arXiv subjects

Yuanzhe Xi

Publications and source records attributed to Yuanzhe Xi.

At least 19 recordsLinked to original sources

arXiv2026

Design Criteria for SGD Preconditioners: Local Conditioning, Noise Floors, and Basin Stability

Stochastic Gradient Descent (SGD) often slows in the late stage of training due to anisotropic curvature and gradient noise. We analyze preconditioned SGD in the geometry induced by a symmetric positive definite matrix $\mathbf{M}$, deriving bounds in which both the convergence rate and the stochastic noise floor are governed by $\mathbf{M}$-dependent quantities: the rate through an effective condition number in the $\mathbf{M}$-metric, and the floor through the product of that condition number and the preconditioned noise level. For nonconvex objectives, we establish a preconditioner-dependent basin-stability guarantee: when smoothness and basin size are measured in the $\mathbf{M}$-norm, the probability that the iterates remain in a well-behaved local region admits an explicit lower bound. This perspective is particularly relevant in Scientific Machine Learning (SciML), where achieving small training loss under stochastic updates is closely tied to physical fidelity, numerical stability, and constraint satisfaction. The framework applies to both diagonal/adaptive and curvature-aware preconditioners and yields a simple design principle: choose $\mathbf{M}$ to improve local conditioning while attenuating noise. Experiments on a quadratic diagnostic and three SciML benchmarks validate the predicted rate-floor behavior.

math.NA↗

arXiv2026

Hierarchical Muon: Tiled Newton-Schulz Updates for Efficient Muon Optimization

Muon-type optimizers construct update directions for dense neural-network weights by applying a finite Newton-Schulz map to momentum-gradient matrices. For an $H \times W$ matrix, with $r=\min\{H,W\}$ and $s=\max\{H,W\}$, $K$ steps of the full-matrix Newton-Schulz update require $O(r^2 s K)$ work and couple all rows and columns through repeated Gram matrix products. We introduce Hierarchical Muon (HiMuon), a tiled Newton-Schulz scheme for Muon-type optimization. HiMuon partitions each momentum-gradient matrix into $T \times T$ tiles, applies the same finite Newton-Schulz map independently to each tile, and reassembles the results. For finite $T$ below the matrix dimensions, HiMuon defines a local matrix-function map rather than a convergent approximation to the full-matrix update: spectral interactions are preserved within tiles and discarded across tile boundaries. For fixed finite $T$, the leading Newton-Schulz work decreases to $O(H W T K)$, and the computation decomposes into independent small dense matrix operations. This structure enables tile-size-dependent GPU kernels, cross-layer batching, memory-bounded chunking, and runtime tile-size schedules. Experiments on transformer training and controlled matrix-function diagnostics show that HiMuon improves optimizer-step efficiency while keeping training behavior close to full-matrix Muon in the tested regimes.

math.NA↗

arXiv2026

Factored Sparse Approximate Inverse Preconditioning via Spectral Optimization

In this paper, we study value selection for fixed-pattern factorized sparse approximate inverse preconditioners. Given a prescribed sparsity pattern for a factor $G,$ we choose its admissible entries by optimizing spectral objectives of the congruent preconditioned operator $P(G)=GAG^T.$ This differs from classical sparse approximate inverse and FSAI constructions, which choose entries through algebraic Frobenius-residual criteria. For symmetric positive definite systems, the spectral target is a cluster near $+1.$ For symmetric indefinite systems, where congruence preserves inertia, we introduce a bimodal loss that drives positive and negative eigenvalues toward separated clusters near $+1$ and $-1,$ while penalizing eigenvalues near zero. To make these objectives practical for large sparse matrices, we derive projected Krylov support-gradients. Lanczos runs provide both a stochastic trace estimate of the spectral objective and a Ritz approximation to the exact gradient. We implement the resulting gradient through a detached Rayleigh surrogate: the Lanczos data are computed without gradient tracking and held fixed, while the backward pass differentiates only recomputed Rayleigh quotients with respect to the admissible entries of $G.$ This avoids differentiating through the Lanczos recurrence while returning a matrix-free gradient on the prescribed support. We also discuss a projected Kernel Polynomial Method rule as a finite polynomial comparison. Experiments on finite-element test problems show that spectral value selection improves fixed-support preconditioners, especially for symmetric indefinite saddle-point systems. We further demonstrate a graph neural network model for predicting admissible factor entries across related matrices.

