论文标题
产品的矩阵浓度不等式
A matrix concentration inequality for products
论文作者
论文摘要
我们为随机矩阵乘积\ begin {equation} \ label {eq：zn} z_n = \ left（i_d-αx_n\ right）\ left（i_d-αx_{i_d-αx_{n-1} {n-1} \ cdots）\ weft（i_d-cdots）{i_d-αxxxx_1{ $ \ left \ {x_k \ right \} _ {k = 1}^{+\ infty} $是一系列有界的独立随机正面正面半芬属矩阵，具有共同的期望$ \ mathbb {e} \ left [x_k \ weft [x_k \ right] =σ$。在这些假设下，我们表明，对于足够小的积极$α$，$ z_n $满足浓度不平等\ begin {equation} \ label {eq：ctbound} \ mathbb {p} \ left（\ left \ left \ left \ left \ welet z_n- \ mathbb {e} \ leq 2d^2 \ cdot \ ext \ left（\ frac {-t^2} {ασ^2} \ right）\ quad \ text {for all} t \ geq 0，\ end e e e e e e e e e equation}其中$σ^2 $表示差异参数。
We present a non-asymptotic concentration inequality for the random matrix product \begin{equation}\label{eq:Zn} Z_n = \left(I_d-αX_n\right)\left(I_d-αX_{n-1}\right)\cdots \left(I_d-αX_1\right), \end{equation} where $\left\{X_k \right\}_{k=1}^{+\infty}$ is a sequence of bounded independent random positive semidefinite matrices with common expectation $\mathbb{E}\left[X_k\right]=Σ$. Under these assumptions, we show that, for small enough positive $α$, $Z_n$ satisfies the concentration inequality \begin{equation}\label{eq:CTbound} \mathbb{P}\left(\left\Vert Z_n-\mathbb{E}\left[Z_n\right]\right\Vert \geq t\right) \leq 2d^2\cdot\exp\left(\frac{-t^2}{ασ^2} \right) \quad \text{for all } t\geq 0, \end{equation} where $σ^2$ denotes a variance parameter.
