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Mixtures of Gaussian (or normal) distributions arise in a variety of application areas. Many heuristics have been proposed for the task of finding the component Gaussians given samples from the mixture, such as the EM algorithm, a…
In this article, we consider the (double) minimization problem $$\min\left\{P(E;\Omega)+\lambda W_p(E,F):~E\subseteq\Omega,~F\subseteq \mathbb{R}^d,~\lvert E\cap F\rvert=0,~ \lvert E\rvert=\lvert F\rvert=1\right\},$$ where $p\geqslant 1$,…
An overview of recent advances in the theory of critical phenomena in $d$-dimensional weakly anisotropic systems is given. On the basis of a generalized shear transformation between anisotropic and isotropic systems, exact and approximate…
In this paper, we extend the previous method for solving inverse problems for steady-state equations using the Generalized Collage Theorem by searching for an approximation that not only minimizes the collage error but also maximizes the…
We study an isoperimetric problem the energy of which contains the perimeter of a set, Coulomb repulsion of the set with itself, and attraction of the set to a background nucleus as a point charge with charge $Z$. For the variational…
We introduce a modified Consensus-Based Optimization model that admits a fully unified and rigorous analysis of its finite-particle dynamics, the associated McKean--Vlasov equation, and their optimization behavior under a single set of…
We introduce notions of a constraint metric approximation and of a constraint stability of a metric approximation. This is done in the language of group equations with coefficients. We give an example of a group which is not constraintly…
Formulating a statistical inverse problem as one of inference in a Bayesian model has great appeal, notably for what this brings in terms of coherence, the interpretability of regularisation penalties, the integration of all uncertainties,…
We investigate variational problems in large-strain magnetoelasticity, both in the static and in the quasistatic setting. The model contemplates a mixed Eulerian-Lagrangian formulation: while deformations are defined on the reference…
We define the parametric closure problem, in which the input is a partially ordered set whose elements have linearly varying weights and the goal is to compute the sequence of minimum-weight lower sets of the partial order as the weights…
We consider the inverse conductivity problem with discontinuous conductivities. We show in a rigorous way, by a convergence analysis, that one can construct a completely discrete minimization problem whose solution is a good approximation…
We introduce a notion of fractional perimeter in an abstract Wiener space, and we show that halfspaces are the only volume-constrained minimisers.
We consider high-dimensional estimation problems where the number of parameters diverges with the sample size. General conditions are established for consistency, uniqueness, and asymptotic normality in both unpenalized and penalized…
One fundamental goal in any learning algorithm is to mitigate its risk for overfitting. Mathematically, this requires that the learning algorithm enjoys a small generalization risk, which is defined either in expectation or in probability.…
We solve explicitly a certain minimization problem for probability measures involving an interaction energy that is repulsive at short distances and attractive at large distances. We complement earlier works by showing that part of the…
This paper studies the Gibbs measure of an interacting particle system with a general interaction kernel at various temperature regimes. We are particularly interested in fine features of the convergence to the mean-field density as the…
The well-posedness and regularity properties of diffusion-aggregation equations, emerging from interacting particle systems, are established on the whole space for bounded interaction force kernels by utilizing a compactness convergence…
We provide a theory to establish the existence of nonzero solutions of perturbed Hammerstein integral equations with deviated arguments, being our main ingredient the theory of fixed point index. Our approach is fairly general and covers a…
There has been increasing interest on summary-free solutions for approximate Bayesian computation (ABC) which replace distances among summaries with discrepancies between the empirical distributions of the observed data and the synthetic…
Many data clustering applications must handle objects that cannot be represented as vectors. In this context, the bag-of-vectors representation describes complex objects through discrete distributions, for which the Wasserstein distance…