Distance-like Functions Research Articles

Strict Kantorovich contractions for Markov chains and Euler schemes with general noise

We study contractions of Markov chains on general metric spaces with respect to some carefully designed distance-like functions, which are comparable to the total variation and the standard Lp-Wasserstein distances for p≥1. We present explicit lower bounds of the corresponding contraction rates. By employing the refined basic coupling and the coupling by reflection, the results are applied to Markov chains whose transitions include additive stochastic noises that are not necessarily isotropic. This can be useful in the study of Euler schemes for SDEs driven by Lévy noises. In particular, motivated by recent works on the use of heavy tailed processes in Markov Chain Monte Carlo, we show that chains driven by the α-stable noise can have better contraction rates than corresponding chains driven by the Gaussian noise, due to the heavy tails of the α-stable distribution.

GP-DMD: a genetic programming variant with dynamic management of diversity

The proper management of diversity is essential to the success of Evolutionary Algorithms. Specifically, methods that explicitly relate the amount of diversity maintained in the population to the stopping criterion and elapsed period of execution, with the aim of attaining a gradual shift from exploration to exploitation, have been particularly successful. However, in the area of Genetic Programming, the performance of this design principle has not been studied. In this paper, a novel Genetic Programming method, Genetic Programming with Dynamic Management of Diversity (GP-DMD), is presented. GP-DMD applies this design principle through a replacement strategy that combines penalties based on distance-like functions with a multi-objective Pareto selection based on accuracy and simplicity. The proposed general method was adapted to the well-established Symbolic Regression benchmark problem using tree-based Genetic Programming. Several state-of-the-art diversity management approaches were considered for the experimental validation, and the results obtained showcase the improvements both in terms of mean square error and size. The effects of GP-DMD on the dynamics of the population are also analyzed, revealing the reasons for its superiority. As in other fields of Evolutionary Computation, this design principle contributes significantly to the area of Genetic Programming.

Higher order distance-like functions and Sobolev spaces

On non-compact Riemannian manifolds, we construct distance-like functions with derivatives controlled up to some order k assuming bounds on the growth of the derivatives of the curvature up to order k−2 and on the decay of the injectivity radius. This construction extends previously known results in various directions, permitting to obtain consequences which are (in a sense) sharp. As a first main application, we give refined conditions guaranteeing the density of compactly supported smooth functions in the Sobolev space Wk,p on the manifold. Contrary to all previously known results this can be obtained also on manifolds with possibly unbounded geometry. In the particular case p=2, making use of the Weitzenböck formula for a Lichnerowicz Laplacian acting on k-covariant totally symmetric tensor fields, we can weaken the assumptions needed to obtain the density property, avoiding any condition on the highest order derivatives of the curvature. Distance-like functions are also used to obtain new disturbed Sobolev inequalities, disturbed Lp-Calderón-Zygmund inequalities and the full Omori-Yau maximum principle for the Hessian under weak assumptions.

Mesh-independent streamline tracing

Following the theory of R-functions, a mesh cell of any type can be implicitly described by one single operator constructed from several distance-like functions which, combined with the transfinite interpolation technique and Darcy's velocities specified at the grid nodes, allows us to compute a velocity vector everywhere in the domain, producing a globally continuous vector field with locally performed calculations. Then, knowing a particle's initial position and using these velocities, one can compute its trajectory, obtaining simple equations to determine the exit point of the particle from a given cell. This alternative approach differs from standard schemes in that there is little dependence on the Eulerian topology for tracing the paths and all computations can be made directly at the physical space for any kind of mesh. Numerical experiments show the applicability of our method for streamline modeling on problems with hybrid grids, domains with non-trivial inner structures, and highly variating flows, among others.

A Parallel Douglas–Rachford Algorithm for Minimizing ROF-like Functionals on Images with Values in Symmetric Hadamard Manifolds

We are interested in restoring images having values in a symmetric Hadamard manifold by minimizing a functional with a quadratic data term and a total variation--like regularizing term. To solve the convex minimization problem, we extend the Douglas--Rachford algorithm and its parallel version to symmetric Hadamard manifolds. The core of the Douglas--Rachford algorithm is reflections of the functions involved in the functional to be minimized. In the Euclidean setting the reflections of convex lower semicontinuous functions are nonexpansive. As a consequence, convergence results for Krasnoselski--Mann iterations imply the convergence of the Douglas--Rachford algorithm. Unfortunately, these general results do not carry over to Hadamard manifolds, where proper convex lower semicontinuous functions can have expansive reflections. However, splitting our restoration functional in an appropriate way, we have only to deal with special functions---namely, several distance-like functions and an indicator function of a special convex set. We prove that the reflections of certain distance-like functions on Hadamard manifolds are nonexpansive, which is an interesting result on its own. Furthermore, the reflection of the involved indicator function is nonexpansive on Hadamard manifolds with constant curvature so that the Douglas--Rachford algorithm converges here. Several numerical examples demonstrate the advantageous performance of the suggested algorithm compared to other existing methods such as the cyclic proximal point algorithm and half-quadratic minimization. Numerical convergence is also observed in our experiments on the Hadamard manifold of symmetric positive definite matrices with the affine invariant metric, which does not have a constant curvature.