math.NA↗

arXiv2026

Hybrid Digital-Analog Approximate Inverse Preconditioning for Krylov Methods

Analog in-memory computing enables highly parallel matrix-vector multiplications with reduced data movement, but the resulting operations are noisy, quantized, and affected by device- and circuit-level non-idealities. This paper studies approximate inverse preconditioning for Krylov subspace methods in a hybrid digital-analog setting. The digital host performs sparse products with the coefficient matrix and the precision-sensitive Krylov operations, while preconditioner applications are performed through analog crossbar matrix-vector multiplications. Since the realized preconditioner is inexact and application-dependent, the outer iteration is formulated as the flexible GMRES method. We show that analog execution changes the usual preconditioner design problem in the sense that a stronger digital preconditioner may be less effective after analog scaling, write noise, input/output perturbations, quantization, and clipping are taken into account. We compare various block Jacobi preconditioning schemes including exact block inverses, sparse approximate inverses, Monte Carlo approximate inverses (MCAI), damping, and nested block Jacobi schemes. Numerical experiments with realistic analog matrix-vector simulations show that analog-aware choices of block size, damping, MCAI construction accuracy, and nesting are important for robust convergence.

math.NA↗

arXiv2026

A Joint Variational Framework for Multimodal X-ray Ptychography and Fluorescence Reconstruction

Recovering high-resolution structural and compositional information from coherent X-ray measurements involves solving coupled, nonlinear, and ill-posed inverse problems. Ptychography reconstructs a complex transmission function from overlapping diffraction patterns, while X-ray fluorescence provides quantitative, element-specific contrast at lower spatial resolution. We formulate a joint variational framework that integrates these two modalities into a single nonlinear least-squares problem with shared spatial variables. This formulation enforces cross-modal consistency between structural and compositional estimates, improving conditioning and promoting stable convergence. The resulting optimization couples complementary contrast mechanisms (i.e., phase and absorption from ptychography, elemental composition from fluorescence) within a unified inverse model. Numerical experiments on simulated data demonstrate that the joint reconstruction achieves faster convergence, sharper and more quantitative reconstructions, and lower relative error compared with separate inversions. The proposed approach illustrates how multimodal variational formulations can enhance stability, resolution, and interpretability in computational X-ray imaging.

math.NA↗

arXiv2026

Preconditioned Truncated Single-Sample Estimators for Scalable Stochastic Optimization

Many large-scale stochastic optimization algorithms involve repeated solutions of linear systems or evaluations of log-determinants. In these regimes, computing exact solutions is often unnecessary; it is more computationally efficient to construct unbiased stochastic estimators with controlled variance. However, classical iterative solvers incur truncation bias, whereas unbiased Krylov-based estimators typically exhibit high variance and numerical instability. To mitigate these issues, we introduce the Preconditioned Truncated Single-Sample (PTSS) estimators--a family of stochastic Krylov methods that integrate preconditioning with truncated Lanczos iterations. PTSS yields low-variance, stable estimators for linear system solutions, log-determinants, and their derivatives. We establish theoretical results on their mean, variance, and concentration properties, explicitly quantifying the variance reduction induced by preconditioning. Numerical experiments confirm that PTSS achieves superior stability and variance control compared with existing unbiased and biased alternatives, providing an efficient framework for stochastic optimization.

