arXiv subjects

Ruhui Jin

Publications and source records attributed to Ruhui Jin.

9 recordsLinked to original sources

arXiv2025

Data selection: at the interface of PDE-based inverse problem and randomized linear algebra

All inverse problems rely on data to recover unknown parameters, yet not all data are equally informative. This raises the central question of data selection. A distinctive challenge in PDE-based inverse problems is their inherently infinite-dimensional nature: both the parameter space and the design space are infinite, which greatly complicates the selection process. Somewhat unexpectedly, randomized numerical linear algebra (RNLA), originally developed in very different contexts, has provided powerful tools for addressing this challenge. These methods are inherently probabilistic, with guarantees typically stating that information is preserved with probability at least 1-p when using N randomly selected, weighted samples. Here, the notion of "information" can take different mathematical forms depending on the setting. In this review, we survey the problem of data selection in PDE-based inverse problems, emphasize its unique infinite-dimensional aspects, and highlight how RNLA strategies have been adapted and applied in this context.

math.NA↗

arXiv2024

Continuous nonlinear adaptive experimental design with gradient flow

In computational inverse problems, the optimal experimental design (OED) problem seeks the best locations in time and space at which to take measurements. We investigate the nonlinear OED problem in the context of continuously-indexed design space for the measurements. In contrast to traditional approaches that select experiments from a finite measurement set, a continuous design space is often a better reflection of practical experimental options, where there is considerable flexibility concerning where and when to take measurements. The continuously-indexed space introduces computational challenges, and we address them by employing gradient-flow and optimal transport techniques, complemented by an adaptive strategy for bi-level optimization. Numerical results on the Lorenz 63 system and Schrodinger equation demonstrate that our solver identifies good measurement times / locations and achieves improved reconstruction of unknown parameters in inverse problems.

math.NA↗

arXiv2024

Unique reconstruction for discretized inverse problems: a random sketching approach via subsampling

Theoretical inverse problems are often studied in an ideal infinite-dimensional setting. The well-posedness theory provides a unique reconstruction of the parameter function, when an infinite amount of data is given. Through the lens of PDE-constrained optimization, this means one attains the zero-loss property of the mismatch function in this setting. This is no longer true in computations when we are limited to finite amount of measurements due to experimental or economical reasons. Consequently, one must compromise the goal, from inferring a function, to a discrete approximation. What is the reconstruction power of a fixed number of data observations? How many parameters can one reconstruct? Here we describe a probabilistic approach, and spell out the interplay of the observation size $(r)$ and the number of parameters to be uniquely identified $(m)$. The technical pillar here is the random sketching strategy, in which the matrix concentration inequality and sampling theory are largely employed. By analyzing a randomly subsampled Hessian matrix, we attain a well-conditioned reconstruction problem with high probability. Our main theory is validated in numerical experiments, using an elliptic inverse problem as an example.

math.NA↗

arXiv2024

Optimal design for linear models via gradient flow

Optimal experimental design (OED) aims to choose the observations in an experiment to be as informative as possible, according to certain statistical criteria. In the linear case (when the observations depend linearly on the unknown parameters), it seeks the optimal weights over rows of the design matrix $\mA$ under certain criteria. Classical OED assumes a discrete design space and thus a design matrix with finite dimensions. In many practical situations, however, the design space is continuous-valued, so that the OED problem is one of optimizing over a continuous-valued design space. The objective becomes a functional over the probability measure, instead of a function of a finite dimensional vector. This change of perspective requires a new set of techniques to optimize over probability measures, and Wasserstein gradient flow becomes a natural candidate. Both the first-order criticality and the convexity properties of the OED objective are presented. Computationally, the Monte Carlo particle method is used to translate the gradient flow equation formulation into a numerical algorithm. This algorithm is applied to two elliptic inverse problems.

math.NA↗

arXiv2023

Scalable symmetric Tucker tensor decomposition

We study the best low-rank Tucker decomposition of symmetric tensors. The motivating application is decomposing higher-order multivariate moments. Moment tensors have special structure and are important to various data science problems. We advocate for projected gradient descent (PGD) method and higher-order eigenvalue decomposition (HOEVD) approximation as computation schemes. Most importantly, we develop scalable adaptations of the basic PGD and HOEVD methods to decompose sample moment tensors. With the help of implicit and streaming techniques, we evade the overhead cost of building and storing the moment tensor. Such reductions make computing the Tucker decomposition realizable for large data instances in high dimensions. Numerical experiments demonstrate the efficiency of the algorithms and the applicability of moment tensor decompositions to real-world datasets. Finally we study the convergence on the Grassmannian manifold, and prove that the update sequence derived by the PGD solver achieves first- and second-order criticality.

