论文标题
在异性固定愿望型基质的非反应浓度上
On the Non-Asymptotic Concentration of Heteroskedastic Wishart-type Matrix
论文作者
论文摘要
本文重点介绍了异性志式型型矩阵的非反应浓度。假设$ z $是$ p_1 $ -by- $ p_2 $随机矩阵和$ z_ {ij} \ sim n（0，σ_{ij}^2）$独立，我们证明了Wishart矩阵偏差的预期光谱规范（即zz^\ top \ right \ | $）在\ begin {equation*}的上限上 \ begin {split} （1 +ε）\ left \ {2σ_cσ_r +σ_c^2 +cσ_rσ_*\ sqrt {\ log（p_1 \ wedge p_2）} +cσ_*_*^2 \ log（p_1 \ end {split} \ end {equation*}其中$σ_c^2：= \ max_j \ sum_ {i = 1}^{p_1} = {ij}^2 $，$σ_r^2：= \ max_i \ sum_i \ sum_ \ sum_ \ sum_ \ sum_ \ sum_ {j = 1}^$} $σ_*^2：= \ max_ {i，j}σ_{ij}^2 $。开发了一个与该上限匹配的最小下限。然后，我们在更一般的分布（例如次高斯和重尾分布）下得出异性恋愿望型型基质的浓度不平等，力矩和尾部边界。接下来，我们考虑$ z $具有homoskedastic列或行的情况（即$σ_{ij} \大约σ_i$或$σ_{ij} \ actionσ_j$），并得出速率 - 最佳的WishArt-wishArt-type-Type型浓度。最后，我们应用开发的工具来识别尖锐的信噪比阈值，以在异性群聚类问题中保持一致的聚类。
This paper focuses on the non-asymptotic concentration of the heteroskedastic Wishart-type matrices. Suppose $Z$ is a $p_1$-by-$p_2$ random matrix and $Z_{ij} \sim N(0,σ_{ij}^2)$ independently, we prove the expected spectral norm of Wishart matrix deviations (i.e., $\mathbb{E} \left\|ZZ^\top - \mathbb{E} ZZ^\top\right\|$) is upper bounded by \begin{equation*} \begin{split} (1+ε)\left\{2σ_Cσ_R + σ_C^2 + Cσ_Rσ_*\sqrt{\log(p_1 \wedge p_2)} + Cσ_*^2\log(p_1 \wedge p_2)\right\}, \end{split} \end{equation*} where $σ_C^2 := \max_j \sum_{i=1}^{p_1}σ_{ij}^2$, $σ_R^2 := \max_i \sum_{j=1}^{p_2}σ_{ij}^2$ and $σ_*^2 := \max_{i,j}σ_{ij}^2$. A minimax lower bound is developed that matches this upper bound. Then, we derive the concentration inequalities, moments, and tail bounds for the heteroskedastic Wishart-type matrix under more general distributions, such as sub-Gaussian and heavy-tailed distributions. Next, we consider the cases where $Z$ has homoskedastic columns or rows (i.e., $σ_{ij} \approx σ_i$ or $σ_{ij} \approx σ_j$) and derive the rate-optimal Wishart-type concentration bounds. Finally, we apply the developed tools to identify the sharp signal-to-noise ratio threshold for consistent clustering in the heteroskedastic clustering problem.