math.NA↗

arXiv2026

HiGP: A high-performance Python package for Gaussian Process

Gaussian Processes (GPs) are flexible, nonparametric Bayesian models widely used for regression and classification because of their ability to capture complex data patterns and quantify predictive uncertainty. However, the O(n^3) computational cost of kernel matrix operations poses a major obstacle to applying GPs at scale. HiGP is a high-performance Python package designed to overcome these scalability limitations through advanced numerical linear algebra and hierarchical kernel representations. It integrates H^2 matrices to achieve near-linear complexity in both storage and computation for spatial datasets, supports on-the-fly kernel evaluation to avoid explicit storage in large-scale problems, and incorporates a robust Adaptive Factorized Nyström (AFN) preconditioner that accelerates convergence of iterative solvers across a broad range of kernel spectra. These computational kernels are implemented in C++ for maximum performance and exposed through Python interfaces, enabling seamless integration with modern machine learning workflows. HiGP also includes analytically derived gradient computations for efficient hyperparameter optimization, avoiding the inefficiencies of automatic differentiation in iterative solvers. By serving as a reusable numerical engine, HiGP complements existing GP frameworks such as GPJax, KeOps, and GaussianProcesses.jl, providing a reliable and scalable computational backbone for large-scale Gaussian Process regression and classification.

cs.LG↗

arXiv2025

Multiscale Neural Networks for Approximating Green's Functions

Neural networks (NNs) have been widely used to solve partial differential equations (PDEs) in the applications of physics, biology, and engineering. One effective approach for solving PDEs with a fixed differential operator is learning Green's functions. However, Green's functions are notoriously difficult to learn due to their poor regularity, which typically requires larger NNs and longer training times. In this paper, we address these challenges by leveraging multiscale NNs to learn Green's functions. Through theoretical analysis using multiscale Barron space methods and experimental validation, we show that the multiscale approach significantly reduces the necessary NN size and accelerates training.

math.NA↗

arXiv2025

Neural Approximate Inverse Preconditioners

In this paper, we propose a data-driven framework for constructing efficient approximate inverse preconditioners for elliptic partial differential equations (PDEs) by learning the Green's function of the underlying operator with neural networks (NNs). The training process integrates four key components: an adaptive multiscale neural architecture ($α$MSNN) that captures hierarchical features across near-, middle-, and far-field regimes; the use of coarse-grid anchor data to ensure physical identifiability; a multi-$\varepsilon$ staged training protocol that progressively refines the Green's function representation across spatial scales; and an overlapping domain decomposition that enables local adaptation while maintaining global consistency. Once trained, the NN-approximated Green's function is directly compressed into either a hierarchical ($\mathcal{H}$-) matrix or a sparse matrix-using only the mesh geometry and the network output. This geometric construction achieves nearly linear complexity in both setup and application while preserving the spectral properties essential for effective preconditioning. Numerical experiments on challenging elliptic PDEs demonstrate that the resulting preconditioners consistently yield fast convergence and small iteration counts.

math.NA↗

arXiv2025

Rethinking Neural-based Matrix Inversion: Why can't, and Where can

Deep neural networks have achieved substantial success across various scientific computing tasks. A pivotal challenge within this domain is the rapid and parallel approximation of matrix inverses, critical for numerous applications. Despite significant progress, there currently exists no universal neural-based method for approximating matrix inversion. This paper presents a theoretical analysis demonstrating the fundamental limitations of neural networks in developing a general matrix inversion model. We expand the class of Lipschitz functions to encompass a wider array of neural network models, thereby refining our theoretical approach. Moreover, we delineate specific conditions under which neural networks can effectively approximate matrix inverses. Our theoretical results are supported by experimental results from diverse matrix datasets, exploring the efficacy of neural networks in addressing the matrix inversion challenge.

cs.LG↗

arXiv2025

Spectral-Refiner: Accurate Fine-Tuning of Spatiotemporal Fourier Neural Operator for Turbulent Flows