Convergence of a Proximal Point Algorithm for Solving Minimization Problems

We introduce and consider a proximal point algorithm for solving minimization problems using the technique of Güler. This proximal point algorithm is obtained by substituting the usual quadratic proximal term by a class of convex nonquadratic distance‐like functions. It can be seen as an extragradient iterative scheme. We prove the convergence rate of this new proximal point method under mild assumptions. Furthermore, it is shown that this estimate rate is better than the available ones.

Interior proximal methods and central paths for convex second-order cone programming

We make a unified analysis of interior proximal methods of solving convex second-order cone programming problems. These methods use a proximal distance with respect to second-order cones which can be produced with an appropriate closed proper univariate function in three ways. Under some mild conditions, the sequence generated is bounded with each limit point being a solution, and global rates of convergence estimates are obtained in terms of objective values. A class of regularized proximal distances is also constructed which can guarantee the global convergence of the sequence to an optimal solution. These results are illustrated with some examples. In addition, we also study the central paths associated with these distance-like functions, and for the linear SOCP we discuss their relations with the sequence generated by the interior proximal methods. From this, we obtain improved convergence results for the sequence for the interior proximal methods using a proximal distance continuous at the boundary of second-order cones.

A New Class of Alternating Proximal Minimization Algorithms with Costs-to-Move

Given two objective functions $f:\mathcal X\mapsto\R\cup\{+\infty\}$ and $g:\mathcal Y\mapsto\R\cup\{+\infty\}$ on abstract spaces $\mathcal X$ and $\mathcal Y$, and a coupling function $c:\mathcal X\times\mathcal Y\mapsto\R^+$, we introduce and study alternative minimization algorithms of the following type: $(x_0,y_0)\in\mathcal X\times\mathcal Y \mbox{ given; } (x_n,y_n)\rightarrow(x_{n+1},y_n)\rightarrow(x_{n+1},y_{n+1}) \mbox{ as follows: }\ \left\{\begin{array}{l} x_{n+1}\in\mbox{argmin} \{f(\xi)+\beta_n c(\xi,y_n)+\alpha_n h(x_n,\xi): \xi\in\mathcal X\},\ y_{n+1}\in\mbox{argmin} \{g(\eta)+\mu_n c(x_{n+1},\eta)+\nu_n k(y_n,\eta): \eta\in\mathcal Y\}. \end{array}\right.$ Their most original feature is the introduction of the terms $h:\mathcal X\times\mathcal X\mapsto\R^+$ and $k:\mathcal Y\times\mathcal Y\mapsto\R^+$ which are costs to change or to move (distance-like functions, relative entropies) accounting for various inertial, friction, or anchoring effects. These algorithms are studied in a general abstract framework. The introduction of the costs to change h and k leads to proximal minimizations with corresponding dissipative effects. As a result, the algorithms enjoy nice convergent properties. Coefficients $\alpha_n$, $\beta_n$, $\mu_n$, $\nu_n$ are nonnegative parameters. When taking $\alpha_n=\nu_n=0$ and quadratic costs on a Hilbert space, one recovers the classical alternating minimization algorithm, which itself is a natural extension of the alternating projection algorithm of von Neumann. A number of new significant results hold in general metric spaces. We pay particular attention to the following cases: (1.) $(\mathcal X,d_{\mathcal X})$ and $(\mathcal Y,d_{\mathcal Y})$ are complete metric spaces and $h\geq d_{\mathcal X}$, $k\geq d_{\mathcal Y}$ (“high local costs to move”); the algorithms then provide sequences that converge to Nash equilibria. (2.) $\mathcal X=\mathcal Y=\mathcal H$ is a Hilbert space, the costs to change are quadratic (“low local costs to move”) and the functions $f,g:\mathcal H\mapsto\R\cup\{+\infty\}$ are closed, convex, proper; then some of the classical convergence theorems for alternating convex minimization algorithms, including those of Acker and Prestel, are properly extended with original proofs.

Geometric finiteness in large families in dimension 3

We prove a diffeomorphism type finiteness theorem in dimension 3 for families C covered by finitely many distance-like functions with bounded twist. C contains families of bounded isoembolic volume and twist. However, the subfamilies of C with fixed volume are not Hausdorff precompact. Our methods do not involve Hausdorff limits and they yield explicit estimates on upper bounds for the number of diffeomorphism types.