Markov Operators Research Articles

Transfunctions Applied to Plans, Markov Operators and Optimal Transport

Abstract A transfunction is a function which maps between sets of finite measures on measurable spaces. In this paper we characterize transfunctions that correspond to Markov operators and to plans; such a transfunction will contain the “instructions” common to several Markov operators and plans. We also define the adjoint of transfunctions in two settings and provide conditions for existence of adjoints. Finally, we develop approximations of identity in each setting and use them to approximate weakly-continuous transfunctions with simple transfunctions; one of these results can be applied to some optimal transport problems to approximate the optimal cost with simple Markov transfunctions.

Quasi-Compactness of Operators for General Markov Chains and Finitely Additive Measures

We study Markov operators T, A, and T* of general Markov chains on an arbitrary measurable space. The operator, T, is defined on the Banach space of all bounded measurable functions. The operator A is defined on the Banach space of all bounded countably additive measures. We construct an operator T*, topologically conjugate to the operator T, acting in the space of all bounded finitely additive measures. We prove the main result of the paper that, in general, a Markov operator T* is quasi-compact if and only if T is quasi-compact. It is proved that the conjugate operator T* is quasi-compact if and only if the Doeblin condition (D) is satisfied. It is shown that the quasi-compactness conditions for all three Markov operators T, A, and T* are equivalent to each other. In addition, we obtain that, for an operator T* to be quasi-compact, it is necessary and sufficient that it does not have invariant purely finitely additive measures. A strong uniform reversible ergodic theorem is proved for the quasi-compact Markov operator T* in the space of finitely additive measures. We give all the proofs for the most general case. A detailed analysis of Lin’s counterexample is provided.

L2-Quasi-compact and hyperbounded Markov operators

L2-Quasi-compact and hyperbounded Markov operators

With Andrzej Lasota There and Back Again

Abstract The paper below is a written version of the 17th Andrzej Lasota Lecture presented on January 12th, 2024 in Katowice. During the lecture we tried to show the impact of Andrzej Lasota’s results on the author’s research concerning various fields of mathematics, including chaos and ergodicity of dynamical systems, Markov operators and semigroups and partial differential equations.

Markov Progressive Framework, a Universal Paradigm for Modeling Long Videos.

Compared to images, video, as an increasingly mainstream visual media, contains more semantic information. For this reason, the computational complexity of video models is an order of magnitude larger than their image-level counterparts, which increases linearly with the square number of frames. Constrained by computational resources, training video models to learn long-term temporal semantics end-to-end is quite a challenge. Currently, the main-stream method is to split a raw video into clips, leading to incomplete fragmentary temporal information flow and failure of modeling long-term semantics. To solve this problem, in this paper, we design the Markov Progressive framework (MaPro), a theoretical framework consisting of the progressive modeling method and a paradigm model tailored for it. Inspired by natural language processing techniques dealing with long sentences, the core idea of MaPro is to find a paradigm model consisting of proposed Markov operators which can be trained in multiple sequential steps and ensure that the multi-step progressive modeling is equivalent to the conventional end-to-end modeling. By training the paradigm model under the progressive method, we are able to model long videos end-to-end with limited resources and ensure the effective transmission of long-term temporal information. We provide detailed implementations of this theoretical system on the mainstream CNN- and Transformer-based models, where they are modified to conform to the Markov paradigm. The theoretical paradigm as a basic model is the lower bound of the model efficiency. With it, we further explore more sophisticated designs for CNN and Transformer-based methods specifically. As a general and robust training method, we experimentally demonstrate that it yields significant performance improvements on different backbones and datasets. As an illustrative example, the proposed method improves the SlowOnly network by 4.1 mAP on Charades and 2.5 top-1 accuracy on Kinetics. And for TimeSformer, MaPro improves its performance on Kinetics by 2.0 top-1 accuracy. Importantly, all these improvements are achieved with a little parameter and computation overhead. We hope the MaPro method can provide the community with new insight into modeling long videos.

