良态Wishart矩阵上的共轭梯度算法几乎是确定的
The conjugate gradient algorithm on well-conditioned Wishart matrices is almost deterministic
摘要：
证明了对于大矩阵，用共轭梯度法求解随机正定线性系所需的迭代次数几乎是确定的。我们处理Wishart矩阵$W = XX^*$的情况下，其中$X$是$n \times m$和$n/m \sim d$的$0 < d < 1$。准确地说，我们证明了大多数选择的误差容限，随着矩阵的大小增加，迭代计数偏离一个明确的确定性值的概率趋于零。此外，对于固定的迭代次数，我们证明了差错向量的范数和残差向量的范数按概率指数快速收敛，按均值收敛和几乎必然收敛。
英文摘要：
We prove that the number of iterations required to solve a random positive definite linear system with the conjugate gradient algorithm is almost deterministic for large matrices. We treat the case of Wishart matrices $W = XX^*$ where $X$ is $n \times m$ and $n/m \sim d$ for $0 < d < 1$. Precisely, we prove that for most choices of error tolerance, as the matrix increases in size, the probability that the iteration count deviates from an explicit deterministic value tends to zero. In addition, for a fixed iteration count, we show that the norm of the error vector and the norm of the residual converge exponentially fast in probability, converge in mean and converge almost surely.
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