arXiv subjects

Alex Gorodetsky

Publications and source records attributed to Alex Gorodetsky.

11 recordsLinked to original sources

arXiv2025

Sequential Bayesian parameter-state estimation in dynamical systems with noisy and incomplete observations via a variational framework

Online joint estimation of a dynamical model's unknown parameters and states with uncertainty quantification is crucial in many applications. For example, digital twins dynamically update their knowledge of model parameters and states to support prediction and decision-making. Reliability and computational speed are vital for DTs. Online parameter-state estimation ensures computational efficiency, while uncertainty quantification is essential for making reliable predictions and decisions. In parameter-state estimation, the joint distribution of the state and model parameters conditioned on the data, termed the joint posterior, provides accurate uncertainty quantification. Because the joint posterior is generally intractable to compute, this paper presents an online variational inference framework to compute its approximation at each time step. The approximation is factorized into a marginal distribution over the model parameters and a state distribution conditioned on the parameters. This factorization enables recursive updates through a two-stage procedure: first, the parameter posterior is approximated via variational inference; second, the state distribution conditioned on the parameters is computed using Gaussian filtering based on the approximate parameter posterior. The algorithmic design is supported by a theorem establishing upper bounds on the joint posterior approximation error. Numerical experiments demonstrate that the proposed method (i) accurately infers both unobserved states and unknown parameters of dynamical and observation models; (ii) remains robust under noisy, partial observations and model discrepancies in a chaotic Lorenz'96 system; and (iii) scales effectively to a high-dimensional state-space system arising from the spatial discretization of a convection-diffusion equation. outperforming the joint ensemble Kalman filter in this setting.

stat.ME↗

arXiv2025

Optimally balancing exploration and exploitation to automate multi-fidelity statistical estimation

Multi-fidelity methods that use an ensemble of models to compute a Monte Carlo estimator of the expectation of a high-fidelity model can significantly reduce computational costs compared to single-model approaches. These methods use oracle statistics, specifically the covariance between models, to optimally allocate samples to each model in the ensemble. However, in practice, the oracle statistics are estimated using additional model evaluations, whose computational cost and induced error are typically ignored. To address this issue, this paper proposes an adaptive algorithm to optimally balance the resources between oracle statistics estimation and final multi-fidelity estimator construction, leveraging ideas from multilevel best linear unbiased estimators in Schaden and Ullmann (2020) and a bandit-learning procedure in Xu et al. (2022). Under mild assumptions, we demonstrate that the multi-fidelity estimator produced by the proposed algorithm exhibits mean-squared error commensurate with that of the best linear unbiased estimator under the optimal allocation computed with oracle statistics. Our theoretical findings are supported by detailed numerical experiments, including a parametric elliptic PDE and an ice-sheet mass-change modeling problem.

stat.CO↗

arXiv2025

Tensor Network Structure Search Via Canonical Dimension Tree Enumeration

Tensor networks provide a powerful framework for compressing multi-dimensional data. The optimal tensor network structure for a given data tensor depends on both data characteristics and specific optimality criteria, making tensor network structure search a challenging problem. Existing solutions typically rely on sampling and compressing numerous candidate structures; these procedures are computationally expensive and therefore limiting for practical applications. We address this challenge by decoupling topology enumeration from rank assignment search. We first represent the search space using canonical dimension trees, which encode potential network topology through nested index partitions and inherently rule out redundant and suboptimal topologies by construction. To mitigate the assessment bottleneck, we introduce a mechanism powered by the precomputation of a singular value map. By archiving the singular values of all feasible tensor matricizations, we transform the evaluation of any candidate dimension tree into a constraint-solving problem. This formulation yields an empirically near-optimal rank assignment via simple metadata lookups, allowing us to compute structural costs and bypass expensive tensor decompositions for all but the final selected candidate. Experimental results show that our approach accelerates the structure search by up to 10x while achieving highly competitive compression ratios, outperforming standard tensor trains and hierarchical tuckers by up to 10x, and matching or exceeding state-of-the-art structure search tools. Notably, our approach scales to larger tensors that are unattainable by prior work. Furthermore, the discovered topologies generalize well to similar data; they achieve compression ratios up to 2.4x better than tensor trains or hierarchical tuckers, while maintaining a search time of approximately 110 seconds for 6D tensors of 1-2GB disk size.

cs.CE↗

arXiv2024

A switching Kalman filter approach to online mitigation and correction of sensor corruption for inertial navigation