Privacy Amplification by Iteration for ADMM with (Strongly) Convex Objective Functions

We examine a private ADMM variant for (strongly) convex objectives which is a primal-dual iterative method. Each iteration has a user with a private function used to update the primal variable, masked by Gaussian noise for local privacy, without directly adding noise to the dual variable. Privacy amplification by iteration explores if noises from later iterations can enhance the privacy guarantee when releasing final variables after the last iteration. Cyffers et al. explored privacy amplification by iteration for the proximal ADMM variant, where a user's entire private function is accessed and noise is added to the primal variable. In contrast, we examine a private ADMM variant requiring just one gradient access to a user's function, but both primal and dual variables must be passed between successive iterations. To apply Balle et al.'s coupling framework to the gradient ADMM variant, we tackle technical challenges with novel ideas. First, we address the non-expansive mapping issue in ADMM iterations by using a customized norm. Second, because the dual variables are not masked with any noise directly, their privacy guarantees are achieved by treating two consecutive noisy ADMM iterations as a Markov operator. Our main result is that the privacy guarantee for the gradient ADMM variant can be amplified proportionally to the number of iterations. For strongly convex objective functions, this amplification exponentially increases with the number of iterations. These amplification results align with the previously studied special case of stochastic gradient descent.

Random invariant densities for markov operator cocycles and random mean ergodic theorem

In this paper, we consider random invariant densities and the mean ergodic theorem for Markov operator cocycles which are applicable to quenched type random dynamical systems. We give necessary and sufficient conditions for the existence of random invariant densities for Markov operator cocycles and establish the mean ergodic theorem for generalized linear operator cocycles over a weakly sequentially complete Banach space. The advantage of the result is that we show the implication of weak precompactness for almost every environment to strong convergence in the global sense.

Characterization of pre-idempotent Copulas

Abstract Copulas C C for which ( C t C ) 2 = C t C {({C}^{t}C)}^{2}={C}^{t}C are called pre-idempotent copulas, of which well-studied examples are idempotent copulas and complete dependence copulas. As such, we shall work mainly with the topology induced by the modified Sobolev norm, with respect to which the class ℛ {\mathcal{ {\mathcal R} }} of pre-idempotent copulas is closed and the class of factorizable copulas is a dense subset of ℛ {\mathcal{ {\mathcal R} }} . Identifying copulas with Markov operators on L 2 {L}^{2} , the one-to-one correspondence between pre-idempotent copulas and partial isometries is one of our main tools. In the same spirit as Darsow and Olsen’s work on idempotent copulas, we obtain an explicit characterization of pre-idempotent copulas, which is split into cases according to the atomicity of its associated σ \sigma -algebras, where the nonatomic case gives all factorizable copulas and the totally atomic case yields conjugates of ordinal sums of copies of the product copula.

Almost everywhere convergence of the iterates of a bi-stochastic Markov operator

Almost everywhere convergence of the iterates of a bi-stochastic Markov operator

Invariant Finitely Additive Measures for General Markov Chains and the Doeblin Condition

In this paper, we consider general Markov chains with discrete time in an arbitrary measurable (phase) space. Markov chains are given by a classical transition function that generates a pair of conjugate linear Markov operators in a Banach space of measurable bounded functions and in a Banach space of bounded finitely additive measures. We study sequences of Cesaro means of powers of Markov operators on the set of finitely additive probability measures. It is proved that the set of all limit measures (points) of such sequences in the weak topology generated by the preconjugate space is non-empty, weakly compact, and all of them are invariant for this operator. We also show that the well-known Doeblin condition (D) for the ergodicity of a Markov chain is equivalent to condition (∗): all invariant finitely additive measures of the Markov chain are countably additive, i.e., there are no invariant purely finitely additive measures. We give all the proofs for the most general case.

