Krylov Subspace Methods

The Levenberg-Marquardt (LM) algorithm is a well-known second-order method for rapid convergence and strong robustness when training small- to medium-sized neural networks (NNs). However, its computational and memory costs increase significantly as the number of parameters in an NN grows. To address this limitation, subspace methods have been proposed, such as the Krylov subspace LM (KSLM) and the hybrid subspace LM (HSLM), making second-order algorithms more efficient. In this work, we evaluate the subspace Levenberg-Marquardt algorithms for regression and classification tasks in neural networks. We compare the performance of subspace LM variants with the classical LM method, as well as other popular first-order algorithms, such as stochastic gradient descent (SGD) and Adam.

Spectral positional encodings (PEs) for \emph{directed} graphs face two obstacles: magnetic Laplacians require an $O(n^3)$ Hermitian eigendecomposition per potential, and their complex eigenvectors are defined only up to unitary gauge, which prior work handles with basis-invariant architectures. We propose learnable spectral PEs of the form $h_θ(A_q)\,R$, where $A_q$ is a normalized magnetic operator, $h_θ$ a learnable scalar spectral response, and $R$ a block of random probes. Because the PE is a \emph{matrix function} of the operator, it is gauge-invariant by construction. We compute it in a Hermitian block Krylov subspace from sparse matrix--vector products only, prove that $k = O(\log(1/\varepsilon))$ block steps suffice uniformly over heat--resolvent response families, and give a covering-number argument for why low-dimensional structured families generalize where free per-eigenvalue weights overfit. On a directed SBM whose symmetrization is uninformative by construction, direction-blind PEs stay at chance while magnetic Krylov PEs converge to the exact-eigendecomposition oracle as the depth grows. The same probes yield gauge-invariant pairwise features with $1/\sqrt{s}$ Monte-Carlo error, and the undirected $q{=}0$ case improves heterophilous benchmarks over no-PE and polynomial baselines.

Graph convolutional networks (GCNs) have demonstrated significant success in capturing complex user-item relationships for collaborative filtering (CF). However, due to their reliance on extensive model training, training-free graph filtering (GF)-based CF methods have emerged as a promising alternative, offering computational efficiency by smoothing graph signals via matrix operations. In particular, polynomial GF-based approaches demonstrate improved accuracy through their ability to design more expressive and flexible filtering functions. Despite these advantages, existing GF methods suffer from a critical memory bottleneck: they necessitate storing the full item similarity graph, incurring prohibitive memory costs for large-scale datasets, which limits their practical applicability. To tackle this challenge, we propose Mem-GF (Memory-efficient GF), a new GF-based CF method that departs from conventional designs by principally leveraging the structure of Krylov subspaces as a core mechanism for approximating polynomial graph filters without explicitly storing the item similarity graph. We theoretically analyze the minimum Krylov subspace size that guarantees lossless approximation. Through extensive experiments, we demonstrate that Mem-GF achieves up to 5.74$\times$ lower memory usage and 4.38$\times$ speedup in runtime, while consistently exceeding the recommendation accuracy of state-of-the-art GF and GCN-based methods. Mem-GF robustly scales to datasets with tens of millions of interactions, establishing itself as a practically viable and theoretically grounded solution for efficient CF.

Federated learning systems must support data deletion requests to comply with privacy regulations, yet retraining from scratch after each deletion is computationally prohibitive. We present HF-KCU, a method that removes a client's contribution by approximating the influence function through conjugate gradient iterations in Krylov subspaces, reducing complexity from O(d^3) to O(kd) where k<<d.A causal weighting mechanism ensures that only clients holding the deleted data receive parameter updates, preventing spurious changes to unaffected clients. Our method is designed to handle bounded adversarial perturbations to the Hessian and gradient, providing graceful degradation under realistic threat models. We validate HF-KCU across convolutional (ResNet-18, SimpleCNN) and transformer (ViT-Lite) architectures on CIFAR-10, MNIST, and Fashion-MNIST. On CIFAR-10 under Dirichlet (alpha=0.5) partitioning, HF-KCU achieves 47.75 times speedup over retraining while maintaining test accuracy within 0.60% of the rational baseline(71.16 vs 71.76 %). Membership inference attacks on the forget set yield success rates of 0.499 matching the retrained model and confirming effective privacy restoration. We provide convergence guarantees showing that the Krylov approximation error decreases as O((k ^1/2-1)/(k^1/2+1)) where k is the Hessian condition number. The causal weighting mechanism ensures surgical updates, where only clients holding deleted data are modified, preserving model quality for unaffected participants and avoiding the instability of gradient-based approaches in asynchronous federated settings. This design provides interpretability as each update is directly traceable to the influence of the deleted data. The method's efficiency and precision make it suitable for production federated systems where deletion requests arrive asynchronously and computational budgets are constrained.

