Generalized Gradient Flow Research Articles

Information geometry of dynamics on graphs and hypergraphs

We introduce a new information-geometric structure associated with the dynamics on discrete objects such as graphs and hypergraphs. The presented setup consists of two dually flat structures built on the vertex and edge spaces, respectively. The former is the conventional duality between density and potential, e.g., the probability density and its logarithmic form induced by a convex thermodynamic function. The latter is the duality between flux and force induced by a convex and symmetric dissipation function, which drives the dynamics of the density. These two are connected topologically by the homological algebraic relation induced by the underlying discrete objects. The generalized gradient flow in this doubly dual flat structure is an extension of the gradient flows on Riemannian manifolds, which include Markov jump processes and nonlinear chemical reaction dynamics as well as the natural gradient. The information-geometric projections on this doubly dual flat structure lead to information-geometric extensions of the Helmholtz–Hodge decomposition and the Otto structure in L2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^{2}$$\\end{document}-Wasserstein geometry. The structure can be extended to non-gradient nonequilibrium flows, from which we also obtain the induced dually flat structure on cycle spaces. This abstract but general framework can broaden the applicability of information geometry to various problems of linear and nonlinear dynamics.

A Cahn–Hilliard–Biot system and its generalized gradient flow structure

In this work, we propose a new model for flow through deformable porous media, where the solid material has two phases with distinct material properties. The two phases of the porous material evolve according to a generalized Ginzburg–Landau energy functional, with additional impact from both elastic and fluid effects, and the coupling between flow and deformation is governed by Biot’s theory. This results in a three-way coupled system which can be seen as an extension of the Cahn–Larché equations with the inclusion of a fluid flowing through the medium. The model covers essential coupling terms for several relevant applications, including solid tumor growth, biogrout, and wood growth simulation. Moreover, we show that this coupled set of equations follow a generalized gradient flow framework. This opens a toolbox of analysis and solvers which can be used for further study of the model. Additionally, we provide a numerical example showing the impact of the flow on the solid phase evolution in comparison to the Cahn–Larché system.

Heterogeneous Domain Adaptation by Information Capturing and Distribution Matching.

Heterogeneous domain adaptation (HDA) is a challenging problem because of the different feature representations in the source and target domains. Most HDA methods search for mapping matrices from the source and target domains to discover latent features for learning. However, these methods barely consider the reconstruction error to measure the information loss during the mapping procedure. In this paper, we propose to jointly capture the information and match the source and target domain distributions in the latent feature space. In the learning model, we propose to minimize the reconstruction loss between the original and reconstructed representations to preserve information during transformation and reduce the Maximum Mean Discrepancy between the source and target domains to align their distributions. The resulting minimization problem involves two projection variables with orthogonal constraints that can be solved by the generalized gradient flow method, which can preserve orthogonal constraints in the computational procedure. We conduct extensive experiments on several image classification datasets to demonstrate that the effectiveness and efficiency of the proposed method are better than those of state-of-the-art HDA methods.

String method for generalized gradient flows: computation of rare events in reversible stochastic processes

Rare transitions in stochastic processes can often be rigorously described via an underlying large deviation principle. Recent breakthroughs in the classification of reversible stochastic processes as gradient flows have led to a connection of large deviation principles to a generalized gradient structure. Here, we show that, as a consequence, metastable transitions in these reversible processes can be interpreted as heteroclinic orbits of the generalized gradient flow. This in turn suggests a numerical algorithm to compute the transition trajectories in configuration space efficiently, based on the string method traditionally restricted only to gradient diffusions.

Finite cutoff AdS5 holography and the generalized gradient flow

Recently proposed double trace deformations of large N holographic CFTs in four dimensions define a one parameter family of quantum field theories, which are interpreted in the bulk dual as living on successive finite radius hypersurfaces. The transformation of variables that turns the equation defining the deformation of a four dimensional large N CFT by such operators into the expression for the radial ADM Hamiltonian in the bulk is found.This prescription clarifies the role of various functions of background fields that appear in the flow equation defining the deformed holographic CFT, and also their relationship to the holographic anomaly.The effect of these deformations can also be seen as triggering a generalized gradient flow for the fields of the induced gravity theory obtained from integrating out the fundamental fields of the holographic CFT. The potential for this gradient flow is found to resemble the two derivative effective action previously derived using holographic renormalization.

