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Effective sample size is a standard summary of Markov chain Monte Carlo output, but it is usually attached to scalar or Euclidean summaries chosen by the analyst. For manifold-valued samples this choice is not canonical: coordinate-wise…
We show that an infinite weighted tree admits a bi-Lipschitz embedding into Hilbert space if and only if it does not contain arbitrarily large complete binary trees with uniformly bounded distortion. We also introduce a new metric invariant…
Generalized sampling is a recently developed linear framework for sampling and reconstruction in separable Hilbert spaces. It allows one to recover any element in any finite-dimensional subspace given finitely many of its samples with…
We connect shift-invariant characteristic kernels to infinitely divisible distributions on $\mathbb{R}^{d}$. Characteristic kernels play an important role in machine learning applications with their kernel means to distinguish any two…
Markov chain (MC) algorithms are ubiquitous in machine learning and statistics and many other disciplines. Typically, these algorithms can be formulated as acceptance rejection methods. In this work we present a novel estimator applicable…
A Kernel Adaptive Metropolis-Hastings algorithm is introduced, for the purpose of sampling from a target distribution with strongly nonlinear support. The algorithm embeds the trajectory of the Markov chain into a reproducing kernel Hilbert…
This paper will provide several classes of strictly stationary, countable-state, irreducible, aperiodic Markov chains that are reversible and have finite second moments, such that the central limit theorem fails to hold. The main purpose is…
Markov Chain Monte Carlo (MCMC) algorithms are routinely used to draw samples from distributions with intractable normalization constants. However, standard MCMC algorithms do not apply to doubly-intractable distributions in which there are…
We introduce discrete time Markov chains that preserve uniform measures on boxed plane partitions. Elementary Markov steps change the size of the box from (a x b x c) to ((a-1) x (b+1) x c) or ((a+1) x (b-1) x c). Algorithmic realization of…
Several new properties of weighted Hilbert transform are obtained. If mu is zero, two Plancherel-like equations and the isotropic properties are derived. For mu is real number, a coerciveness is derived and two iterative sequences are…
We address high dimensional covariance estimation for elliptical distributed samples, which are also known as spherically invariant random vectors (SIRV) or compound-Gaussian processes. Specifically we consider shrinkage methods that are…
In this work we obtain mixing (and in some cases sharp mixing rates) for a reasonable large class of invertible systems preserving an infinite measure. The examples considered here are the invertible analogue of both Markov and non Markov…
We introduce a randomized iterative fragmentation procedure for finite metric spaces, which is guaranteed to result in a polynomially large subset that is $D$-equivalent to an ultrametric, where $D\in (2,\infty)$ is a prescribed target…
In this paper, we describe families of those bounded linear operators on a separable Hilbert space that are simultaneously unitarily equivalent to integral operators on $L_2(R)$ with bounded and arbitrarily smooth Carleman kernels. The main…
It is often said that control and estimation problems are in duality. Recently, in (Aubin-Frankowski,2021), we found new reproducing kernels in Linear-Quadratic optimal control by focusing on the Hilbert space of controlled trajectories,…
The paper offers versions of Hilbert's Irreducibility Theorem for the lifting of points in a cyclic subgroup of an algebraic group to a ramified cover. A version of Bertini Theorem in this context is also obtained.
In the analysis of Markov chains and processes, it is sometimes convenient to replace an unbounded state space with a "truncated" bounded state space. When such a replacement is made, one often wants to know whether the equilibrium behavior…
The cutoff phenomena for Markovian dynamics have been observed and rigorously verified for a multitude of models, particularly for Glauber-type dynamics on spin systems. However, prior studies have barely considered irreversible chains. In…
We study optimal solutions to an abstract optimization problem for measures, which is a generalization of classical variational problems in information theory and statistical physics. In the classical problems, information and relative…
We introduce a general algorithm for the computation of the scale functions of a spectrally negative L\'evy process $X$, based on a natural weak approximation of $X$ via upwards skip-free continuous-time Markov chains with stationary…