles, but also with the hope that with further progress of the methods, they will become competitive for automatically creating robot behaviors of interest. However, current methods are limited with respect to the (Kolmogorov) complexity of evolved behavior. Using the evolution of robot trajectories as an example, we show that by adding four features, namely (1) freezing of previously evolved structure, (2) temporal scaffolding, (3) a homogeneous transfer function for output nodes, and (4) mutations that create new pathways to outputs, to standard methods for the evolution of neural networks, we can achieve an approximately linear growth of the complexity of behavior over thousands of generations. Overall, evolved complexity is up to two orders of magnitude over that achieved by standard methods in the experiments reported here, with the major limiting factor for further growth being the available run time. Thus, the set of methods proposed here promises to be a useful addition to various current neuroevolution methods.

cs.NE↗

arXiv2019

Topology and curvature of metric spaces

We develop a new concept of non-positive curvature for metric spaces, based on intersection patterns of closed balls. In contrast to the synthetic approaches of Alexandrov and Buesemann, our concept also applies to metric spaces that might be discrete. The natural comparison spaces that emerge from our discussion are no longer Euclidean spaces, but rather tripod spaces. These tripod spaces include the hyperconvex spaces which have trivial Cech homology. This suggests a link of our geometrical method to the topological method of persistent homology in topological data analysis. We also investigate the geometry of general tripod spaces.

math.MG↗

arXiv2019

Curvature-based Methods for Brain Network Analysis

The human brain forms functional networks on all spatial scales. Modern fMRI scanners allow to resolve functional brain data in high resolutions, allowing to study large-scale networks that relate to cognitive processes. The analysis of such networks forms a cornerstone of experimental neuroscience. Due to the immense size and complexity of the underlying data sets, efficient evaluation and visualization remain a challenge for data analysis. In this study, we combine recent advances in experimental neuroscience and applied mathematics to perform a mathematical characterization of complex networks constructed from fMRI data. We use task-related edge densities [Lohmann et al., 2016] for constructing networks of task-related changes in synchronization. This construction captures the dynamic formation of patterns of neuronal activity and therefore represents efficiently the connectivity structure between brain regions. Using geometric methods that utilize Forman-Ricci curvature as an edge-based network characteristic [Weber et al., 2017], we perform a mathematical analysis of the resulting complex networks. We motivate the use of edge-based characteristics to evaluate the network structure with geometric methods. The geometric features could aid in understanding the connectivity and interplay of brain regions in cognitive processes.

q-bio.NC↗

arXiv2019

The Geometry of Maximum Principles and a Bernstein Theorem in Codimension 2

Moser's Bernstein theorem \cite{moser61} says that an entire minimal graph of codimension 1 with bounded slope must be a hyperplane. An analogous result for arbitrary codimension is not true, by an example of Lawson-Osserman. Here, we show that Moser's theorem nevertheless extends to codimension 2, i.e., a minimal $p$-submanifold in $R^{p+2}$, which is the graph of a smooth function defined on the entire $R^p$ with bounded slope, must be a $p$-plane. Our method depends on convexity properties of Grassmannians which come into play as the targets of the (harmonic) Gauss maps of our minimal submanifolds. In fact, we develop a general method to construct subsets of complete Riemannian manifolds that cannot contain images of non-constant harmonic maps from compact manifolds. When applied to Grassmannians, for codimension 2, it yields an appropriate domain that cannot contain nontrivial images of such harmonic maps. Our conclusion will then be reached by an application of Allard's theorem to handle the issue that the minimal submanifolds in question are not compact.

math.DG↗

arXiv2019

$α$-Dirac-harmonic maps from closed surfaces

$α$-Dirac-harmonic maps are variations of Dirac-harmonic maps, analogous to $α$-harmonic maps that were introduced by Sacks-Uhlenbeck to attack the existence problem for harmonic maps from surfaces. For $α>1$, the latter are known to satisfy a Palais-Smale condtion, and so, the technique of Sacks-Uhlenbeck consists in constructing $α$-harmonic maps for $α>1$ and then letting $α\to 1$. The extension of this scheme to Dirac-harmonic maps meets with several difficulties, and in this paper, we start attacking those. We first prove the existence of nontrivial perturbed $α$-Dirac-harmonic maps when the target manifold has nonpositive curvature. The regularity theorem then shows that they are actually smooth. By $\varepsilon$-regularity and suitable perturbations, we can then show that such a sequence of perturbed $α$-Dirac-harmonic maps converges to a smooth nontrivial $α$-Dirac-harmonic map.

math.DG↗

arXiv2019

Unique Information and Secret Key Decompositions

The unique information ($UI$) is an information measure that quantifies a deviation from the Blackwell order. We have recently shown that this quantity is an upper bound on the one-way secret key rate. In this paper, we prove a triangle inequality for the $UI$, which implies that the $UI$ is never greater than one of the best known upper bounds on the two-way secret key rate. We conjecture that the $UI$ lower bounds the two-way rate and discuss implications of the conjecture.