Q-Heat flow and the gradient flow of the Renyi entropy in the p-Wasserstein space

Based on the idea of a recent paper by Ambrosio–Gigli–Savaré (2014) [5], we show that the L2-gradient flow of the q-Cheeger energy, called q-heat flow, solves a generalized gradient flow problem of the Renyi entropy functional in the p-Wasserstein. For that, a further study of the q-heat flow is presented including a condition for its mass preservation. Under a convexity assumption on the upper gradient, which holds for all q≥2, one gets uniqueness of the gradient flow and the two flows can be identified. Smooth solutions of the q-heat flow are solutions to the parabolic q-Laplace equation, i.e. ∂tft=Δqft.

Gradient structures for the thermomechanics of shape-memory materials

Abstract We investigate the variational structure of a phenomenological model for the coupled thermomechanical behavior of shape-memory polycrystalline materials. The nonisothermal evolution of the medium is reformulated as a generalized gradient flow of the entropy with respect to an entropy-production potential. Based on this reformulation, a semi-implicit time-discretization of the fully coupled thermomechanical problem is presented and proved to be unconditionally stable and convergent. The flexibility and robustness of the numerical method is assessed via both uniaxial and multiaxial computational tests.

Generalized gradient flow and singularities of the Riemannian distance function

Generalized gradient flow and singularities of the Riemannian distance function

Generalized gradient flow equation and its application to super Yang-Mills theory

We generalize the gradient flow equation for field theories with nonlinearly realized symmetry. Applying the formalism to super Yang-Mills theory, we construct a supersymmetric extension of the gradient flow equation. It can be shown that the super gauge symmetry is preserved in the gradient flow. Furthermore, choosing an appropriate modification term to damp the gauge degrees of freedom, we obtain a gradient flow equation which is closed within the Wess-Zumino gauge.

The regularity of the distance function propagates along minimizing geodesics

We consider the distance function from the boundary of an open bounded set Ω⊂Rn associated to a Riemannian metric with C1,1 coefficients. We show that the C1,1 regularity propagates, towards the boundary ∂Ω, along the distance minimizing geodesics. Hence, we show that the cut-locus is invariant with respect to the generalized gradient flow associated to the distance function and that it has the same homotopy type as Ω.

Singular gradient flow of the distance function and homotopy equivalence

Let \(M\) be a Riemannian manifold and let \(\varOmega \) be a bounded open subset of \(M\). It is well known that significant information about the geometry of \(\varOmega \) is encoded into the properties of the distance, \(d_{\partial \varOmega }\), from the boundary of \(\varOmega \). Here, we show that the generalized gradient flow associated with the distance preserves singularities, that is, if \(x_0\) is a singular point of \(d_{\partial \varOmega }\) then the generalized characteristic starting at \(x_0\) stays singular for all times. As an application, we deduce that the singular set of \(d_{\partial \varOmega }\) has the same homotopy type as \(\varOmega \).

Mumford–Shah Model for One-to-One Edge Matching

This paper presents a new algorithm based on the Mumford-Shah model for simultaneously detecting the edge features of two images and jointly estimating a consistent set of transformations to match them. Compared to the current asymmetric methods in the literature, this fully symmetric method allows one to determine one-to-one correspondences between the edge features of two images. The entire variational model is realized in a multiscale framework of the finite element approximation. The optimization process is guided by an estimation minimization-type algorithm and an adaptive generalized gradient flow to guarantee a fast and smooth relaxation. The algorithm is tested on T1 and T2 magnetic resonance image data to study the parameter setting. We also present promising results of four applications of the proposed algorithm: interobject monomodal registration, retinal image registration, matching digital photographs of neurosurgery with its volume data, and motion estimation for frame interpolation.

A System of Equations Describing Quasi-Symmetric Mesoscale Phenomena.

An approximated system of equations describing mesoscale (horizontal length scale 105 m) phenomena is constructed. The wind speed of the mesoscale phenomena considered is assumed to be quasi-symmetric along the horizontal stream lines. The quasi-symmetry means that the variation of wind speed in the tangential direction of horizontal stream lines is one order smaller than that of the mesoscale.As a result of the quasi-symmetry, the primary flow is in a generalized gradient balance with geopotential φ. The primary flow (i.e., the generalized gradient flow) is parallel to the iso-φ lines, and determined by the derivatives of φ up to the second. The secondary flow, which constitutes the secondary circulation in the vertical plane perpendicular to the primary flow, is algebraically determined by the derivatives of φ up to the third. The constructed system of equations, which include the primary and secondary flows, exactly conserves the energy, while the potential vorticity is only approximately conserved. The potential vorticity is exactly conserved, when or where the local directional angle of iso-φ lines is vertically and temporally uniform. When or where the flow is large-scale, then or there the present system is reduced to the quasi-geostrophic system. Furthermore, for quasi-circular flows, the present equations are reduced to the balanced vortex equations.