Derivations and KMS-Symmetric Quantum Markov Semigroups

We prove that the generator of the L^2 implementation of a KMS-symmetric quantum Markov semigroup can be expressed as the square of a derivation with values in a Hilbert bimodule, extending earlier results by Cipriani and Sauvageot for tracially symmetric semigroups and the second-named author for GNS-symmetric semigroups. This result hinges on the introduction of a new completely positive map on the algebra of bounded operators on the GNS Hilbert space. This transformation maps symmetric Markov operators to symmetric Markov operators and is essential to obtain the required inner product on the Hilbert bimodule.

Continuous Dependence in a Problem of Convergence of Random Iteration

We show that an invariant measure of a Markov operator that is contracting in the Hutchison distance, acting on a space of Borel probability measures on a Polish space, depends continuously on the given operator. In addition, we establish an estimate for a distance between invariant measures. Some applications to the weak limit of iterates of random-valued functions (in particular, the so-called random affine maps, occuring, e.g., in perpetuities analysis) are also given.

On a [formula omitted]-Mixing property for Entangled Markov Chains

We study a mixing condition for entangled Markov chains, so-called ψ-mixing property. We prove that every entangled Markov chain, whose Markov operators satisfy the Markov–Dobrushin condition, is ψ-mixing. Moreover, we show the restriction of the underlying QMC to the diagonal algebra gives rise to a classical mixing Markov chains with the same exponential rate of convergence.

On Harmonic Hilbert Spaces on Compact Abelian Groups

Harmonic Hilbert spaces on locally compact abelian groups are reproducing kernel Hilbert spaces (RKHSs) of continuous functions constructed by Fourier transform of weighted L^2 spaces on the dual group. It is known that for suitably chosen subadditive weights, every such space is a Banach algebra with respect to pointwise multiplication of functions. In this paper, we study RKHSs associated with subconvolutive functions on the dual group. Sufficient conditions are established for these spaces to be symmetric Banach ^*-algebras with respect to pointwise multiplication and complex conjugation of functions (here referred to as RKHAs). In addition, we study aspects of the spectra and state spaces of RKHAs. Sufficient conditions are established for an RKHA on a compact abelian group G to have the same spectrum as the C^*-algebra of continuous functions on G. We also consider one-parameter families of RKHSs associated with semigroups of self-adjoint Markov operators on L^2(G), and show that in this setting subconvolutivity is a necessary and sufficient condition for these spaces to have RKHA structure. Finally, we establish embedding relationships between RKHAs and a class of Fourier–Wermer algebras that includes spaces of dominating mixed smoothness used in high-dimensional function approximation.

Convergence properties of data augmentation algorithms for high-dimensional robit regression

The logistic and probit link functions are the most common choices for regression models with a binary response. However, these choices are not robust to the presence of outliers/unexpected observations. The robit link function, which is equal to the inverse CDF of the Student’s t-distribution, provides a robust alternative to the probit and logistic link functions. A multivariate normal prior for the regression coefficients is the standard choice for Bayesian inference in robit regression models. The resulting posterior density is intractable and a Data Augmentation (DA) Markov chain is used to generate approximate samples from the desired posterior distribution. Establishing geometric ergodicity for this DA Markov chain is important as it provides theoretical guarantees for asymptotic validity of MCMC standard errors for desired posterior expectations/quantiles. Previous work [16] established geometric ergodicity of this robit DA Markov chain assuming (i) the sample size n dominates the number of predictors p, and (ii) an additional constraint which requires the sample size to be bounded above by a fixed constant which depends on the design matrix X. In particular, modern high-dimensional settings where n<p are not considered. In this work, we show that the robit DA Markov chain is trace-class (i.e., the eigenvalues of the corresponding Markov operator are summable) for arbitrary choices of the sample size n, the number of predictors p, the design matrix X, and the prior mean and variance parameters. The trace-class property implies geometric ergodicity. Moreover, this property allows us to conclude that the sandwich robit chain (obtained by inserting an inexpensive extra step in between the two steps of the DA chain) is strictly better than the robit DA chain in an appropriate sense, and enables the use of recent methods to estimate the spectral gap of trace class DA Markov chains.

