Minimum-variance portfolio optimizations rely on accurate covariance estimator to obtain optimal portfolios. However, it usually suffers from large error from sample covariance matrix when the sample size $n$ is not significantly larger than the number of assets $p$. We analyze the random matrix aspects of portfolio optimization and identify the order of errors in sample optimal portfolio weight and show portfolio risk are underestimated when using samples. We also provide LoCoV (low dimension covariance voting) algorithm to reduce error inherited from random samples. From various experiments, LoCoV is shown to outperform the classical method by a large margin.