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Jürgen Jost

Publications and source records attributed to Jürgen Jost.

At least 127 records · Page 7Linked to original sources
arXiv2016

Ergodicity of scalar stochastic differential equations with Hölder continuous coefficients

It is well-known that for a one dimensional stochastic differential equation driven by Brownian noise, with coefficient functions satisfying the assumptions of the Yamada-Watanabe theorem \cite{yamada1,yamada2} and the Feller test for explosions \cite{feller51,feller54}, there exists a unique stationary distribution with respect to the Markov semigroup of transition probabilities. We consider systems on a restricted domain $D$ of the phase space $\mathbb{R}$ and study the rate of convergence to the stationary distribution. Using a geometrical approach that uses the so called {\it free energy function} on the density function space, we prove that the density functions, which are solutions of the Fokker-Planck equation, converge to the stationary density function exponentially under the Kullback-Leibler {divergence}, thus also in the total variation norm. The results show that there is a relation between the Bakry-Emery curvature dimension condition and the dissipativity condition of the transformed system under the Fisher-Lamperti transformation. Several applications are discussed, including the Cox-Ingersoll-Ross model and the Ait-Sahalia model in finance and the Wright-Fisher model in population genetics.

math.PR↗
arXiv2016

Forman-Ricci flow for change detection in large dynamic data sets

We present a viable solution to the challenging question of change detection in complex networks inferred from large dynamic data sets. Building on Forman's discretization of the classical notion of Ricci curvature, we introduce a novel geometric method to characterize different types of real-world networks with an emphasis on peer-to-peer networks. Furthermore we adapt the classical Ricci flow that already proved to be a powerful tool in image processing and graphics, to the case of undirected and weighted networks. The application of the proposed method on peer-to-peer networks yields insights into topological properties and the structure of their underlying data.

cs.SI↗
arXiv2016

Forman curvature for complex networks

We adapt Forman's discretization of Ricci curvature to the case of undirected networks, both weighted and unweighted, and investigate the measure in a variety of model and real-world networks. We find that most nodes and edges in model and real networks have a negative curvature. Furthermore, the distribution of Forman curvature of nodes and edges is narrow in random and small-world networks, while the distribution is broad in scale-free and real-world networks. In most networks, Forman curvature is found to display significant negative correlation with degree and centrality measures. However, Forman curvature is uncorrelated with clustering coefficient in most networks. Importantly, we find that both model and real networks are vulnerable to targeted deletion of nodes with highly negative Forman curvature. Our results suggest that Forman curvature can be employed to gain novel insights on the organization of complex networks.

q-bio.MN↗
arXiv2015

A hierarchical extension scheme for solutions of the Wright-Fisher model

We develop a global and hierarchical scheme for the forward Kolmogorov (Fokker-Planck) equation of the diffusion approximation of the Wright-Fisher model of population genetics. That model describes the random genetic drift of several alleles at the same locus in a population. The key of our scheme is to connect the solutions before and after the loss of an allele. Whereas in an approach via stochastic processes or partial differential equations, such a loss of an allele leads to a boundary singularity, from a biological or geometric perspective, this is a natural process that can be analyzed in detail. Our method depends on evolution equations for the moments of the process and a careful analysis of the boundary flux.

math.AP↗
arXiv2015

Super Riemann surfaces, metrics, and gravitinos

The underlying even manifold of a super Riemann surface is a Riemann surface with a spinor valued differential form called gravitino. Consequently infinitesimal deformations of super Riemann surfaces are certain infinitesimal deformations of the Riemann surface and the gravitino. Furthermore the action functional of non-linear super symmetric sigma models, the action functional underlying string theory, can be obtained from a geometric action functional on super Riemann surfaces. All invariances of the super symmetric action functional are explained in super geometric terms and the action functional is a functional on the moduli space of super Riemann surfaces.

math-ph↗
arXiv2015

Representation of Markov chains by random maps: existence and regularity conditions

We systematically investigate the problem of representing Markov chains by families of random maps, and which regularity of these maps can be achieved depending on the properties of the probability measures. Our key idea is to use techniques from optimal transport to select optimal such maps. Optimal transport theory also tells us how convexity properties of the supports of the measures translate into regularity properties of the maps via Legendre transforms. Thus, from this scheme, we cannot only deduce the representation by measurable random maps, but we can also obtain conditions for the representation by continuous random maps. Finally, we present conditions for the representation of Markov chain by random diffeomorphisms.

math.DS↗
arXiv2014

The uniqueness of hierarchically extended backward solutions of the Wright-Fisher model

