compute block lewis weights

This work addresses the problem of efficiently and accurately computing ℓ_p-Lewis weights (for p ≥ 4) of a matrix, which quantify the importance of its rows. By alternating between primal and dual formulations of the underlying optimization problem and integrating leverage score iteration with a locally relative smooth gradient descent method, the authors propose a novel algorithm that significantly reduces computational overhead while maintaining high accuracy. Specifically, the proposed approach improves the iteration complexity from O(p³ log(m/ε)) to O(p² log(m/ε)), achieving the current best-known bound on the number of iterations required for ε-approximate Lewis weight computation.

本文解决了计算矩阵的Lewis权重问题，通过固定点迭代法对所有p>2的情况提供了一种高精度计算方法。

Computing exact leverage scores for Kronecker-product structured matrices in large-scale least-squares problems is computationally prohibitive, while existing approximation methods incur statistical bias and high overhead. Method: We propose the first efficient exact leverage score algorithm tailored to Kronecker-structured matrices. Leveraging the inherent tensor structure, our method designs a near-linear-time framework for exact leverage score computation and sampling—bypassing costly full-matrix SVD or biased sketching approximations. Contribution/Results: Theoretically and empirically, our algorithm achieves significantly lower sampling error than state-of-the-art approximate methods (e.g., FJLT- or CountSketch-accelerated approaches), while maintaining substantially lower time complexity than full SVD. This work establishes the first scalable, exact, and efficient leverage score sampling scheme for Kronecker-structured matrices, enabling improved structured random projections and large-scale regression.

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 multi-level low-rank (MLR) matrix approximation problem under the Frobenius norm, tackling three core challenges: hierarchical structural partitioning (row/column stratification), rank allocation (optimizing individual block ranks under a total storage budget), and joint factor fitting. We propose the first end-to-end joint optimization framework for MLR matrices, unifying structural design, rank assignment, and factor learning within a single model. Our approach employs hierarchical block-diagonal parameterization, alternating optimization, and a constrained rank allocation algorithm to achieve coordinated optimization. The resulting approximation preserves matrix-vector multiplication complexity at O(n). Empirical evaluation on multiple benchmark datasets shows that our method reduces approximation error by 35% on average compared to single-level low-rank baselines, significantly improving both accuracy and storage efficiency. The implementation is publicly available.

This work addresses the grouped distributionally robust (GDR) least squares problem, which seeks to minimize the worst-case loss across multiple data groups. The authors introduce block Lewis weights—a novel geometric tool—to reformulate the problem as a specially weighted least squares instance. By integrating an accelerated proximal algorithm with a structured linear system solver tailored for systems of the form \(A^\top B A\), they achieve an efficient solution method that unifies optimization frameworks for both average and robust losses. The proposed approach outperforms interior-point methods at moderate accuracy levels. Theoretically, it attains a \((1+\varepsilon)\)-approximate solution using only \(\widetilde{O}(\min\{\mathrm{rank}(A), m\}^{1/3} \varepsilon^{-2/3})\) linear system solves, yielding the current best-known guarantee for the special case of \(\ell_\infty\) regression.

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.

本文使用线性关系的机制研究线性代数中的优化问题，通过广义最小二乘问题实现伪逆，并提出一种截断伪逆方法解决低秩逼近问题。

This work addresses the challenging problems of entrywise low-rank matrix approximation for $p \neq 2$ and efficient approximation of $p \rightarrow q$ matrix norms. The authors propose a novel algorithmic framework based on the Sherali–Adams convex programming hierarchy combined with global correlation rounding. For the first time, they deliver a polynomial-time approximation scheme for any even integer $p > 2$ with fixed rank $k$, and achieve the first nontrivial additive approximation algorithm in the regime $p < 2 < q$, thereby overcoming limitations inherent to traditional sketching and column selection techniques. The proposed method significantly outperforms existing approaches—including $(3+\varepsilon)$-approximation, bi-criteria, and additive schemes—and represents a substantial theoretical and algorithmic advance in the field.

This study addresses weighted fair division and discrepancy theory under matroid constraints, proposing strongly polynomial-time algorithms grounded in local exchange theorems and constructive proofs. Key contributions include establishing weighted matroid partitionability and extending the Beck-Fiala framework to matroid settings, yielding logarithmic discrepancy bounds. The work achieves EF1 and additive approximation guarantees for fair allocation and scheduling optimization while refuting the weighted carpool conjecture. By systematically resolving fairness and discrepancy control challenges in constrained environments, this research provides novel theoretical foundations and efficient algorithmic tools for related combinatorial optimization problems.