math.NA↗

arXiv2021

Space-time reduced-order modeling for uncertainty quantification

This work focuses on the space-time reduced-order modeling (ROM) method for solving large-scale uncertainty quantification (UQ) problems with multiple random coefficients. In contrast with the traditional space ROM approach, which performs dimension reduction in the spatial dimension, the space-time ROM approach performs dimension reduction on both the spatial and temporal domains, and thus enables accurate approximate solutions at a low cost. We incorporate the space-time ROM strategy with various classical stochastic UQ propagation methods such as stochastic Galerkin and Monte Carlo. Numerical results demonstrate that our methodology has significant computational advantages compared to state-of-the-art ROM approaches. By testing the approximation errors, we show that there is no obvious loss of simulation accuracy for space-time ROM given its high computational efficiency.

math.NA↗

arXiv2021

Tensor-structured sketching for constrained least squares

Constrained least squares problems arise in many applications. Their memory and computation costs are expensive in practice involving high-dimensional input data. We employ the so-called "sketching" strategy to project the least squares problem onto a space of a much lower "sketching dimension" via a random sketching matrix. The key idea of sketching is to reduce the dimension of the problem as much as possible while maintaining the approximation accuracy. Tensor structure is often present in the data matrices of least squares, including linearized inverse problems and tensor decompositions. In this work, we utilize a general class of row-wise tensorized sub-Gaussian matrices as sketching matrices in constrained optimizations for the sketching design's compatibility with tensor structures. We provide theoretical guarantees on the sketching dimension in terms of error criterion and probability failure rate. In the context of unconstrained linear regressions, we obtain an optimal estimate for the sketching dimension. For optimization problems with general constraint sets, we show that the sketching dimension depends on a statistical complexity that characterizes the geometry of the underlying problems. Our theories are demonstrated in a few concrete examples, including unconstrained linear regression and sparse recovery problems.

math.OC↗

arXiv2020

Faster Johnson-Lindenstrauss Transforms via Kronecker Products

The Kronecker product is an important matrix operation with a wide range of applications in supporting fast linear transforms, including signal processing, graph theory, quantum computing and deep learning. In this work, we introduce a generalization of the fast Johnson-Lindenstrauss projection for embedding vectors with Kronecker product structure, the Kronecker fast Johnson-Lindenstrauss transform (KFJLT). The KFJLT reduces the embedding cost to an exponential factor of the standard fast Johnson-Lindenstrauss transform (FJLT)'s cost when applied to vectors with Kronecker structure, by avoiding explicitly forming the full Kronecker products. We prove that this computational gain comes with only a small price in embedding power: given $N = \prod_{k=1}^d n_k$, consider a finite set of $p$ points in a tensor product of $d$ constituent Euclidean spaces $\bigotimes_{k=d}^{1}\mathbb{R}^{n_k} \subset \mathbb{R}^{N}$. With high probability, a random KFJLT matrix of dimension $N \times m$ embeds the set of points up to multiplicative distortion $(1\pm \varepsilon)$ provided by $m \gtrsim \varepsilon^{-2} \cdot \log^{2d - 1} (p) \cdot \log N$. We conclude by describing a direct application of the KFJLT to the efficient solution of large-scale Kronecker-structured least squares problems for fitting the CP tensor decomposition.

cs.IT↗

arXiv2018

SUNLayer: Stable denoising with generative networks

Deep neural networks are often used to implement powerful generative models for real-world data. Notable applications include image denoising, as well as other classical inverse problems like compressed sensing and super-resolution. To provide a rigorous but simplified analysis of generative models, in this work, we introduce an elegant theoretical framework based on spherical harmonics, namely \textbf{SUNLayer}. Our theoretical framework identifies explicit conditions on activation functions that guarantee denoising under local optimization. Numerical experiments examine the theoretical properties on commonly used activation functions and demonstrate their stable denoising performance.

cs.LG↗