低秩近似认证的信息极限
Information Limits of Low-Rank Approximation Certification
浏览论文内容
中文总结 AI 辅助
本研究刻画低秩近似误差认证的查询成本，提出重用验证响应的嵌套路径方法，并证明其最优性，优化验证预算得到N^{1/3}和N^{2/3}的成本阶。
中文摘要 AI 辅助
低秩近似可能需要额外的矩阵-向量乘积来验证其误差是否满足规定的容差。我们针对相对矩阵误差和均方输出误差两种情形，刻画了这种认证成本。对于单个近似矩阵候选，我们确定了在允许失败概率趋于零时精确的维度均匀极小极大查询常数。我们的主要结果关注随着近似空间扩展而重用验证响应。对于独立于验证构建的候选族，一个批次即可支持整个嵌套路径，而无需随着检查次数增加查询预算。在W条路径中，利用共享残差能量的集中界产生了√log(W+1)的依赖关系。一个匹配的下界确立了其在固定内部误差目标和足够小的分离间隙下的最优性。最后，我们在相同的分散谱族上比较了两个均匀有效的证书。在每个规则族内优化验证预算，产生验证和构建超出真实目标的成本分别为N^{1/3}和N^{2/3}阶。代码可在该https URL获取。
英文摘要
Low-rank approximation can require additional matrix--vector products to verify that its error meets a prescribed tolerance. We characterize this certification cost for both relative matrix error and mean-square output error. For a single approximation matrix candidate, we determine the exact dimension-uniform minimax query constant as the allowed failure probability vanishes. Our main result concerns reusing validation responses as the approximation space expands. For a candidate family constructed independently of validation, one batch supports an entire nested path without increasing the query budget with the number of checks. Across \(W\) paths, a concentration bound exploiting shared residual energy yields a \(\sqrt{\log(W+1)}\) dependence. A matching lower bound establishes its optimality for fixed interior error targets and sufficiently small separation gaps. Finally, we compare two uniformly valid certificates on the same dispersed-spectrum family. Optimizing the validation budget within each rule family yields costs of orders \(N^{1/3}\) and \(N^{2/3}\) for validation and construction beyond the true target. Code is available at https://anonymous.4open.science/r/Low-rank-approximation-1275/