The diffusion approximation of the Wright-Fisher model of population genetics leads to partial differentiable equations, the so-called Kolmogorov equations, with an operator that degenerates at the boundary. Standard tools do not apply, and in fact, solutions lack regularity properties. In this paper, we develop a regularising blow-up scheme for a certain class of solutions of the backward Kolmogorov equation, the iteratively extended global solutions presented in \cite{THJ5}, and establish their uniqueness. As the model describes the random genetic drift of several alleles at the same locus from a backward perspective, the singularities result from the loss of an allele. While in an analytical approach, this causes substantial difficulties, from a biological or geometric perspective, this is a natural process that can be analyzed in detail. The presented scheme regularises the solution via a tailored successive transformation of the domain.

math.AP↗
arXiv2014

Spectral classes of regular, random, and empirical graphs

We define a (pseudo-)distance between graphs based on the spectrum of the normalized Laplacian, which is easy to compute or to estimate numerically. It can therefore serve as a rough classification of large empirical graphs into families that share the same asymptotic behavior of the spectrum so that the distance of two graphs from the same family is bounded by $\mathcal{O}(1/n)$ in terms of size $n$ of their vertex sets. Numerical experiments demonstrate that the spectral distance provides a practically useful measure of graph dissimilarity.

math.SP↗
arXiv2014

A hierarchical extension scheme for backward solutions of the Wright-Fisher model

We develop an iterative global solution scheme for the backward Kolmogorov equation of the diffusion approximation of the Wright-Fisher model of population genetics. That model describes the random genetic drift of several alleles at the same locus in a population from a backward perspective. The key of our scheme is to connect the solutions before and after the loss of an allele. Whereas in an approach via stochastic processes or partial differential equations, such a loss of an allele leads to a boundary singularity, from a biological or geometric perspective, this is a natural process that can be analyzed in detail. A clarification of the role of the boundary resolves certain uniqueness issues and enlucidates the construction of hierarchical solutions.

math.AP↗
arXiv2014

Reconsidering unique information: Towards a multivariate information decomposition

The information that two random variables $Y$, $Z$ contain about a third random variable $X$ can have aspects of shared information (contained in both $Y$ and $Z$), of complementary information (only available from $(Y,Z)$ together) and of unique information (contained exclusively in either $Y$ or $Z$). Here, we study measures $\widetilde{SI}$ of shared, $\widetilde{UI}$ unique and $\widetilde{CI}$ complementary information introduced by Bertschinger et al., which are motivated from a decision theoretic perspective. We find that in most cases the intuitive rule that more variables contain more information applies, with the exception that $\widetilde{SI}$ and $\widetilde{CI}$ information are not monotone in the target variable $X$. Additionally, we show that it is not possible to extend the bivariate information decomposition into $\widetilde{SI}$, $\widetilde{UI}$ and $\widetilde{CI}$ to a non-negative decomposition on the partial information lattice of Williams and Beer. Nevertheless, the quantities $\widetilde{UI}$, $\widetilde{SI}$ and $\widetilde{CI}$ have a well-defined interpretation, even in the multivariate setting.

cs.IT↗
arXiv2014

A notion of nonpositive curvature for general metric spaces

We introduce a new definition of nonpositive curvature in metric spaces and study its relationship to the existing notions of nonpositive curvature in comparison geometry. The main feature of our definition is that it applies to all metric spaces and does not rely on geodesics. Moreover, a scaled and a relaxed version of our definition are appropriate in discrete metric spaces, and are believed to be of interest in geometric data analysis.

math.MG↗
arXiv2014

Natural statistics of binaural sounds

Binaural sound localization is usually considered a discrimination task, where interaural time (ITD) and level (ILD) disparities at pure frequency channels are utilized to identify a position of a sound source. In natural conditions binaural circuits are exposed to a stimulation by sound waves originating from multiple, often moving and overlapping sources. Therefore statistics of binaural cues depend on acoustic properties and the spatial configuration of the environment. In order to process binaural sounds efficiently, the auditory system should be adapted to naturally encountered cue distributions. Statistics of cues encountered naturally and their dependence on the physical properties of an auditory scene have not been studied before. Here, we performed binaural recordings of three auditory scenes with varying spatial properties. We have analyzed empirical cue distributions from each scene by fitting them with parametric probability density functions which allowed for an easy comparison of different scenes. Higher order statistics of binaural waveforms were analyzed by performing Independent Component Analysis (ICA) and studying properties of learned basis functions. Obtained results can be related to known neuronal mechanisms and suggest how binaural hearing can be understood in terms of adaptation to the natural signal statistics.