Rates of convergence for chains of expansive Markov Operators

Abstract We provide conditions that guarantee local rates of convergence in distribution of iterated random functions that are not nonexpansive mappings in locally compact Hadamard spaces. Our results are applied to stochastic instances of common algorithms in optimization, stochastic tomography for X-FEL imaging and a stochastic algorithm for the computation of Fréchet means in model spaces for phylogenetic trees.

Spatial populations with seed-banks in random environment: III. Convergence towards mono-type equilibrium

We consider the spatially inhomogeneous Moran model with seed-banks introduced in [20]. Populations comprising active and dormant individuals are spatially structured in colonies labeled by Zd,d≥1. The population sizes are sampled from a translation-invariant, ergodic, uniformly elliptic field that constitutes a static random environment. Individuals carry one of two types: ♡ and ♠. Dormant individual resides in what is called a seed-bank. Active individuals exchange type from the seed-bank of their own colony, and resample type by choosing a parent uniformly at random from the distinct active populations according to a symmetric migration kernel. In [20] by exploiting a dual process given by an interacting coalescing particle system, we showed that the spatial system exhibits a dichotomy between clustering (mono-type equilibrium) and coexistence (multi-type equilibrium). In this paper, we identify the domain of attraction for each mono-type equilibrium in the clustering regime for an arbitrary fixed environment. Furthermore, we show that in dimensions d≤2, when the migration kernel is recurrent, for almost all realization of the environment, the system with an initially consistent type-distribution converges weakly to a mono-type equilibrium in which the probability of fixation to the all type-♡ configuration does not depend on the environment. An explicit formula for the fixation probability is given in terms of an annealed average of the type-♡ densities in the active and the dormant population, biased by the ratio of the two population sizes at the target colony. Primary techniques employed in the proofs include stochastic duality and the environment process viewed from particle, introduced in [24] for random walk in random environment on a strip. A spectral analysis of Markov operator yields quenched weak convergence of the environment process associated with the single-particle dual process to a reversible ergodic distribution, which we transfer to the spatial system of populations by using duality.

Thermodynamic Formalism for General Iterated Function Systems with Measures.

This paper introduces a theory of Thermodynamic Formalism for Iterated Function Systems with Measures (IFSm). We study the spectral properties of the Transfer and Markov operators associated to a IFSm. We introduce variational formulations for the topological entropy of holonomic measures and the topological pressure of IFSm given by a potential. A definition of equilibrium state is then natural and we prove its existence for any continuous potential. We show, in this setting, a uniqueness result for the equilibrium state requiring only the Gâteaux differentiability of the pressure functional.

Existence of invariant idempotent measures by contractivity of idempotent Markov operators

We prove that the idempotent Markov operator generated by contractive max plus normalized iterated function system (IFS) is also a contractive map w.r.t. natural metrics on the space of idempotent measures. This gives alternative proofs of the existence of invariant idempotent measures for such IFSs.

Non-Markovian dynamics under time-translation symmetry

A dynamical symmetry is employed to determine the structure of the quantum non-Markovian time-local master equation. Such a structure is composed from two components: scalar kinetic coefficients and the standard quantum Markovian operator form. The kinetic coefficients are generally time-dependent and incorporate information on the kinematics and memory effects, while the operators manifest the dynamical symmetry. Specifically, we focus on time-translation symmetric dynamics, where the Lindblad jump operators constitute the eigenoperators of the free dynamics. This symmetry is motivated by thermodynamic microscopic considerations, where strict energy conservation between system and environment imposes the time-translation symmetry. The construction is generalized to other symmetries, and to driven quantum systems. The formalism is illustrated by three exactly solvable non-Markovian models, where the exact reduced description exhibits a dynamical symmetric structure. The formal structure of the master equation leads to a first principle calculation of the exact kinetic coefficients. This opens the possibility to simulate in a modular fashion non-Markovian dynamics.