Recent advancements in operator-type neural networks have shown promising results in approximating the solutions of spatiotemporal Partial Differential Equations (PDEs). However, these neural networks often entail considerable training expenses, and may not always achieve the desired accuracy required in many scientific and engineering disciplines. In this paper, we propose a new learning framework to address these issues. A new spatiotemporal adaptation is proposed to generalize any Fourier Neural Operator (FNO) variant to learn maps between Bochner spaces, which can perform an arbitrary-length temporal super-resolution for the first time. To better exploit this capacity, a new paradigm is proposed to refine the commonly adopted end-to-end neural operator training and evaluations with the help from the wisdom from traditional numerical PDE theory and techniques. Specifically, in the learning problems for the turbulent flow modeled by the Navier-Stokes Equations (NSE), the proposed paradigm trains an FNO only for a few epochs. Then, only the newly proposed spatiotemporal spectral convolution layer is fine-tuned without the frequency truncation. The spectral fine-tuning loss function uses a negative Sobolev norm for the first time in operator learning, defined through a reliable functional-type a posteriori error estimator whose evaluation is exact thanks to the Parseval identity. Moreover, unlike the difficult nonconvex optimization problems in the end-to-end training, this fine-tuning loss is convex. Numerical experiments on commonly used NSE benchmarks demonstrate significant improvements in both computational efficiency and accuracy, compared to end-to-end evaluation and traditional numerical PDE solvers under certain conditions. The source code is publicly available at https://github.com/scaomath/torch-cfd.

cs.LG↗

arXiv2025

Mixed Precision Orthogonalization-Free Projection Methods for Eigenvalue and Singular Value Problems

Mixed-precision arithmetic offers significant computational advantages for large-scale matrix computation tasks, yet preserving accuracy and stability in eigenvalue problems and the singular value decomposition (SVD) remains challenging. This paper introduces an approach that eliminates orthogonalization requirements in traditional Rayleigh-Ritz projection methods. The proposed method employs non-orthogonal bases computed at reduced precision, resulting in bases computed without inner-products. A primary focus is on maintaining the linear independence of the basis vectors. Through extensive evaluation with both synthetic test cases and real-world applications, we demonstrate that the proposed approach achieves the desired accuracy while fully taking advantage of mixed-precision arithmetic.

math.NA↗

arXiv2025

Posterior Covariance Structures in Gaussian Processes

In this paper, we present a comprehensive analysis of the posterior covariance field in Gaussian processes, with applications to the posterior covariance matrix. The analysis is based on the Gaussian prior covariance but the approach also applies to other covariance kernels. Our geometric analysis reveals how the Gaussian kernel's bandwidth parameter and the spatial distribution of the observations influence the posterior covariance as well as the corresponding covariance matrix, enabling straightforward identification of areas with high or low covariance in magnitude. Drawing inspiration from the a posteriori error estimation techniques in adaptive finite element methods, we also propose several estimators to efficiently measure the absolute posterior covariance field, which can be used for efficient covariance matrix approximation and preconditioning. We conduct a wide range of experiments to illustrate our theoretical findings and their practical applications.

stat.ML↗

arXiv2025

Preconditioned Additive Gaussian Processes with Fourier Acceleration

Gaussian processes (GPs) are crucial in machine learning for quantifying uncertainty in predictions. However, their associated covariance matrices, defined by kernel functions, are typically dense and large-scale, posing significant computational challenges. This paper introduces a matrix-free method that utilizes the Non-equispaced Fast Fourier Transform (NFFT) to achieve nearly linear complexity in the multiplication of kernel matrices and their derivatives with vectors for a predetermined accuracy level. To address high-dimensional problems, we propose an additive kernel approach. Each sub-kernel in this approach captures lower-order feature interactions, allowing for the efficient application of the NFFT method and potentially increasing accuracy across various real-world datasets. Additionally, we implement a preconditioning strategy that accelerates hyperparameter tuning, further improving the efficiency and effectiveness of GPs.

