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Computation of optimal transport on discrete metric measure spaces
No full textErbar M, Rumpf M, Schmitzer B, Simon S. Computation of optimal transport on discrete metric measure spaces. Numer. Math. 2020;144(1):157-200
Going Beyond Counting First Authors in Author Co-citation Analysis
Full text linkThe present study examines one of the fundamental aspects of author co-citation analysis (ACA) - the way co-citation
counts are defined. Co-citation counting provides the data on which all subsequent statistical analyses and mappings
are based, and we compare ACA results based on two different types of co-citation counting - the traditional type that
only counts the first one among a cited work's authors on the one hand and a non-traditional type that takes into
account the first 5 authors of a cited work on the other hand. Results indicate that the picture produced through this non-traditional author co-citation counting contains more coherent author groups and is therefore considerably clearer. However, this picture represents fewer specialties in the research field being studied than that produced through the traditional first-author co-citation counting when the same number of top-ranked authors is selected and analyzed. Reasons for these effects are discussed
Gradient Flows, Metastability and Interacting Particle Systems
Full text linkMany stochastic models exhibit a phenomenon called metastability. The first goal of this thesis is to study this phenomenon for certain classes of interacting particle systems. The second goal of this thesis is the following. Many models that are expected to exhibit metastable behaviour consist of a large number of particles. Thus, their dynamics takes place in a high-dimensional configuration space. It is then a typical idea to describe the system on the macroscopic level by introducing a macroscopic order parameter. In the case of high-dimensional diffusion systems, the empirical distribution turns out to be a suitable order parameter. Hence, the macroscopic level is given by the infinite-dimensional space of probability measures. Therefore, in order to study the macroscopic behaviour, it is useful to have the structure of a Riemannian manifold on the space of probability measure. It is known that the so-called Wasserstein formalism provides such a structure. The second goal of this thesis is to extend this Wasserstein formalism to a certain class of diffusion equations, and to use this formalism to build a rigorous bridge between the microscopic and the macroscopic level in the case of local mean-field interacting diffusions.
The outline of this thesis is as follows.
In Chapter 2 we study the metastable behaviour of three modifications of the standard, two-dimensional Ising model. The first model is an anisotropic version of the Ising model, where the interaction energy takes different values on vertical and horizontal bonds. The second model adds next-nearest-neighbour attraction to the standard Ising model. In the third model, the magnetic field is assumed to have different alternating signs on even and on odd rows.
In Chapter 3 we first establish a gradient flow representation for evolution equations that depend on a non-evolving parameter. These equations are connected to a local mean-field interacting spin system. We then use the gradient flow representation to prove a large deviation principle and a law of large numbers for the empirical process associated to this system. This is done by using the so-called Fathi-Sandier-Serfaty approach.
In Chapter 4 we consider a system of N mean-field interacting diffusions that are driven by a single-site potential of the form z↦z^4/4-z^2/2. The strength of the noise is measured by ε>0, and the strength of the interaction by J>1. Choosing the empirical mean, P, as the macroscopic order parameter, we show that the resulting macroscopic Hamiltonian admits two global minima, one at -m^*0. We are interested in the transition time to the hyperplane P^{-1}(m^*), when the initial configuration is close to P^{-1}(-m^*). The main result is a formula for this transition time, which is reminiscent of the celebrated Eyring-Kramers formula up to a multiplicative error term that tends to 1 as N→∞ and ε↘0. We also provide some estimates on this transition time in the case ε=1 and for a large class of single-site potentials.
In Chapter 5 we again consider the system of Chapter 4 in the case ε=1 and for a large class of single-site potentials. This time, instead of the empirical mean, we choose the empirical distribution as the order parameter. We then prove some results about the ergodicity and the basins of attraction in the macroscopic energy landscape
Optimal Transport for Measure and Image Interpolation and for Information Design
Full text linkIn this work, we exploit the versatility and stability of optimal-transport–based methods to address challenges arising in computer vision, machine learning, and financial mathematics.
