eigenspace perturbation analysis

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 addresses the lack of theoretical characterization in existing kernel methods for machine learning regarding the residual structure and energy stability of multichannel signals in complex systems. The authors propose an analytical framework grounded in operator defect identities, introducing the novel concept of “telescopic energy residuals.” By integrating iterative products with a λₙ-relaxed Kaczmarz scheme, they establish admissibility conditions for residuals and derive prior energy bounds. For the first time, this framework incorporates operator defect theory into kernel methods and kernel principal component analysis (KPCA), rigorously proving explicit convergence of generalized algorithms, a residual energy decomposition theorem, and stability criteria under noise. The approach significantly extends infinite-dimensional Kaczmarz theory to broader applications in machine learning.

This work investigates the impact of noise—particularly differential privacy (DP) noise—on spectral-norm approximation of low-rank matrices, with a focus on precise characterization of subspace directional distortion: Frobenius-norm error bounds fail to capture worst-case angular deviation, whereas spectral-norm guarantees are strongest for downstream applications. Methodologically, we transcend the classical Eckart–Young–Mirsky theorem by deriving high-probability spectral perturbation bounds, achieving an up-to-√n improvement in error upper bounds. We introduce a novel contour-based bootstrap technique from complex analysis, extended to matrix exponentials and polynomial spectral functions, and jointly leverage eigenvalue gaps and matrix condition numbers for fine-grained error analysis. Our theoretical advances resolve, for the first time, the long-standing open problem of subspace stability in DP-PCA. Empirical evaluation on real-world datasets confirms that our spectral bounds substantially outperform Frobenius-based analyses, yielding the strongest known utility guarantees for privacy-preserving low-rank learning.

This work investigates the impact of matrix concatenation operations on singular value spectra, aiming to bridge a theoretical gap in the structural stability of SVDs under concatenation. Addressing the central question—“How are the singular values of a concatenated matrix determined by those of its constituent submatrices?”—we extend Weyl’s inequality to block-wise concatenation for the first time, establishing a quantitative analytical framework grounded in matrix perturbation theory and norm inequalities. We derive computable upper bounds on singular value deviations and rigorously prove that dominant singular values remain stable when the operator norms of the submatrices are comparable. This result provides theoretical guarantees and principled design guidance for low-rank approximation, robust matrix clustering, and compression algorithms.

Parameter estimation in high-dimensional structured generalized linear models suffers from low efficiency, particularly under realistic design matrices exhibiting anisotropy and strong correlations. Method: This paper introduces a novel spectral estimation framework based on Approximate Message Passing (AMP). Contribution/Results: We provide the first exact asymptotic characterization of spectral estimators under correlated Gaussian designs. We identify a universally optimal covariance-adaptive preprocessing strategy, partially resolving a long-standing conjecture on optimal spectral estimation for rotationally invariant models. Theoretically and empirically, our approach substantially reduces sample complexity and achieves provably statistically optimal estimation accuracy—outperforming existing heuristic methods on canonical designs from computational imaging and genomics.

This study addresses a fundamental challenge in system identification: distinguishing spurious eigenvalues arising from limited data from those genuinely reflecting the underlying system dynamics. To this end, the paper introduces—for the first time—the probabilistic sampling pseudospectrum \( P(\lambda) \) and its computationally efficient estimator \( \hat{P}(\lambda) \). By leveraging resampling and statistical inference, this framework quantifies the uncertainty of eigenvalues across the complex plane. The proposed approach provides a general and rigorous statistical criterion for data-driven methods such as Dynamic Mode Decomposition and subspace identification, substantially enhancing the reliability of identifying true dynamical modes from noisy, finite-length observations.

Classical spectral perturbation theory, such as the Davis–Kahan theorem, fails to capture the systematic geometric bias in the leading eigenspace under heterogeneous noise. This work studies a signal-plus-noise model with heteroscedastic and sparse random perturbations, and for the first time identifies and quantifies a deterministic geometric bias induced by the alignment between the signal structure and the noise variance profile. The total perturbation is decomposed into three components: a signal-to-noise ratio term, a stochastic fluctuation term, and this geometric bias term. Leveraging the quadratic vector equation (QVE) framework and refined isotropic local laws, the paper establishes non-asymptotic perturbation bounds in both operator norm and ℓ²→ℓ^∞ norm. The resulting upper bounds are nearly optimal and substantially outperform classical theory in predicting eigenspace perturbations under heterogeneous noise.

This work addresses spectral pollution in extended dynamic mode decomposition (EDMD) caused by non-invariant dictionaries, which distorts the eigenvalues of the Koopman operator. To resolve this issue, the authors propose a projected Koopman approximation framework that constructs compatible subspaces via forward-intersection chains, enabling accurate preservation of Koopman eigenpairs associated with nonzero eigenvalues without requiring dictionary invariance. The approach integrates coordinate-wise and $L^2(\mu)$-orthogonal projection geometries, leveraging singular value decomposition to formulate a filtered EDMD variant. Numerical experiments on the Kronecker flow, polynomial systems, and the Van der Pol oscillator demonstrate significant suppression of spectral pollution: the coordinate projector recovers the local equilibrium spectrum independently of the sampling measure, while the $L^2(\mu)$ projector accurately approximates the limit-cycle spectrum.

This work addresses the lack of theoretical guarantees for the reliability of Koopman eigenpairs computed from noisy data in data-driven spectral analysis. It introduces, for the first time, shadowing trajectory theory combined with backward error analysis to interpret the residual of eigenpairs obtained via Extended Dynamic Mode Decomposition (EDMD) as an operator perturbation of the original dynamical system. The study rigorously proves that this approximate solution corresponds exactly to a pseudo-trajectory shadowed by a true system trajectory. By establishing a precise connection among residuals, operator perturbations, and system trajectories, the paper constructs a backward stability framework for assessing Koopman eigenpairs, thereby providing a novel theoretical foundation for the credibility of data-driven methods in noisy environments.

This work addresses the gap between algorithmic prototypes and efficient implementations in scientific research by proposing a lightweight approach to translate statistical and machine learning algorithms—such as kernel ridge regression and stochastic gradient descent matrix factorization—from mathematical formulations into readable, high-performance C++ code. Leveraging the Eigen template library for core linear algebra operations—including kernel matrix construction, regularized solvers, and vectorized updates—the implementation seamlessly integrates into the Python ecosystem via pybind11, enabling efficient interoperability with NumPy arrays. The project provides concise, reproducible code examples that encapsulate common computational patterns in research, significantly lowering the barrier for researchers to adopt C++ for high-performance development while balancing performance, readability, and usability.