子采样矩阵的积累及其在统计计算中的应用
Accumulation of Sub-Sampling Matrices with Applications to Statistical Computation
AI总结：
本文提出积累子采样方法，通过累积多个子采样投影，提升统计效率并控制有效矩阵大小，应用于统计模型中实现计算效率的提升。
摘要：
通过适当选择采样概率，基于采样的随机投影可用于实现大规模统计方法，显著降低计算成本的同时保持低统计误差。然而，计算最优采样概率本身往往很昂贵，在实践中通常采用次优方案。这通常导致时间与空间成本的增加，因为需要更多的子样本，从而导致投影矩阵变大，使推断过程更具计算强度。在本文中，我们扩展了基于采样的随机投影框架，并提出了一种新的投影方法，即积累子采样。通过仔细累积多个此类投影，积累子采样在保持统计效率的同时，控制整个统计计算中的有效矩阵大小。在理论方面，我们量化了子采样方案的质量如何影响近似矩阵乘积和正定矩阵的误差，并展示了所提出的累积策略如何减轻这种影响。此外，我们将我们的方法应用于涉及大量矩阵运算的统计模型，如谱聚类中的特征分解和核岭回归中的矩阵求逆，并证明了减少有效矩阵大小带来了显著的计算节省。在一系列问题上的数值实验进一步表明，与现有随机投影基线相比，我们的方法在次优采样方案下始终提高了计算效率。
英文摘要：
With appropriately chosen sampling probabilities, sampling-based random projection can be used to implement large-scale statistical methods, substantially reducing computational cost while maintaining low statistical error. However, computing optimal sampling probabilities is often itself expensive, and in practice one typically resorts to suboptimal schemes. This generally leads to increased time and space costs, as more subsamples are required and the resulting projection matrices become larger, thereby making the inference procedure more computationally demanding. In this paper, we extend the framework of sampling-based random projection and propose a new projection method, \emph{accumulative sub-sampling}. By carefully accumulating multiple such projections, accumulative sub-sampling improves statistical efficiency while controlling the effective matrix size throughout the statistical computation. On the theoretical side, we quantify how the quality of the subsampling scheme affects the error in approximating matrix products and positive semidefinite matrices, and show how the proposed accumulation strategy mitigates this effect. Moreover, we apply our method to statistical models involving intensive matrix operations, such as eigendecomposition in spectral clustering and matrix inversion in kernel ridge regression, and demonstrate that reducing the effective matrix size leads to substantial computational savings. Numerical experiments across a range of problems further show that our approach consistently improves computational efficiency compared to existing random projection baselines under suboptimal sampling schemes.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