This paper introduces a novel approach to detect and address faulty or corrupted external sensors in the context of inertial navigation by leveraging a switching Kalman Filter combined with parameter augmentation. Instead of discarding the corrupted data, the proposed method retains and processes it, running multiple observation models simultaneously and evaluating their likelihoods to accurately identify the true state of the system. We demonstrate the effectiveness of this approach to both identify the moment that a sensor becomes faulty and to correct for the resulting sensor behavior to maintain accurate estimates. We demonstrate our approach on an application of balloon navigation in the atmosphere and shuttle reentry. The results show that our method can accurately recover the true system state even in the presence of significant sensor bias, thereby improving the robustness and reliability of state estimation systems under challenging conditions. We also provide a statistical analysis of problem settings to determine when and where our method is most accurate and where it fails.

eess.SY↗

arXiv2024

Low-rank Bayesian matrix completion via geodesic Hamiltonian Monte Carlo on Stiefel manifolds

We present a new sampling-based approach for enabling efficient computation of low-rank Bayesian matrix completion and quantifying the associated uncertainty. Firstly, we design a new prior model based on the singular-value-decomposition (SVD) parametrization of low-rank matrices. Our prior is analogous to the seminal nuclear-norm regularization used in non-Bayesian setting and enforces orthogonality in the factor matrices by constraining them to Stiefel manifolds. Then, we design a geodesic Hamiltonian Monte Carlo (-within-Gibbs) algorithm for generating posterior samples of the SVD factor matrices. We demonstrate that our approach resolves the sampling difficulties encountered by standard Gibbs samplers for the common two-matrix factorization used in matrix completion. More importantly, the geodesic Hamiltonian sampler allows for sampling in cases with more general likelihoods than the typical Gaussian likelihood and Gaussian prior assumptions adopted in most of the existing Bayesian matrix completion literature. We demonstrate an applications of our approach to fit the categorical data of a mice protein dataset and the MovieLens recommendation problem. Numerical examples demonstrate superior sampling performance, including better mixing and faster convergence to a stationary distribution. Moreover, they demonstrate improved accuracy on the two real-world benchmark problems we considered.

stat.ML↗

arXiv2024

Language Model Powered Digital Biology with BRAD

Recent advancements in Large Language Models (LLMs) are transforming biology, computer science, engineering, and every day life. However, integrating the wide array of computational tools, databases, and scientific literature continues to pose a challenge to biological research. LLMs are well-suited for unstructured integration, efficient information retrieval, and automating standard workflows and actions from these diverse resources. To harness these capabilities in bioinformatics, we present a prototype Bioinformatics Retrieval Augmented Digital assistant (BRAD). BRAD is a chatbot and agentic system that integrates a variety of bioinformatics tools. The Python package implements an AI \texttt{Agent} that is powered by LLMs and connects to a local file system, online databases, and a user's software. The \texttt{Agent} is highly configurable, enabling tasks such as Retrieval-Augmented Generation, searches across bioinformatics databases, and the execution of software pipelines. BRAD's coordinated integration of bioinformatics tools delivers a context-aware and semi-autonomous system that extends beyond the capabilities of conventional LLM-based chatbots. A graphical user interface (GUI) provides an intuitive interface to the system.

cs.AI↗

arXiv2021

MFNets: Data efficient all-at-once learning of multifidelity surrogates as directed networks of information sources

We present an approach for constructing a surrogate from ensembles of information sources of varying cost and accuracy. The multifidelity surrogate encodes connections between information sources as a directed acyclic graph, and is trained via gradient-based minimization of a nonlinear least squares objective. While the vast majority of state-of-the-art assumes hierarchical connections between information sources, our approach works with flexibly structured information sources that may not admit a strict hierarchy. The formulation has two advantages: (1) increased data efficiency due to parsimonious multifidelity networks that can be tailored to the application; and (2) no constraints on the training data -- we can combine noisy, non-nested evaluations of the information sources. Numerical examples ranging from synthetic to physics-based computational mechanics simulations indicate the error in our approach can be orders-of-magnitude smaller, particularly in the low-data regime, than single-fidelity and hierarchical multifidelity approaches.

cs.LG↗

arXiv2020

Efficient MCMC Sampling for Bayesian Matrix Factorization by Breaking Posterior Symmetries

