arXiv subjects

Attila Lovas

Publications and source records attributed to Attila Lovas.

17 recordsLinked to original sources

arXiv2025

Sharp Mixing Rates for Markov Chains on General Spaces with Unbounded Random Environments

We consider Markov chains on general state spaces in stationary random environment which are defined by a random mapping that is contractive up to a bounded perturbation. We prove their convergence to a limiting law, providing convergence rates. We also show that these processes are strongly mixing and estimate their mixing coefficients. Our results significantly extend those available in the literature. In particular, for some additive autoregressive processes with exogenous covariates we achieve mixing rates that are optimal up to logarithmic factors.

math.PR↗

arXiv2025

Transition of $α$-mixing in Random Iterations with Applications in Queuing Theory

Nonlinear time series models with exogenous regressors are essential in econometrics, queuing theory, and machine learning, though their statistical analysis remains incomplete. Key results, such as the law of large numbers and the functional central limit theorem, are known for weakly dependent variables. We demonstrate the transfer of mixing properties from the exogenous regressor to the response via coupling arguments. Additionally, we study Markov chains in random environments with drift and minorization conditions, even under non-stationary environments with favorable mixing properties, and apply this framework to single-server queuing models.

math.ST↗

arXiv2025

Mixing properties of some Markov chains models in random environments

Markov chains in random environments (MCREs) have recently attracted renewed interest, as these processes naturally arise in many applications, such as econometrics and machine learning. Although specific asymptotic results, such as the law of the large numbers and central limit theorems, have been obtained for some of these models, their annealed dependence properties, such as strong mixing properties, are not well understood in general. We derive strong mixing properties for a wide range of MCREs that satisfy some drift/small set conditions, with general assumptions on the corresponding stochastic parameters and the mixing properties of the environments. We then demonstrate the wide range of applications of our results in time series analysis and stochastic gradient Langevin dynamics, with fewer restrictions than those found in existing literature.

math.PR↗

arXiv2024

On the strong stability of ergodic iterations

We revisit processes generated by iterated random functions driven by a stationary and ergodic sequence. Such a process is called strongly stable if a random initialization exists, for which the process is stationary and ergodic, and for any other initialization, the difference between the two processes converges to zero almost surely. Under some mild conditions on the corresponding recursive map, without any condition on the driving sequence, we show the strong stability of iterations. Several applications are surveyed such as generalized autoregression and queuing. Furthermore, new results are deduced for Langevin-type iterations with dependent noise and for multitype branching processes.

math.PR↗

arXiv2023

Functional Central Limit Theorem and Strong Law of Large Numbers for Stochastic Gradient Langevin Dynamics

We study the mixing properties of an important optimization algorithm of machine learning: the stochastic gradient Langevin dynamics (SGLD) with a fixed step size. The data stream is not assumed to be independent hence the SGLD is not a Markov chain, merely a \emph{Markov chain in a random environment}, which complicates the mathematical treatment considerably. We derive a strong law of large numbers and a functional central limit theorem for SGLD.

math.PR↗

arXiv2023

Taming neural networks with TUSLA: Non-convex learning via adaptive stochastic gradient Langevin algorithms

Artificial neural networks (ANNs) are typically highly nonlinear systems which are finely tuned via the optimization of their associated, non-convex loss functions. In many cases, the gradient of any such loss function has superlinear growth, making the use of the widely-accepted (stochastic) gradient descent methods, which are based on Euler numerical schemes, problematic. We offer a new learning algorithm based on an appropriately constructed variant of the popular stochastic gradient Langevin dynamics (SGLD), which is called tamed unadjusted stochastic Langevin algorithm (TUSLA). We also provide a nonasymptotic analysis of the new algorithm's convergence properties in the context of non-convex learning problems with the use of ANNs. Thus, we provide finite-time guarantees for TUSLA to find approximate minimizers of both empirical and population risks. The roots of the TUSLA algorithm are based on the taming technology for diffusion processes with superlinear coefficients as developed in \citet{tamed-euler, SabanisAoAP} and for MCMC algorithms in \citet{tula}. Numerical experiments are presented which confirm the theoretical findings and illustrate the need for the use of the new algorithm in comparison to vanilla SGLD within the framework of ANNs.

