Motivated by the popularity of stochastic rounding in the context of machine
learning and the training of large-scale deep neural network models, we
consider stochastic nearness rounding of real matrices $\mathbf{A}$ with many
more rows than columns. We provide novel theoretical evidence, supported by
extensive experimental evaluation that, with high probability, the smallest
singular value of a stochastically rounded matrix is well bounded away from
zero -- regardless of how close $\mathbf{A}$ is to being rank deficient and
even if $\mathbf{A}$ is rank-deficient. In other words, stochastic rounding
\textit{implicitly regularizes} tall and skinny matrices $\mathbf{A}$ so that
the rounded version has full column rank. Our proofs leverage powerful results
in random matrix theory, and the idea that stochastic rounding errors do not
concentrate in low-dimensional column spaces.

通过随机舍入来对实型矩阵进行随机接近处理，研究了其对矩阵的奇异值和秩不足性的影响，并得出结论：高概率下，经过随机舍入后的矩阵的最小奇异值与零点的距离有明确的下界，即使原矩阵接近秩不足也是如此。换言之，随机舍入隐式地对高而瘦的矩阵进行正则化，使舍入后的矩阵具有完全列秩。