spectral radius computation

This work addresses the problem of locally approximating the leading eigenvector of a symmetric bounded matrix while querying only a small number of its entries. It proposes the first local computation algorithm for this task, operating in a preprocessing-and-query model and achieving a preprocessing complexity of Õ(1/ε⁴) and a per-coordinate query complexity of Õ(1/ε²), under the condition that |λ_min(A)| = O(λ_max(A)). The study establishes the first tight, error-dependent upper and lower bounds on query complexity for this problem. Furthermore, it demonstrates the practical impact of the proposed method by applying it to sparsest cut and max-cut problems in dense graph models, significantly enhancing the efficiency of local spectral methods.

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.

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.

To address the poor scalability of conventional real sum-of-squares (RSOS) methods—which rely on large-scale semidefinite programming (SDP) for computing lower bounds on the minimum of real homogeneous polynomials over the unit sphere—this paper proposes a purely spectral (non-SDP) convergent hierarchy. The key innovation is the first rigorous reduction of real spherical optimization to Hermitian optimization, enabling the construction of a sequence of lower bounds via minimal eigenvalue computations alone, within the Hermitian sum-of-squares (HSOS) framework. This approach naturally extends to estimating the spectral norm of real tensors, thereby opening a new pathway for spectral methods in general constrained real optimization. Numerical experiments and asymptotic analysis demonstrate substantial improvements over RSOS and other baseline methods; moreover, the framework yields a computable, convergent hierarchy for the spectral norm.

This work addresses the problem of establishing lower bounds on the smallest singular value of random matrices whose entries are low-degree polynomials of a small number of base random variables—bypassing fundamental limitations of classical smoothed analysis under strong anti-concentration assumptions. The method introduces a novel anti-concentration framework grounded in the well-conditionedness of polynomial mappings’ Jacobians, integrating hierarchical ε-nets, higher-order matrix lifting, and spectral analysis of linear operators. This yields the first singular-value criterion applicable to matrices with strong algebraic dependencies. The contribution resolves long-standing open problems—including power-sum decomposition and robust subspace entanglement certification—by providing the first smoothed-analysis guarantees for such settings. Crucially, it extends the scope of smoothed analysis from fully independent random matrices to non-robust, algebraically dependent ensembles, thereby furnishing new theoretical tools and foundations for algorithmic smoothed analysis.

This work addresses the spectral-norm robustness of low-rank pseudoinverse approximations to matrix inversion under observational noise. To handle noisy perturbations, we introduce, for the first time, a compact non-asymptotic perturbation bound based on contour integral techniques applied to the non-holomorphic function $f(z) = 1/z$, overcoming the asymptotic and loose nature of classical bounds—achieving up to $sqrt{n}$-fold improvement in theoretical accuracy. The bound explicitly characterizes error dependence on eigenvalue gaps, spectral decay, and alignment between noise and low-curvature directions. By integrating matrix perturbation theory with spectral analysis, our framework enables quantifiable robustness modeling. Experiments demonstrate that the derived bound tightly tracks empirical errors and significantly outperforms existing estimates across diverse synthetic and real-world datasets. This provides a new theoretical guarantee for efficient, spectrum-aware matrix computation in noisy environments.

Conventional trigonometric closed-form methods for computing eigenvalues of 3×3 real symmetric matrices suffer from numerical instability in the presence of multiple eigenvalues. Method: This paper proposes a robust closed-form solution leveraging the trace, deviatoric invariants, and discriminant—integrating Cardano–Viète algebraic principles, invariant analysis, and error propagation theory. We derive, for the first time, a tight forward error bound for closed-form eigenvalue computation of 3×3 matrices and design a high-precision J₂ algorithm that ensures controllable error under well-conditioned eigenvector bases. Contribution/Results: Experimental evaluation demonstrates accuracy comparable to LAPACK, with approximately 10× speedup; observed errors strictly satisfy the theoretical forward error bound. The method significantly enhances computational efficiency and robustness in high-reliability applications requiring guaranteed numerical stability.

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.

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 paper addresses two structured matrix recovery problems: (1) recovering a low-rank Toeplitz matrix from noisy observations, and (2) reconstructing the column space of a Fourier matrix from a single observation. Both problems are challenging due to the incompatibility of jointly optimizing structural constraints—Toeplitz (or Hankel), low-rankness, and subspace structure. To overcome this, we introduce Gradient-MUSIC—a novel unifying framework integrating spectral estimation with constrained optimization—into structured matrix recovery for the first time. We establish a minimax-optimal error bound under noise: ‖T − T̂‖₂ ≤ C√r ‖E‖₂. Moreover, we provide the first quantitative characterization linking spectral estimation accuracy to structured matrix recovery performance. The method naturally extends to Hankel matrices and exhibits both computational efficiency and strong robustness to noise.