cs.IT↗

arXiv2018

Forman-Ricci Curvature for Hypergraphs

In contrast to graph-based models for complex networks, hypergraphs are more general structures going beyond binary relations of graphs. For graphs, statistics gauging different aspects of their structures have been devised and there is undergoing research for devising them for hypergraphs. Forman-Ricci curvature is a statistics for graphs, which is based on Riemannian geometry, and that stresses the relational character of vertices in a network through the analysis of edges rather than vertices. In spite of the different applications of this curvature, it has not yet been formulated for hypergraphs. Here we devise the Forman-Ricci curvature for directed and undirected hypergraphs, where the curvature for graphs is a particular case. We report its upper and lower bounds and the respective bounds for the graph case. The curvature quantifies the trade-off between hyperedge(arc) size and the degree of participation of hyperedge(arc) vertices in other hyperedges(arcs). We calculated the curvature for two large networks: Wikipedia vote network and \emph{Escherichia coli} metabolic network. In the first case the curvature is ruled by hyperedge size, while in the second by hyperedge degree. We found that the number of users involved in Wikipedia elections goes hand-in-hand with the participation of experienced users. The curvature values of the metabolic network allowed detecting redundant and bottle neck reactions. It is found that ADP phosphorilation is the metabolic bottle neck reaction but that the reverse reaction is not that central for the metabolism.

cs.DM↗

arXiv2018

Discrete Ricci curvatures for directed networks

A goal in network science is the geometrical characterization of complex networks. In this direction, we have recently introduced Forman's discretization of Ricci curvature to the realm of undirected networks. Investigation of this edge-centric network measure, Forman-Ricci curvature, in diverse model and real-world undirected networks revealed that the curvature measure captures several aspects of the organization of undirected complex networks. However, many important real-world networks are inherently directed in nature, and the definition of the Forman-Ricci curvature for undirected networks is unsuitable for the analysis of such directed networks. Hence, we here extend the Forman-Ricci curvature for undirected networks to the case of directed networks. The simple mathematical formula for the Forman-Ricci curvature of a directed edge elegantly incorporates vertex weights, edge weights and edge direction. Furthermore we have compared the Forman-Ricci curvature with the adaptation to directed networks of another discrete notion of Ricci curvature, namely, the well established Ollivier-Ricci curvature. However, the two above-mentioned curvature measures do not account for higher-order correlations between vertices. To this end, we adjusted Forman's original definition of Ricci curvature to account for directed simplicial complexes and also explored the potential of this new, augmented type of Forman-Ricci curvature, in directed complex networks.

math.MG↗

arXiv2018

Hypergraph Laplace Operators for Chemical Reaction Networks

We generalize the normalized combinatorial Laplace operator for graphs by defining two Laplace operators for hypergraphs that can be useful in the study of chemical reaction networks. We also investigate some properties of their spectra.

math.SP↗

arXiv2018

Comparative analysis of two discretizations of Ricci curvature for complex networks

We have performed an empirical comparison of two distinct notions of discrete Ricci curvature for graphs or networks, namely, the Forman-Ricci curvature and Ollivier-Ricci curvature. Importantly, these two discretizations of the Ricci curvature were developed based on different properties of the classical smooth notion, and thus, the two notions shed light on different aspects of network structure and behavior. Nevertheless, our extensive computational analysis in a wide range of both model and real-world networks shows that the two discretizations of Ricci curvature are highly correlated in many networks. Moreover, we show that if one considers the augmented Forman-Ricci curvature which also accounts for the two-dimensional simplicial complexes arising in graphs, the observed correlation between the two discretizations is even higher, especially, in real networks. Besides the potential theoretical implications of these observations, the close relationship between the two discretizations has practical implications whereby Forman-Ricci curvature can be employed in place of Ollivier-Ricci curvature for faster computation in larger real-world networks whenever coarse analysis suffices.

math.DG↗

arXiv2018

Partial Regularity for a Nonlinear Sigma Model with Gravitino in Higher Dimensions

We study the regularity problem of the nonlinear sigma model with gravitino fields in higher dimensions. After setting up the geometric model, we derive the Euler--Lagrange equations and consider the regularity of weak solutions defined in suitable Sobolev spaces. We show that any weak solution is actually smooth under some smallness assumption for certain Morrey norms. By assuming some higher integrability of the vector spinor, we can show a partial regularity result for stationary solutions, provided the gravitino is critical, which means that the corresponding supercurrent vanishes. Moreover, in dimension less than 6, partial regularity holds for stationary solutions with respect to general gravitino fields.

math.DG↗

arXiv2017

On extractable shared information

We consider the problem of quantifying the information shared by a pair of random variables $X_{1},X_{2}$ about another variable $S$. We propose a new measure of shared information, called extractable shared information, that is left monotonic; that is, the information shared about $S$ is bounded from below by the information shared about $f(S)$ for any function $f$. We show that our measure leads to a new nonnegative decomposition of the mutual information $I(S;X_1X_2)$ into shared, complementary and unique components. We study properties of this decomposition and show that a left monotonic shared information is not compatible with a Blackwell interpretation of unique information. We also discuss whether it is possible to have a decomposition in which both shared and unique information are left monotonic.

cs.IT↗