cs.LG↗

arXiv2022

Ergodic aspects of trading with threshold strategies

To profit from price oscillations, investors frequently use threshold-type strategies where changes in the portfolio position are triggered by some indicators reaching prescribed levels. In this paper, we investigate threshold-type strategies in the context of ergodic control. We make the first steps towards their optimization by proving the ergodic properties of related functionals. Assuming Markovian price increments satisfying a minorization condition and (one-sided) boundedness we show, in particular, that for given thresholds, the distribution of the gains converges in the long run. We also extend recent results on the stability of overshoots of random walks from the i.i.d.\ increment case to Markovian increments, under suitable conditions.

math.PR↗

arXiv2020

Markov chains in random environment with applications in queueing theory and machine learning

We prove the existence of limiting distributions for a large class of Markov chains on a general state space in a random environment. We assume suitable versions of the standard drift and minorization conditions. In particular, the system dynamics should be contractive on the average with respect to the Lyapunov function and large enough small sets should exist with large enough minorization constants. We also establish that a law of large numbers holds for bounded functionals of the process. Applications to queuing systems, to machine learning algorithms and to autoregressive processes are presented.

math.PR↗

arXiv2020

Ergodic theorems for queuing systems with dependent inter-arrival times

We study a G/GI/1 single-server queuing model with i.i.d.\ service times that are independent of a stationary process of inter-arrival times. We show that the distribution of the waiting time converges to a stationary law as time tends to infinity provided that inter-arrival times satisfy a Gärtner-Ellis type condition. A convergence rate is given and a law of large numbers established. These results provide tools for the statistical analysis of such systems, transcending the standard case with independent inter-arrival times.

math.PR↗

arXiv2020

Quantum Aitchison geometry

Multiplying a likelihood function with a positive number makes no difference in Bayesian statistical inference, therefore after normalization the likelihood function in many cases can be considered as probability distribution. This idea led Aitchison to define a vector space structure on the probability simplex in 1986. Pawlowsky-Glahn and Egozcue gave a statistically relevant scalar product on this space in 2001, endowing the probability simplex with a Hilbert space structure. In this paper we present the noncommutative counterpart of this geometry. We introduce a real Hilbert space structure on the quantum mechanical finite dimensional state space. We show that the scalar product in quantum setting respects the tensor product structure and can be expressed in terms of modular operators and Hamilton operators. Using the quantum analogue of the log-ratio transformation it turns out that all the newly introduced operations emerge naturally in the language of Gibbs states. We show an orthonormal basis in the state space and study the introduced geometry on the space of qubits in details.

math-ph↗

arXiv2019

On the notion of quantum copulas

Working with multivariate probability distributions Sklar introduced the notion of copula in 1959, which turned out to be a key concept to understand the structure of distributions of composite systems. Roughly speaking Sklar proved that a joint distribution can be represented with its marginals and a copula. The main goal of this paper is to present a quantum analogue of the notion of copula. Our main theorem states that for any state of a composite quantum system there exists a unique copula such that they are connected to each other by invertible matrices, moreover, they are both separable or both entangled. So considering copulas instead of states is a separability preserving transformation and efficiently decreases the dimension of the state space of composite systems. The method how we prove these results draws attention to the fact that theorem for states can be achieved by considering the states as quantum channels and using theorems for channels and different kind of positive maps.

math-ph↗

arXiv2017

Volume of the space of qubit-qubit channels and state transformations under random quantum channels

The simplest building blocks for quantum computations are the qubit-qubit quantum channels. In this paper, we analyze the structure of these channels via their Choi representation. The restriction of a quantum channel to the space of classical states (i.e. probability distributions) is called the underlying classical channel. The structure of quantum channels over a fixed classical channel is studied, the volume of general and unital qubit channels with respect to the Lebesgue measure is computed and explicit formulas are presented for the distribution of the volume of quantum channels over given classical channels. We study the state transformation under uniformly random quantum channels. If one applies a uniformly random quantum channel (general or unital) to a given qubit state, the distribution of the resulted quantum states is presented.

