arXiv subjects

Abylay Zhumekenov

Publications and source records attributed to Abylay Zhumekenov.

6 recordsLinked to original sources

arXiv2026

Particle Filtering for a Class of State-Space Models with Low and Degenerate Observational Noise

We consider the discrete-time filtering problem in scenarios where the observation noise is low or degenerate. We focus on the case where the observation equation is a linear function of the state and the data involve additive noise. However, we place minimal assumptions on the hidden state process. For such a class of models we derive new particle filters (PFs) with the key property that their performance is robust to the size of the observation noise. As a consequence, the developed PFs are well-defined in the limiting case of degenerate observation noise. Indicatively, we prove (under assumptions) that the PF applied in this low noise setting inherits the properties of the PF used in the degenerate case. We extend our framework to the case where the hidden states are drawn from a diffusion process. We illustrate our algorithms numerically on several examples.

stat.CO↗

arXiv2026

Sequential Markov Chain Monte Carlo for Filtering of State-Space Models with Low or Degenerate Observation Noise

We consider the discrete-time filtering problem in scenarios where the observation noise is degenerate or low. More precisely, one is given access to a discrete time observation sequence which at any time $k$ depends only on the state of an unobserved Markov chain. We specifically assume that the functional relationship between observations and hidden Markov chain has either degenerate or low noise. In this article, under suitable assumptions, we derive the filtering density and its recursions for this class of problems on a specific sequence of manifolds defined through the observation function. We then design sequential Markov chain Monte Carlo methods to approximate the filter serially in time. For a certain linear observation model, we show that using sequential Markov chain Monte Carlo for low noise will converge as the noise disappears to that of using sequential Markov chain Monte Carlo for degenerate noise. We illustrate the performance of our methodology on several challenging stochastic models arising in statistics and applied mathematics.

stat.CO↗

arXiv2026

Score-Based Martingale Posteriors for Deep Neural Networks

In this paper we investigate the efficacy of the score-based martingale posteriors (SMP) (Cui & Walker, 2025; Fong et al., 2023) in the context of modern and large-scale machine learning problems and its potential for meaningful uncertainty quantification. SMPs work with a stochastic gradient ascent-type recursion on the parameter space of stochastic models and construct a martingale on the parameter space. Under simple mathematical assumptions, the recursion can be built so that the parameters form a martingale sequence which possesses a limiting, in time, random variable, the latter of which can be simulated very quickly, in contrast to Monte Carlo-based methods such as Markov chain Monte Carlo. In this expository paper we explore the SMP for inferring the parameters of deep neural networks (DNNs) and, where feasible, compare our results to the state-of-the-art Monte Carlo methods aimed at inferring conventional Bayesian posteriors.

stat.CO↗

arXiv2026

Parallel Selected Inversion for Space-Time Gaussian Markov Random Fields

Performing Bayesian inference on large spatio-temporal models requires extracting inverse elements of large sparse precision matrices for marginal variances, as well as estimating model hyperparameters. Although direct matrix factorizations can be used for the inversion, such methods fail to scale well for distributed problems when run on large computing clusters. On the contrary, Krylov subspace methods for the selected inversion have been gaining traction. We propose a parallel hybrid approach based on domain decomposition, which extends the Rao-Blackwellized Monte Carlo estimator for distributed precision matrices. Our approach exploits the strength of Krylov subspace methods as global solvers and efficiency of direct factorizations as base case solvers to compute the marginal variances and the derivatives required for hyperparameter estimation using a divide-and-conquer strategy. By introducing subdomain overlaps, one can achieve greater accuracy at an increased computational effort with little to no additional communication. We demonstrate the speed improvements and efficient hyperparameter inference on both simulated models and a massive US daily temperature data.

stat.CO↗

arXiv2025

Asymptotic Variance in the Central Limit Theorem for Multilevel Markovian Stochastic Approximation

In this note we consider the finite-dimensional parameter estimation problem associated to inverse problems. In such scenarios, one seeks to maximize the marginal likelihood associated to a Bayesian model. This latter model is connected to the solution of partial or ordinary differential equation. As such, there are two primary difficulties in maximizing the marginal likelihood (i) that the solution of differential equation is not always analytically tractable and (ii) neither is the marginal likelihood. Typically (i) is dealt with using a numerical solution of the differential equation, leading to a numerical bias and (ii) has been well studied in the literature using, for instance, Markovian stochastic approximation. It is well-known that to reduce the computational effort to obtain the maximal value of the parameter, one can use a hierarchy of solutions of the differential equation and combine with stochastic gradient methods. Several approaches do exactly this. In this paper we consider the asymptotic variance in the central limit theorem, associated to known estimates and find bounds on the asymptotic variance in terms of the precision of the solution of the differential equation. The significance of these bounds are the that they provide missing theoretical guidelines on how to set simulation parameters; that is, these appear to be the first mathematical results which help to run the methods efficiently in practice.

math.NA↗

arXiv2019

Fourier Neural Networks: A Comparative Study

We review neural network architectures which were motivated by Fourier series and integrals and which are referred to as Fourier neural networks. These networks are empirically evaluated in synthetic and real-world tasks. Neither of them outperforms the standard neural network with sigmoid activation function in the real-world tasks. All neural networks, both Fourier and the standard one, empirically demonstrate lower approximation error than the truncated Fourier series when it comes to an approximation of a known function of multiple variables.

cs.NE↗