Single-cell RNA sequencing (scRNA-seq) can determine gene expression in numerous individual cells simultaneously, promoting progress in the biomedical sciences. However, scRNA-seq data are high-dimensional with substantial technical noise, including dropouts. During analysis of scRNA-seq data, such noise engenders a statistical problem known as the curse of dimensionality (COD). Based on high-dimensional statistics, we herein formulate a noise reduction method, RECODE (resolution of the curse of dimensionality), for high-dimensional data with random sampling noise. We show that RECODE consistently resolves COD in relevant scRNA-seq data with unique molecular identifiers. RECODE does not involve dimension reduction and recovers expression values for all genes, including lowly expressed genes, realizing precise delineation of cell fate transitions and identification of rare cells with all gene information. Compared with representative imputation methods, RECODE employs different principles and exhibits superior overall performance in cell-clustering, expression value recovery, and single-cell-level analysis. The RECODE algorithm is parameter-free, data-driven, deterministic, and high-speed, and its applicability can be predicted based on the variance normalization performance. We propose RECODE as a powerful strategy for preprocessing noisy high-dimensional data.