处理奇异协方差估计的随机矩阵理论方法
A Random Matrix--Theoretic Approach to Handling Singular Covariance Estimates
摘要：
在许多实际情况下，我们希望从不足的数据量中估计一组变量的协方差矩阵。更具体地说，如果我们有一组$N$独立，$M$维度随机向量的相同分布的测量值，则最大似然估计是样本协方差矩阵。在这里，我们考虑的情况下，其中$N<M$，使得这个估计是奇异的，因此从根本上坏。我们提出了一种全新的方法来处理这种情况。令$X$为$M\times N$数据矩阵，其中列为随机向量与协方差矩阵$Σ$的$N$独立实现。不失一般性，我们可以假设随机变量的均值为零。我们希望从$X$估计$Σ$。令$K$为经典样本协方差矩阵。固定参数$1\leq L\leq N$，并考虑具有Haar概率测度的$L\times M$随机酉矩阵的集合$\{Φ\}$。前和后分别将$K$乘以$Φ$和$Φ$的共轭转置，以产生非奇异的$L\times L$降维协方差估计。由$\mathrm{cov}_L(K)$表示的$Σ$的新估计通过以下方式获得：a）通过分别与$Φ$的共轭转置和与$Φ$的前乘和后乘，将减少的协方差估计投影出（到$M\times M$），以及b）在酉系综上取期望。另一个新的估计（这次是针对$Σ^{-1}$的），$\mathrm{invcov}_L(K)$，通过a）求约化协方差估计的逆，b）通过分别与$Φ$的共轭转置和与$Φ$的前乘和后乘，将逆投影出来（到$M\times M$），以及c）在酉系综上取期望。我们有一个封闭的解析表达式$\mathrm{invcov}_L(K)$和$\mathrm{cov}_L(K)$在其特征值分解。
英文摘要：
In many practical situations we would like to estimate the covariance matrix of a set of variables from an insufficient amount of data. More specifically, if we have a set of $N$ independent, identically distributed measurements of an $M$ dimensional random vector the maximum likelihood estimate is the sample covariance matrix. Here we consider the case where $N<M$ such that this estimate is singular and therefore fundamentally bad. We present a radically new approach to deal with this situation. Let $X$ be the $M\times N$ data matrix, where the columns are the $N$ independent realizations of the random vector with covariance matrix $Σ$. Without loss of generality, we can assume that the random variables have zero mean. We would like to estimate $Σ$ from $X$. Let $K$ be the classical sample covariance matrix. Fix a parameter $1\leq L\leq N$ and consider an ensemble of $L\times M$ random unitary matrices, $\{Φ\}$, having Haar probability measure. Pre and post multiply $K$ by $Φ$, and by the conjugate transpose of $Φ$ respectively, to produce a non--singular $L\times L$ reduced dimension covariance estimate. A new estimate for $Σ$, denoted by $\mathrm{cov}_L(K)$, is obtained by a) projecting the reduced covariance estimate out (to $M\times M$) through pre and post multiplication by the conjugate transpose of $Φ$, and by $Φ$ respectively, and b) taking the expectation over the unitary ensemble. Another new estimate (this time for $Σ^{-1}$), $\mathrm{invcov}_L(K)$, is obtained by a) inverting the reduced covariance estimate, b) projecting the inverse out (to $M\times M$) through pre and post multiplication by the conjugate transpose of $Φ$, and by $Φ$ respectively, and c) taking the expectation over the unitary ensemble. We have a closed analytical expression for $\mathrm{invcov}_L(K)$ and $\mathrm{cov}_L(K)$ in terms of its eigenvalue decomposition.
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