均值矩阵可对角化的预测
Predicting Diagonalizability of a Mean Matrix
浏览论文内容
中文总结 AI 辅助
该研究解决了从二元随机矩阵的递增独立同分布样本预测均值矩阵可对角化的问题，证明固定维度有界均值的半代数性质可几乎必然预测，扩展至无界观测时维度≥2则不可预测。
中文摘要 AI 辅助
Wu和Santhanam提出：能否从二元随机矩阵的递增独立同分布样本中，几乎仅以有限误差判定未知均值矩阵是否可对角化？我们对实数域或复数域上的可对角化均给出肯定回答。核心发现是一条通用原则：固定维度的有界均值参数的每个半代数性质，最终几乎必然可预测。我们给出自包含的收缩置信集证明，以及由多项式符号检验得到的显式预测器。Tarski–Seidenberg量词消去表明，实数和复数可对角化轨迹均为半代数，尽管它们既非闭集也非开集。我们进一步利用Marcinkiewicz–Zygmund强大数定律，将正结果扩展到任意固定有限阶r>1矩的无界观测。结合Dembo–Peres拓扑准则，得到鲜明对比：在所有仅可积的矩阵律类中，当维度至少为2时，可对角化并非最终几乎必然可预测。该构造对固定维度有效，但未给出实际复杂度界。
英文摘要
Wu and Santhanam asked whether one can determine, from an increasing i.i.d. sample of binary random matrices, whether the unknown mean matrix is diagonalizable while making only finitely many errors almost surely. We answer this question affirmatively, for diagonalizability over either $\mathbb{R}$ or $\mathbb{C}$. The main observation is a general principle: every semialgebraic property of a fixed-dimensional bounded mean parameter is eventually almost surely predictable. We give a self-contained shrinking-confidence-set proof and an explicit predictor obtained from polynomial sign tests. Tarski--Seidenberg quantifier elimination shows that both the real- and complex-diagonalizable loci are semialgebraic, despite being neither closed nor open. We further extend the positive result to unbounded observations with any fixed finite moment of order $r>1$, using the Marcinkiewicz--Zygmund strong law. Combined with the Dembo--Peres topological criterion, this yields a sharp contrast: over the class of all merely integrable matrix laws, diagonalizability is not eventually almost surely predictable when the dimension is at least two. The construction is effective for fixed dimension, although no practical complexity bound is claimed.