Bayesian low-rank matrix factorization techniques have become an essential tool for relational data analysis and matrix completion. A standard approach is to assign zero-mean Gaussian priors on the columns or rows of factor matrices to create a conjugate system. This choice of prior leads to simple implementations; however it also causes symmetries in the posterior distribution that can severely reduce the efficiency of Markov-chain Monte-Carlo (MCMC) sampling approaches. In this paper, we propose a simple modification to the prior choice that provably breaks these symmetries and maintains/improves accuracy. Specifically, we provide conditions that the Gaussian prior mean and covariance must satisfy so the posterior does not exhibit invariances that yield sampling difficulties. For example, we show that using non-zero linearly independent prior means significantly lowers the autocorrelation of MCMC samples, and can also lead to lower reconstruction errors.

stat.ML↗

arXiv2020

Bayesian System ID: Optimal management of parameter, model, and measurement uncertainty

We evaluate the robustness of a probabilistic formulation of system identification (ID) to sparse, noisy, and indirect data. Specifically, we compare estimators of future system behavior derived from the Bayesian posterior of a learning problem to several commonly used least squares-based optimization objectives used in system ID. Our comparisons indicate that the log posterior has improved geometric properties compared with the objective function surfaces of traditional methods that include differentially constrained least squares and least squares reconstructions of discrete time steppers like dynamic mode decomposition (DMD). These properties allow it to be both more sensitive to new data and less affected by multiple minima --- overall yielding a more robust approach. Our theoretical results indicate that least squares and regularized least squares methods like dynamic mode decomposition and sparse identification of nonlinear dynamics (SINDy) can be derived from the probabilistic formulation by assuming noiseless measurements. We also analyze the computational complexity of a Gaussian filter-based approximate marginal Markov Chain Monte Carlo scheme that we use to obtain the Bayesian posterior for both linear and nonlinear problems. We then empirically demonstrate that obtaining the marginal posterior of the parameter dynamics and making predictions by extracting optimal estimators (e.g., mean, median, mode) yields orders of magnitude improvement over the aforementioned approaches. We attribute this performance to the fact that the Bayesian approach captures parameter, model, and measurement uncertainties, whereas the other methods typically neglect at least one type of uncertainty.

stat.ML↗

arXiv2019

Adaptive Multi-index Collocation for Uncertainty Quantification and Sensitivity Analysis

In this paper, we present an adaptive algorithm to construct response surface approximations of high-fidelity models using a hierarchy of lower fidelity models. Our algorithm is based on multi-index stochastic collocation and automatically balances physical discretization error and response surface error to construct an approximation of model outputs. This surrogate can be used for uncertainty quantification (UQ) and sensitivity analysis (SA) at a fraction of the cost of a purely high-fidelity approach. We demonstrate the effectiveness of our algorithm on a canonical test problem from the UQ literature and a complex multi-physics model that simulates the performance of an integrated nozzle for an unmanned aerospace vehicle. We find that, when the input-output response is sufficiently smooth, our algorithm produces approximations that can be over two orders of magnitude more accurate than single fidelity approximations for a fixed computational budget.

math.NA↗

arXiv2019

Randomized Functional Sparse Tucker Tensor for Compression and Fast Visualization of Scientific Data

We propose a strategy to compress and store large volumes of scientific data represented on unstructured grids. Approaches utilizing tensor decompositions for data compression have already been proposed. Here, data on a structured grid is stored as a tensor which is then subjected to appropriate decomposition in suitable tensor formats. Such decompositions are based on generalization of singular value decomposition to tensors and capture essential features in the data with storage cost lower by orders of magnitude. However, tensor based data compression is limited by the fact that one can only consider scientific data represented on structured grids. In case of data on unstructured meshes, we propose to consider data as realizations of a function that is based on functional view of the tensor thus avoiding such limitations. The key is to efficiently estimate the parameters of the function whose complexity is small compared to the cardinality of the dataset (otherwise there is no compression). Here, we introduce the set of functional sparse Tucker tensors and propose a method to construct approximation in this set such that the resulting compact functional tensor can be rapidly evaluated to recover the original data. The compression procedure consists of three steps. In the first step, we consider a fraction of the original dataset for interpolation on a structured grid followed by sequentially truncated higher order singular value decomposition to get a compressed version of the interpolated data.We then fit singular vectors on a set of functional basis using sparse approximation to obtain corresponding functional sparse Tucker tensor representation. Finally, we re-evaluate the coefficients of this functional tensor using randomized least squares at a reduced computational complexity. This strategy leads to compression ratio of orders of magnitude on combustion simulation datasets.

math.NA↗