基于RDT的最大平均子矩阵值上界
RDT based upper bounds on the largest average submatrix values
浏览论文内容
中文总结 AI 辅助
针对线性区域的最大平均子矩阵问题，提出随机对偶理论框架，获得闭式上界，并证明提升变体更优，且与复制和次线性结果一致。
中文摘要 AI 辅助
我们研究了经典最大平均子矩阵问题的统计变体。对于小子矩阵（其维度小于与原矩阵的线性比例），该问题通常已被充分理解，并被认为存在统计-计算差距（SCG）。然而，在线性区域中，对信息论和算法方面的分析处理仍然具有挑战性，且至今没有任何数学上严格的结果接近证明或否定该设置中SCG的存在。聚焦于线性区域，我们在几个关键方向上取得了重大进展：1）我们开发了一个通用的随机对偶理论（RDT）框架来刻画最大平均子矩阵值。2）使用普通RDT变体，我们获得了作为维度比例参数显式函数的闭式上界。3）我们证明了在子矩阵维度的某个线性范围内，提升的RDT变体严格优于普通RDT。4）对于维度比例常数趋近于零的小子矩阵，我们证明了我们的结果与文献[34]中的复制结果（通过一步复制对称破缺获得）以及文献[18,45]中的次线性结果相匹配。
英文摘要
We study statistical variants of the classical largest average submatrix problem. For small submatrices (where the dimension is less than linearly proportional to the original matrix), the problem is typically well understood and believed to exhibit the statistical-computational gap (SCG). However, analytical treatment of both the information-theoretic and algorithmic aspects of the linear regime remains challenging, and no mathematically rigorous results have yet arrived anywhere close to proving or disproving SCG existence in this setting. Focusing on the linear regime, we make strong progress in several key directions: 1) We develop a generic Random Duality Theory (RDT) framework to characterize largest average submatrix values. 2) Using the plain RDT variant, we obtain closed-form upper bounds as explicit functions of dimensionality proportionality parameters. 3) We demonstrate that a lifted RDT variant strictly improves upon the plain RDT within a certain linear range of submatrix dimensions. 4) For small submatrices where the dimensional proportionality constants approach zero, we prove that our results match both the replica results from [34] (obtained via one-step replica symmetry breaking) and the sublinear results from [18,45].