q-bio.NC↗
arXiv2014

Quantifying unique information

We propose new measures of shared information, unique information and synergistic information that can be used to decompose the multi-information of a pair of random variables $(Y,Z)$ with a third random variable $X$. Our measures are motivated by an operational idea of unique information which suggests that shared information and unique information should depend only on the pair marginal distributions of $(X,Y)$ and $(X,Z)$. Although this invariance property has not been studied before, it is satisfied by other proposed measures of shared information. The invariance property does not uniquely determine our new measures, but it implies that the functions that we define are bounds to any other measures satisfying the same invariance property. We study properties of our measures and compare them to other candidate measures.

cs.IT↗
arXiv2013

Information geometry and sufficient statistics

Information geometry provides a geometric approach to families of statistical models. The key geometric structures are the Fisher quadratic form and the Amari-Chentsov tensor. In statistics, the notion of sufficient statistic expresses the criterion for passing from one model to another without loss of information. This leads to the question how the geometric structures behave under such sufficient statistics. While this is well studied in the finite sample size case, in the infinite case, we encounter technical problems concerning the appropriate topologies. Here, we introduce notions of parametrized measure models and tensor fields on them that exhibit the right behavior under statistical transformations. Within this framework, we can then handle the topological issues and show that the Fisher metric and the Amari-Chentsov tensor on statistical models in the class of symmetric 2-tensor fields and 3-tensor fields can be uniquely (up to a constant) characterized by their invariance under sufficient statistics, thereby achieving a full generalization of the original result of Chentsov to infinite sample sizes. More generally, we decompose Markov morphisms between statistical models in terms of statistics. In particular, a monotonicity result for the Fisher information naturally follows.

math.ST↗
arXiv2013

Ollivier's Ricci curvature, local clustering and curvature dimension inequalities on graphs

In this paper, we explore the relationship between one of the most elementary and important properties of graphs, the presence and relative frequency of triangles, and a combinatorial notion of Ricci curvature. We employ a definition of generalized Ricci curvature proposed by Ollivier in a general framework of Markov processes and metric spaces and applied in graph theory by Lin-Yau. In analogy with curvature notions in Riemannian geometry, we interpret this Ricci curvature as a control on the amount of overlap between neighborhoods of two neighboring vertices. It is therefore naturally related to the presence of triangles containing those vertices, or more precisely, the local clustering coefficient, that is, the relative proportion of connected neighbors among all the neighbors of a vertex. This suggests to derive lower Ricci curvature bounds on graphs in terms of such local clustering coefficients. We also study curvature dimension inequalities on graphs, building upon previous work of several authors.

math.CO↗
arXiv2013

Ollivier-Ricci curvature and the spectrum of the normalized graph Laplace operator

We prove the following estimate for the spectrum of the normalized Laplace operator $Δ$ on a finite graph $G$, \begin{equation*}1- (1- k[t])^{\frac{1}{t}}\leq λ_1 \leq \cdots \leq λ_{N-1}\leq 1+ (1- k[t])^{\frac{1}{t}}, \,\forall \,\,\text{integers}\,\, t\geq 1. \end{equation*} Here $k[t]$ is a lower bound for the Ollivier-Ricci curvature on the neighborhood graph $G[t]$, which was introduced by Bauer-Jost. In particular, when $t=1$ this is Ollivier's estimates $k\leq λ_1\leq \ldots \leq λ_{N-1}\leq 2-k$. For sufficiently large $t$ we show that, unless $G$ is bipartite, our estimates for $λ_1$ and $λ_{N-1}$ are always nontrivial and improve Ollivier's estimates for all graphs with $k\leq 0$. By definition neighborhood graphs are weighted graphs which may have loops. To understand the Ollivier-Ricci curvature on neighborhood graphs, we generalize a sharp estimate of the Ricci curvature given by Jost-Liu to weighted graphs with loops and relate it to the relative local frequency of triangles and loops.

math.CO↗
arXiv2013

Geometric analysis aspects of infinite semiplanar graphs with nonnegative curvature

In the present paper, we apply Alexandrov geometry methods to study geometric analysis aspects of infinite semiplanar graphs with nonnegative combinatorial curvature in the sense of Higuchi. We obtain the metric classification of these graphs and construct the graphs embedded in the projective plane minus one point. Moreover, we show the volume doubling property and the Poincaré inequality on such graphs. The quadratic volume growth of these graphs implies the parabolicity. In addition, we prove the polynomial growth harmonic function theorem analogous to the case of Riemannian manifolds.

math.MG↗