Chapter 1 introduces a practical framework for spline interpolation over probability measures, utilizing the Riemannian geometry of Wasserstein space. The model defines distributional splines, continuous-time trajectories of distributions that balance smoothness and transport efficiency. We prove existence of minimizers and Γ-convergence for selected discretizations. An efficient Nesterov-accelerated solver is presented, which scales to practical problem sizes and supports data modalities such as images and latent embeddings. This framework extends traditional machine learning techniques to data represented as probability distributions, including textures, single-cell genomics, and stochastic processes. We evaluate the model on three tasks: (i) generative texture video synthesis from a few exemplar frames, producing temporally coherent textures; (ii) latent-space interpolation with variational autoencoders (VAEs), yielding smooth, controllable interpolation and extrapolation paths; and (iii) Wasserstein regression with distribution-valued responses. Across these applications, the model demonstrates flexibility, stable optimization, and high qualitative performance.
Chapter 2 generalizes this framework to unbalanced distributions, where the total mass can vary over time. We define a spline objective that combines three key components: (i) a diffeomorphic flow for geometric deformation, (ii) optimal transport for mass displacement, and (iii) a penalized source term to model mass creation and absorption. The resulting model jointly captures transport, deformation, and mass variation. Unlike the balanced case, this formulation is not Riemannian-consistent, but rather a pragmatic extension designed to handle scenarios with varying sample sizes and structural changes, common in computer vision and machine learning tasks.
Chapter 3 applies optimal transport to a class of information design problems, focusing on learning optimal information policies. Using entropy-regularized optimal transport, solved via the Sinkhorn algorithm, we obtain scalable, numerically stable solutions. The entropy term introduces a temperature parameter that smooths the objective, improving conditioning and enabling fast matrix-vector-product iterations. As the temperature parameter ε approaches zero, the regularized solution converges to the original, unregularized problem. We demonstrate the approach through Bayesian persuasion in the context of the monopolist's problem, where strategic disclosure policies are learned to optimize decisions. The method shows that optimal information policies can improve expected utility relative to traditional approaches, even under varying dataset sizes and levels of model misspecification. Practical guidance is provided for deployment, including stochastic gradient descent and model selection routines such as early stopping and validation of ε
Quantum optimal transport for AF-C*-algebras
Full text linkWe introduce quantum optimal transport of states on tracial AF-C*-algebras to study non-spatial transport of quantum information, and view it as the pointwise case of a general parametrised one. We define quantum optimal transport distances as dynamic transport distances in a tracial but non-ergodic and infinite-dimensional quantum setting, called AF-C*-setting. We further extend foundational results of Carlen and Maas to the AF-C*-setting and develop a theory of quantum optimal transport yielding non-spatial lower Ricci bounds suitable for meaningful geometric analysis. Essential for our discussion is a coarse graining process arising from the underlying metric geometry as encoding scheme of the given tracial AF-C*-algebra. In the logarithmic mean setting, we apply the coarse graining process to show equivalence of the EVI_λ-gradient flow property for quantum relative entropy, its strong geodesic λ-convexity, a, possibly infinite-dimensional, Bakry-Émery condition, and a Hessian lower bound condition. We then define lower Ricci bounds of our quantum gradients using any one of said equivalent conditions, give sufficient conditions for lower Ricci bounds of direct sum quantum gradients and, assuming lower Ricci bounds, derive functional inequalities HWI_λ, MLSI_λ and TW_λ in the AF-C*-setting alongside their chain of implications. Fundamental example classes give quantum optimal transport of normal states on hyperfinite factors of type I and II with both non-negative and strictly positive lower Ricci bounds. An application is given by first and second quantisation of spectral triples
Variations on the Author