math-ph↗

arXiv2017

Invariance of separability probability over reduced states in 4x4 bipartite systems

The geometric separability probability of composite quantum systems is extensively studied in the last decades. One of most simple but strikingly difficult problem is to compute the separability probability of qubit-qubit and rebit-rebit quantum states with respect to the Hilbert-Schmidt measure. A lot of numerical simulations confirm the P(rebit-rebit)=29/64 and P(qubit-qubit)=8/33 conjectured probabilities. Milz and Strunz studied the separability probability with respect to given subsystems. They conjectured that the separability probability of qubit-qubit (and qubit-qutrit) depends on sum of single qubit subsystems (D), moreover it depends just on the Bloch radii (r) of D and it is constant in r. Using the Peres-Horodecki criterion for separability we give mathematical proof for the P(rebit-rebit)=29/64 probability and we present an integral formula for the complex case which hopefully will help to prove the P(qubit-qubit)=8/33 probability too. We prove Milz and Strunz's conjecture for rebit-rebit and qubit-qubit states. The case, when the state space is endowed with the volume form generated by the operator monotone function f(x)=sqrt(x) is studied too in detail. We show, that even in this setting the Milz and Strunz's conjecture holds and we give an integral formula for separability probability according to this measure.

math-ph↗

arXiv2016

Volume of the space of qubit channels and some new results about the distribution of the quantum Dobrushin coefficient

The simplest building blocks for quantum computations are the qbit-qbit quantum channels. In this paper we analyse the structure of these channels via their Choi representation. The restriction of a quantum channel to the space of classical states (i.e. probability distributions) is called the underlying classical channel. The structure of quantum channels over a fixed classical channel is studied, the volume of general and unital qubit channels over real and complex state spaces with respect to the Lebesgue measure is computed and explicit formulas are presented for the distribution of the volume of quantum channels over given classical channels. Moreover an algorithm is presented to generate uniformly distributed channels with respect to the Lebesgue measure, which enables further studies. With this algorithm the distribution of trace-distance contraction coefficient (Dobrushin) is investigated numerically by Monte-Carlo simulations, which leads to some conjectures and points out the strange behaviour of the real state space.

math-ph↗

arXiv2015

Numerical reconstruction of pulsatile blood flow from 4D computer tomography angiography data

We present a novel numerical algorithm developed to reconstuct pulsatile blood flow from ECG-gated CT angiography data. A block-based optimization method was constructed to solve the inverse problem corresponding to the Riccati-type ordinary differential equation that can be deduced from conservation principles and Hooke's law. Local flow rate for 5 patients was computed in 10cm long aorta segments that are located 1cm below the heart. The wave form of the local flow rate curves seems to be realistic. Our approach is suitable for estimating characteristics of pulsatile blood flow in aorta based on ECG gated CT scan thereby contributing to more accurate description of several cardiovascular lesions.

math.NA↗

arXiv2015

Refinement of Robertson-type uncertainty principles with geometric interpretation

A generalisation of the classical covariance for quantum mechanical observables has previously been presented by Gibilisco, Hiai and Petz. Gibilisco and Isola has proved that the usual quantum covariance gives the sharpest inequalities for the determinants of covariance matrices. We introduce a new generalisation of the classical covariance which gives better inequalities than the classical one furthermore it has a direct geometric interpretation.

math-ph↗

arXiv2015

Measure on gauge invariant symmetric norms

The concept of a gauge invariant symmetric random norm is elaborated in this paper. We introduce norm processes and show that this kind of stochastic processes are closely related to gauge invariant symmetric random norms. We construct a gauge invariant symmetric random norm on the plain. We define two different extensions of these random norms to higher (even infinite) dimensions. We calculate numerically unit spheres of expected norms in two and three dimensions for the constructed random norm.

math.FA↗