cs.LG↗

arXiv2024

Straggler-tolerant stationary methods for linear systems

In this paper, we consider the iterative solution of linear algebraic equations under the condition that matrix-vector products with the coefficient matrix are computed only partially. At the same time, non-computed entries are set to zeros. We assume that both the number of computed entries and their associated row index set are random variables, with the row index set sampled uniformly given the number of computed entries. This model of computations is realized in hybrid cloud computing architectures following the controller-worker distributed model under the influence of straggling workers. We propose straggler-tolerant Richardson iteration scheme and Chebyshev semi-iterative schemes, and prove sufficient conditions for their convergence in expectation. Numerical experiments verify the presented theoretical results as well as the effectiveness of the proposed schemes on a few sparse matrix problems.

math.NA↗

arXiv2024

Efficient Two-Stage Gaussian Process Regression Via Automatic Kernel Search and Subsampling

Gaussian Process Regression (GPR) is widely used in statistics and machine learning for prediction tasks requiring uncertainty measures. Its efficacy depends on the appropriate specification of the mean function, covariance kernel function, and associated hyperparameters. Severe misspecifications can lead to inaccurate results and problematic consequences, especially in safety-critical applications. However, a systematic approach to handle these misspecifications is lacking in the literature. In this work, we propose a general framework to address these issues. Firstly, we introduce a flexible two-stage GPR framework that separates mean prediction and uncertainty quantification (UQ) to prevent mean misspecification, which can introduce bias into the model. Secondly, kernel function misspecification is addressed through a novel automatic kernel search algorithm, supported by theoretical analysis, that selects the optimal kernel from a candidate set. Additionally, we propose a subsampling-based warm-start strategy for hyperparameter initialization to improve efficiency and avoid hyperparameter misspecification. With much lower computational cost, our subsampling-based strategy can yield competitive or better performance than training exclusively on the full dataset. Combining all these components, we recommend two GPR methods-exact and scalable-designed to match available computational resources and specific UQ requirements. Extensive evaluation on real-world datasets, including UCI benchmarks and a safety-critical medical case study, demonstrates the robustness and precision of our methods.

cs.LG↗

arXiv2024

Anderson Acceleration with Truncated Gram-Schmidt

Anderson Acceleration (AA) is a popular algorithm designed to enhance the convergence of fixed-point iterations. In this paper, we introduce a variant of AA based on a Truncated Gram-Schmidt process (AATGS) which has a few advantages over the classical AA. In particular, an attractive feature of AATGS is that its iterates obey a three-term recurrence in the situation when it is applied to solving symmetric linear problems and this can lead to a considerable reduction of memory and computational costs. We analyze the convergence of AATGS in both full-depth and limited-depth scenarios and establish its equivalence to the classical AA in the linear case. We also report on the effectiveness of AATGS through a set of numerical experiments, ranging from solving nonlinear partial differential equations to tackling nonlinear optimization problems. In particular, the performance of the method is compared with that of the classical AA algorithms.

math.NA↗

arXiv2024

An Adaptive Factorized Nyström Preconditioner for Regularized Kernel Matrices

The spectrum of a kernel matrix significantly depends on the parameter values of the kernel function used to define the kernel matrix. This makes it challenging to design a preconditioner for a regularized kernel matrix that is robust across different parameter values. This paper proposes the Adaptive Factorized Nyström (AFN) preconditioner. The preconditioner is designed for the case where the rank k of the Nyström approximation is large, i.e., for kernel function parameters that lead to kernel matrices with eigenvalues that decay slowly. AFN deliberately chooses a well-conditioned submatrix to solve with and corrects a Nyström approximation with a factorized sparse approximate matrix inverse. This makes AFN efficient for kernel matrices with large numerical ranks. AFN also adaptively chooses the size of this submatrix to balance accuracy and cost.

math.NA↗