krylov subspace methods

Computing low-pass graph filters over parametric graph families is computationally expensive due to repeated eigenvalue decompositions. Method: This paper introduces a novel low-frequency subspace interpolation method grounded in Riemannian geometry on the Grassmann manifold. It pioneers the use of normal coordinate interpolation for low-frequency subspace estimation, provides a theoretical error bound, incorporates a similarity correction mechanism to accommodate dynamic topology evolution, and constructs a dot-product-based graph-family-enhanced message-passing framework that enables node-feature-driven adaptive graph structure updates. Results: Experiments demonstrate substantial reduction in eigendecomposition overhead and improved message-passing performance on node classification tasks, effectively supporting dynamic graph modeling.

This work addresses key linear algebraic tasks—matrix-vector multiplication, matrix-matrix multiplication, spectral norm estimation, and all-entries summation—on $n imes n$ kernel matrices. We propose a unified approximation framework based on kernel density estimation (KDE), achieving $(1+varepsilon)$-relative error guarantees. Our core insight is to reduce each matrix operation to a set of KDE queries, thereby circumventing the standard $O(n^2)$ time barrier and attaining complexities nearly matching the optimal KDE runtime—for instance, $widetilde{O}(n/varepsilon^2)$ for all-entries summation. Theoretically, we establish the first conditional quadratic-time lower bounds for multiple kernel matrix problems and prove that our KDE-based paradigm achieves tight trade-offs between accuracy and efficiency. Empirically, our method significantly outperforms existing acceleration techniques on high-dimensional datasets.

This work addresses the challenge of efficiently computing leading eigenvectors in dynamic graphs, where frequent updates to the adjacency or Laplacian matrix render traditional eigendecomposition methods computationally prohibitive. To overcome this limitation, the authors propose a fast spectral embedding update framework based on Rayleigh-Ritz projection. By leveraging eigenvector perturbation analysis, the method constructs a low-dimensional approximate invariant subspace that preserves high approximation accuracy while substantially reducing computational and memory costs. Experimental results demonstrate that the proposed approach outperforms existing techniques in both the quality of leading eigenvector approximation and performance on downstream tasks—such as influential node identification and node clustering—offering a compelling balance between efficiency and accuracy.

This work proposes AutoSpec, a novel framework for automatically discovering efficient iterative spectral algorithms tailored to large-scale numerical linear algebra and optimization tasks. By uniquely integrating self-supervised learning with executable matrix polynomial recursions, AutoSpec leverages coarse spectral information to drive a neural network that adaptively predicts recursion coefficients, thereby constructing task-specific iterative solvers. The key innovation lies in the fusion of classical minimax approximation theory with data-driven learning, enabling effective generalization from small-scale synthetic problems to real-world large-scale operators. Experimental results demonstrate that the learned algorithms achieve several orders of magnitude improvement over baselines in both accuracy and convergence speed on real matrices, while exhibiting near-equiripple minimax behavior characteristic of optimal spectral approximations.

Existing data-driven Koopman operator methods struggle to ensure approximate invariance of subspaces under the operator in non-Euclidean settings, limiting predictive accuracy. This work addresses this challenge by extending principal vector–guided subspace pruning to reproducing kernel Hilbert spaces (RKHS) for the first time. By precisely computing principal angles and vectors in RKHS, we introduce Kernel-SPV and its computationally efficient Nyström approximation–based variant, Approximate Kernel-SPV. These approaches overcome the limitations of traditional Euclidean formulations, significantly enhancing the invariance of Koopman-invariant subspaces while maintaining scalability and substantially improving prediction accuracy.

This work addresses the challenges in multivariate polynomial models arising from the exponential growth of coefficient tensors with model order, the limited expressivity of existing tensor decomposition methods, and their sensitivity to feature ordering. To overcome these limitations, the authors propose an efficient optimization framework based on matrix product operators (MPOs). The approach leverages learnable MPO-based feature embeddings and compact polynomial weight tensors to achieve permutation-invariant function approximation. Furthermore, structured operators—such as projection, convolution, and masking—are incorporated to explicitly model weight symmetries, thereby transcending the expressivity bottlenecks of conventional tensor decompositions. Experimental results demonstrate that the proposed framework significantly outperforms existing tensor-decomposition-based polynomial models on both regression and classification benchmarks, offering a highly expressive, flexible, and computationally efficient solution for polynomial approximation.

This work addresses the problem of maintaining a consistent low-rank approximation for matrices whose rows arrive in a streaming fashion, with the dual objective of preserving approximation accuracy while minimizing recourse—the total amount of change in the output sequence over time. The authors propose a novel recursive subspace update algorithm that, at each time step, produces a near-optimal rank-$k$ approximation. They establish the first tight theoretical bounds characterizing the trade-off between recourse and both additive and multiplicative approximation errors. Leveraging techniques from streaming matrix processing, low-rank approximation, and condition number analysis, their theoretical framework significantly sharpens upper bounds under certain conditions. Empirical evaluations on real-world datasets demonstrate the algorithm’s efficiency and stability in practice.

This work addresses the challenge in graph signal processing where spectral-based filtering methods are often inapplicable due to incomplete knowledge of the full graph topology. To overcome this limitation, we propose the first data-driven algebraic framework for subgraph filtering, constructing a distance-aware Laplacian-based subgraph filtering algebra that defines a structured and controllable class of filters capable of approximating full-graph filters. Leveraging statistical learning theory, we establish risk bounds on the approximation performance under least-squares loss, providing rigorous theoretical guarantees. Empirical evaluations demonstrate that our approach significantly outperforms polynomial filters, distribution-agnostic operators, and end-to-end numerical learning baselines on real-world datasets.

This work proposes Clifford Kolmogorov-Arnold Networks (ClKAN) to address the exponential computational complexity inherent in function approximation within high-dimensional Clifford algebra spaces. By extending the Kolmogorov-Arnold representation theorem to the Clifford algebra domain for the first time, ClKAN mitigates the curse of dimensionality through randomized quasi-Monte Carlo grid generation and incorporates a tailored batch normalization mechanism that adapts to variable-domain inputs. Experimental evaluations demonstrate the efficacy of ClKAN on both synthetic benchmarks and physics-inspired tasks, showcasing its capability to enable efficient and flexible approximation of Clifford-valued functions. The method has been successfully applied to real-world scientific discovery and engineering scenarios, highlighting its practical utility and robustness in complex, high-dimensional settings.

This work addresses the long-standing lack of a clear geometric interpretation for pivoted Cholesky decomposition in kernel methods. From the perspective of reproducing kernel Hilbert spaces (RKHS), it establishes that the pivot selection in pivoted Cholesky is equivalent to farthest-point sampling under the kernel-induced metric, while the construction of the Cholesky factors implicitly performs a Gram–Schmidt orthogonalization process. By revealing this geometric equivalence between pivoted Cholesky and farthest-point sampling, the study provides an intuitive theoretical explanation accompanied by an exceptionally concise mathematical derivation. Furthermore, a lightweight Python implementation is provided, effectively bridging the gap between theoretical analysis and practical application in kernel-based learning.