Reliable adaptive beamforming is critical for large microphone arrays operating in highly dynamic acoustic environments. In scenarios characterized by fast-moving talkers and interferers, the available sample support for estimating the spatial correlation matrix is often snapshot-deficient. This deficiency degrades the White Noise Gain (WNG), leading to severe target signal cancellation. To ensure stable and robust beamforming, we previously proposed an adaptive diagonal loading method that leverages the Kantorovich inequality to guarantee the WNG remains strictly within specified bounds. However, accurately determining the smallest necessary loading level requires calculating the extreme eigenvalues of the spatial correlation matrix, a computationally expensive $\mathcal{O}(M^3)$ operation for large arrays. In this paper, we introduce a highly efficient $\mathcal{O}(kM^2)$ estimation technique using Lanczos iterations to build a small Krylov subspace. By projecting the correlation matrix onto a tridiagonal matrix of dimension $k \ll M$, we extract Ritz values that rapidly converge to the exact extreme eigenvalues. Our evaluations demonstrate that this Lanczos-accelerated approach achieves performance identical to exact Eigenvalue Decomposition (EVD), ensuring optimal interference suppression and strict WNG adherence at a fraction of the computational cost.

The convergence of Krylov-based linear iterative solvers applied to parametric partial differential equations (PDEs) is often highly sensitive to the domain, its discretization, the location/values of the applied Dirichlet/Neumann boundary conditions, body forces and material properties, among others. We have previously introduced hybridization of classical linear iterative solvers with neural operators for specific geometries, but they tend to not perform well on geometries not previously seen during training. We partially addressed this challenge by introducing the deep operator network Geo-DeepONet and hybridizing it with Krylov-based iterative linear solvers, which, despite learning effectively across arbitrary unstructured meshes without requiring retraining, led to only modest reductions in iterations compared to state-of-the-art preconditioners. In this study we introduce Neural Subspace Proper Orthogonal Decomposition (NSPOD), a multigrid-like deep operator network-based preconditioner which can dramatically reduce the number of iterations needed for convergence in Krylov-based linear iterative solvers, even when compared to state-of-the-art methods such as algebraic multigrid preconditioners. We demonstrate its efficiency via numerical experiments on a linearized version of solid mechanics PDEs applied to unstructured domains obtained from complex CAD geometries. We expect that the findings in this study lead to more efficient hybrid preconditioners that can match, or possibly even surpass, the convergence properties of the current gold standard preconditioning methods for solid mechanics PDEs.

Random-feature neural networks (RFNNs), including architectures with fixed hidden layers and analytically determined output weights, offer fast training but often suffer from issues due to dense representations of the hidden layer activation. Their reliance on dense feature mappings and least squares solvers can limit scalability and numerical stability, particularly for high-dimensional or stiff systems. Specifically, the activation matrix is observed to be low-rank and extremely ill-conditioned. In this work, we propose a sparse framework for RFNNs that integrates structured sparsity into the hidden layer activations that increases the rank and employs Sparse Singular Value Decomposition (sSVD) for solving the resulting linear least squares problem scalably and efficiently while catering to the bad condition number. We explore the theory behind Lanczos-Golub-Kahan Bidiagonalization technique for sparse SVD and conduct some experiments to identify some limitations and justify the requirement for orthogonalization step in our application. Then, we demonstrate that the proposed method maintains or improves solution accuracy for solving the benchmark one-dimensional steady convection-diffusion equations case having stronger advection, while achieving substantial gains in training efficiency and robustness compared to standard dense implementations.

The quest for an algorithm that solves an $n\times n$ linear system in $O(n^2)$ time complexity, or $O(n^2 \text{poly}(1/ε))$ when solving up to $ε$ relative error, is a long-standing open problem in numerical linear algebra and theoretical computer science. There are two predominant paradigms for measuring relative error: forward error (i.e., distance from the output to the optimum solution) and backward error (i.e., distance to the nearest problem solved by the output). In most prior studies, convergence of iterative linear system solvers is measured via various notions of forward error, and as a result, depends heavily on the conditioning of the input. Yet, the numerical analysis literature has long advocated for backward error as the more practically relevant notion of approximation. In this work, we show that -- surprisingly -- the classical and simple Richardson iteration incurs at most $1/k$ (relative) backward error after $k$ iterations on any positive semidefinite (PSD) linear system, irrespective of its condition number. This universal convergence rate implies an $O(n^2/ε)$ complexity algorithm for solving a PSD linear system to $ε$ backward error, and we establish similar or better complexity when using a variety of Krylov solvers beyond Richardson. Then, by directly minimizing backward error over a Krylov subspace, we attain an even faster $O(1/k^2)$ universal rate, and we turn this into an efficient algorithm, MINBERR, with complexity $O(n^2/\sqrtε)$. Finally, we extend this approach via normal equations to solving general linear systems in $O(n^2\log(n)/ε)$ time complexity. We report strong numerical performance of our algorithms on benchmark problems.

Mixed-effects models are widely used to model data with complex grouping structures and high-cardinality categorical predictor variables. However, for high-dimensional crossed random effects, current standard computations relying on Cholesky decompositions can become prohibitively slow. In this work, we present Krylov subspace-based methods that address existing computational bottlenecks, and we analyze them both theoretically and empirically. In particular, we derive new results on the convergence and accuracy of the preconditioned stochastic Lanczos quadrature and conjugate gradient methods for mixed-effects models, and we develop scalable methods for calculating predictive variances. In experiments with simulated and real-world data, the proposed methods yield speedups of several orders of magnitude and are more computationally robust than Cholesky-based computations, while maintaining essentially the same accuracy.