condition number analysis

Existing iterative solvers for large-scale linear systems suffer from strong dependence on the global condition number and coarse-grained complexity analyses. Method: We introduce the *spectral tail condition number* $kappa_ell$, a new fine-grained spectral measure, and develop a refined time-complexity framework. Our approach formally defines $kappa_ell$, integrates it with the Sketch-and-Project paradigm, Nesterov acceleration, determinant point process sampling, and universality theory for Gaussian matrices, thereby exposing an intrinsic connection between iteration complexity and the matrix multiplication exponent $omega$. Contribution/Results: Our analysis achieves a sharper separation between deterministic and randomized algorithms, yielding an $ ilde{O}(kappa_ell n^2 log(1/varepsilon))$ bound for computing an $varepsilon$-accurate solution—valid for $ell$ up to $O(n^{0.729})$. This significantly improves the fine-grained analysis of the conjugate gradient method and establishes a novel theoretical benchmark for iterative algorithm design.

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 efficiency bottlenecks in solving large-scale linear systems and approximating matrix norms. We propose a multilevel randomized sketching preconditioned iterative method, integrating Nyström low-rank approximation, sparse random sketching, and multilevel preconditioning. It establishes the first multilevel sketched preconditioning framework grounded in the natural average condition number. Theoretical contributions include: (1) optimal complexity $ ilde{O}(n^2 + d_lambda^omega)$ for solving regularized linear systems; (2) accelerated complexity $ ilde{O}(n^{2.065} + k^omega)$ for systems with $k$ outlying singular values; and (3) Schatten-$p$ norm approximation—particularly the nuclear norm—at $ ilde{O}(n^{2.11})$, improving upon the prior best $ ilde{O}(n^{2.18})$. These advances significantly enhance computational efficiency for key subproblems in applications such as Gaussian process regression.

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.

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 study investigates the low-degree polynomial approximation of the leading eigenpair of random symmetric matrices. Focusing on the Spiked Gaussian Orthogonal Ensemble (GOE) and standard GOE models, it integrates exact spectral methods with the low-degree algorithmic framework. By leveraging the extremal properties of Chebyshev polynomials and random matrix theory, this work rectifies prevailing misconceptions regarding the required number of iterations in classical power methods. It establishes a critical degree threshold for approximating the leading eigenpair and derives an exact expression for the asymptotic overlap. The resulting theoretical predictions significantly improve upon existing bounds, offering new insights into the fundamental limits of polynomial-based algorithms for random matrix computations.

This work addresses the problem of efficiently maintaining the rank, a column basis, and a maximum-rank submatrix of a matrix under dynamic updates—either to individual entries or entire columns—and applies these techniques to the dynamic maximum matching problem in graphs. The paper presents the first dynamic algorithm whose update time depends on the current rank \( r \) rather than the matrix dimension \( n \). By integrating sparse update strategies with rank-sensitive complexity analysis, it achieves an amortized update time of \( \tilde{O}(r^{1.405}) \) for single-entry modifications and \( \tilde{O}(r^{1.528} + z) \) for column updates, where \( z \) denotes the number of changed entries. This approach is the first to simultaneously support dynamic maintenance of rank, basis, and maximum-rank submatrix, yielding an edge update time of \( \tilde{O}(|M|^{1.405}) \) for dynamic graph matching and significantly improving upon prior methods.

This study addresses the challenge of optimizing the upper bound of the matrix multiplication exponent ω by reconstructing the combinatorial loss analysis framework to expand the solution space. Furthermore, it innovatively integrates machine learning with the AlphaEvolve evolutionary search algorithm. Through this hybrid optimization strategy, the upper bound of ω is successfully reduced to 2.371177. This achievement not only overcomes existing theoretical bottlenecks but also surpasses previous state-of-the-art records. Consequently, this work establishes a novel algorithmic paradigm and provides critical theoretical support for research on matrix multiplication complexity, significantly advancing progress in the field.

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 addresses a critical limitation of Laplacian spectral sparsification: while it preserves quadratic forms, it offers little control over the adjacency spectral radius—the largest eigenvalue of the adjacency matrix—which plays a pivotal role in NIMFA epidemic thresholds and spectral clustering. The paper establishes, for the first time, tight error bounds on the adjacency spectral radius under Laplacian sparsification. It introduces a novel analytical framework combining Perron–Frobenius monotonicity, eigenvector delocalization, and resolvent perturbation theory, integrated with effective resistance sampling and Matrix Bernstein inequalities. The resulting deterministic bound is |λ₁(A_H) − λ₁(A_G)| ≤ ε(2Δ − λ₁), and a high-probability bound of O(εΔ/√c). For Erdős–Rényi graphs, regular expanders, and stochastic block models, refined bounds of O(εΔ√(log n)/√n + ε²Δ²/δ_gap) are derived, with tightness proven for regular graphs.