estimate operator norm

This project addresses efficient randomized algorithms for three fundamental matrix computations under resource constraints: (1) low-rank positive semidefinite (PSD) matrix approximation, aiming for exact low-rank reconstruction with minimal element accesses; (2) estimation of implicit matrix properties—trace, diagonal entries, and row norms—using only matrix-vector products; and (3) numerically robust floating-point solutions to overdetermined linear least-squares problems. Methodologically, we propose a randomized pivoted Cholesky decomposition to drastically reduce element accesses in PSD approximation; introduce a novel leave-one-out framework for property estimation, achieving optimal sample complexity; and design the first randomized least-squares solver that simultaneously attains asymptotic acceleration—O(nd log d) time for n × d matrices—and numerical backward stability, matching the accuracy of direct methods. These contributions advance both theoretical guarantees and practical efficiency in large-scale, memory- or query-constrained numerical linear algebra.

This work addresses the problem of estimating tight upper bounds on the spectral norm of matrices with rapidly decaying spectra—common in deep learning and inverse problems—using only matrix-vector products. Conventional methods such as the power iteration yield loose, theoretically unguaranteed bounds for such matrices. To overcome this, we propose the Counterbalance estimator: a randomized scheme leveraging matrix-vector queries, augmented with statistical bias correction and sharp concentration analysis. It is the first method to deliver provably tight upper bounds with rigorous probabilistic guarantees. Theoretically, its estimation error converges faster as spectral decay accelerates. Empirically, it significantly outperforms baseline approaches on both synthetic and real-world datasets; for matrices with fast spectral decay, it reduces upper-bound error by several orders of magnitude.

Traditional neural networks suffer from limited interpretability and weak theoretical foundations. Method: This paper proposes a novel machine learning paradigm grounded in infinite-dimensional Hilbert spaces, centering on linear operators. It integrates reproducing kernel Hilbert spaces (RKHS), spectral operator learning, wavelet representations, scattering transforms, and Koopman operator theory to formulate learning tasks as sampling, approximation, and dynamical inference in infinite-dimensional function spaces. Contribution/Results: We establish the first unified Hilbert-space-theoretic framework bridging spectral learning and symbolic reasoning. The approach significantly enhances mathematical rigor and model interpretability by grounding learning in well-defined functional-analytic principles. Moreover, it provides a rigorous mathematical foundation and new methodological pathways for deep interdisciplinary integration between signal processing and machine learning—enabling principled analysis of structured data, hierarchical feature extraction, and nonlinear dynamical system modeling.

This paper addresses exact recovery of low-rank matrices from an extremely small number of noisy observations. To overcome the limitations of existing methods—which rely on strong spectral assumptions such as small condition numbers and large singular value gaps—we propose the first polynomial-time algorithm that guarantees high-probability exact recovery under only three fundamental assumptions: low rank, delocalization of singular vectors, and sufficiently random sampling. Our key innovation is a contour integral analysis framework, synergistically integrating ∞-norm error bounds with random sampling theory to break the theoretical barrier of zero-error reconstruction under noise. Under bounded-precision noise, we establish, for the first time without imposing additional spectral structural assumptions, rigorous guarantees of perfect matrix recovery. This significantly extends both the applicability boundary and robustness of low-rank matrix completion.

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 study addresses the average-case discrepancy problem for Gaussian orthogonal ensemble matrices, seeking sign assignments to control the operator norm in both offline and online settings. Methodologically, it proposes a recentering-and-rounding algorithm for the offline setting and a Frobenius-greedy algorithm for the online setting. By combining probabilistic analysis with rotational symmetry techniques, the approach reduces Frobenius norm control to operator norm guarantees. The core contribution lies in precisely characterizing, for the first time, the phase transition thresholds of stable algorithms in the proportional regime: τ = Ω(1/(κ² log(1/κ))) is required offline, while τ > π/(4κ²) is needed online. Furthermore, matching lower bounds are established, revealing a fundamental gap between achievable algorithmic performance and the satisfiability threshold.

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.

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 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 investigates computational lower bounds for approximating nontrivial norms of higher-order tensors, such as the spectral norm. We introduce a general framework that systematically formalizes the detection–estimation gap as a mechanism for establishing computational hardness, realized through the low-degree polynomial method under the low-degree conjecture. Applying this framework to the symmetric tensor spectral norm, we prove that any degree-$D$ algorithm with $D \leq c_d(\log p)^2$ incurs an approximation distortion of at least $p^{d/4 - 1/2}/\mathrm{polylog}(p)$. This lower bound matches existing upper bounds up to polylogarithmic factors in several important regimes, thereby revealing an inherent computational barrier characterized by the exponent $d/4 - 1/2$.