Full text link“Variations on the Author” discusses two of Eduardo Coutinho’s recent films (Um Dia na Vida, from 2010, and Últimas Conversas, posthumously released in 2015) and their contribution to the general question of documentary authorship. The director’s filmography is characterized by a consistent yet self-effacing form of authorial self-inscription: Coutinho often features as an interviewer that rather than express opinions propels discourses; an interviewer that is good at listening. This mode of self-inscription characterizes him as an author who is not expressive but who is nonetheless markedly present on the screen. In Um Dia na Vida, however, Coutinho is completely absent form the image, while Últimas Conversas, on the contrary, includes a confessional prologue that moves the director from the margins to the center of his films. This article examines the ways in which these works stand out in the filmography of a director who offers new insights into the notion of cinematic authorship
Appropriate Similarity Measures for Author Cocitation Analysis
Full text linkWe provide a number of new insights into the methodological discussion about author cocitation analysis. We first argue that the use of the Pearson correlation for measuring the similarity between authors’ cocitation profiles is not very satisfactory. We then discuss what kind of similarity measures may be used as an alternative to the Pearson correlation. We consider three similarity measures in particular. One is the well-known cosine. The other two similarity measures have not been used before in the bibliometric literature. Finally, we show by means of an example that our findings have a high practical relevance.information science;Pearson correlation;cosine;similarity measure;author cocitation analysis
Heat flow aspects of synthetic Ricci bounds in the extended Kato class
Full text linkThis thesis studies heat flows acting on different objects on possibly singular spaces that admit synthetic lower Ricci curvature bounds by constants, functions, or signed measures. Geometric properties of such spaces and probabilistic features of diffusion processes on these are related to functional inequalities for the involved semigroups. Moreover, heat flow methods are used to set up a second order calculus in the general presence of such measure-valued lower Ricci bounds.
First, for a given RCD space, we prove the equivalence of the following synthetic characterizations (with respect to a given lower semicontinuous function k) of the "Ricci curvature at every point being bounded from below by k": geodesic semiconvexity of the relative entropy, the evolution variational inequality, Bochner’s inequality, gradient bounds for the functional heat flow, transport estimates, and the pathwise coupling property.
Second, on arbitrary weighted Riemannian manifolds, we prove the equivalence of the previous pathwise coupling property with respect to k and pointwise lower boundedness of the Bakry–Émery Ricci tensor by k, only assuming continuity of k. Under an additional exponential integrability condition on k, which holds if k is in the functional Kato class of the weighted manifold, we prove conservativeness and Bismut–Elworthy–Li’s derivative formula.
Third, we extend the second order calculus for RCD spaces from Gigli to Dirichlet spaces which are tamed by a signed extended Kato class measure in the sense of Erbar, Rigoni, Sturm and Tamanini. Inter alia, nonsmooth analogues of Hessians, covariant and exterior derivatives, and the Ricci curvature are defined. Employing these objects, in turn, we define heat flows on 1-forms and vector fields and, along with their basic properties, prove domination of the latter by certain semigroups acting on functions.
Fourth, again in the setting of RCD spaces, we obtain improved functional inequalities and regularization properties of the heat flow on 1-forms. The spectrum of its generator, the Hodge Laplacian, is studied as well. Finally, we construct a heat kernel for this heat flow and prove Gaussian upper bounds on its pointwise operator norm
Dispelling the Myths Behind First-author Citation Counts
Full text linkWe conducted a full-scale evaluative citation analysis study of scholars in the XML research field to explore just how different from each other author rankings resulting from different citation counting methods actually are, and to demonstrate the capability of emerging data and tools on the Web in supporting more realistic citation counting methods. Our results contest some common arguments for the continued
use of first-author citation counts in the evaluation of scholars, such as high correlations between author rankings by first-author citation counts and other citation
counting methods, and high costs of using more realistic citation counting methods that are not well-supported by the ISI databases. It is argued that increasingly available digital full text research papers make it possible for citation analysis studies to go beyond what the ISI databases have directly supported and to employ more
sophisticated methods
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